REVIEW
Leptogenesis and Planck-scale black hole remnants in a Pati-Salam cosmology
High Energy Physics - Phenomenology · Submitted 18 Sep 2026 · 87 references
Pith's verdict: Well-executed PBH scenario with a clean central tension, but load-bearing assumptions (pre-inflationary PS breaking, non-restoration, monochromaticity, remnants) keep it in the proof-of-concept regime rather than a robust prediction.
T0 · AI review
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Referee report
The paper
The paper claims
A single primordial black hole mass of about 10^6 grams can explain dark matter, the baryon asymmetry, and the Pati-Salam breaking scale together, but only if thermal leptogenesis is included.
The reading
This paper explores leptogenesis and dark matter from primordial black hole (PBH) evaporation within a Pati-Salam grand unified theory. The key insight is that the right-handed neutrino mass is tied to the Pati-Salam breaking scale, so the black hole's ability to emit those neutrinos is not free. Two regimes emerge: lighter PBHs drive non-thermal leptogenesis, while heavier ones require thermal leptogenesis and cause entropy dilution. If quantum gravity leaves stable Planck-scale remnants from each black hole, the remnant abundance scales as the inverse 5/2 power of the initial mass. The paper shows that non-thermal leptogenesis and Planck-scale remnant dark matter are mutually exclusive: the light PBHs needed for non-thermal leptogenesis overproduce remnants. However, when thermal leptogenesis contributes, a single initial PBH mass near $10^{6}$ grams simultaneously yields the observed dark matter density, the heavy neutrino mass required for the baryon asymmetry, and sets a lower bound on the Pati-Salam breaking scale. The scenario predicts gravitational wave signals from PBH Poisson fluctuations within reach of future experiments like the Einstein Telescope.
What carries the argument
The machinery combines: (1) the Pati-Salam gauge structure, which ties the right-handed neutrino mass M_N1 to the SU(2)_R breaking scale v_R via M_N1 = y_R v_R, and requires pre-inflationary symmetry breaking to avoid monopole overproduction; (2) primordial black hole formation from a near-inflection point in the inflaton potential, producing a monochromatic PBH population; (3) Hawking evaporation with greybody factors, tracked via Boltzmann equations to compute non-thermal RHN production and entropy injection; (4) the remnant counting argument: each PBH leaves one stable Planck-scale remnant, so the remnant energy density at evaporation is just (m_rem/M_in) times the total energy density; (5) the sphaleron freeze-out condition for PBH-only leptogenesis (T_ev > 130 GeV gives M_in ≲ 4.2×$10^{5}$ g); and (6) the relic abundance calculation that sets Ω_rem ∝ $M_in^{{-5/2}}$. The central identity is the remnant fraction scaling: f_rem = r (1.1×$10^{6}$ g / M_in)^{5/2} for r=m_rem/M_Pl.
Extended reading notes
Core claim
The central claim is that Planck-scale black hole remnants and PBH-only non-thermal leptogenesis are incompatible, but the tension is resolved when a thermal contribution to leptogenesis is included. In that case, a single initial PBH mass around 1.2 × $10^{6}$ grams can simultaneously: (i) produce Planck-scale remnants that account for all of dark matter, (ii) provide the right-handed neutrino mass (~4.4 × $10^{11}$ GeV) that gives the observed baryon asymmetry through leptogenesis, and (iii) be consistent with the Pati-Salam breaking scale (≳ 4 × $10^{12}$ GeV). The paper demonstrates this by deriving the relic abundance of remnants as a function of initial PBH mass and showing that the sphaleron freeze-out bound (M_in ≲ 4.2 × $10^{5}$ g) for PBH-only leptogenesis is violated by the mass required for remnants to be all of dark matter (M_in ≃ 1.1 × $10^{6}$ g for r=1). When thermal production of right-handed neutrinos is included, the asymmetry is generated earlier and the sphaleron bound is relaxed, allowing the heavier PBH mass to work.
Load-bearing premise
The Pati-Salam gauge symmetry must be broken before inflation and never restored afterward, which requires the SU(2)_R breaking scale v_R to be larger than the PBH formation temperature T_f ~ 4×$10^{15}$ GeV (1 g / M_in)^{1/2}. If the symmetry were ever restored, monopoles would reappear and the right-handed neutrino mass relation would change, invalidating the basis for the leptogenesis calculation.
Editorial extensions
If this is right
- If the scenario is correct, the observed dark matter could consist entirely of stable Planck-scale remnants, with no need for WIMPs or axions, provided the initial PBH mass is near 10^6 g.
- The Pati-Salam breaking scale is predicted to be at least around 4×10^12 GeV (for the benchmark PBH mass), which is testable through future proton decay searches and neutrino mass patterns.
- The non-thermal leptogenesis regime (PBH mass < about 100 g) is excluded as a viable source of baryogenesis if remnants are all of dark matter, because it would overproduce remnants.
- A stochastic gravitational wave background from Poisson fluctuations of the PBH distribution is expected at frequencies around tens of hertz for the mass that explains dark matter, within reach of the Einstein Telescope and Cosmic Explorer.
- High-frequency graviton emission from PBH evaporation is constrained by future CMB measurements of N_eff, providing a complementary probe of the PBH mass and abundance.
Reading between the lines
- The paper leaves implicit that the pre-inflationary Pati-Salam breaking assumption (Eq. 2.14) can be tested by null detection of monopoles in cosmic rays or by future proton decay experiments, which would place independent bounds on v_R.
- Because the remnant-to-dark-matter ratio scales as M_in^{-5/2}, the mass window near 10^6 g is narrow; a factor of 2 change in M_in would force r away from O(1), so the scenario makes a sharp prediction that the PBH mass is within a factor of a few of 10^6 g if remnants are all dark matter and r is natural.
- The mutual exclusivity of remnants and non-thermal leptogenesis could be avoided if remnants are not stable (e.g., they decay), but the paper assumes stability motivated by loop quantum gravity; an unstable remnant would also affect the CMB and BBN, providing another test.
- The gravitational wave signal from Poisson fluctuations is the only observable that probes the initial PBH abundance β, which is otherwise free; a detection would break the degeneracy and test the PBH domination assumption that underlies the remnant abundance calculation.
Original abstract
We study leptogenesis and Planck-scale remnant dark matter from primordial black hole (PBH) evaporation in a minimal Pati-Salam cosmology, with the gauge symmetry broken before inflation so that magnetic monopoles are diluted away. A singlet inflaton with a near-inflection potential enhances the curvature power spectrum on small scales, producing a narrow black hole population that briefly dominates the energy density. The Pati-Salam embedding ties the right-handed neutrino masses to the SU(2)_R breaking scale, and cosmological consistency forces both the scalar-sector Yukawa coupling and the heavy neutrino mass well below that scale. Two regimes emerge, depending on whether the black holes are hot enough to emit the lightest right-handed neutrino. Lighter PBHs drive non-thermal leptogenesis, while heavier ones require a thermal asymmetry and also affect it through entropy dilution. If quantum gravitational backreaction halts evaporation at the Planck scale, each PBH leaves a stable remnant whose present abundance scales as the inverse five-halves power of the initial mass. Suppressing remnants requires heavier black holes, but the lighter PBHs needed for non-thermal leptogenesis overproduce remnant dark matter. Thus, remnant dark matter and non-thermal leptogenesis are mutually exclusive. When the thermal contribution is included, a single initial PBH mass near 10^6 g accommodates the observed dark matter abundance, the heavy neutrino mass required by the baryon asymmetry, and the Pati-Salam breaking scale. The scenario predicts a stochastic gravitational wave background from Poisson fluctuations of the black hole distribution, within reach of future experiments, alongside high-frequency graviton emission constrained by future measurements of the effective number of relativistic species.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We study leptogenesis and Planck-scale remnant dark matter from primordial black hole (PBH) evaporation in a minimal Pati-Salam cosmology, with the gauge symmetry broken before inflation so that magnetic monopoles are diluted away. A singlet inflaton with a near-inflection potential enhances the curvature power spectrum on small scales, producing a narrow black hole population that briefly dominates the energy density. The Pati-Salam embedding ties the right-handed neutrino masses to the SU(2)_R breaking scale, and cosmological consistency forces both the scalar-sector Yukawa coupling and the heavy neutrino mass well below that scale. Two regimes emerge, depending on whether the black holes are hot enough to emit the lightest right-handed neutrino. Lighter PBHs drive non-thermal leptogenesis, while heavier ones require a thermal asymmetry and also affect it through entropy dilution. If quantum gravitational backreaction halts evaporation at the Planck scale, each PBH leaves a stable remnant whose present abundance scales as the inverse five-halves power of the initial mass. Suppressing remnants requires heavier black holes, but the lighter PBHs needed for non-thermal leptogenesis overproduce remnant dark matter. Thus, remnant dark matter and non-thermal leptogenesis are mutually exclusive. When the thermal contribution is included, a single initial PBH mass near 10^6 g accommodates the observed dark matter abundance, the heavy neutrino mass required by the baryon asymmetry, and the Pati-Salam breaking scale. The scenario predicts a stochastic gravitational wave background from Poisson fluctuations of the black hole distribution, within reach of future experiments, alongside high-frequency graviton emission constrained by future measurements of the effective number of relativistic species.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study leptogenesis and Planck-scale remnant dark matter from primordial black hole (PBH) evaporation in a minimal Pati-Salam cosmology, with the gauge symmetry broken before inflation so that magnetic monopoles are diluted away. | contribution 0.99 | 0.78 | 0.04 | asserted 0.97 |
| s1 | A singlet inflaton with a near-inflection potential enhances the curvature power spectrum on small scales, producing a narrow black hole population that briefly dominates the energy density. | method 0.67 | 0.63 | 0.03 | asserted 1.00 |
| s2 | The Pati-Salam embedding ties the right-handed neutrino masses to the SU(2)_R breaking scale, and cosmological consistency forces both the scalar-sector Yukawa coupling and the heavy neutrino mass well below that scale. | result 0.80 | 0.72 | 0.05 | asserted 1.00 |
| s3 | Two regimes emerge, depending on whether the black holes are hot enough to emit the lightest right-handed neutrino. | result 0.84 | 0.77 | 0.03 | asserted 1.00 |
| s4 | Lighter PBHs drive non-thermal leptogenesis, while heavier ones require a thermal asymmetry and also affect it through entropy dilution. | result 0.92 | 0.81 | 0.06 | asserted 1.00 |
| s5 | If quantum gravitational backreaction halts evaporation at the Planck scale, each PBH leaves a stable remnant whose present abundance scales as the inverse five-halves power of the initial mass. | result 0.94 | 0.66 | 0.06 | asserted 0.76 |
| s6 | Suppressing remnants requires heavier black holes, but the lighter PBHs needed for non-thermal leptogenesis overproduce remnant dark matter. | result 0.59 | 0.81 | 0.14 | asserted 1.00 |
| s7 | Thus, remnant dark matter and non-thermal leptogenesis are mutually exclusive. | interpretation 0.55 | 0.83 | 0.41 | asserted 1.00 |
| s8 | When the thermal contribution is included, a single initial PBH mass near 10^6 g accommodates the observed dark matter abundance, the heavy neutrino mass required by the baryon asymmetry, and the Pati-Salam breaking scale. | result 1.00 | 0.87 | 0.38 | asserted 0.99 |
| s9 | The scenario predicts a stochastic gravitational wave background from Poisson fluctuations of the black hole distribution, within reach of future experiments, alongside high-frequency graviton emission constrained by future measurements of the effective number of relativistic species. | result 0.85 | 0.78 | 0.07 | asserted 0.90 |
What the paper reports
At the two benchmarks considered in this work it gives $y_R≲0.11$ for $M_in≃1.2×10^6$ g with $M_N_1≃4.4×10^11$ GeV, and $y_R≲6.5×10^-4$ for $M_in≃1.3×10^2$ g with $M_N_1≃2.5×10^11$ GeV.
Their energy density shows a sharp decrease at $a/a_0∼10^-23$ in the top left panel for the initial mass of PBHs $M_in=1 g$ with $β=10^-4$, and at $a/a_0∼10^-21$ in the bottom left panel for $M_in=10 g$ with the same value of $β$.
As shown by the evolution of the total number density of RHNs, around $a/a_0 ∼3 ×10^-23$ the production of non-thermal RHN from PBH overtakes the thermal contribution due to the entropy injection into the thermal bath during PBH evaporation.
Therefore, using Eq. ([ref]), the corresponding upper bound on the initial mass of PBH is found as M_in ≲4.2 ×10^5 g.
Requiring the remnants to account for the entire observed dark matter abundance gives $M_in≃1.1×10^6 g$.
For the chosen initial conditions, PBHs come to dominate the energy density of the Universe before completely evaporating.
For $M_in = 10 g$ case, the minimum initial PBH abundance required for the PBH dominated epoch is $β_min ≃6.8 ×10^-7 g$.
For the benchmark $M_in≃1.2×10^6$ g relevant to remnant dark matter it requires $v_R≳4×10^12$ GeV, while for $M_in≃1.3×10^2$ g it requires $v_R≳3.8×10^14$ GeV.
Nevertheless, as the Universe continues to evolve, the thermal contribution eventually dominates the total RHN number density evolution.
The sterile neutrino number density is estimated as n_N_1(t_ev) ≃15 ζ(3)g_N_1 /π^4g_*(T_BH) M^2_in/M^2_p n_BH(t_f) (a_f/a_ev)^3 ≃1.9 ×10^23 GeV^3 (10 g/M_in )^5 .
Furthermore, in the combined scenario, the chosen benchmark value of $M_in ∼10^6 g$ simultaneously reproduces the correct dark matter relic abundance through Planck scale PBH remnants.
For BP2, this requires only $r≃1.2$.
Since neither the baryon asymmetry nor the remnant abundance depends appreciably on the initial abundance $β$, this background is the only observable sensitive to it and a detection would fix this free parameter of the scenario.
Therefore, n_BH(t_f) (a_f/a_ev)^3 = ρ_r(t_ev)/M_in= π^2/30g_*(T_ev) T_ev^4/M_in ≃2.33 ×10^2 M_p^10/M_in^7.
The baryon asymmetry can be generated in two distinct regimes depending on whether the black holes are hot enough to emit the lightest right-handed neutrino.
Conditions under which these results were obtained
assumption or scope
We assume that D-parity is broken at a sufficiently high scale such that the left-handed triplet is substantially heavier than the $SU(2)_R$ lb breaking scale.
assumption or scope
Throughout this work, we assume that PS symmetry breaking occurs prior to the onset of inflation.
assumption or scope
Throughout this work we restrict attention to histories in which the black holes come to dominate the total energy density before evaporating, $β≥β_min$ with $β_min$ given in Eq. ([ref]).
assumption or scope
We assume throughout that reheating following inflation completes before the scale $k_f$ re-enters the horizon, so that formation occurs in a radiation-dominated background; as noted in Sec. [ref], we take the minimal case in which the two epochs are close, $T_RH≃T_f$, since a prolonged radiation era between reheating and formation would strengthen the non-restoration bound of Eq. ([ref]) by the factor $T_RH/T_f$.
assumption or scope
We work in the single-flavour approximation and retain only inverse decays in the washout. $ΔL=2$ scatterings are neglected, which is justified for the RHN masses considered here, $M_N_1≲10^12$ GeV, at the temperatures at which the asymmetry is generated.
assumption or scope
We therefore work in the type-I seesaw-dominated limit, in which the contribution from $Δ_L$ to the light-neutrino mass matrix can be neglected.
assumption or scope
Consistency of this assumption requires that the inflationary Hubble scale to remain below the characteristic PS-breaking scale [equation] so that the Pati–Salam symmetry is not restored during inflation.
assumption or scope
In what follows we adopt the minimal assumption that reheating completes shortly before the scale $k_f$ re-enters the horizon, so that $T_RH≃T_f$ and Eq. ([ref]) is controlled by the formation temperature; for a more prolonged radiation-dominated era between reheating and formation the bound on $v_R$ strengthens by the factor $T_RH/T_f$.
assumption or scope
In the present work we adopt this monochromatic approximation, which captures the dominant dynamics and allows a transparent analytic connection between inflationary parameters, PBH mass, evaporation temperature, and the baryon asymmetry derived in subsequent sections.
assumption or scope
It applies only to the limiting case in which the entire $B-L$ asymmetry is carried by right-handed neutrinos emitted by the black holes, and it is therefore shown as a constraint only in the left panel of Fig. [ref].
assumption or scope
The analysis mandates that the Pati-Salam gauge symmetry is broken prior to the onset of inflation.
assumption or scope
This temporal sequencing imposes strict cosmological consistency conditions on the model: the Pati-Salam symmetry must not be thermally restored at any point in the post-inflationary history, and the initial black hole temperature must be sufficiently high to kinematically permit the unsuppressed emission of the massive right-handed neutrinos.
assumption or scope
Being a gauge singlet, $ϕ$ is inert under $G_PS$ and therefore plays no role in the symmetry-breaking chain of Eq. ([ref]) and the Pati-Salam transition is assumed to have completed before the onset of the inflationary phase described here, as discussed in Sec. [ref].
assumption or scope
Throughout the numerical analysis we adopt $g_*(T_BH)=g_*,s(T_BH)=106.75$ and $g_N_1=2$.
assumption or scope
In the limiting case where the lepton asymmetry is generated solely by the non-thermal RHNs produced from PBH evaporation, one needs to require that PBH evaporation should be completed before the EW sphalerons freeze out in order for the generated lepton asymmetry to be efficiently converted into the baryon asymmetry.
assumption or scope
If the black holes dominate the energy density before evaporating, essentially all of the ambient energy at $t_ev$ originates from the black holes themselves, and therefore the fraction of the relic density of PBH remnants at the moment of evaporation is given by Ω_rem (t_ev) = ρ_rem(t_ev)/ρ_tot(t_ev) = m_rem n_BH(t_ev)/M_in n_BH(t_ev) = m_rem/M_in = r M_Pl/M_in ≃2.17×10^-5 r(1 g/M_in) , up to corrections of order $m_rem/M_in$.
assumption or scope
This is the central simplification afforded by the PBH-dominated history assumed throughout, and it is the reason we restricted attention to that case in Sec. [ref].
assumption or scope
In the limiting case in which the baryon asymmetry is generated predominantly by RHNs emitted from PBHs, the resulting $B-L$ asymmetry must be produced and made available to the sphaleron processes before they become inefficient at $T_EW≃130 GeV$.
assumption or scope
We assume that one linear combination remains light and is identified with the Standard Model Higgs doublet.
assumption or scope
The relevant requirement is therefore that the post-inflationary plasma never reaches a temperature high enough to restore the PS symmetry. i.e. [equation] up to an order-one factor reflecting the ratio of the critical temperature to the VEV in the PS scalar potential.
Measurements
| Quantity | Group | Value | Where | Checked by code |
|---|---|---|---|---|
| sphaleron processes before they become inefficient at T_EW≃ | 130 GeV p 0.79 | Introduction | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Leptogenesis) |
|
| sphaleron freeze-out then yields the upper bound M_in≲ | 4.2×10^5 g p 0.98 | Introduction | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Introduction, Leptogenesis) |
|
| the entire observed dark matter abundance gives M_in≃ | 1.1×10^6 g p 0.98 | Introduction | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves) |
|
| value is larger than the upper bound M_in≲ | 4.2×10^5 g p 0.99 | Introduction | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Introduction, Leptogenesis) |
|
| this relation reduces parametrically to [equation] Taking m_ν≃ | 0.05 eV p 0.87 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Heavy RHN Production from PBH, Inflation and primordial black hole formation, Leptogenesis) |
|
| at formation is [equation] so that T_f/H_f ≃ | 1.6×10^2 p 0.96 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves) |
|
| formation is [equation] so that T_f/H_f ≃1.6×10^2 (M_in/ | 1 g p 0.96 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text | |
| Eq. ([ref]) throughout the mass range 0.4 g≲M_in≲ | 9.7×10^8 g p 0.97 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Leptogenesis) |
|
| weaker than Eq. ([ref]) throughout the mass range | 0.4 g p 0.97 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Inflation and primordial black hole formation, Leptogenesis, Pati–Salam framework and cosmological consistency conditions) |
|
| that range is itself the statement H_f < | 2.5×10^-5 p 0.98 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Leptogenesis, Pati–Salam framework and cosmological consistency conditions) |
|
| < 2.5×10^-5M_p, which Eq. ([ref]) reproduces as M_in≳ | 0.4 g p 0.98 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Inflation and primordial black hole formation, Leptogenesis, Pati–Salam framework and cosmological consistency conditions) |
|
| For the benchmark M_in≃ | 1.2×10^6 g p 0.98 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves, Pati–Salam framework and cosmological consistency conditions) |
|
| relevant to remnant dark matter it requires v_R≳ | 4×10^12 GeV p 0.98 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Leptogenesis) |
|
| matter it requires v_R≳4×10^12 GeV, while for M_in≃ | 1.3×10^2 g p 0.98 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Pati–Salam framework and cosmological consistency conditions) |
|
| GeV, while for M_in≃1.3×10^2 g it requires v_R≳ | 3.8×10^14 GeV p 0.98 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| in this work it gives y_R≲0.11 for M_in≃ | 1.2×10^6 g p 0.99 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves, Pati–Salam framework and cosmological consistency conditions) |
|
| it gives y_R≲0.11 for M_in≃1.2×10^6 g with M_N_1≃ | 4.4×10^11 GeV p 0.99 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| for M_in≃1.2×10^6 g with M_N_1≃4.4×10^11 GeV, and y_R≲ | 6.5×10^-4 p 0.99 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| g with M_N_1≃4.4×10^11 GeV, and y_R≲6.5×10^-4 for M_in≃ | 1.3×10^2 g p 0.99 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Pati–Salam framework and cosmological consistency conditions) |
|
| GeV, and y_R≲6.5×10^-4 for M_in≃1.3×10^2 g with M_N_1≃ | 2.5×10^11 GeV p 0.99 | Pati–Salam framework and cosmological consistency conditions | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Leptogenesis) |
|
| given by a_f/a_0 ≃T_0/T_f (g_*,s(T_0)/g_*,s (T_f) )^1/3 ≃ | 1.8 ×10^-29 p 0.96 | Inflation and primordial black hole formation | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| a_f/a_0 ≃T_0/T_f (g_*,s(T_0)/g_*,s (T_f) )^1/3 ≃1.8 ×10^-29 (M_in/ | 1 g p 0.96 | Inflation and primordial black hole formation | consistent number matches the authors' source text | |
| β_min ≃(H_ev/ √(2) H_f)^1/2 = 1/2^1/4 .T_ev|_MD/T_f = | 7 ×10^-2 p 0.97 | PBH evaporation and non-resonant leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Conclusion, Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves) |
|
| 1/2^1/4 .T_ev|_MD/T_f = 7 ×10^-2 (g_*(T_BH)/γ)^1/2 M_p/M_in ≃ | 6.8 ×10^-6 p 0.97 | PBH evaporation and non-resonant leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Leptogenesis, PBH evaporation and non-resonant leptogenesis) |
|
| 7 ×10^-2 (g_*(T_BH)/γ)^1/2 M_p/M_in ≃6.8 ×10^-6 (0.2/γ)^1/2 (g_*(T_BH)/100)^1/2 | 1 g p 0.97 | PBH evaporation and non-resonant leptogenesis | consistent number matches the authors' source text | |
| dominates the energy density before evaporation if β≳ | 6.8 ×10^-6 p 0.98 | PBH evaporation and non-resonant leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Leptogenesis, PBH evaporation and non-resonant leptogenesis) |
|
| Hence, a | 1 g p 0.98 | PBH evaporation and non-resonant leptogenesis | consistent number matches the authors' source text | |
| panel for the initial mass of PBHs M_in= | 1 g p 0.97 | PBH evaporation and non-resonant leptogenesis | consistent number matches the authors' source text | |
| a/a_0∼10^-21 in the bottom left panel for M_in= | 10 g p 0.97 | PBH evaporation and non-resonant leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Conclusion, Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves) |
|
| is Γ_N_1 =(y_ν^†y_ν) M_N_1/8π with y_ν^2 ∼m_νM_N_1/v^2, m_ν∼ | 0.05 eV p 0.81 | Heavy RHN Production from PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Inflation and primordial black hole formation, Leptogenesis, Pati–Salam framework and cosmological consistency conditions) |
|
| M_N_1/8π with y_ν^2 ∼m_νM_N_1/v^2, m_ν∼0.05 eV, and v= | 174 GeV p 0.81 | Heavy RHN Production from PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Leptogenesis) |
|
| Therefore, n_BH(t_f) (a_f/a_ev)^3 = ρ_r(t_ev)/M_in= π^2/30g_*(T_ev) T_ev^4/M_in ≃ | 2.33 ×10^2 p 0.97 | Heavy RHN Production from PBH | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| n_N_1(t_ev) ≃15 ζ(3)g_N_1 /π^4g_*(T_BH) M^2_in/M^2_p n_BH(t_f) (a_f/a_ev)^3 ≃ | 1.9 ×10^23 GeV p 0.97 | Heavy RHN Production from PBH | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| /π^4g_*(T_BH) M^2_in/M^2_p n_BH(t_f) (a_f/a_ev)^3 ≃1.9 ×10^23 GeV^3 | 10 g p 0.97 | Heavy RHN Production from PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Conclusion, Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves) |
|
| total number density of RHNs, around a/a_0 ∼ | 3 ×10^-23 p 0.98 | Heavy RHN Production from PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Heavy RHN Production from PBH, PBH evaporation and non-resonant leptogenesis) |
|
| justified for the RHN masses considered here, M_N_1≲10^ | 12 GeV p 0.92 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Pati–Salam framework and cosmological consistency conditions) |
|
| asymmetry given by [equation] where we take m_ν= | 0.05 eV p 0.94 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Heavy RHN Production from PBH, Inflation and primordial black hole formation, Pati–Salam framework and cosmological consistency conditions) |
|
| where we take m_ν= 0.05 eV and υ= | 174 GeV p 0.94 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Heavy RHN Production from PBH) |
|
| = 7.04 Y_B, and the observed value Y_B^obs≃ | 8.7×10^-11 p 0.99 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Introduction) |
|
| Y_B, and the observed value Y_B^obs≃8.7×10^-11 requires Y^f_B-L≃ | 2.5×10^-10 p 0.99 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Pati–Salam framework and cosmological consistency conditions) |
|
| for the PBH dominated epoch is β_min ≃ | 6.8 ×10^-7 g p 0.98 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (PBH evaporation and non-resonant leptogenesis) |
|
| For M_in = | 10 g p 0.98 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Conclusion, Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves) |
|
| the PBH dominated epoch to satisfy T_ev|_MD ≳T_EW≃ | 130 GeV p 0.98 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Introduction, Leptogenesis) |
|
| mass of PBH is found as M_in ≲ | 4.2 ×10^5 g p 0.99 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Introduction) |
|
| initial mass of PBH, 0.4 g ≲M_in ≲ | 9.7 ×10^8 g p 0.94 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Pati–Salam framework and cosmological consistency conditions) |
|
| allowed range of the initial mass of PBH | 0.4 g p 0.94 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Inflation and primordial black hole formation, Leptogenesis, Pati–Salam framework and cosmological consistency conditions) |
|
| that smaller initial PBH masses, 0.4 ≲M_in ≲10^ | 2 g p 0.97 | Leptogenesis | consistent number matches the authors' source text | |
| for the initial PBH mass, 0.5 ≲M_in ≲ | 10 g p 0.97 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Conclusion, Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves) |
|
| Moreover, for approximately M_in ≳10^ | 2 g p 0.96 | Leptogenesis | consistent number matches the authors' source text | |
| scenario, the chosen benchmark value of M_in ∼10^ | 6 g p 0.96 | Leptogenesis | consistent number matches the authors' source text | |
| the sphalerons in the standard way at T_EW≃ | 130 GeV p 0.98 | Leptogenesis | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Introduction, Leptogenesis) |
|
| n_BH(t_ev)/M_in n_BH(t_ev) = m_rem/M_in = r M_Pl/M_in ≃ | 2.17×10^-5 p 0.97 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH) |
|
| n_BH(t_ev) = m_rem/M_in = r M_Pl/M_in ≃2.17×10^-5 r | 1 g p 0.97 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text | |
| 30/π^2 g_*(T_ev) T_ev T_0^3 ( g_*,s(T_ev)/g_*,s(T_0) ) ≃ | 5.8 ×10^-10 p 0.98 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| g_*,s(T_ev)/g_*,s(T_0) ) ≃5.8 ×10^-10 f_rem (GeV/T_ev) ≃ | 1.6 ×10^-20 p 0.98 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| have used g_*,s(T_0) ≃3.91, g_*,s(T_ev) ≃g_*(T_ev), T_0 ≃ | 2.34 ×10^-13 GeV p 0.98 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| T_0 ≃2.34 ×10^-13 GeV and ρ_cr,0 ≃8 h^ | 2 ×10^-47 GeV p 0.98 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| ≃5.8 ×10^-10 f_rem (GeV/T_ev) ≃1.6 ×10^-20 f_rem (M_in/ | 1 g p 0.98 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text | |
| M_in/1 g) ≃ | 1.1 ×10^6 p 0.63 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves, Introduction) |
|
| M_in/ | 1 g p 0.63 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text | |
| M_in ≃ | 1.1 ×10^6 g p 0.97 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves, Introduction) |
|
| of the benchmark BP2 of Table [ref], M_in≃ | 1.2×10^6 g p 0.99 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves, Pati–Salam framework and cosmological consistency conditions) |
|
| matter abundance, for any initial mass above ≃ | 1.1×10^6 g p 0.98 | Dark Matter from Planck Scale Remnants of PBH | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Gravitational Waves, Introduction) |
|
| the signal associated with the gram-scale to 10^ | 6 g p 0.94 | Gravitational Waves | consistent number matches the authors' source text | |
| considered here redshifts to frequencies of order 10^4–10^ | 8 Hz p 0.94 | Gravitational Waves | consistent number matches the authors' source text | |
| for example BP1 gives f≃35 Hz at M_in= | 1.2×10^6 g p 0.98 | Gravitational Waves | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Pati–Salam framework and cosmological consistency conditions) |
|
| mean black hole separation, corresponding today to f≃ | 1.7 kHz p 0.98 | Gravitational Waves | consistent number matches the authors' source text not repeated same number stated elsewhere in the paper (stated once) |
|
| hole separation, corresponding today to f≃1.7 kHz (M_in/10^ | 4 g p 0.98 | Gravitational Waves | consistent number matches the authors' source text | |
| tens of hertz, for example BP1 gives f≃ | 35 Hz p 0.98 | Gravitational Waves | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (PBH evaporation and non-resonant leptogenesis, Pati–Salam framework and cosmological consistency conditions) |
|
| number of relativistic species at nucleosynthesis imposes β≲ | 1.1×10^-6 p 0.97 | Gravitational Waves | consistent number matches the authors' source text consistent same number stated elsewhere in the paper (Dark Matter from Planck Scale Remnants of PBH, Introduction) |
|
| number of relativistic species at nucleosynthesis imposes β≲1.1×10^-6(M_in/10^ | 4 g p 0.97 | Gravitational Waves | consistent number matches the authors' source text | |
| initial PBH masses, M_in = 1, 10^4, 10^ | 8 g p 0.73 | Gravitational Waves | consistent number matches the authors' source text | |
| Below roughly 10^ | 2 g p 0.97 | Conclusion | consistent number matches the authors' source text |
Limitations the authors state
- We use Eq. ([ref]) only as an order-of-magnitude estimate of the enhancement.
- It is important to be precise about the scope of Eq. ([ref]).
- It applies only to the limiting case in which the entire $B-L$ asymmetry is carried by right-handed neutrinos emitted by the black holes, and it is therefore shown as a constraint only in the left panel of Fig. [ref].
- In the vicinity of an inflection point the slow-roll hierarchy is by construction violated, and an accurate spectrum requires numerical solution of the Mukhanov–Sasaki equation [ref] , which typically shifts the peak amplitude by an $𝒪(1)$ to $𝒪(10)$ factor.
- The condition in Eq. ([ref]) is not, by itself, sufficient to guarantee that the broken PS phase is maintained throughout the subsequent cosmological evolution.
The full text, sentence by sentence
336 body sentences from the LaTeX source, each with the classifier's labels. The number is the probability it gave the label.
| section | sentence | role | the authors' own work | states the main conclusion | states a condition | quantitative |
|---|---|---|---|---|---|---|
| Introduction | The origin of the baryon asymmetry of the Universe (BAU) and the fundamental nature of dark matter (DM) remain two of the most profound and persistent open problems in modern cosmology and particle physics. | background 0.82 | 0.04 | 0.05 | 0.06 | 0.06 |
| Introduction | Astronomical observations of the cosmic microwave background and the primordial elemental abundances derived from Big Bang Nucleosynthesis (BBN) indicate a precise baryon-to-entropy ratio of $Y_B^obs ≃8.7 ×10^-11$, [ref] . | background 1.00 | 0.07 | 0.03 | 0.09 | 0.99 |
| Introduction | While the Standard Model (SM) of particle physics successfully accommodates both CP violation through the Cabibbo-Kobayashi-Maskawa (CKM) matrix [ref] and baryon number violation through non-perturbative electroweak sphaleron transitions, it falls quantitatively short of explaining the observed asymmetry. | background 0.97 | 0.12 | 0.05 | 0.12 | 0.40 |
| Introduction | The electroweak phase transition, given the measured mass of the Higgs boson, is a smooth crossover rather than a first-order phase transition, thereby failing to provide the required departure from thermal equilibrium dictated by the Sakharov conditions [ref] , and the inherent CP violation is insufficient by several orders of magnitude [ref] . | background 0.99 | 0.11 | 0.04 | 0.18 | 0.82 |
| Introduction | Furthermore, the Standard Model entirely lacks a viable, non-baryonic cold dark matter candidate to account for the roughly $27%$ of the Universe's energy budget. | background 0.98 | 0.07 | 0.04 | 0.09 | 0.96 |
| Introduction | Leptogenesis presents an elegant, minimal, and highly motivated solution to the baryogenesis problem by linking the generation of the baryon asymmetry to the out-of-equilibrium, CP-violating decays of heavy right-handed neutrinos (RHNs) [ref] . | background 1.00 | 0.06 | 0.05 | 0.11 | 0.05 |
| Introduction | In the canonical thermal leptogenesis paradigm, these heavy states are populated by the thermal bath of the early Universe [ref] . | background 0.99 | 0.05 | 0.03 | 0.42 | 0.04 |
| Introduction | Their subsequent CP-violating decays generate a primary lepton asymmetry, which is partially reprocessed into a net baryon number by baryon-plus-lepton ($B+L$) violating electroweak sphalerons processes which convert the $B-L$ asymmetry into a baryon asymmetry before they become ineffective at temperatures of order the electroweak scale. | background 0.99 | 0.06 | 0.04 | 0.38 | 0.17 |
| Introduction | This framework is deeply connected to the neutrino sector, as it simultaneously explains the extreme smallness of the active light neutrino masses through the type-I seesaw mechanism [ref] . | background 1.00 | 0.09 | 0.04 | 0.08 | 0.11 |
| Introduction | To provide a rigorous, symmetry-motivated origin for these right-handed neutrinos, the leptogenesis mechanism can be naturally embedded within the Pati-Salam gauge structure, $SU(4)_C ×SU(2)_L ×SU(2)_R$ [ref] . | background 0.53 | 0.18 | 0.06 | 0.25 | 0.08 |
| Introduction | In the Pati-Salam framework, quarks and leptons are fundamentally unified by treating lepton number as the fourth color. | background 0.64 | 0.10 | 0.05 | 0.16 | 0.10 |
| Introduction | Right-handed neutrinos arise automatically and inevitably as intrinsic components of the $SU(2)_R$ fermion doublets, making their existence a structural prediction of the gauge geometry rather than an ad hoc phenomenological extension of the Standard Model. | setup 0.55 | 0.15 | 0.08 | 0.19 | 0.06 |
| Introduction | Furthermore, the Majorana masses of these heavy right-handed neutrinos are directly tied to the $SU(2)_R$ symmetry-breaking scale through the vacuum expectation value of an extended scalar multiplet. | setup 0.81 | 0.29 | 0.09 | 0.27 | 0.05 |
| Introduction | This structural feature removes the arbitrariness often present in generic leptogenesis models, explicitly linking the high-energy mass scale of the right handed neutrinos to the grand unified symmetry-breaking scale. | setup 0.43 | 0.23 | 0.12 | 0.18 | 0.07 |
| Introduction | In standard hierarchical thermal leptogenesis, however, the viable parameter space is tightly restricted by the Davidson–Ibarra bound [ref] . | background 0.98 | 0.07 | 0.06 | 0.50 | 0.10 |
| Introduction | Achieving the observed baryon asymmetry within the standard thermal scenario therefore generally requires a sufficiently heavy $N_1$ and a correspondingly high temperature for its efficient production. | background 0.96 | 0.12 | 0.07 | 0.54 | 0.16 |
| Introduction | In a Pati–Salam cosmology, however, such high temperatures can be problematic if they restore the broken gauge symmetry after inflation and triggering the overproduction of dangerous topological relics like magnetic monopoles, which would subsequently overclose the Universe. | limitation 0.32 | 0.15 | 0.10 | 0.74 | 0.04 |
| Introduction | These considerations motivate a non-thermal source of RHNs that does not rely on their efficient production from the thermal plasma. | other 0.48 | 0.16 | 0.12 | 0.28 | 0.08 |
| Introduction | An alternative and highly compelling pathway that cleanly bypasses these high-temperature tensions is the non-thermal production of heavy right-handed neutrinos via the Hawking evaporation of primordial black holes. | setup 0.38 | 0.16 | 0.18 | 0.21 | 0.05 |
| Introduction | Primordial black holes can form in the early Universe following the horizon re-entry of enhanced small-scale curvature perturbations [ref] , which are typically generated during a transient ultra-slow-roll phase of cosmic inflation. | background 0.99 | 0.08 | 0.04 | 0.43 | 0.05 |
| Introduction | Depending on their initial mass fraction, primordial black holes can temporarily dominate the total energy density of the Universe, ushering in an early matter-dominated epoch [ref] . | background 0.99 | 0.11 | 0.06 | 0.58 | 0.07 |
| Introduction | As they evaporate through Hawking radiation, they inject massive amounts of entropy into the surrounding plasma [ref] and emit massive particles democratically, regardless of their specific Standard Model gauge couplings. | background 0.94 | 0.11 | 0.05 | 0.34 | 0.08 |
| Introduction | This democratic emission provides a genuinely non-thermal source for right-handed neutrinos [ref] and hence leptogenesis. | background 0.55 | 0.20 | 0.13 | 0.23 | 0.05 |
| Introduction | This mechanism therefore relaxes the need for the thermal plasma to populate the heavy RHN sector at temperatures of order $M_N_1$ allowing for successful baryogenesis at significantly lower reheating temperatures [ref] . | interpretation 0.77 | 0.24 | 0.18 | 0.35 | 0.52 |
| Introduction | In this manuscript, primordial black hole-assisted non-thermal leptogenesis is embedded within a minimal Pati-Salam cosmology. | setup 0.87 | 0.69 | 0.09 | 0.42 | 0.04 |
| Introduction | The analysis mandates that the Pati-Salam gauge symmetry is broken prior to the onset of inflation. | setup 1.00 | 0.53 | 0.11 | 0.96 | 0.06 |
| Introduction | This pre-inflationary breaking ensures that any dangerous topological defects, such as magnetic monopoles generated during the phase transition, are exponentially diluted by the inflationary expansion and do not disrupt subsequent cosmological evolution. | setup 0.96 | 0.35 | 0.07 | 0.80 | 0.07 |
| Introduction | This temporal sequencing imposes strict cosmological consistency conditions on the model: the Pati-Salam symmetry must not be thermally restored at any point in the post-inflationary history, and the initial black hole temperature must be sufficiently high to kinematically permit the unsuppressed emission of the massive right-handed neutrinos. | setup 0.93 | 0.56 | 0.12 | 0.96 | 0.06 |
| Introduction | A particularly important constraint arises from the interplay between PBH evaporation and electroweak sphaleron conversion. | limitation 0.23 | 0.46 | 0.07 | 0.28 | 0.05 |
| Introduction | In the limiting case in which the baryon asymmetry is generated predominantly by RHNs emitted from PBHs, the resulting $B-L$ asymmetry must be produced and made available to the sphaleron processes before they become inefficient at $T_EW≃130 GeV$. | setup 0.42 | 0.52 | 0.09 | 0.95 | 0.79 |
| Introduction | Requiring PBH evaporation to be completed before sphaleron freeze-out then yields the upper bound $M_in≲4.2×10^5 g ,$ for the initial PBH mass. | result 0.89 | 0.80 | 0.12 | 0.87 | 0.98 |
| Introduction | This bound is strongest for the PBH-dominated, PBH-only leptogenesis scenario. | interpretation 0.83 | 0.79 | 0.10 | 0.90 | 0.65 |
| Introduction | When thermally produced RHNs contribute to the asymmetry, part of the $B-L$ asymmetry is generated before PBH evaporation and the constraint on the evaporation time is correspondingly relaxed. | interpretation 0.58 | 0.68 | 0.11 | 0.75 | 0.08 |
| Introduction | Beyond the generation of the baryon asymmetry, we also investigate the possibility that quantum gravitational backreaction can halt the Hawking evaporation process as the black hole mass approaches the fundamental Planck scale. | setup 0.60 | 0.66 | 0.09 | 0.37 | 0.06 |
| Introduction | Theories such as Loop Quantum Gravity and various non-commutative geometries suggest that the event horizon may be replaced by a quantum-corrected structure, preventing the formation of a physical singularity and halting the evaporation. | background 0.91 | 0.06 | 0.07 | 0.55 | 0.04 |
| Introduction | This process leaves behind stable, non-radiating Planck-scale remnants that serve as highly motivated cold dark matter candidates [ref] . | background 0.81 | 0.14 | 0.13 | 0.29 | 0.08 |
| Introduction | Because these remnants are incredibly compact, massive compared to elementary particles, and interact exclusively through gravitational forces, they represent an idealized dark matter state that easily evades all conventional direct detection constraints. | interpretation 0.39 | 0.15 | 0.07 | 0.54 | 0.85 |
| Introduction | Their relic abundance is therefore determined by the number density of PBHs together with the remnant mass, and can be directly related to the initial PBH mass. | setup 0.60 | 0.39 | 0.12 | 0.31 | 0.05 |
| Introduction | By analytically tracking the relic density of these Planck-scale remnants, we find that there is a critical and inescapable phenomenological tension between the requirements of leptogenesis and dark matter production. | result 0.62 | 0.95 | 0.86 | 0.40 | 0.13 |
| Introduction | For the remnants to account for the entirety of the observed dark matter density, the required initial mass of the primordial black holes strictly exceeds the upper mass bound permitted by electroweak sphaleron conversion. | result 0.68 | 0.89 | 0.75 | 0.78 | 0.83 |
| Introduction | Requiring the remnants to account for the entire observed dark matter abundance gives $M_in≃1.1×10^6 g$. | result 0.99 | 0.90 | 0.29 | 0.73 | 0.98 |
| Introduction | This value is larger than the upper bound $M_in≲4.2×10^5 g$ obtained when the baryon asymmetry is generated solely by PBH-produced RHNs and must be processed by electroweak sphalerons before their freeze-out. | result 0.71 | 0.79 | 0.41 | 0.92 | 0.99 |
| Introduction | Consequently, within this hierarchical non-resonant setup considered here, PBH remnants cannot simultaneously constitute all of the dark matter and realize the PBH-only leptogenesis limit. | interpretation 0.49 | 0.89 | 0.79 | 0.92 | 0.17 |
| Introduction | The tension can instead be alleviated when thermally produced RHNs contribute to the $B-L$ asymmetry, or when the remnants constitute only a subdominant fraction of the observed dark matter density. | interpretation 0.76 | 0.81 | 0.35 | 0.89 | 0.07 |
| Introduction | Finally, although the stochastic gravitational wave background associated with the pre-inflationary Pati–Salam phase transition is exponentially diluted during inflation, the primordial black hole population can give rise to several distinct gravitational-wave-related signatures. | other 0.55 | 0.20 | 0.06 | 0.43 | 0.36 |
| Introduction | These include scalar-induced gravitational waves sourced by the enhanced curvature perturbations responsible for PBH formation, tensor perturbations associated with the inhomogeneous PBH distribution and its subsequent evolution, and gravitational radiation generated during PBH evaporation through processes such as direct Hawking emission and bremsstrahlun [ref] . | background 0.39 | 0.18 | 0.04 | 0.14 | 0.05 |
| Introduction | In this work, we focus specifically on the stochastic background of gravitons produced directly through Hawking evaporation of the PBHs. | other 0.37 | 0.62 | 0.06 | 0.89 | 0.04 |
| Introduction | The resulting ultra-high-frequency gravitational waves provide a complementary probe of the PBH population and of the underlying cosmological history, and may be relevant for future constraints on additional relativistic degrees of freedom through their contribution to the radiation energy density. | interpretation 0.90 | 0.32 | 0.09 | 0.21 | 0.05 |
| Introduction | These signatures span a vast range of frequencies, providing distinct, testable predictions for future gravitational wave observatories such as the Einstein Telescope (ET) [ref] , Cosmic Explorer (CE) [ref] , and next-generation cosmic microwave background surveys [ref] targeting the effective number of relativistic species. | interpretation 0.53 | 0.23 | 0.05 | 0.18 | 0.09 |
| Introduction | This paper is organized as follows: we summarize the relevant aspects of the PS model and its connection to PBH-assisted leptogenesis in section [ref]. | other 1.00 | 0.20 | 0.03 | 0.08 | 0.04 |
| Introduction | Moreover, we discuss the PBH formation scenario during inflation in section [ref], while we review the basics of PBH evaporation and the non-resonant leptogenesis scenario in section [ref]. | other 1.00 | 0.35 | 0.03 | 0.12 | 0.05 |
| Introduction | In section [ref], we further investigate the Planck scale PBH remnants as a dark matter candidate. | other 0.99 | 0.66 | 0.07 | 0.11 | 0.04 |
| Introduction | In section [ref], we point out that our scenario can be tested by the existing and future gravitational wave experiments and finally conclude in section [ref]. | other 0.96 | 0.48 | 0.05 | 0.10 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | We embed the PBH-assisted leptogenesis scenario studied in this work within the Pati–Salam (PS) gauge framework [equation] originally proposed in Refs. [ref] . | setup 0.86 | 0.51 | 0.04 | 0.49 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | In this framework, quarks and leptons are fundamentally unified by treating lepton number as the fourth color. | setup 0.95 | 0.13 | 0.04 | 0.21 | 0.08 |
| Pati–Salam framework and cosmological consistency conditions | For each generation, the left-handed and right-handed fermion representations transform as [equation] where the numbers denote representations under $SU(4)_C$, $SU(2)_L$, and $SU(2)_R$, respectively. | setup 1.00 | 0.16 | 0.04 | 0.31 | 0.07 |
| Pati–Salam framework and cosmological consistency conditions | Because the right-handed multiplet $F_R = (u_R, d_R, ν_R, e_R)$ explicitly contains the right-handed neutrino $ν_R ≡N_R$ without additional assumptions, the existence of heavy Majorana right-handed neutrinos is a rigorous structural prediction of the gauge symmetry rather than an ad hoc extension of the Standard Model. | setup 0.87 | 0.23 | 0.09 | 0.22 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | Thus, the PS framework provides a gauge-theoretic origin for the right-handed neutrinos required for the seesaw mechanism and leptogenesis. | interpretation 0.42 | 0.20 | 0.09 | 0.14 | 0.03 |
| Pati–Salam framework and cosmological consistency conditions | We consider the symmetry-breaking chain [equation] where the first step occurs at a scale $v_R$ associated with the breaking of $SU(2)_R$. | setup 1.00 | 0.48 | 0.04 | 0.65 | 0.05 |
| Pati–Salam framework and cosmological consistency conditions | The first stage involves the breaking of the PS gauge symmetry, including $SU(4)_C→SU(3)_C×U(1)_B-L$ and $SU(2)_R→U(1)_R$, with the hypercharge generator $Y = T_3R + (B-L)/2$ arising from the appropriate combination of the $SU(2)_R$ and $B-L$ generators. | setup 0.98 | 0.24 | 0.04 | 0.46 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | The parameter $v_R$ denotes the vacuum expectation value (VEV) of the scalar multiplet responsible for $SU(2)_R$ breaking. | setup 1.00 | 0.32 | 0.04 | 0.23 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | This breaking is driven by a scalar multiplet $Δ_R ∼(10,1,3)$. | setup 1.00 | 0.41 | 0.03 | 0.46 | 0.11 |
| Pati–Salam framework and cosmological consistency conditions | The Majorana mass term for the right-handed neutrinos arises from a Yukawa interaction of the form [equation] | setup 1.00 | 0.45 | 0.03 | 0.22 | 0.03 |
| Pati–Salam framework and cosmological consistency conditions | When the neutral component acquires a vacuum expectation value $⟨Δ_R ⟩= v_R$, the Yukawa interaction generates a Majorana mass matrix for the right-handed neutrinos, [equation] with $y_R$ is a dimensionless Yukawa coupling matrix and Eq. ([ref]) is understood as a matrix relation, up to an $𝒪(1)$ normalization factor arising from the triplet convention, which we absorb into the definition of $y_R$. | setup 1.00 | 0.67 | 0.04 | 0.81 | 0.11 |
| Pati–Salam framework and cosmological consistency conditions | Thus, the PS framework directly correlates the RHN mass scale with the scale of $SU(2)_R$ breaking. | setup 0.64 | 0.47 | 0.11 | 0.21 | 0.11 |
| Pati–Salam framework and cosmological consistency conditions | In the hierarchical regime relevant for the leptogenesis analysis, we identify the lightest RHN mass as [equation] where $y_R^(1)$ denotes the corresponding eigenvalue in the basis in which the RHN mass matrix is diagonal. | setup 1.00 | 0.79 | 0.05 | 0.87 | 0.08 |
| Pati–Salam framework and cosmological consistency conditions | In the numerical analysis below, we therefore regard $y_R^(1)$ as a derived quantity once $M_N_1$ and $v_R$ are specified. | setup 0.70 | 0.82 | 0.04 | 0.62 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | Due to the underlying left-right discrete symmetry (D-parity) of the Pati–Salam group, the scalar sector naturally contains a left-handed triplet $Δ_L ∼(10,3,1)$ alongside the right-handed triplet $Δ_R ∼(10,1,3)$. | setup 0.88 | 0.25 | 0.03 | 0.21 | 0.09 |
| Pati–Salam framework and cosmological consistency conditions | If $Δ_L$ acquires an induced vacuum expectation value $v_L$, it yields an additional type-II seesaw contribution to the light-neutrino mass matrix. | setup 0.67 | 0.30 | 0.04 | 0.48 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | We assume that D-parity is broken at a sufficiently high scale such that the left-handed triplet is substantially heavier than the $SU(2)_R$ lb breaking scale. | setup 1.00 | 0.79 | 0.05 | 0.98 | 0.90 |
| Pati–Salam framework and cosmological consistency conditions | Its induced VEV is then suppressed, schematically as $v_L∝v_EW^2/M_Δ_L,$ up to model-dependent scalar couplings. | setup 0.89 | 0.50 | 0.04 | 0.76 | 0.21 |
| Pati–Salam framework and cosmological consistency conditions | We therefore work in the type-I seesaw-dominated limit, in which the contribution from $Δ_L$ to the light-neutrino mass matrix can be neglected. | setup 0.99 | 0.85 | 0.06 | 0.97 | 0.28 |
| Pati–Salam framework and cosmological consistency conditions | This assumption fixes the neutrino-mass relation used in the leptogenesis analysis below without introducing an additional type-II contribution. | setup 0.93 | 0.79 | 0.05 | 0.78 | 0.05 |
| Pati–Salam framework and cosmological consistency conditions | Below the PS-breaking scale, the effective theory consists of the Standard Model gauge group supplemented by heavy singlet neutrinos of mass $M_N$ (the RHNs, now $SU(2)_L$ singlets). | setup 1.00 | 0.32 | 0.04 | 0.80 | 0.09 |
| Pati–Salam framework and cosmological consistency conditions | Electroweak symmetry breaking is realized through a scalar bi-doublet transforming as $(1,2,2)$ under $G_PS$, which contains the Standard Model Higgs doublet after $SU(2)_R$ breaking. | setup 0.99 | 0.29 | 0.04 | 0.58 | 0.16 |
| Pati–Salam framework and cosmological consistency conditions | We assume that one linear combination remains light and is identified with the Standard Model Higgs doublet. | setup 1.00 | 0.70 | 0.03 | 0.95 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | The Dirac neutrino mass matrix takes the form [equation] where $y_ν$ is the Dirac Yukawa coupling matrix and $v_EW=246$ GeV is the electroweak vacuum expectation value. | setup 1.00 | 0.49 | 0.03 | 0.32 | 0.90 |
| Pati–Salam framework and cosmological consistency conditions | Light neutrino masses arise through the type-I seesaw mechanism [ref] , [equation] | background 0.59 | 0.16 | 0.04 | 0.30 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | For the single-flavour approximation employed in the Boltzmann analysis, this relation reduces parametrically to [equation] Taking $m_ν≃0.05 eV$ as a representative light-neutrino mass scale gives [equation] This relation is used below to determine the effective Dirac Yukawa coupling entering the RHN decay width and the leptogenesis dynamics. | setup 0.77 | 0.82 | 0.04 | 0.94 | 0.87 |
| Pati–Salam framework and cosmological consistency conditions | Throughout this work, we assume that PS symmetry breaking occurs prior to the onset of inflation. | setup 1.00 | 0.51 | 0.05 | 0.98 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | Any monopoles or other topological relics produced during the PS phase transition are therefore diluted away by inflation and do not affect the subsequent cosmological evolution. | setup 1.00 | 0.35 | 0.07 | 0.81 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | Consistency of this assumption requires that the inflationary Hubble scale to remain below the characteristic PS-breaking scale [equation] so that the Pati–Salam symmetry is not restored during inflation. | setup 0.98 | 0.54 | 0.06 | 0.97 | 0.84 |
| Pati–Salam framework and cosmological consistency conditions | After inflation the Universe is described by the Standard Model gauge group, with the right-handed neutrinos present in the spectrum as massive states. | setup 1.00 | 0.32 | 0.05 | 0.54 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | The condition in Eq. ([ref]) is not, by itself, sufficient to guarantee that the broken PS phase is maintained throughout the subsequent cosmological evolution. | limitation 0.64 | 0.63 | 0.07 | 0.71 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | Since the PS-breaking transition occurs before inflation, any topological defects regenerated after inflation would no longer be diluted by subsequent inflation. | setup 0.68 | 0.35 | 0.06 | 0.56 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | The relevant requirement is therefore that the post-inflationary plasma never reaches a temperature high enough to restore the PS symmetry. i.e. [equation] up to an order-one factor reflecting the ratio of the critical temperature to the VEV in the PS scalar potential. | setup 0.88 | 0.59 | 0.07 | 0.95 | 0.13 |
| Pati–Salam framework and cosmological consistency conditions | Here $T_RH$ is the reheating temperature and $T_f$ is the plasma temperature at PBH formation. | setup 1.00 | 0.23 | 0.03 | 0.16 | 0.05 |
| Pati–Salam framework and cosmological consistency conditions | Since the PBH-forming modes re-enter the horizon during radiation domination, formation necessarily postdates the completion of reheating and $T_RH>T_f$ in general. | setup 0.88 | 0.50 | 0.04 | 0.63 | 0.93 |
| Pati–Salam framework and cosmological consistency conditions | In what follows we adopt the minimal assumption that reheating completes shortly before the scale $k_f$ re-enters the horizon, so that $T_RH≃T_f$ and Eq. ([ref]) is controlled by the formation temperature; for a more prolonged radiation-dominated era between reheating and formation the bound on $v_R$ strengthens by the factor $T_RH/T_f$. | setup 0.94 | 0.76 | 0.04 | 0.97 | 0.57 |
| Pati–Salam framework and cosmological consistency conditions | In the cosmological history considered here, the hottest epoch relevant to the present mechanism is the moment of PBH formation, at which the plasma temperature is given by [equation] Following PBH formation, the radiation bath cools as $T∝a^-1$ during radiation domination and subsequently as $T∝a^-3/8$ during the PBH-dominated epoch, before approaching the standard radiation-dominated scaling after evaporation. | setup 0.98 | 0.64 | 0.05 | 0.75 | 0.24 |
| Pati–Salam framework and cosmological consistency conditions | To strictly avoid the thermal regeneration of the PS symmetry, the breaking scale must remain larger than this formation temperature: [equation] | setup 0.92 | 0.59 | 0.06 | 0.95 | 0.78 |
| Pati–Salam framework and cosmological consistency conditions | Because $T_ev<T_f$ throughout the mass range considered, Eq. ([ref]) automatically implies the corresponding non-restoration condition at evaporation. | interpretation 0.54 | 0.75 | 0.05 | 0.88 | 0.70 |
| Pati–Salam framework and cosmological consistency conditions | Using $M_in = 4πγM_p^2/H_f$ with $γ≃0.2$, the Hubble rate at formation is [equation] so that $T_f/H_f ≃1.6×10^2 (M_in/1 g)^1/2$. | result 0.55 | 0.85 | 0.05 | 0.82 | 0.96 |
| Pati–Salam framework and cosmological consistency conditions | Since $H_inf≥H_f$, the constraint $v_R > H_inf$ is weaker than Eq. ([ref]) throughout the mass range $0.4 g≲M_in≲9.7×10^8$ g permitted by the CMB and BBN bounds [ref] . | interpretation 0.52 | 0.83 | 0.07 | 0.92 | 0.97 |
| Pati–Salam framework and cosmological consistency conditions | We note in passing that the lower end of that range is itself the statement $H_f < 2.5×10^-5M_p$, which Eq. ([ref]) reproduces as $M_in≳0.4$ g. | interpretation 0.47 | 0.79 | 0.05 | 0.74 | 0.98 |
| Pati–Salam framework and cosmological consistency conditions | Equation ([ref]) is a lower bound on the symmetry-breaking scale that tightens as the PBH mass decreases. | interpretation 0.57 | 0.82 | 0.10 | 0.54 | 0.39 |
| Pati–Salam framework and cosmological consistency conditions | For the benchmark $M_in≃1.2×10^6$ g relevant to remnant dark matter it requires $v_R≳4×10^12$ GeV, while for $M_in≃1.3×10^2$ g it requires $v_R≳3.8×10^14$ GeV. | result 0.98 | 0.88 | 0.10 | 0.88 | 0.98 |
| Pati–Salam framework and cosmological consistency conditions | Sub-gram PBHs push $v_R$ to within an order of magnitude of the Planck scale and are therefore disfavoured within a pre-inflationary breaking scenario. | interpretation 0.87 | 0.85 | 0.15 | 0.78 | 0.94 |
| Pati–Salam framework and cosmological consistency conditions | The heavy right-handed neutrinos governing the generation of the lepton asymmetry | setup 0.96 | 0.13 | 0.04 | 0.20 | 0.03 |
| Pati–Salam framework and cosmological consistency conditions | can originate from two distinct channels, each constrained differently by the Pati–Salam architecture. | setup 0.66 | 0.44 | 0.09 | 0.23 | 0.15 |
| Pati–Salam framework and cosmological consistency conditions | First, Hawking emission from the PBHs is efficient only when the emitted species is light compared with the instantaneous black hole temperature [ref] , [equation] where $M_p≃2.435×10^18$ GeV is the reduced Planck mass. | setup 0.65 | 0.42 | 0.05 | 0.91 | 0.89 |
| Pati–Salam framework and cosmological consistency conditions | Combining Eq. ([ref]) with Eq. ([ref]), the natural kinematic threshold for unsuppressed emission, [equation] translates into [equation] We emphasise that Eq. ([ref]) is an efficiency criterion and not a hard kinematic cut. | setup 0.36 | 0.76 | 0.06 | 0.89 | 0.13 |
| Pati–Salam framework and cosmological consistency conditions | Emission above threshold is Boltzmann suppressed rather than forbidden, and this suppression is retained exactly in our numerical treatment through the function $ℱ(z)$ of Eq. ([ref]) with $z=M_N_1/T_BH$. | method 0.90 | 0.86 | 0.06 | 0.79 | 0.11 |
| Pati–Salam framework and cosmological consistency conditions | Second, if efficient thermal production of $N_1$ is to occur, | setup 0.52 | 0.14 | 0.04 | 0.93 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | it demands that the plasma reach a temperature comparable to the RHN mass i.e. $M_N_1≲T_f$. | setup 0.80 | 0.53 | 0.06 | 0.94 | 0.93 |
| Pati–Salam framework and cosmological consistency conditions | Combined with the non-restoration requirement in Eq. ([ref]), $v_R≳T_f$, these conditions firmly bracket the Yukawa coupling of the PS scalar sector: [equation] | result 0.74 | 0.76 | 0.07 | 0.88 | 0.79 |
| Pati–Salam framework and cosmological consistency conditions | The content of Eq. ([ref]) lies in its numerical value at the benchmarks rather than in the parametric statement $y_R≲1$, which follows trivially from combining $v_R≳T_f$ with $M_N_1≲T_f$. | interpretation 0.85 | 0.78 | 0.08 | 0.73 | 0.53 |
| Pati–Salam framework and cosmological consistency conditions | At the two benchmarks considered in this work it gives $y_R≲0.11$ for $M_in≃1.2×10^6$ g with $M_N_1≃4.4×10^11$ GeV, and $y_R≲6.5×10^-4$ for $M_in≃1.3×10^2$ g with $M_N_1≃2.5×10^11$ GeV. | result 1.00 | 0.93 | 0.06 | 0.59 | 0.99 |
| Pati–Salam framework and cosmological consistency conditions | We note that the requirement $M_N_1≲T_f$ applies only when the thermal channel is invoked; in the PBH-only limit it should be dropped, and $y_R$ is then bounded only by the ratio $M_N_1/v_R$ with $M_N_1$ a free parameter. | interpretation 0.33 | 0.81 | 0.10 | 0.94 | 0.18 |
| Pati–Salam framework and cosmological consistency conditions | The RHN mass is thus required to sit well below the $SU(2)_R$ breaking scale, which is a genuine prediction of embedding the mechanism in a pre-inflationary Pati–Salam cosmology rather than an assumption. | interpretation 0.82 | 0.83 | 0.27 | 0.49 | 0.68 |
| Pati–Salam framework and cosmological consistency conditions | In presenting numerical results we therefore treat $y_R$ as a derived quantity fixed by the pair $(M_in, M_N_1)$, and not as a free parameter set to unity. | method 0.73 | 0.89 | 0.11 | 0.68 | 0.12 |
| Pati–Salam framework and cosmological consistency conditions | Two further conditions delimit the scenario. | setup 0.56 | 0.32 | 0.03 | 0.52 | 0.04 |
| Pati–Salam framework and cosmological consistency conditions | The first concerns the initial PBH abundance $β=ρ_BH/ρ_rad|_T_f$. | setup 1.00 | 0.43 | 0.03 | 0.73 | 0.09 |
| Pati–Salam framework and cosmological consistency conditions | Throughout this work we restrict attention to histories in which the black holes come to dominate the total energy density before evaporating, $β≥β_min$ with $β_min$ given in Eq. ([ref]). | setup 0.83 | 0.82 | 0.05 | 0.98 | 0.72 |
| Pati–Salam framework and cosmological consistency conditions | We emphasise that this is a choice of cosmological history rather than a consistency requirement of the model. | setup 0.61 | 0.46 | 0.05 | 0.85 | 0.05 |
| Pati–Salam framework and cosmological consistency conditions | It is, however, the case in which the ambient entropy of the post-evaporation plasma is generated entirely by the black holes themselves rather than inherited from the primordial inflationary bath, so that the reheating scale $T_ev|_MD$ is fixed purely by the black hole lifetime and the relic abundance of Planck-scale remnants becomes independent of $β$. | setup 0.47 | 0.72 | 0.10 | 0.87 | 0.14 |
| Pati–Salam framework and cosmological consistency conditions | The complementary regime $β<β_min$, in which evaporation occurs during radiation domination, is not excluded and is revisited in Sec. [ref], where it is shown to provide one of the two ways of relieving the tension between remnant dark matter and sphaleron reprocessing. | other 0.34 | 0.83 | 0.11 | 0.66 | 0.19 |
| Pati–Salam framework and cosmological consistency conditions | Since $β_min∝M_in^-1$, domination becomes progressively easier as the PBH mass is increased, so that the heavier PBHs required by the remnant dark matter condition demand the least fine-tuned curvature enhancement. | interpretation 0.92 | 0.82 | 0.10 | 0.61 | 0.76 |
| Pati–Salam framework and cosmological consistency conditions | The second is that the mass loss driving the evaporation, computed from the Hawking spectrum with greybody corrections [ref] , terminate early enough for the accumulated $(B-L)$ asymmetry to be reprocessed by the electroweak sphalerons. | setup 0.52 | 0.44 | 0.09 | 0.93 | 0.08 |
| Pati–Salam framework and cosmological consistency conditions | When the thermal channel contributes, the asymmetry is already present in the plasma well before evaporation and this restriction is correspondingly relaxed. | interpretation 0.63 | 0.69 | 0.10 | 0.86 | 0.19 |
| Pati–Salam framework and cosmological consistency conditions | Table [ref] collects the two benchmark points used throughout this work and the values taken by each of the conditions discussed above. | setup 0.48 | 0.82 | 0.05 | 0.20 | 0.06 |
| Pati–Salam framework and cosmological consistency conditions | BP1 realizes the PBH-only leptogenesis limit; BP2 is the mass required for Planck-scale remnants to saturate the observed dark matter abundance, and fails the sphaleron condition, as anticipated in Sec. [ref] and quantified in Sec. [ref]. | interpretation 0.46 | 0.86 | 0.21 | 0.69 | 0.22 |
| Pati–Salam framework and cosmological consistency conditions | Collecting the above, the Pati–Salam embedding correlates the symmetry-breaking scale with the PBH mass through the interdependent relations: [equation] so that the breaking scale, the inflationary scale, and the PBH mass are not independent inputs. | interpretation 0.51 | 0.79 | 0.13 | 0.63 | 0.10 |
| Pati–Salam framework and cosmological consistency conditions | The structure of the allowed region is controlled by the single relation $v_R ≳T_f ∝M_in^-1/2$, which permits low breaking scales only for heavy PBHs, precisely the regime selected by the remnant dark matter requirement analyzed later in this work. | interpretation 0.57 | 0.83 | 0.18 | 0.90 | 0.54 |
| Pati–Salam framework and cosmological consistency conditions | Within this window $v_R$ is constrained by cosmological consistency and by the leptogenesis requirements rather than by specific gauge unification assumptions [ref] . | interpretation 0.82 | 0.65 | 0.14 | 0.82 | 0.12 |
| Pati–Salam framework and cosmological consistency conditions | The Pati–Salam structure therefore determines the kinematically consistent region and ties the RHN mass to a gauge symmetry breaking scale, but does not by itself fix the normalisation of the baryon asymmetry. | interpretation 0.73 | 0.71 | 0.15 | 0.60 | 0.05 |
| Inflation and primordial black hole formation | We assume that PBHs originate from enhanced curvature perturbations generated during inflation. | setup 1.00 | 0.38 | 0.04 | 0.95 | 0.03 |
| Inflation and primordial black hole formation | The inflaton sector consists of a single real gauge singlet scalar field $ϕ$, minimally coupled to gravity, with canonical kinetic term and potential $V(ϕ)$. | setup 1.00 | 0.45 | 0.03 | 0.87 | 0.08 |
| Inflation and primordial black hole formation | Being a gauge singlet, $ϕ$ is inert under $G_PS$ and therefore plays no role in the symmetry-breaking chain of Eq. ([ref]) and the Pati-Salam transition is assumed to have completed before the onset of the inflationary phase described here, as discussed in Sec. [ref]. | setup 1.00 | 0.43 | 0.04 | 0.96 | 0.05 |
| Inflation and primordial black hole formation | To make the origin of the required small-scale enhancement explicit, we consider a potential possessing a localized near-inflection feature, a construction widely employed to generate PBHs within single-field inflation [ref] . | setup 0.72 | 0.44 | 0.04 | 0.50 | 0.05 |
| Inflation and primordial black hole formation | Expanding about the feature at $ϕ=ϕ_0$, | setup 0.60 | 0.54 | 0.02 | 0.44 | 0.06 |
| Inflation and primordial black hole formation | where $V_0$ sets the overall inflationary energy density, $ϕ_0$ denotes the approximate inflection point, and $a$ and $b$ are dimensionless parameters controlling deviations from exact flatness. | setup 1.00 | 0.40 | 0.02 | 0.53 | 0.06 |
| Inflation and primordial black hole formation | Equation ([ref]) is a local expansion valid in the neighbourhood of the feature. | setup 0.83 | 0.43 | 0.03 | 0.91 | 0.04 |
| Inflation and primordial black hole formation | The global completion of the potential, which must possess a minimum in which inflation terminates and reheating proceeds, is model dependent and plays no role in the small-scale enhancement studied here; explicit realizations may be found in Refs. [ref] . | limitation 0.54 | 0.22 | 0.04 | 0.88 | 0.05 |
| Inflation and primordial black hole formation | Since Eq. ([ref]) contains no quadratic term, $V”(ϕ_0)=0$ holds identically, and the near-inflection condition | setup 0.97 | 0.71 | 0.03 | 0.91 | 0.21 |
| Inflation and primordial black hole formation | reduces to the single requirement $|a|≪1$, leading to a transient ultra-slow-roll phase during which the inflaton velocity $ϕ̇$ is strongly suppressed. | setup 0.74 | 0.75 | 0.05 | 0.91 | 0.79 |
| Inflation and primordial black hole formation | Since $V'(ϕ_0)=V_0 a/M_p$ and $V(ϕ_0)=V_0$, the potential slow-roll parameter,, which controls the inflaton kinetic energy through $ϕ̇^2 ≃2ϵ_V V$, at the feature is fixed directly by $a$, [equation] so that the parameter $a$ is not merely small but quantitatively determined by the required amplitude of the enhancement, as we make explicit below. | setup 0.69 | 0.83 | 0.05 | 0.77 | 0.18 |
| Inflation and primordial black hole formation | The cubic coefficient $b$ controls the width of the feature in field space and hence the width of the resulting peak in $k$-space, and is chosen such that the enhancement is narrow enough for the monochromatic approximation adopted here. | setup 0.76 | 0.86 | 0.04 | 0.91 | 0.06 |
| Inflation and primordial black hole formation | The gauge-invariant curvature perturbation $ζ$, defined on uniform-density hypersurfaces, has dimensionless power spectrum | setup 0.99 | 0.15 | 0.02 | 0.16 | 0.04 |
| Inflation and primordial black hole formation | where $k$ is the comoving wavenumber and $ζ_k$ denotes the Fourier mode. | setup 0.99 | 0.08 | 0.02 | 0.09 | 0.03 |
| Inflation and primordial black hole formation | In standard slow-roll inflation this spectrum is evaluated at horizon exit $k=aH$, approximately [ref] | background 0.81 | 0.11 | 0.02 | 0.75 | 0.06 |
| Inflation and primordial black hole formation | We use Eq. ([ref]) only as an order-of-magnitude estimate of the enhancement. | limitation 0.88 | 0.81 | 0.04 | 0.94 | 0.20 |
| Inflation and primordial black hole formation | In the vicinity of an inflection point the slow-roll hierarchy is by construction violated, and an accurate spectrum requires numerical solution of the Mukhanov–Sasaki equation [ref] , which typically shifts the peak amplitude by an $𝒪(1)$ to $𝒪(10)$ factor. | limitation 0.82 | 0.52 | 0.05 | 0.69 | 0.90 |
| Inflation and primordial black hole formation | Because the PBH abundance depends on $𝒫_ζ$ only logarithmically, this does not affect the conclusions drawn below. | limitation 0.56 | 0.73 | 0.07 | 0.84 | 0.21 |
| Inflation and primordial black hole formation | Near the inflection point the suppression of $V'$ reduces $ϵ_V$ by several orders of magnitude while $V≃V_0$ remains essentially constant across the feature. | setup 0.56 | 0.71 | 0.05 | 0.78 | 0.52 |
| Inflation and primordial black hole formation | A temporary reduction from $ϵ_V ∼10^-3$, consistent with the bound $r ≃16 ϵ_V < 0.036$ on CMB scales [ref] , to $ϵ_V ∼10^-10$ enhances the curvature spectrum from | result 0.53 | 0.69 | 0.04 | 0.76 | 0.98 |
| Inflation and primordial black hole formation | at the small scale $k_f$ associated with PBH formation. | setup 0.91 | 0.44 | 0.03 | 0.59 | 0.07 |
| Inflation and primordial black hole formation | Because the feature is localized in field space, this enhancement is confined to ultraviolet scales and leaves large-scale observables unaffected. | interpretation 0.76 | 0.65 | 0.07 | 0.79 | 0.09 |
| Inflation and primordial black hole formation | After inflation, enhanced modes re-enter the Hubble horizon during radiation domination. | setup 0.91 | 0.33 | 0.03 | 0.44 | 0.04 |
| Inflation and primordial black hole formation | We assume throughout that reheating following inflation completes before the scale $k_f$ re-enters the horizon, so that formation occurs in a radiation-dominated background; as noted in Sec. [ref], we take the minimal case in which the two epochs are close, $T_RH≃T_f$, since a prolonged radiation era between reheating and formation would strengthen the non-restoration bound of Eq. ([ref]) by the factor $T_RH/T_f$. | setup 1.00 | 0.72 | 0.04 | 0.98 | 0.73 |
| Inflation and primordial black hole formation | The comoving curvature perturbation $ζ$ is related to the density contrast $δ$ at horizon crossing by $δ≃(4/9)ζ$ in radiation domination. | background 0.70 | 0.18 | 0.03 | 0.79 | 0.73 |
| Inflation and primordial black hole formation | Regions collapse to form PBHs if their density contrast exceeds a threshold $δ_c ∼0.4$ [ref] . | background 0.63 | 0.13 | 0.04 | 0.87 | 0.95 |
| Inflation and primordial black hole formation | The initial PBH mass at the formation temperature $T_f$ is approximately related to the horizon mass at re-entry, [equation] where the efficiency of collapse is $γ≃0.2$ [ref] and the total energy density at the formation temperature is $ρ_tot(T_f) = 3 M_p^2 H^2(T_f)$, so that Eq. ([ref]) reduces to $M_in=4πγM_p^2/H_f$ as used in Eq. ([ref]). | setup 0.58 | 0.66 | 0.03 | 0.85 | 0.88 |
| Inflation and primordial black hole formation | Therefore, the corresponding formation temperature $T_f$ is related to $M_in$ as: [equation] For this calculation, the PBH formation time $t_f$ is determined by [equation] Equivalently, the comoving scale associated with mass $M_in$ becomes [equation] with the corresponding scale factor given by a_f/a_0 ≃T_0/T_f (g_*,s(T_0)/g_*,s (T_f) )^1/3 ≃1.8 ×10^-29 (M_in/1 g)^1/2 . | method 0.56 | 0.82 | 0.03 | 0.76 | 0.96 |
| Inflation and primordial black hole formation | From Eq. ([ref]), one can realize that the comoving scale $k_f$ lies many orders of magnitude beyond the range of scales probed by the CMB. | interpretation 0.61 | 0.81 | 0.04 | 0.26 | 0.83 |
| Inflation and primordial black hole formation | where $σ^2(M_in)$ is the variance of the smoothed density contrast. | setup 0.90 | 0.19 | 0.02 | 0.23 | 0.06 |
| Inflation and primordial black hole formation | For Gaussian perturbations and $σ≪δ_c$, the integral gives | method 0.65 | 0.55 | 0.02 | 0.93 | 0.35 |
| Inflation and primordial black hole formation | up to order-one window-function factors, the numerical coefficient following from $δ≃(4/9)ζ$ for a narrow peak. | method 0.33 | 0.43 | 0.03 | 0.88 | 0.81 |
| Inflation and primordial black hole formation | Equations ([ref]) and ([ref]) determine the curvature amplitude required to realize a given initial abundance. | method 0.80 | 0.64 | 0.04 | 0.26 | 0.06 |
| Inflation and primordial black hole formation | The relevant range of $β$ in this work is set at the lower end by the PBH-domination requirement of Eq. ([ref]), and at the upper end by the requirement that the induced gravitational wave background not spoil the effective number of relativistic species at nucleosynthesis, discussed in Sec. [ref]. | setup 0.58 | 0.80 | 0.04 | 0.94 | 0.07 |
| Inflation and primordial black hole formation | Achieving $β∼10^-8$–$10^-7$ therefore requires $σ∼0.05$–$0.1$, corresponding to | result 0.77 | 0.75 | 0.04 | 0.75 | 0.95 |
| Inflation and primordial black hole formation | Although this amplitude is several orders of magnitude larger than the CMB value, it is phenomenologically viable provided the enhancement is sufficiently narrow in $k$-space. | setup 0.49 | 0.39 | 0.06 | 0.95 | 0.93 |
| Inflation and primordial black hole formation | Narrow features avoid constraints from CMB spectral distortions and pulsar timing arrays [ref] , and are well approximated by a monochromatic PBH mass function. | setup 0.47 | 0.25 | 0.05 | 0.90 | 0.11 |
| Inflation and primordial black hole formation | In the present work we adopt this monochromatic approximation, which captures the dominant dynamics and allows a transparent analytic connection between inflationary parameters, PBH mass, evaporation temperature, and the baryon asymmetry derived in subsequent sections. | setup 0.57 | 0.88 | 0.05 | 0.97 | 0.05 |
| PBH evaporation and non-resonant leptogenesis | After formation, PBHs can emit all particles in the spectrum through Hawking evaporation provided their masses satisfy $m_i ≲T_BH$ where [equation] with the reduced Planck mass is $M_p ≃2.435 ×10^18 GeV ≃4.33 ×10^-6 g$. | setup 0.82 | 0.37 | 0.04 | 0.92 | 0.97 |
| PBH evaporation and non-resonant leptogenesis | For a Schwarzschild black hole, the greybody factor in the geometric optics (GO) limit is approximated as $σ_s_i = (27/64π) M_BH^2/M_p^4$. | setup 0.61 | 0.40 | 0.03 | 0.94 | 0.55 |
| PBH evaporation and non-resonant leptogenesis | Consequently, the spectrum simplifies to [ref] [equation] with the sign in the denominator distinguishing bosonic from fermionic emission. | setup 0.44 | 0.63 | 0.03 | 0.49 | 0.11 |
| PBH evaporation and non-resonant leptogenesis | Defining the comoving energy densities of radiation and PBH as $ρ̃_r = a^4 ρ_r$ and $ρ̃_BH = a^3 ρ_BH$, the Boltzmann equations governing their evolution are [ref] [equation] where $ℰ_SM≡∑_i∈SM g_iℰ_i (z_i)$ accounts for the evaporation contributions from Standard Model particles, excluding right-handed neutrinos. | setup 0.84 | 0.72 | 0.03 | 0.76 | 0.08 |
| PBH evaporation and non-resonant leptogenesis | The total PBH evaporation function is $ℰ(M_BH) ≡∑_i g_i ℰ_i (z_i)$ with $z_i=m_i/T_BH$ and [equation] We denote the evaporation function by $ℰ$ rather than $ε$ to avoid confusion with the CP asymmetry parameter $ε_1$ introduced in Sec. [ref]. | setup 0.98 | 0.78 | 0.03 | 0.30 | 0.14 |
| PBH evaporation and non-resonant leptogenesis | In the GO limit ($ψ_s_i=1$), PBH mass evolution follows [equation] with the PBH lifetime given by [equation] Since most of the PBH energy is emitted near the end of evaporation, the evaporation temperature, $T_ev$, depends on whether the Universe is PBH dominated (matter-dominated, MD) or radiation-dominated (RD) prior to evaporation [equation] [equation] | setup 0.53 | 0.63 | 0.04 | 0.90 | 0.16 |
| PBH evaporation and non-resonant leptogenesis | the ratio following from $H_ev=1/2τ_BH$ during radiation domination as against $H_ev=2/3τ_BH$ during matter domination. | setup 0.73 | 0.50 | 0.03 | 0.72 | 0.64 |
| PBH evaporation and non-resonant leptogenesis | An important quantity is the ratio of the PBH lifetime to the formation time, $t_f$, defined in Eq. ([ref]), τ_BH/t_f ≃38/g_*(T_BH) (M_in/M_p)^2 ≫1 , indicating that the PBH lifetime is much longer than the formation time. | setup 0.77 | 0.70 | 0.04 | 0.78 | 0.97 |
| PBH evaporation and non-resonant leptogenesis | Consequently, an early PBH dominated era can be readily realized. | interpretation 0.72 | 0.72 | 0.08 | 0.26 | 0.06 |
| PBH evaporation and non-resonant leptogenesis | The evaporation time is given by $t_ev= τ_BH + t_f$, which can be approximated as $t_ev ≃τ_BH$, since $t_f ≪τ_BH$. | setup 0.71 | 0.73 | 0.04 | 0.94 | 0.91 |
| PBH evaporation and non-resonant leptogenesis | If the Universe indeed undergoes an early MD epoch driven by PBHs, we denote by $a_eeq$ the scale factor at the epoch of early radiation–PBH equality. | setup 1.00 | 0.53 | 0.03 | 0.81 | 0.05 |
| PBH evaporation and non-resonant leptogenesis | At the early equality time and the PBH formation time, the energy densities satisfy 3 M_p^2 H^2_eeq = ρ_BH(T_eeq) + ρ_r(T_eeq) = 2 ρ_BH(T_eeq) , and 3 M_p^2 H^2_f = ρ_BH(T_f) + ρ_r (T_f) =ρ_BH(T_eeq) [ (a_f/a_eeq)^-3 + (a_f/a_eeq)^-4] . | setup 0.51 | 0.78 | 0.03 | 0.83 | 0.75 |
| PBH evaporation and non-resonant leptogenesis | Comparing them, one can obtain H^2_eeq/H^2_f = 2/[ (a_f/a_eeq)^-3 + (a_f/a_eeq)^-4] ≈2 (a_f/a_eeq)^4 . | method 0.54 | 0.90 | 0.04 | 0.79 | 0.90 |
| PBH evaporation and non-resonant leptogenesis | Thus, the definition of $β$ can be rewritten as β= ρ_BH (T_f)/ρ_tot(T_f) = ρ_BH(T_eeq) (a_f/a_eeq)^-3/ρ_BH(T_eeq) [ (a_f/a_eeq)^-3 + (a_f/a_eeq)^-4] = 1/1+(a_f/a_eeq)^-1 , which is simplified to $1/β= 1+ (a_eeq/a_f)$. | method 0.42 | 0.88 | 0.04 | 0.67 | 0.35 |
| PBH evaporation and non-resonant leptogenesis | If $β≪1$, Eq. ([ref]) becomes H_eeq/H_f ≈√(2) β^2 . | result 0.62 | 0.86 | 0.04 | 0.94 | 0.87 |
| PBH evaporation and non-resonant leptogenesis | Therefore, depending on the initial PBH energy density fraction, PBH evaporation may occur either during the RD epoch or during the early PBH-driven MD epoch. | interpretation 0.49 | 0.76 | 0.08 | 0.59 | 0.16 |
| PBH evaporation and non-resonant leptogenesis | Imposing the condition for the existence of the PBH dominated era, namely $H_ev≤H_eeq$, the minimum initial PBH abundance is obtained as β_min ≃(H_ev/ √(2) H_f)^1/2 = 1/2^1/4 .T_ev|_MD/T_f = 7 ×10^-2 (g_*(T_BH)/γ)^1/2 M_p/M_in ≃6.8 ×10^-6 (0.2/γ)^1/2 (g_*(T_BH)/100)^1/2(1 g/M_in). | result 0.88 | 0.87 | 0.05 | 0.90 | 0.97 |
| PBH evaporation and non-resonant leptogenesis | Hence, a $1 g$ PBH dominates the energy density before evaporation if $β≳6.8 ×10^-6$. | result 0.98 | 0.78 | 0.08 | 0.84 | 0.98 |
| PBH evaporation and non-resonant leptogenesis | As we assume a monochromatic PBH mass spectrum, the comoving PBH number density remains conserved: [equation] where the initial number density $n_BH(t_f)$ is given by [equation] In addition to Eq. ([ref]), the temperature evolution can be written in a simple form by taking into account the non-conservation of entropy due to PBH evaporation dT/d ln(a) = - T (1+ T/3 g_*,s(T) d g_*,s (T)/dT )^-1 [1- 1/4 g_*(T)/g_*,s(T) 1/ρ̃_rad d ρ̃_rad/d ln(a) ] . | setup 0.52 | 0.77 | 0.04 | 0.94 | 0.14 |
| PBH evaporation and non-resonant leptogenesis | Equation ([ref]) is the relation that carries the entropy injected by the black holes into the plasma, and it is through this equation that the dilution of any pre-existing asymmetry is captured; we return to this point in Sec. [ref]. | method 0.42 | 0.69 | 0.05 | 0.20 | 0.04 |
| PBH evaporation and non-resonant leptogenesis | Throughout the numerical analysis we adopt $g_*(T_BH)=g_*,s(T_BH)=106.75$ and $g_N_1=2$. | setup 0.86 | 0.89 | 0.02 | 0.96 | 0.89 |
| PBH evaporation and non-resonant leptogenesis | In Fig. [ref], we show the numerical results obtained by simultaneously solving the Boltzmann equations in Eq. ([ref]) and the plasma temperature evolution equation, Eq. ([ref]). | method 0.88 | 0.92 | 0.03 | 0.21 | 0.06 |
| PBH evaporation and non-resonant leptogenesis | The left panels show the evolution of the comoving energy densities of radiation (red), and PBH (black) as functions of the scale factor $a$. | result 0.82 | 0.87 | 0.03 | 0.14 | 0.07 |
| PBH evaporation and non-resonant leptogenesis | For the chosen initial conditions, PBHs come to dominate the energy density of the Universe before completely evaporating. | result 0.99 | 0.89 | 0.11 | 0.68 | 0.54 |
| PBH evaporation and non-resonant leptogenesis | Their energy density shows a sharp decrease at $a/a_0∼10^-23$ in the top left panel for the initial mass of PBHs $M_in=1 g$ with $β=10^-4$, and at $a/a_0∼10^-21$ in the bottom left panel for $M_in=10 g$ with the same value of $β$. | result 1.00 | 0.91 | 0.05 | 0.43 | 0.97 |
| PBH evaporation and non-resonant leptogenesis | Since the PBH energy density redshifts as non-relativistic matter, their comoving energy density remains constant throughout the evolution. | interpretation 0.60 | 0.51 | 0.05 | 0.62 | 0.07 |
| PBH evaporation and non-resonant leptogenesis | During the MD epoch, the radiation energy density scales as $a^-3/2$, implying that the comoving radiation energy density grows as $a^3/2$. | interpretation 0.39 | 0.61 | 0.05 | 0.61 | 0.32 |
| PBH evaporation and non-resonant leptogenesis | The right panels display the evolution of the plasma temperature as a function of the scale factor $a$. | result 0.70 | 0.86 | 0.03 | 0.07 | 0.04 |
| PBH evaporation and non-resonant leptogenesis | A distinct kink appears in the temperature evolution at approximately $a/a_0∼10^-23$ (top) and $10^-21$ (bottom), denoting the onset of the entropy injection from PBH evaporation into the thermal plasma. | result 0.82 | 0.90 | 0.07 | 0.19 | 0.94 |
| PBH evaporation and non-resonant leptogenesis | Following the completion of PBH evaporation, the Universe returns to the standard RD evolution, and the plasma temperature resumes its usual scaling, $T ∝a^-1$. | result 0.83 | 0.79 | 0.06 | 0.36 | 0.15 |
| Heavy RHN Production from PBH | We now compute the baryon asymmetry generated by heavy right-handed neutrinos ($N_1$) emitted during PBH evaporation. | method 0.83 | 0.81 | 0.05 | 0.21 | 0.06 |
| Heavy RHN Production from PBH | The Boltzmann equation tracking the evolution of RHNs produced from PBH evaporation is: [equation] which has to be solved along with Eq. ([ref]) simultaneously. | method 0.95 | 0.72 | 0.03 | 0.22 | 0.05 |
| Heavy RHN Production from PBH | Here $ñ_N_1 ≡n_N_1 a^3$ is the comoving number density of RHNs, and $Γ_N_1^BH$ is the decay width corrected by an average inverse time dilatation factor as given by [ref] Γ_N_1^BH ≡⟨M_N_1/E_N_1⟩_BH Γ_N_1 ≃K_1(M_N_1/T_BH)/K_2(M_N_1/T_BH) Γ_N_1 , where the total RHN decay width is $Γ_N_1 =(y_ν^†y_ν) M_N_1/8π$ with $y_ν^2 ∼m_νM_N_1/v^2$, $m_ν∼0.05$ eV, and $v=174$ GeV. | setup 0.76 | 0.53 | 0.04 | 0.59 | 0.81 |
| Heavy RHN Production from PBH | The momentum-integrated sterile neutrino emission rate from PBH evaporation is: [equation] where $x=E/T_BH$ and $z = m_N_1/T_BH$. | method 0.65 | 0.70 | 0.03 | 0.24 | 0.08 |
| Heavy RHN Production from PBH | In the GO limit, the emission rate simplifies to [ref] : [equation] where $ℱ(z) = [z Li_2(e^-z) + Li_3(e^-z)]$, with $Li_n$ denoting the polylog function of order $n$. | method 0.61 | 0.44 | 0.03 | 0.90 | 0.26 |
| Heavy RHN Production from PBH | In the high-temperature limit ($z≪1$), $ℱ(z) →ζ(3) ≈1.20206$. | method 0.42 | 0.43 | 0.03 | 0.89 | 0.97 |
| Heavy RHN Production from PBH | For our numerical analysis, we utilize the "" package [ref] , incorporating modified graybody factors. | method 1.00 | 0.83 | 0.02 | 0.27 | 0.04 |
| Heavy RHN Production from PBH | Using the sudden decay approximation with $ρ_BH(t_ev) = ρ_rad(t_ev)$, we find the relation, ρ_BH(t_ev)= ρ_BH(t_f) (a_f/a_ev)^3= ρ_r(t_ev) = π^2/30g_*(T_ev) T_ev^4 . | method 0.83 | 0.93 | 0.04 | 0.94 | 0.32 |
| Heavy RHN Production from PBH | The ratio of scale factors $a_f/a_ev$ is determined using entropy conservation, along with Eqs. ([ref]) and ([ref]), and is given by: a_f/a_ev = T_ev|_MD/T_f (g_*,s(T_ev)/g_*,s(T_f))^1/3 ≃1.9 M_p/M_in. | method 0.51 | 0.88 | 0.04 | 0.79 | 0.95 |
| Heavy RHN Production from PBH | Therefore, n_BH(t_f) (a_f/a_ev)^3 = ρ_r(t_ev)/M_in= π^2/30g_*(T_ev) T_ev^4/M_in ≃2.33 ×10^2 M_p^10/M_in^7. | result 0.86 | 0.92 | 0.04 | 0.58 | 0.97 |
| Heavy RHN Production from PBH | The sterile neutrino number density is estimated as n_N_1(t_ev) ≃15 ζ(3)g_N_1 /π^4g_*(T_BH) M^2_in/M^2_p n_BH(t_f) (a_f/a_ev)^3 ≃1.9 ×10^23 GeV^3 (10 g/M_in )^5 . | result 0.89 | 0.92 | 0.06 | 0.62 | 0.97 |
| Heavy RHN Production from PBH | Since the number density of non-thermally produced RHNs is inversely proportional to the initial PBH mass, the production of the non-thermal RHNs is consequently suppressed as the initial PBH mass increases. | interpretation 0.61 | 0.86 | 0.11 | 0.29 | 0.57 |
| Heavy RHN Production from PBH | In Fig. [ref], we compare the different contributions to the production of RHNs: (i) thermal production only in the absence of PBHs (black dotted line), (ii) non-thermal production solely from PBH evaporation (blue dashed line), (iii) non-thermal production from PBH evaporation in the GO limit (green dashed line), and (iv) the combined thermal and non-thermal contributions (blue solid line). | result 0.56 | 0.90 | 0.06 | 0.28 | 0.25 |
| Heavy RHN Production from PBH | The excellent agreement between the green dashed line, which represents the numerical solution obtained in the GO limit, and the analytic estimation in Eq. ([ref]) validates our numerical implementation of PBH-induced RHN production. | interpretation 0.44 | 0.91 | 0.07 | 0.57 | 0.47 |
| Heavy RHN Production from PBH | As shown by the evolution of the total number density of RHNs, around $a/a_0 ∼3 ×10^-23$ the production of non-thermal RHN from PBH overtakes the thermal contribution due to the entropy injection into the thermal bath during PBH evaporation. | result 0.97 | 0.92 | 0.10 | 0.22 | 0.98 |
| Heavy RHN Production from PBH | Nevertheless, as the Universe continues to evolve, the thermal contribution eventually dominates the total RHN number density evolution. | result 0.97 | 0.88 | 0.10 | 0.17 | 0.81 |
| Heavy RHN Production from PBH | Furthermore, increasing $M_in$ suppresses the non-thermal production of RHNs, in agreement with the analytic scaling shown in Eq. ([ref]). | result 0.84 | 0.89 | 0.10 | 0.35 | 0.43 |
| Leptogenesis | In order to track the lepton asymmetry, we need to account for the comoving number density of $N_1$ produced thermally from the plasma, in addition to the non-thermal production via PBH evaporation described by Eq. ([ref]). | method 0.73 | 0.63 | 0.05 | 0.44 | 0.05 |
| Leptogenesis | The Boltzmann equations governing the cosmological evolution of the thermally produced RHN number density and the $(B-L)$ asymmetry generated through the decay of the RHNs are written as [ref] [equation] where $Γ_N_1^TH= Γ_N_1 K_1(M_N_1/T)/K_2(M_N_1/T)$ is the thermally averaged decay rate and $n_N_1^eq$ is the equilibrium number density of the RHNs as given by n_N_1^eq = g_N_1/2 π^2 M_N_1^2 T K_2(M_N_1/T) . | method 0.61 | 0.61 | 0.04 | 0.36 | 0.12 |
| Leptogenesis | Here, $ε_1$ is the CP asymmetry parameter generated by $N_1$, $Br_N_1→l$ denotes the branching ratio of RHN into leptons, $Br_N_1→l=1$, and $W_1$ represents the washout term due to inverse decays W_1 = 1/4 Γ_N_1^TH K_2(M_N_1/T) (M_N_1/T)^2 . | setup 0.95 | 0.55 | 0.03 | 0.79 | 0.73 |
| Leptogenesis | We work in the single-flavour approximation and retain only inverse decays in the washout. $ΔL=2$ scatterings are neglected, which is justified for the RHN masses considered here, $M_N_1≲10^12$ GeV, at the temperatures at which the asymmetry is generated. | setup 0.75 | 0.84 | 0.04 | 0.98 | 0.92 |
| Leptogenesis | Considering the interference of tree level diagram with one-loop diagrams the $CP$ asymmetry parameter is written as [ref] [equation] where the functions $f_v(x)$ and $f_s(x)$ arise from the one-loop vertex and self-energy corrections, respectively, and are given by [equation] Using the Davidson–Ibarra bound, one can find the simple expression for the $CP$ asymmetry given by [equation] where we take $m_ν= 0.05 eV$ and $υ=174 GeV$ in the last step, and where we work throughout in the hierarchical, non-resonant regime $M_N_1≪M_N_2,3$. | method 0.51 | 0.64 | 0.04 | 0.94 | 0.94 |
| Leptogenesis | The final $(B-L)$ asymmetry is converted into a baryon asymmetry by the electroweak sphalerons according to [equation] where $Y^f_B-L=ñ_B-L/(s a^3)$ is evaluated once PBH evaporation is complete and the asymmetry has saturated. | method 0.49 | 0.57 | 0.07 | 0.79 | 0.09 |
| Leptogenesis | We emphasise that no separate dilution factor is required in Eq. ([ref]) as the entropy density $s$ is taken from the full numerical solution of Eq. ([ref]), and therefore already includes the entropy injected by the evaporating black holes as well as the standard reduction in $g_*,s$ between leptogenesis and recombination. | method 0.88 | 0.84 | 0.09 | 0.79 | 0.11 |
| Leptogenesis | A fixed dilution factor of the form $g_*(T_lep)/g_*,s(T_0)$, appropriate to a standard thermal history, would not capture the PBH entropy release, which is the dominant dilution effect in the present scenario. | method 0.73 | 0.64 | 0.12 | 0.52 | 0.58 |
| Leptogenesis | For reference, the corresponding baryon-to-photon ratio today is $η_B = 7.04 Y_B$, and the observed value $Y_B^obs≃8.7×10^-11$ requires $Y^f_B-L≃2.5×10^-10$. | setup 0.53 | 0.50 | 0.05 | 0.26 | 0.99 |
| Leptogenesis | In Fig. [ref], we show the evolution of the yields of non-thermally produced RHNs from PBH evaporation together with the $(B-L)$ asymmetry as a function of the scale factor for four different initial PBH abundances, $β$. | result 0.75 | 0.90 | 0.05 | 0.19 | 0.09 |
| Leptogenesis | For $M_in = 10 g$ case, the minimum initial PBH abundance required for the PBH dominated epoch is $β_min ≃6.8 ×10^-7 g$. | result 0.98 | 0.89 | 0.06 | 0.73 | 0.98 |
| Leptogenesis | As a consequence, for $β=10^-6$ (lower right panel), the PBH dominated era is shorter than in the cases with larger initial abundances, leading to reduced production of non-thermal RHNs. | interpretation 0.52 | 0.90 | 0.10 | 0.61 | 0.95 |
| Leptogenesis | The figure clearly shows that the dependence of the final $(B-L)$ asymmetry on $β$ is relatively weak once the Universe undergoes the PBH dominated epoch. | interpretation 0.60 | 0.92 | 0.14 | 0.75 | 0.45 |
| Leptogenesis | The $(B-L)$ asymmetry initially grows due to the thermal contribution of RHNs and subsequently diminishes due to the washout effect induced by the inverse decays. | result 0.81 | 0.68 | 0.09 | 0.25 | 0.06 |
| Leptogenesis | During the PBH evaporation, the entropy injection into the thermal bath causes an additional dilution of the asymmetry. | result 0.77 | 0.76 | 0.10 | 0.38 | 0.07 |
| Leptogenesis | After the PBHs have completely evaporated, the $(B-L)$ asymmetry saturates reaching a constant value. | result 0.97 | 0.78 | 0.12 | 0.37 | 0.05 |
| Leptogenesis | The dilution effect due to the entropy injection becomes more prominent for massive PBHs as they evaporate over a longer duration. | result 0.82 | 0.77 | 0.11 | 0.42 | 0.81 |
| Leptogenesis | In this work, we consider the scenario in which the lepton asymmetry generated by decay of RHNs, including both thermal and non-thermal contributions described by Eq. ([ref]), is converted into observed baryon asymmetry through the $(B-L)$-violating EW sphaleron transitions. | setup 0.90 | 0.73 | 0.07 | 0.62 | 0.06 |
| Leptogenesis | In the limiting case where the lepton asymmetry is generated solely by the non-thermal RHNs produced from PBH evaporation, one needs to require that PBH evaporation should be completed before the EW sphalerons freeze out in order for the generated lepton asymmetry to be efficiently converted into the baryon asymmetry. | setup 0.58 | 0.63 | 0.11 | 0.96 | 0.17 |
| Leptogenesis | This requires the PBH evaporation temperature during the PBH dominated epoch to satisfy $T_ev|_MD ≳T_EW≃130 GeV$. | setup 0.53 | 0.67 | 0.07 | 0.95 | 0.98 |
| Leptogenesis | Therefore, using Eq. ([ref]), the corresponding upper bound on the initial mass of PBH is found as M_in ≲4.2 ×10^5 g. | result 0.98 | 0.91 | 0.11 | 0.64 | 0.99 |
| Leptogenesis | In Fig. [ref], we perform a parameter scan to identify the region consistent with the observed baryon asymmetry by randomly varying the initial PBH mass and the RHN mass over the ranges $M_in ∈[0.4, 10^9] g$ and $M_N_1 ∈[10^5, 10^13] GeV$, respectively. | method 0.99 | 0.94 | 0.05 | 0.81 | 0.79 |
| Leptogenesis | For each parameter point, the initial PBH abundance, $β$, is chosen to satisfy the condition for the PBH dominated epoch. | method 0.97 | 0.92 | 0.05 | 0.91 | 0.06 |
| Leptogenesis | The allowed range of the initial mass of PBH, $0.4 g ≲M_in ≲9.7 ×10^8 g$ [ref] , is indicated by gray shaded regions. | result 0.81 | 0.83 | 0.07 | 0.73 | 0.94 |
| Leptogenesis | As a reference, the upper bound on $M_in$ derived from Eq. ([ref]), corresponding to the requirement that PBH evaporation be completed before the EW sphalerons freeze out, is shown by the gray dashed line. | method 0.43 | 0.87 | 0.05 | 0.80 | 0.11 |
| Leptogenesis | The left panel of Fig. [ref] shows the parameter space capable of generating the observed baryon asymmetry when only the PBH contribution to leptogenesis is taken into account, whereas the right panel includes both the thermal and PBH contributions. | result 0.40 | 0.92 | 0.10 | 0.70 | 0.19 |
| Leptogenesis | One can see that smaller initial PBH masses, $0.4 ≲M_in ≲10^2 g$[Ref. [ref] found a narrower allowed range for the initial PBH mass, $0.5 ≲M_in ≲10 g$, based on the requirement of reproducing the observed baryon asymmetry from PBH evaporation.], are more efficient to generate the baryon asymmetry in the PBH only scenario than in the case where both production mechanisms are included. | result 0.52 | 0.71 | 0.16 | 0.46 | 0.97 |
| Leptogenesis | Moreover, for approximately $M_in ≳10^2 g$, leptogenesis is predominantly driven by the thermal production of RHNs, while the contribution from PBH evaporation becomes subdominant. | result 0.52 | 0.89 | 0.15 | 0.79 | 0.96 |
| Leptogenesis | In Fig. [ref], we show the evolution of the relevant yields for two different scenarios: considering only the PBH contribution (left) and the combined thermal and PBH contributions (right), respectively. | result 0.77 | 0.88 | 0.08 | 0.29 | 0.20 |
| Leptogenesis | The corresponding benchmark parameter values are selected from the parameter scan presented in Fig. [ref], and correspond to the benchmark points BP1 and BP2 of Table [ref]. | method 0.94 | 0.86 | 0.05 | 0.33 | 0.11 |
| Leptogenesis | Furthermore, in the combined scenario, the chosen benchmark value of $M_in ∼10^6 g$ simultaneously reproduces the correct dark matter relic abundance through Planck scale PBH remnants. | result 0.90 | 0.89 | 0.28 | 0.51 | 0.96 |
| Leptogenesis | The corresponding initial PBH mass is consistent with the value obtained in Eq. ([ref]). | result 0.73 | 0.84 | 0.09 | 0.29 | 0.46 |
| Leptogenesis | It is important to be precise about the scope of Eq. ([ref]). | limitation 0.84 | 0.56 | 0.04 | 0.46 | 0.03 |
| Leptogenesis | It applies only to the limiting case in which the entire $B-L$ asymmetry is carried by right-handed neutrinos emitted by the black holes, and it is therefore shown as a constraint only in the left panel of Fig. [ref]. | limitation 0.83 | 0.84 | 0.08 | 0.97 | 0.09 |
| Leptogenesis | When the thermal channel contributes, the asymmetry is generated at $T∼M_N_1≫T_EW$, long before evaporation, and is reprocessed by the sphalerons in the standard way at $T_EW≃130$ GeV. | setup 0.42 | 0.52 | 0.10 | 0.81 | 0.98 |
| Leptogenesis | Evaporation at a later time, $T_ev|_MD<T_EW$, then no longer destroys the asymmetry but merely dilutes it: the baryon number per comoving volume is conserved once the sphalerons have frozen out, while the entropy is increased by the black hole decay. | interpretation 0.85 | 0.72 | 0.13 | 0.84 | 0.59 |
| Leptogenesis | This dilution is fully included in the numerical solution through Eq. ([ref]) and is visible as the drop in $Y_B-L$ near $a/a_0∼10^-14$ in the right panel of Fig. [ref]. | result 0.42 | 0.93 | 0.07 | 0.21 | 0.90 |
| Leptogenesis | We note in passing that the sphaleron freeze-out temperature is set by the electroweak crossover, through the exponential suppression of the sphaleron rate once the Higgs acquires its vacuum expectation value, rather than by a competition between the sphaleron rate and the expansion rate; the modified expansion history during the PBH-dominated epoch therefore does not shift $T_EW$ appreciably. | interpretation 0.67 | 0.61 | 0.06 | 0.74 | 0.27 |
| Dark Matter from Planck Scale Remnants of PBH | While the semiclassical Hawking evaporation predicts a complete mass loss, quantum gravitational backreaction is expected to become significant as the black hole mass approaches the Planck scale $M_Pl=(8π)^1/2M_p ≃1.22×10^19 GeV≃2.17 ×10^-5$ g. | setup 0.71 | 0.13 | 0.05 | 0.72 | 0.98 |
| Dark Matter from Planck Scale Remnants of PBH | Theories such as Loop Quantum Gravity suggest that the event horizon may be replaced by a quantum-corrected structure, preventing the singularity and halting evaporation. | background 0.59 | 0.08 | 0.05 | 0.59 | 0.06 |
| Dark Matter from Planck Scale Remnants of PBH | As a result, a stable, non-radiating remnant may form with mass $m_rem = r M_Pl$, where $r ≥1$ is an order-one coefficient, whose precise value is not predicted by any of the candidate frameworks and which we therefore retain as a free parameter throughout. | setup 0.95 | 0.44 | 0.06 | 0.89 | 0.43 |
| Dark Matter from Planck Scale Remnants of PBH | Such remnants are natural candidates for cold dark matter [ref] . | background 0.68 | 0.12 | 0.07 | 0.18 | 0.08 |
| Dark Matter from Planck Scale Remnants of PBH | They are compact, massive compared to elementary particles, and interact predominantly through gravity, rendering them effectively invisible to both direct detection and astrophysical searches. | background 0.45 | 0.12 | 0.06 | 0.34 | 0.86 |
| Dark Matter from Planck Scale Remnants of PBH | To assess whether these remnants can account for the observed dark matter abundance today, we must calculate their relic density within our PBH-assisted leptogenesis scenario. | method 0.39 | 0.24 | 0.06 | 0.28 | 0.05 |
| Dark Matter from Planck Scale Remnants of PBH | The abundance at the moment of evaporation follows from a simple counting argument. | method 0.83 | 0.75 | 0.06 | 0.24 | 0.03 |
| Dark Matter from Planck Scale Remnants of PBH | Each evaporating black hole leaves behind exactly one remnant, so $n_rem(t_ev) = n_BH(t_ev)$, while the energy released into the plasma is $(M_in-m_rem)n_BH(t_ev)$ per unit volume. | setup 0.51 | 0.69 | 0.09 | 0.82 | 0.36 |
| Dark Matter from Planck Scale Remnants of PBH | If the black holes dominate the energy density before evaporating, essentially all of the ambient energy at $t_ev$ originates from the black holes themselves, and therefore the fraction of the relic density of PBH remnants at the moment of evaporation is given by Ω_rem (t_ev) = ρ_rem(t_ev)/ρ_tot(t_ev) = m_rem n_BH(t_ev)/M_in n_BH(t_ev) = m_rem/M_in = r M_Pl/M_in ≃2.17×10^-5 r(1 g/M_in) , up to corrections of order $m_rem/M_in$. | result 0.85 | 0.90 | 0.13 | 0.96 | 0.97 |
| Dark Matter from Planck Scale Remnants of PBH | The remnant fraction at evaporation is thus nothing other than the fraction of the initial PBH mass that survives, and in particular it is entirely independent of the initial abundance $β$. | result 0.61 | 0.88 | 0.21 | 0.81 | 0.13 |
| Dark Matter from Planck Scale Remnants of PBH | This is the central simplification afforded by the PBH-dominated history assumed throughout, and it is the reason we restricted attention to that case in Sec. [ref]. | interpretation 0.38 | 0.89 | 0.09 | 0.96 | 0.04 |
| Dark Matter from Planck Scale Remnants of PBH | It is useful to verify Eq. ([ref]) against the explicit expansion history. | method 0.90 | 0.44 | 0.02 | 0.11 | 0.04 |
| Dark Matter from Planck Scale Remnants of PBH | Writing $ρ_rem(t_ev) = m_rem n_BH(t_f)(a_f/a_ev)^3$ with $n_BH(t_f)$ from Eq. ([ref]), and $ρ_tot = 3M_p^2H^2$ at each epoch, one has Ω_rem(t_ev) = β m_rem/M_in (H^2_f/H^2_ev)(a_f/a_ev)^3 . | method 0.72 | 0.84 | 0.04 | 0.40 | 0.16 |
| Dark Matter from Planck Scale Remnants of PBH | During a PBH-dominated epoch $3M_p^2H^2 ≃ρ_BH = β ρ_tot(t_f)(a_f/a)^3$, so that [equation] and Eq. ([ref]) reduces to Eq. ([ref]), as it must. | setup 0.41 | 0.58 | 0.04 | 0.84 | 0.17 |
| Dark Matter from Planck Scale Remnants of PBH | Defining the fraction of the PBH remnants in the total DM density today as $f_rem ≡ρ_rem,0/ρ_DM,0$, and noting that $ρ_rem ∝a^-3$, Eq. ([ref]) can equivalently be written as Ω_rem (t_ev) = f_rem ρ_DM, 0/ρ_tot(t_ev) (a_0/a_ev)^3 . | setup 0.65 | 0.82 | 0.04 | 0.72 | 0.12 |
| Dark Matter from Planck Scale Remnants of PBH | The ratio of the scale factor at evaporation to the present value is approximately: [equation] where we have used Eq. ([ref]). | method 0.65 | 0.81 | 0.04 | 0.70 | 0.13 |
| Dark Matter from Planck Scale Remnants of PBH | Taking $ρ_tot(t_ev) = π^2/30 g_*(T_ev) T_ev^4$ and $Ω_DM, 0 h^2=0.12$, $Ω_rem (t_ev)$ can be simplified as Ω_rem (t_ev) = f_rem ρ_cr,0 Ω_DM, 0 30/π^2 g_*(T_ev) T_ev T_0^3 ( g_*,s(T_ev)/g_*,s(T_0) ) ≃5.8 ×10^-10 f_rem (GeV/T_ev) ≃1.6 ×10^-20 f_rem (M_in/1 g)^3/2 , where we have used $g_*,s(T_0) ≃3.91$, $g_*,s(T_ev) ≃g_*(T_ev)$, $T_0 ≃2.34 ×10^-13 GeV$ and $ρ_cr,0 ≃8 h^2 ×10^-47 GeV^4$. | method 0.57 | 0.89 | 0.04 | 0.92 | 0.98 |
| Dark Matter from Planck Scale Remnants of PBH | Combining Eqs. ([ref]) and ([ref]), the initial PBH mass is determined as a function of $r/f_rem$ | result 0.60 | 0.87 | 0.06 | 0.19 | 0.08 |
| Dark Matter from Planck Scale Remnants of PBH | (M_in/1 g) ≃1.1 ×10^6 (r/f_rem)^2/5 , or equivalently [equation] Assuming that PBH remnants constitute the total DM density today, i.e., $f_rem=1$, with $r=1$, the required initial PBH mass in the PBH dominated scenario becomes | result 0.92 | 0.80 | 0.09 | 0.95 | 0.63 |
| Dark Matter from Planck Scale Remnants of PBH | $M_in ≃1.1 ×10^6 g$. | result 0.98 | 0.74 | 0.07 | 0.37 | 0.97 |
| Dark Matter from Planck Scale Remnants of PBH | This is within ten percent of the benchmark BP2 of Table [ref], $M_in≃1.2×10^6$ g, which was chosen from the leptogenesis scan of Fig. [ref] and corresponds to $f_rem≃0.8 r$. | result 0.73 | 0.81 | 0.08 | 0.52 | 0.99 |
| Dark Matter from Planck Scale Remnants of PBH | The coefficient $r$ is not predicted by any of the frameworks in which remnants arise. | limitation 0.51 | 0.21 | 0.09 | 0.37 | 0.05 |
| Dark Matter from Planck Scale Remnants of PBH | Hence it can be tuned according to the initial mass to achieve the correct DM relic density. | interpretation 0.59 | 0.43 | 0.14 | 0.64 | 0.05 |
| Dark Matter from Planck Scale Remnants of PBH | Setting $f_rem=1$ in Eq. ([ref]) gives [equation] Thus remnants can account for the entire dark matter abundance, for any initial mass above $≃1.1×10^6$ g by a suitable choice of $r≥1$. | result 0.76 | 0.78 | 0.18 | 0.93 | 0.98 |
| Dark Matter from Planck Scale Remnants of PBH | For BP2, this requires only $r≃1.2$. | result 0.95 | 0.84 | 0.09 | 0.72 | 0.97 |
| Dark Matter from Planck Scale Remnants of PBH | Here, it is worth mentioning that this freedom is one-sided as for lighter black holes, it would demand $r<1$ which is not allowed and hence the remnants overclose the Universe. | interpretation 0.50 | 0.73 | 0.21 | 0.91 | 0.36 |
| Dark Matter from Planck Scale Remnants of PBH | Because the remnant fraction falls as the five-halves power of the initial mass, a reduction of the mass by only ten percent is enough to overclose the Universe for any permitted $r$, while an increase of the same size leaves a remnant abundance too small to account for the dark matter unless $r$ departs appreciably from unity. | interpretation 0.69 | 0.81 | 0.22 | 0.85 | 0.94 |
| Dark Matter from Planck Scale Remnants of PBH | Thus, the naturalness of an order-unity coefficient bounds the initial mass from above as well. | interpretation 0.84 | 0.80 | 0.21 | 0.89 | 0.17 |
| Gravitational Waves | Because Pati–Salam symmetry breaking occurs prior to inflation, any stochastic gravitational wave background generated by the associated first-order phase transition [ref] is exponentially diluted alongside the magnetic monopoles, its energy density redshifting as $e^-4N$ once the relevant modes are pushed outside the horizon. | result 0.37 | 0.62 | 0.09 | 0.79 | 0.52 |
| Gravitational Waves | The same conclusion applies to the domain walls that accompany D-parity breaking [ref] . | result 0.73 | 0.58 | 0.07 | 0.46 | 0.06 |
| Gravitational Waves | Topological defects of the one type that would survive, namely cosmic strings, do not arise here: for the breaking chain of Eq. ([ref]) the relevant homotopy group is $π_1(G_PS/H)=π_0(H)$, which vanishes because the unbroken group $H=SU(3)_C×U(1)_Y$ is connected. | result 0.54 | 0.76 | 0.08 | 0.60 | 0.15 |
| Gravitational Waves | The gravitational wave phenomenology of this framework is therefore carried not by the gauge sector but by the primordial black hole population itself, through three distinct second-order channels: the scalar-induced background associated with the curvature perturbations that form the black holes, the tensor modes sourced by the Poisson fluctuations of the black hole distribution, and the direct emission of gravitons during Hawking evaporation. | interpretation 0.49 | 0.76 | 0.19 | 0.41 | 0.22 |
| Gravitational Waves | The first channel is the scalar-induced background generated at second order by the enhanced curvature perturbations required for PBH formation [ref] . | result 0.42 | 0.38 | 0.04 | 0.42 | 0.10 |
| Gravitational Waves | An amplitude $𝒫_ζ(k_f)∼𝒪(10^-2)$ of the kind discussed in Sec. [ref] generically sources a tensor spectrum of comparable prominence, with $Ω_GW∼𝒫_ζ^2$ near the peak. | result 0.78 | 0.55 | 0.06 | 0.74 | 0.96 |
| Gravitational Waves | Since the peak tracks the formation scale $k_f$ of Eq. ([ref]), and $k_f∝M_in^-1/2$, the signal associated with the gram-scale to $10^6$ g black holes considered here redshifts to frequencies of order $10^4$–$10^8$ Hz, with the entropy released during evaporation both lowering the peak frequency and suppressing the amplitude. | result 0.83 | 0.70 | 0.06 | 0.76 | 0.94 |
| Gravitational Waves | This lies far above the band of any current or planned interferometer, and its detection would require the high-frequency detector concepts currently under development [ref] , the present-day peak frequency being [ref] [equation] | interpretation 0.49 | 0.59 | 0.05 | 0.42 | 0.71 |
| Gravitational Waves | The second channel is more promising observationally. | interpretation 0.90 | 0.39 | 0.08 | 0.14 | 0.73 |
| Gravitational Waves | Because primordial black holes form at rare peaks of the density field, their spatial distribution carries Poisson fluctuations, which behave as isocurvature perturbations at formation, are converted into curvature perturbations once the black holes dominate, and source tensor modes at the abrupt end of the matter-dominated epoch [ref] . | background 0.61 | 0.32 | 0.07 | 0.54 | 0.05 |
| Gravitational Waves | This background peaks at the comoving scale of the mean black hole separation, corresponding today to $f≃1.7 kHz (M_in/10^4 g)^-5/6$, which for the heavier masses selected by the remnant dark matter condition of Eq. ([ref]) falls in the range of tens of hertz, for example BP1 gives $f≃35$ Hz at $M_in=1.2×10^6$ g, within the reach of the Einstein Telescope [ref] and Cosmic Explorer [ref] . | result 0.91 | 0.80 | 0.08 | 0.79 | 0.98 |
| Gravitational Waves | Its amplitude scales steeply with the initial abundance, and the requirement that it not spoil the effective number of relativistic species at nucleosynthesis imposes $β≲1.1×10^-6(M_in/10^4 g)^-17/24$ [ref] , a bound satisfied by the benchmarks of Sec. [ref]. | result 0.68 | 0.73 | 0.06 | 0.91 | 0.97 |
| Gravitational Waves | Since the baryon asymmetry depends only weakly on $β$ once the PBHs dominate, while the remnant abundance is entirely independent of $β$, the initial PBH abundance remains a free parameter of the scenario. | interpretation 0.56 | 0.74 | 0.12 | 0.82 | 0.28 |
| Gravitational Waves | It is constrained only from below by the PBH domination requirement of Eq. ([ref]) and from above by the requirement of successful BBN. | limitation 0.38 | 0.57 | 0.07 | 0.82 | 0.14 |
| Gravitational Waves | The Poisson-induced stochastic gravitational wave background is the only observable sensitive to the $β$, and the corresponding signal may therefore lie anywhere between the two limits. | interpretation 0.77 | 0.55 | 0.11 | 0.54 | 0.13 |
| Gravitational Waves | This makes it, in principle, a very good discriminator of the scenarios as a detection at the Einstein Telescope or Cosmic Explorer would fix the one remaining free parameter of the framework. | interpretation 0.88 | 0.32 | 0.10 | 0.21 | 0.12 |
| Gravitational Waves | A quantitative computation of both gravitational wave spectra, together with the resulting constraints on the $(M_in,β)$ plane and their interplay with the electroweak sphaleron condition of Eq. ([ref]), lies beyond the scope of the present work and is left for a dedicated study. | future work 0.89 | 0.15 | 0.04 | 0.91 | 0.05 |
| Gravitational Waves | In addition to the induced stochastic gravitational wave background, PBHs also generate a gravitational wave signal through the direct emission of gravitons during Hawking evaporation. | background 0.78 | 0.18 | 0.04 | 0.12 | 0.05 |
| Gravitational Waves | This contribution, together with graviton bremsstrahlung from the decay of massive scalar particles, has been investigated in Refs. [ref] . | background 1.00 | 0.08 | 0.04 | 0.12 | 0.05 |
| Gravitational Waves | However, the resulting gravitational wave spectrum peaks in the ultra-high frequency regime, well beyond the sensitivity of the current and planned gravitational wave detectors. | limitation 0.41 | 0.46 | 0.07 | 0.29 | 0.73 |
| Gravitational Waves | For example, the peak frequency is determined by the initial mass of PBH and is given by f ≃1.6 ×10^13 Hz (M_in/1 g)^1/2 . | result 0.48 | 0.40 | 0.04 | 0.28 | 0.95 |
| Gravitational Waves | This scaling is opposite to that of Eq. ([ref]) because heavier black holes emit softer gravitons, but they evaporate later and the emitted gravitons are therefore less redshifted. | interpretation 0.88 | 0.59 | 0.05 | 0.29 | 0.46 |
| Gravitational Waves | As a consequence, increasing the initial PBH mass shifts the peak of the gravitational wave towards higher frequencies. | result 0.54 | 0.66 | 0.06 | 0.17 | 0.62 |
| Gravitational Waves | For illustration, Fig. [ref] shows the characteristic gravitational wave strain from direct evaporation of PBH as a function of frequency $f$ for three representative initial PBH masses, $M_in = 1, 10^4, 10^8 g$, displayed by the red, blue and green solid curves, respectively. | result 0.96 | 0.75 | 0.03 | 0.25 | 0.73 |
| Gravitational Waves | The shaded regions represent the limits from various proposed GW detectors, including LISA [ref] , the Big Bang Observer (BBO) [ref] , CE [ref] , DECIGO [ref] , and the advanced LIGO [ref] . | background 0.57 | 0.28 | 0.03 | 0.16 | 0.08 |
| Gravitational Waves | We also display the current and projected sensitivities of several proposed high-frequency gravitational wave detection techniques, including optically levitated sensors, enhanced magnetic conversion (EMC), and the inverse Gertsenshtein effect, as well as the experiments JURA, ALPS II, OSQAR, IAXO, and CAST. | background 0.62 | 0.61 | 0.03 | 0.11 | 0.05 |
| Gravitational Waves | These sensitivity curves are adopted from Ref. [ref] . | background 0.64 | 0.15 | 0.02 | 0.18 | 0.04 |
| Gravitational Waves | The brown curve illustrates the parameter space that can be probed by resonant cavities [ref] . | background 0.31 | 0.31 | 0.03 | 0.18 | 0.05 |
| Gravitational Waves | Although the gravitational wave signals from PBH evaporation remain inaccessible to conventional interferometric detectors, future resonant cavity experiments with significantly improved sensitivity could probe the GW spectra produced by sufficiently light PBHs. | interpretation 0.67 | 0.38 | 0.07 | 0.58 | 0.22 |
| Gravitational Waves | Since the stochastic gravitational wave background behaves as an additional relativistic component in the Universe, it contributes to the effective number of neutrino species, $N_eff$, through [equation] where $Ω_GW$ is the total gravitational wave energy density integrated over frequency. | setup 0.58 | 0.28 | 0.06 | 0.40 | 0.07 |
| Gravitational Waves | Consequently, constraints on $ΔN_eff$ can be translated into the upper bounds on the parameter space of $h_c-f$ plane. | interpretation 0.47 | 0.55 | 0.08 | 0.29 | 0.08 |
| Gravitational Waves | The Planck collaboration has already put a constraint $ΔN_eff < 0.30$ at 95$%$ confidence level [ref] . | background 1.00 | 0.08 | 0.05 | 0.24 | 0.98 |
| Gravitational Waves | Future experiments, including CMB-S4 [ref] , Euclid [ref] , and potentially cosmic-variance-limited (CVL) CMB polarization experiments are anticipated to improve the constraint on $ΔN_eff$ to $≲0.06$, $≲0.013$, and $≃3.1 ×10^-6$, respectively [ref] . | background 0.67 | 0.16 | 0.05 | 0.20 | 0.96 |
| Gravitational Waves | These corresponding constraints are depicted with dark yellow lines in Fig. [ref]. | other 0.47 | 0.75 | 0.04 | 0.14 | 0.07 |
| Gravitational Waves | The stochastic gravitational wave background originating from the direct evaporation of PBH aligns with the current $ΔN_eff$ constraint from Planck. | result 0.83 | 0.76 | 0.10 | 0.25 | 0.52 |
| Gravitational Waves | Furthermore, a significant region of the parameter space could be tested by future CMB observations, particularly if next-generation measurements achieve the projected sensitivity to $ΔN_eff$. | interpretation 0.52 | 0.31 | 0.08 | 0.59 | 0.06 |
| Conclusion | We have examined non-thermal leptogenesis and Planck-scale remnant dark matter driven by primordial black hole evaporation in a minimal Pati–Salam cosmology, with the gauge symmetry broken before inflation so that the monopoles produced at the transition are diluted away. | setup 0.55 | 0.88 | 0.11 | 0.93 | 0.04 |
| Conclusion | The motivation for the embedding is that the right-handed neutrinos are then no longer optional rather they are components of the $SU(2)_R$ doublets, and their Majorana masses are tied to the breaking scale through $M_N=y_Rv_R$. | setup 0.90 | 0.47 | 0.14 | 0.41 | 0.08 |
| Conclusion | The heavy neutrino mass is therefore not a free parameter but a derived quantity, constrained by the same conditions that govern the black hole population. | interpretation 0.56 | 0.84 | 0.24 | 0.48 | 0.07 |
| Conclusion | Requiring that the Pati–Salam symmetry be neither restored during inflation nor regenerated by the hot plasma that follows, and that the black holes be hot enough to emit the heavy neutrinos, forces the Yukawa coupling of the Pati–Salam scalar sector well below unity and the heavy neutrino mass well below the $SU(2)_R$ breaking scale. | result 0.89 | 0.85 | 0.30 | 0.93 | 0.84 |
| Conclusion | This is a prediction of the embedding rather than an assumption, and it is the principal way in which the gauge structure constrains what would otherwise be an unconstrained mechanism. | interpretation 0.97 | 0.79 | 0.21 | 0.18 | 0.07 |
| Conclusion | The baryon asymmetry can be generated in two distinct regimes depending on whether the black holes are hot enough to emit the lightest right-handed neutrino. | result 0.93 | 0.84 | 0.29 | 0.59 | 0.12 |
| Conclusion | Below roughly $10^2$ g Hawking emission populates the heavy neutrino sector efficiently and the asymmetry is genuinely non-thermal and the requirement that it be reprocessed by the electroweak sphalerons before they freeze out gives an upper bound on the intial mass. | result 0.81 | 0.84 | 0.24 | 0.91 | 0.97 |
| Conclusion | For heavier black holes the emission of RHNs remain Boltzmann suppressed and leptogenesis also relies on the thermal mechanism to produce the asymmetry and in this scenario the black holes evaporation also cause an entropy dilution of the asymmetry generated long before they evaporate. | result 0.68 | 0.82 | 0.21 | 0.89 | 0.12 |
| Conclusion | These two regimes respond very differently to the possibility that quantum gravitational backreaction halts the evaporation at the Planck scale. | interpretation 0.56 | 0.73 | 0.21 | 0.46 | 0.20 |
| Conclusion | Each black hole then leaves a single stable remnant, and the surviving mass fraction is simply the ratio of the remnant mass to the initial mass, independent of the initial abundance. | result 0.81 | 0.76 | 0.21 | 0.85 | 0.25 |
| Conclusion | The present-day remnant fraction consequently scales as the inverse five-halves power of the initial mass, so that suppressing it demands heavier black holes rather than lighter ones. | result 0.86 | 0.87 | 0.40 | 0.55 | 0.87 |
| Conclusion | The gram-scale masses obligatory for non-thermal leptogenesis then overproduce the dark matter. | result 0.45 | 0.86 | 0.49 | 0.41 | 0.34 |
| Conclusion | Thus Planck-scale remnants and PBH-only leptogenesis are mutually exclusive. | result 0.57 | 0.89 | 0.76 | 0.44 | 0.11 |
| Conclusion | However the thermal regime solves this problem. | interpretation 0.76 | 0.84 | 0.49 | 0.72 | 0.09 |
| Conclusion | In this scenario a single initial PBH mass of order $10^6$, gives rise to the observed dark matter abundance, selects the heavy neutrino mass that reproduces the baryon asymmetry, and sets the lower limit on the Pati–Salam breaking scale. | result 0.80 | 0.88 | 0.78 | 0.68 | 0.93 |
| Conclusion | Here the heavy neutrino mass scale is light enough to be produced from the thermal plasma. | interpretation 0.58 | 0.78 | 0.17 | 0.82 | 0.36 |
| Conclusion | The scenario therefore has the freedom to accommodate both the baryon asymmetry and the dark matter at a single point in parameter space. | interpretation 0.71 | 0.88 | 0.75 | 0.34 | 0.06 |
| Conclusion | The gauge sector leaves no gravitational wave signal: the pre-inflationary phase transition is inflated away and the breaking chain produces no cosmic strings. | result 0.77 | 0.74 | 0.11 | 0.62 | 0.06 |
| Conclusion | Gravitons radiated directly during evaporation peak in the ultra-high-frequency band, beyond the reach of any detector, and are constrained only through their contribution to $ΔN_eff$, where next-generation CMB measurements will probe a significant region of parameter space. | result 0.52 | 0.76 | 0.11 | 0.69 | 0.52 |
| Conclusion | The observationally promising channel is the tensor background sourced by the Poisson fluctuations of the black hole distribution, which for the masses selected by the remnant condition falls within the anticipated sensitivity of the future GW experiments. | interpretation 0.76 | 0.81 | 0.19 | 0.82 | 0.18 |
| Conclusion | Since neither the baryon asymmetry nor the remnant abundance depends appreciably on the initial abundance $β$, this background is the only observable sensitive to it and a detection would fix this free parameter of the scenario. | interpretation 0.91 | 0.87 | 0.19 | 0.84 | 0.22 |
| Acknowledgments | S.M. acknowledges support from the IIT Goa Startup Grant [2025/SG/SM/057]. | other 1.00 | 0.10 | 0.01 | 0.04 | 0.47 |
| Acknowledgments | A.C. acknowledges VIT for support. | other 1.00 | 0.07 | 0.01 | 0.03 | 0.02 |
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author = "Baumann, Daniel and Steinhardt, Paul J. and Turok, Neil", title = "Primordial Black Hole Baryogenesis", eprint = "hep-th/0703250", archivePrefix = "arXiv", reportNumber = "PUPT-2229", month = "3", year = "2007"
This mechanism therefore relaxes the need for the thermal plasma to populate the heavy RHN sector at temperatures of order $M_N_1$ allowing for successful baryogenesis at significantly lower reheating temperatures [ref] .
author = "Pati, Jogesh C. and Salam, Abdus", title = "Lepton Number as the Fourth Color", reportNumber = "IC-74-7", doi = "10.1103/PhysRevD.10.275", journal = "Phys. Rev. D", volume = "10", pages = "275–289", year = "1974", note = "[Erratum
To provide a rigorous, symmetry-motivated origin for these right-handed neutrinos, the leptogenesis mechanism can be naturally embedded within the Pati-Salam gauge structure, $SU(4)_C ×SU(2)_L ×SU(2)_R$ [ref] .
author = "Baumann, Daniel and Steinhardt, Paul J. and Takahashi, Keitaro and Ichiki, Kiyotomo", title = "Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations", eprint = "hep-th/0703290", archivePrefix = "arXiv", doi = "10.
The first channel is the scalar-induced background generated at second order by the enhanced curvature perturbations required for PBH formation [ref] .
author = "Perez-Gonzalez, Yuber F. and Turner, Jessica", title = "Assessing the tension between a black hole dominated early universe and leptogenesis", eprint = "2010.03565", archivePrefix = "arXiv", primaryClass = "hep-ph", reportNumber =
The Boltzmann equations governing the cosmological evolution of the thermally produced RHN number density and the $(B-L)$ asymmetry generated through the decay of the RHNs are written as [ref] [equation] where $Γ_N_1^TH= Γ_N_1 K_1(M_N_1/T)/K_2(M_N_1/T)$ is the thermally averaged decay rate and $n_N_1^eq$ is the equilibrium number density of the RHNs as given by n_N_1^eq = g_N_1/2 π^2 M_N_1^2 T K_2(M_N_1/T) .
author = "Ong, Yen Chin", title = "The Case For Black Hole Remnants: A Review", eprint = "2412.00322", archivePrefix = "arXiv", primaryClass = "gr-qc", month = "11", year = "2024"
This process leaves behind stable, non-radiating Planck-scale remnants that serve as highly motivated cold dark matter candidates [ref] .
author = "Pati, Jogesh C. and Salam, Abdus", title = "Unified Lepton-Hadron Symmetry and a Gauge Theory of the Basic Interactions", reportNumber = "IC-73-41-INT-REP", doi = "10.1103/PhysRevD.8.1240", journal = "Phys. Rev. D", volume = "8",
To provide a rigorous, symmetry-motivated origin for these right-handed neutrinos, the leptogenesis mechanism can be naturally embedded within the Pati-Salam gauge structure, $SU(4)_C ×SU(2)_L ×SU(2)_R$ [ref] .
author = "Gavela, M. B. and Hernandez, P. and Orloff, J. and Pene, O. and Quimbay, C.", title = "Standard model CP violation and baryon asymmetry. Part 2: Finite temperature", eprint = "hep-ph/9406289", archivePrefix = "arXiv", reportNumber
The electroweak phase transition, given the measured mass of the Higgs boson, is a smooth crossover rather than a first-order phase transition, thereby failing to provide the required departure from thermal equilibrium dictated by the Sakharov conditions [ref] , and the inherent CP violation is insufficient by several orders of magnitude [ref] .
author = "Perez-Gonzalez, Yuber F. and Turner, Jessica", title = "Assessing the tension between a black hole dominated early universe and leptogenesis", eprint = "2010.03565", archivePrefix = "arXiv", primaryClass = "hep-ph", reportNumber =
Here $ñ_N_1 ≡n_N_1 a^3$ is the comoving number density of RHNs, and $Γ_N_1^BH$ is the decay width corrected by an average inverse time dilatation factor as given by [ref] Γ_N_1^BH ≡⟨M_N_1/E_N_1⟩_BH Γ_N_1 ≃K_1(M_N_1/T_BH)/K_2(M_N_1/T_BH) Γ_N_1 , where the total RHN decay width is $Γ_N_1 =(y_ν^†y_ν) M_N_1/8π$ with $y_ν^2 ∼m_νM_N_1/v^2$, $m_ν∼0.05$ eV, and $v=174$ GeV.
author = "Carr, Bernard and Kuhnel, Florian", title = "Primordial Black Holes as Dark Matter: Recent Developments", eprint = "2006.02838", archivePrefix = "arXiv", primaryClass = "astro-ph.CO", doi = "10.1146/annurev-nucl-050520-125911", jo
This process leaves behind stable, non-radiating Planck-scale remnants that serve as highly motivated cold dark matter candidates [ref] .
author = "Domènech, Guillem and Sasaki, Misao", title = "Gravitational wave hints black hole remnants as dark matter", eprint = "2303.07661", archivePrefix = "arXiv", primaryClass = "gr-qc", reportNumber = "YITP-23-30", doi = "10.1088/1361-
This process leaves behind stable, non-radiating Planck-scale remnants that serve as highly motivated cold dark matter candidates [ref] .
author = "Granelli, Alessandro and Moffat, Kristian and Perez-Gonzalez, Yuber F. and Schulz, Holger and Turner, Jessica", title = "ULYSSES: Universal LeptogeneSiS Equation Solver", eprint = "2007.09150", archivePrefix = "arXiv", primaryClas
For our numerical analysis, we utilize the "" package [ref] , incorporating modified graybody factors.
author = "Abazajian, Kevork N. and others", collaboration = "CMB-S4", title = "CMB-S4 Science Book, First Edition", eprint = "1610.02743", archivePrefix = "arXiv", primaryClass = "astro-ph.CO", reportNumber = "FERMILAB-FN-1024-A-AE", doi =
Future experiments, including CMB-S4 [ref] , Euclid [ref] , and potentially cosmic-variance-limited (CVL) CMB polarization experiments are anticipated to improve the constraint on $ΔN_eff$ to $≲0.06$, $≲0.013$, and $≃3.1 ×10^-6$, respectively [ref] .
author = "Gavela, M. B. and Hernandez, P. and Orloff, J. and Pene, O.", title = "Standard model CP violation and baryon asymmetry", eprint = "hep-ph/9312215", archivePrefix = "arXiv", reportNumber = "CERN-TH-7081-93, LPTHE-ORSAY-93-48, HUTP
The electroweak phase transition, given the measured mass of the Higgs boson, is a smooth crossover rather than a first-order phase transition, thereby failing to provide the required departure from thermal equilibrium dictated by the Sakharov conditions [ref] , and the inherent CP violation is insufficient by several orders of magnitude [ref] .
title = Primordial black hole evaporation and dark matter production. I. Solely Hawking radiation, author = Cheek, Andrew and Heurtier, Lucien and Perez-Gonzalez, Yuber F. and Turner, Jessica, journal = Phys. Rev. D, volume = 105, issue = 1
In the GO limit, the emission rate simplifies to [ref] : [equation] where $ℱ(z) = [z Li_2(e^-z) + Li_3(e^-z)]$, with $Li_n$ denoting the polylog function of order $n$.
author = "Hawking, S. W.", editor = "Gibbons, G. W. and Hawking, S. W.", title = "Particle Creation by Black Holes", doi = "10.1007/BF02345020", journal = "Commun. Math. Phys.", volume = "43", pages = "199–220", year = "1975", note = "[Erra
First, Hawking emission from the PBHs is efficient only when the emitted species is light compared with the instantaneous black hole temperature [ref] , [equation] where $M_p≃2.435×10^18$ GeV is the reduced Planck mass.
author = "Chluba, Jens and Erickcek, Adrienne L. and Ben-Dayan, Ido", title = "Probing the inflaton: Small-scale power spectrum constraints from measurements of the CMB energy spectrum", eprint = "1203.2681", archivePrefix = "arXiv", primar
Narrow features avoid constraints from CMB spectral distortions and pulsar timing arrays [ref] , and are well approximated by a monochromatic PBH mass function.
author = "Lehmann, Benjamin V. and Johnson, Christian and Profumo, Stefano and Schwemberger, Thomas", title = "Direct detection of primordial black hole relics as dark matter", eprint = "1906.06348", archivePrefix = "arXiv", primaryClass =
This process leaves behind stable, non-radiating Planck-scale remnants that serve as highly motivated cold dark matter candidates [ref] .
author = "Mohapatra, Rabindra N. and Senjanovic, Goran", title = "Neutrino Mass and Spontaneous Parity Nonconservation", reportNumber = "MDDP-TR-80-060, MDDP-PP-80-105, CCNY-HEP-79-10", doi = "10.1103/PhysRevLett.44.912", journal = "Phys. R
This framework is deeply connected to the neutrino sector, as it simultaneously explains the extreme smallness of the active light neutrino masses through the type-I seesaw mechanism [ref] .
What was examined, and what was not
sources read
- arXiv listing abstract v1
- arXiv full text v1, LaTeX source: 10 sections, 336 body sentences
not examined
- figures and tables
- equation content
- supplementary material
- relations to other papers
levels run
- P abstract
- S sections
- T body sentences
- C citations
not run
- R cross-paper relations
References
Discussion
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