Implication of preheating on gravity assisted baryogenesis in $R^2$-Higgs inflation

Christoph Englert; Mariano Quir\'os; Tanmoy Modak; Yann Cado

REVIEW 4 major objections 5 minor 85 references

Implication of preheating on gravity assisted baryogenesis in $R^2$-Higgs inflation

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper establishes that preheating, driven mostly by Goldstone bosons, determines the reheating temperature and thereby pins the baryogenesis operator scale $\Lambda$ to a narrow window around $2.2 \times 10^{-5}\, M_{\rm P}$ for…

desk verdict A real step forward that turns the reheating temperature from a free parameter into a specific Λ window for gravity-assisted baryogenesis, but the quoted precision outruns the linear-order, backreaction-free preheating calculation. read the letter →

arxiv 2411.11128 v1 pith:II5HXHIF submitted 2024-11-17 hep-ph astro-ph.COgr-qc

Yann Cado , Christoph Englert , Tanmoy Modak , Mariano Quirós This is my paper · ORCID

classification hep-phastro-ph.COgr-qc

keywords R2-HiggsinflationpreheatinggravityassistedbaryogenesishypermagnetichelicitybaryonasymmetryoftheUniverseelectroweakcrossoverreheatingtemperaturedimension-sixoperator

topics Matter-Antimatter Asymmetry

open problems Matter-Antimatter Asymmetry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that in $R^2$-Higgs inflation the reheating temperature after inflation, rather than being a free parameter, is fixed by preheating dynamics, and that this temperature determines the scale $\Lambda$ of the CP-violating operator $(R/\Lambda^2) B_{\mu\nu}\tilde{B}^{\mu\nu}$ that produces the baryon asymmetry. Working in the $R^2$-like regime with non-minimal Higgs coupling $\xi_H$ between 1 and 10, the authors compute particle production of the inflaton, Higgs, $W/Z$ bosons, photon and Goldstone bosons in a linearized doubly-covariant treatment. They find that the Goldstone bosons preheat the Universe first, at about $N \approx 3$ for $\xi_H \approx 1$ and $N \approx 1.8$ for $\xi_H \approx 10$, giving reheating temperatures of $5 \times 10^{14}$ GeV and $6 \times 10^{14}$ GeV respectively. Feeding these into the baryogenesis transport equations, they conclude that the observed baryon asymmetry requires $\Lambda \approx 2.2\,(2.6) \times 10^{-5}\, M_{\rm P}$, with allowed windows $2.07\!-\!2.30 \times 10^{-5}\, M_{\rm P}$ and $2.52\!-\!2.76 \times 10^{-5}\, M_{\rm P}$. If right, this turns a previously free high-energy scale into a sharp, testable prediction of the model.

What carries the argument

The load-bearing object is the five-field scalar-tensor system obtained from the $R^2 + \xi_H |\Phi|^2 R + (R/\Lambda^2) B \tilde{F}$ action after Weyl rescaling, with field-space metric $G_{IJ} = \mathrm{diag}(1, e^{-\sqrt{2/3}\,\varphi/M_{\rm P}}, \ldots)$. The argument runs on the linearized mode equations for the Mukhanov-Sasaki variables $X^I$, the transverse $Z/W$/photon modes with effective frequencies $\omega^2 = k^2 + a^2 m^2 + \zeta^\lambda(\tau,k)$, and the Goldstone equations that acquire friction terms $E_{(I)}(\tau,k)$ and spike-like frequency enhancements at Higgs zero-crossings. These spikes drive exponential Goldstone production and set the preheating completion time; the completion density gives $T_{\rm rh}$ through $\rho_{\rm rh} = g_{\rm rh} \pi^2 T_{\rm rh}^4/30$, which then fixes the hypermagnetic helicity entering the baryon asymmetry formula. The baryogenesis side uses the anomaly relation between baryon number, hypermagnetic helicity and sphaleron washout, with constraints $R_m > 1$ and $T_{\rm CPI} < 10^5$ GeV bounding the viable $\Lambda$.

What would settle it

A fully nonlinear lattice simulation of preheating in $R^2$-Higgs inflation with $\xi_H = 1$ and $10$, tracking Higgs, Goldstone, $Z$, $W$, and photon fields together with rescattering, would directly test the completion times $N_{\rm rh} \approx 3.05$ and $1.83$ and the resulting $T_{\rm rh}$ values of $5 \times 10^{14}$ and $6 \times 10^{14}$ GeV; a materially different $T_{\rm rh}$ would move the required $\Lambda$ out of the quoted windows. A second check is whether the Goldstone friction spikes survive once backreaction on the Higgs condensate $h_0$ is included.

Watch

Extended reading notes

Core claim

The central claim is that the dimension-six gravity-assisted baryogenesis operator $(R/\Lambda^2) B_{\mu\nu}\tilde{B}^{\mu\nu}$ in $R^2$-Higgs inflation must have its cutoff $\Lambda$ in a specific narrow window, because the preheating stage fixes the reheating temperature: $\Lambda \sim 2.2 \times 10^{-5}\, M_{\rm P}$ for $\xi_H \approx 1$ and $\sim 2.6 \times 10^{-5}\, M_{\rm P}$ for $\xi_H \approx 10$, with the full baryogenesis-viable ranges $2.07 \times 10^{-5}\, M_{\rm P} \lesssim \Lambda \lesssim 2.30 \times 10^{-5}\, M_{\rm P}$ and $2.52 \times 10^{-5}\, M_{\rm P} \lesssim \Lambda \lesssim 2.76 \times 10^{-5}\, M_{\rm P}$. Preheating is dominated by Goldstone bosons, which reheat faster than the Higgs, transverse gauge bosons or inflaton quanta; for $\xi_H \approx 10$ the $Z$ and Higgs sectors also contribute, while for very small $\xi_H$ preheating does not occur unless $\Lambda$ is small enough to overproduce the asymmetry. The reheating temperature enters the baryogenesis calculation both through the hypermagnetic helicity at the start of the electroweak crossover and through the magnetic Reynolds number and chiral-plasma-instability constraints that must be satisfied for the helicity to survive.

Load-bearing premise

The quoted reheating temperatures and completion times come from a linear-order calculation that ignores backreaction and treats preheating as complete when the produced energy density equals the background density; if nonlinear effects deplete or delay the produced fields, the reheating temperature and hence the $\Lambda$ window shift.

Editorial extensions

If this is right

For $\xi_H \approx 1$, successful baryogenesis in this model requires $\Lambda$ between $2.07$ and $2.30 \times 10^{-5}\, M_{\rm P}$, and for $\xi_H \approx 10$ between $2.52$ and $2.76 \times 10^{-5}\, M_{\rm P}$.
Larger $\xi_H$ leads to earlier preheating, a higher reheating temperature, and therefore a larger $\Lambda$ to match the observed $\eta_B = 8.7 \times 10^{-11}$.
Goldstone bosons, not the inflaton or the Higgs, are the fastest preheating channel in the mixed $R^2$-Higgs regime, and they set the reheating temperature in both benchmark points.
In the deep $R^2$-like regime ($\xi_H \sim 10^{-3}$), no efficient preheating occurs unless $\Lambda$ is small, but such small values overproduce the baryon asymmetry, so that regime is disfavored for this mechanism.
The predicted $\Lambda$ is of order $10^{-5}\, M_{\rm P}$, indicating a high-scale CP-violation source and no need for new degrees of freedom beyond the Standard Model in this baryogenesis channel.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A fully nonlinear lattice treatment including backreaction and rescattering would test whether the Goldstone-driven completion times $N_{\rm rh} \approx 3.05$ and $1.83$ are stable; if the true completion occurs later or earlier, the quoted $\Lambda$ windows would shift correspondingly.
The same doubly-covariant machinery could be applied to baryogenesis operators built from $W \tilde{W}$ or to other $f(R,\Phi)$ regimes, producing a $\Lambda$-versus-$\xi_H$ exclusion plane that collider and cosmological probes could in principle confront.
Because the hypermagnetic helicity must survive until the electroweak crossover, the model implies a helical hypermagnetic field at $T \sim 135$ GeV whose magnitude is tied to $\Lambda$; future primordial-magnetic-field or cosmic-microwave-background bounds could provide an independent test of this window.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The one thing to know: this paper actually computes the reheating temperature from preheating in R2-Higgs inflation and uses it to pin down the scale Λ of the gravity-assisted baryogenesis operator. The headline numbers, Λ ~ 2.2 (2.6) × 10^-5 M_P for ξ_H ~ 1 (10), are the first concrete targets from this mechanism. That's a real step beyond their earlier work, which left T_rh as a free parameter.

What's done well: the formalism is thorough. They derive linearized EoMs for the full SU(2)×U(1) sector with the doubly-covariant treatment, and they properly handle the Goldstone bosons in Coulomb gauge, avoiding the unitary gauge breakdown at Higgs zero crossings. The result that Goldstone preheating sets the reheating temperature for both benchmark points is a genuine finding, and the connection between ξ_H, T_rh, and Λ (larger ξ_H → earlier preheating → higher T_rh → larger Λ) is physically sensible and clearly explained. The Reynolds number and chiral plasma instability constraints are standard and applied carefully.

Now the soft spots, which are the usual preheating story but actually matter here because the paper sells precision. The values in Table II come from a linear-order, backreaction-free calculation, and "completion" is defined as ρ_q = ρ_inf, precisely where linear theory breaks down. The authors acknowledge this and claim the choice of convention doesn't significantly affect results, but they don't show it. Worse, the decay suppression factor in Eq. (7.3) is applied to Z/W bosons but not to the Goldstone fields that actually determine T_rh for BP b and BP c. So the central numbers inherit an unquantified error. And the Λ window is obtained by inverting Eq. (8.8) to match the measured η_B, with f_θ_W spanning several orders of magnitude—so it's a fit, not an independent prediction. None of this kills the qualitative picture, but the abstract's "most accurate estimation to date" overstates what the calculation delivers.

Who should read it: anyone working on preheating, baryogenesis, or R2-Higgs inflation. It deserves a serious referee—the calculation is non-trivial, the literature is engaged honestly, and the limitations are mostly stated, even if the abstract oversells. My recommendation: send it to peer review, but ask the authors to quantify the completion-convention sensitivity, include decay effects for the Goldstone fields, and either soften the precision claim or provide error bars.

Referee Report

4 major / 5 minor

Summary. This paper studies preheating in R^2-Higgs inflation using a doubly-covariant linear perturbation formalism that includes the inflaton, the Higgs, the three Goldstone bosons, and the full SU(2)_L x U(1)_Y gauge sector in Coulomb gauge. For three benchmark points (BP a, b, c) it computes the energy densities produced in each sector, identifies which fields can preheat the Universe, and reads off the reheating time, scale factor, energy density, and temperature at the moment rho_q = rho_inf. These quantities are then fed into the gravity-assisted baryogenesis formula, Eq. (8.8), to derive the allowed window for the dimension-six operator scale Lambda: Lambda ~ 2.07-2.30 x 10^-5 M_P for xi_H=1 and Lambda ~ 2.52-2.76 x 10^-5 M_P for xi_H=10. The central quantitative claim is that preheating dynamics fixes the scale Lambda needed to reproduce the observed baryon asymmetry.

Significance. If the central claim holds, the paper makes a useful step by connecting the preheating epoch in R^2-Higgs inflation to the scale of a dimension-six CP-violating operator, converting a previously free parameter into a constrained numerical range. The formal derivation is a strength: the equations of motion for background and perturbations, the quantization with Bunch-Davies vacuum subtraction, and the treatment of Goldstone bosons in a gauge where the unitary-gauge pathology at Higgs zero-crossings is avoided are all presented explicitly. The finding that Goldstone bosons preheat faster than Higgs or gauge sectors for xi_H ~ 1-10 is also a concrete, checkable result. The numerical Lambda window, however, is not yet robust because it rests on a completion criterion at the boundary of validity of the linear approximation and on an ad hoc decay treatment for only some of the produced species. If these uncertainties are quantified, the paper would be a significant advance for gravity-assisted baryogenesis models.

major comments (4)

[Sec. 4, Table II, Eq. (8.8)] The preheating completion time used in Table II is defined by rho_q = rho_inf, a point at which the paper itself states that linear theory is not reliable: Sec. 4 notes that 'the linear analysis is not reliable when rho_q approaches rho_inf' and that the continued growth beyond N ~ 3 'should be shut off once decay, backreaction and rescattering are taken into account.' Since N_rh, a_rh, rho_rh, and T_rh for BP b and BP c are read exactly at this point and enter Eq. (8.8) through (T_rh a_rh)^3 as well as the Reynolds and CPI constraints in Eqs. (8.13) and (8.18), an error in the completion time directly rescales the entropy denominator and moves the quoted Lambda window. The claim that the results are 'not significantly impacted' by the choice of completion convention needs a quantitative comparison, for example evaluating T_rh and a_rh at rho_q = 0.1 rho_inf as in Ref. [53], rather than a heuristic statement.
[Sec. 7, Eq. (7.3), Table II] The decay correction exp(-integral Gamma) in Eq. (7.3) is applied only to the Z and W boson energy densities (Fig. 10), but the fields that actually set preheating completion in Table II are Goldstone bosons: phi3 and phi4 for BP b and phi2 for BP c. The paper's argument that Goldstone decays into gauge bosons are kinematically disallowed and that decays into fermions proceed too slowly is qualitative and not quantified with integrated depletion factors or a comparison to the Hubble rate. Because the reheating temperature is determined by these Goldstone fields, the omission of an analogous decay factor is a load-bearing gap in the derivation of the Lambda window.
[Sec. 8, Eqs. (8.8), (8.10), (8.19)] The quoted Lambda range is obtained by solving Eq. (8.8) for Lambda such that eta_B equals the measured value; it is therefore a fitted range rather than an independent prediction. This is acceptable as a constraint, but the paper should state this explicitly and propagate the uncertainties of f_theta_W (Eq. (8.10)), T_step, and Delta_T (Eq. (8.5)) into the final range. As written, Eq. (8.19) quotes two significant digits while f_theta_W spans two orders of magnitude, and no error budget is given for the many input parameters that enter Eq. (8.8).
[Sec. 7, Eq. (7.6)] The identification of rho_inf(arh) with the thermal bath energy density in Eq. (7.6) assumes instantaneous and complete thermalization of the produced Goldstone and gauge energy densities. No thermalization rate or efficiency factor is estimated, and the paper defers the perturbative reheating calculation to future work. Since T_rh is the central input to the baryogenesis computation, this assumption should be either justified with a quantitative estimate or explicitly included as a source of uncertainty in the Lambda window.

minor comments (5)

[Sec. 7, Table II] The values in Table II are labeled approximate; it would help to state which figures are rounded and how a_rh is derived from N_rh for each benchmark point.
[Sec. 4, Fig. 3] The text says 'For BP a and BP b, both rho_q(phi) and rho_q(h) are much smaller than rho_inf', but the corresponding panel does not show rho_inf; adding the background line to all panels or stating its value in the caption would improve readability.
[Sec. 8, Eq. (8.8)] The notation HY is used for the hypermagnetic helicity at arh, while H(t) denotes the Hubble rate; this is not confusing in context but the two should be distinguished in symbols or by a sentence.
[Fig. 13] The caption contains the typo 'see text tor detail' and should read 'see text for details'.
[Secs. 3.1.2, 5.1] The notation for W boson mode functions is inconsistent: Eq. (5.6b) uses w^lambda_k while the text sometimes writes fW; unifying the notation would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quoted Lambda window is obtained by matching the observed baryon asymmetry, and the preheating inputs are computed independently of that target.

full rationale

The paper's central numbers come from two separate calculations. Table II (Nrh, arh, rhorho, Trh) is obtained by solving the linearized mode equations of Secs. 4-5 for the three benchmark points; the equations contain no input from the observed baryon asymmetry. The baryogenesis step (Sec. 8) then uses the transport formula of Eq. (8.8), taken from independent prior work, and scans Lambda so that the resulting eta_B matches the CMB value (8.2). Solving for a free model parameter (Lambda) from an observable is standard parameter determination rather than a derivation that reduces to its own input; the paper does not claim to predict eta_B without using it as input. The Lambda-dependence enters the helical hypermagnetic field production in Sec. 6, which is computed from first-order mode equations, so the fit is not inserted by definition elsewhere. Self-citations to Ref. [1] supply benchmark points and context, but the load-bearing reheating and baryogenesis calculations are performed in this paper with the cited external transport machinery. The admitted limitations - linear theory breaking down near completion, the ad hoc decay factor (7.3), and qualitative treatment of the Schwinger effect - are accuracy and systematic-error concerns, not circular steps. No equation was found in which a predicted quantity is identical by construction to a fitted input or to a self-citation.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper does not introduce new particles or fields. The R^2 scalaron is the standard dual scalar degree of freedom of f(R) gravity, and the CP-odd dimension-six operator is an input interaction inherited from previous work. The central claim rests instead on chosen benchmark parameters, imported baryogenesis formulas, and the linear-order preheating approximation.

free parameters (6)

xi_R = BP a: 2.35e9, BP b: 2.55e9, BP c: 2.2e9
Non-minimal coupling of the R^2 term. Values are benchmark choices from prior Planck-consistent analyses, not derived in this paper.
xi_H = BP a: 1e-3, BP b: 1, BP c: 10
Non-minimal coupling of the Higgs to the Ricci scalar. Controls which fields preheat and directly affects the claimed Lambda window.
Initial field values phi(t_in), h0(t_in) = BP a: 5.5 M_P, 2e-4 M_P; BP b: 5.5 M_P, 8.94e-4 M_P; BP c: 5.4 M_P, 5.00e-3 M_P
Background initial conditions for the inflaton and Higgs fields, chosen so that the scalar amplitude, spectral index, and tensor-to-scalar ratio agree with Planck 2018 within 95% CL.
Lambda = 2.07e-5 to 2.76e-5 M_P
The headline scale is obtained by requiring the baryogenesis formula, Eq. (8.8), to match the observed eta_B. It is therefore fitted to the target observable rather than predicted from first principles.
f_theta_W = 5.6e-4 to 0.32
Electroweak crossover parameter from Refs. [15, 16]. The broad range is used to bracket the acceptable Lambda values.
T_step and Delta_T = T_step in 155-160 GeV, Delta_T in 5-20 GeV
Parameters of the smooth step function used to model the temperature-dependent weak mixing angle in Eq. (8.5).

assumptions (5)

domain assumption The dimension-six operator (R/Lambda^2) B F-tilde is the sole CP-violating source driving helical hypermagnetic fields and the baryon asymmetry.
The paper assumes this operator is present and dominant without deriving it from an ultraviolet completion.
ad hoc to paper Linear perturbation theory remains reliable up to the preheating completion time defined by rho_q = rho_inf.
The authors define completion of preheating at rho_q = rho_inf in Sec. 4, while explicitly acknowledging in Sec. 7 and Sec. 9 that nonlinear backreaction and rescattering are neglected and can shut off preheating for some species.
domain assumption The Kamada-Long baryogenesis transport equations and the lattice-based sphaleron rate apply to this model without modification.
Eqs. (8.6)-(8.8) are imported from Refs. [15, 79], including the frozen temperature of 135 GeV and the gamma_W,sph fit.
domain assumption The electroweak weak mixing angle evolution is accurately described by the smooth tanh step function in Eq. (8.5).
The step function and its parameter ranges are taken from analytic and lattice results in Refs. [76, 77] and are not re-derived here.
domain assumption Fermion backreaction and the Schwinger effect can be neglected for the benchmark points used for baryogenesis.
The paper excludes BP a because the Schwinger effect may destroy gauge preheating, but keeps BP b and BP c without a quantitative fermion backreaction calculation.

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Pith. "Pith review of Implication of preheating on gravity assisted baryogenesis in $R^2$-Higgs inflation." pith.science (2026). https://pith.science/paper/II5HXHIF

@misc{pith2026241111128,
  author       = {Pith},
  title        = {Pith review of: Implication of preheating on gravity assisted baryogenesis in $R^2$-Higgs inflation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/II5HXHIF}},
  note         = {Machine review of arXiv:2411.11128}
}

abstract

We investigate the impact of preheating on baryogenesis in $R^2$-Higgs inflation. In this scenario, the inclusion of a dimension-six operator ${(R/ \Lambda^2)} B_{\mu\nu} \widetilde{B}^{\mu\nu} $ abundantly generates helical hypermagnetic fields during inflation, leading to a baryon asymmetric Universe at the electroweak crossover. Focusing on the $R^2$-like regime, we first derive the relevant dynamics of preheating using a doubly-covariant formalism. We find that preheating can happen for the Higgs, transverse gauge and Goldstone bosons, however, it is dependent on the value of the non-minimal coupling $\xi_H$ between the Standard Model Higgs field and the Ricci scalar. We identify the preheating temperature to determine the appropriate scale $\Lambda$ for driving baryogenesis, which is around $\Lambda \sim 2.2 \, (2.6) \times 10^{-5}\, M_{\rm P}$ for $\xi_H \sim 1 \, (10)$. Our results represent the most accurate estimation of the scale of gravity induced baryogenesis in $R^2$-Higgs inflation to date. Areas for further improvement are identified.

Figures

Figures reproduced from arXiv: 2411.11128 by the authors.

**Figure 1.** The effective masses m2 eff,(I) (black) for ϕ (left) and h (right) as in Eq. (4.7) and their respective contributions (4.8) for BPa (upper row), BPb (middle row) and BPc (lower row). See text for details. The normalization v1kv ∗′ 1k − v ′ 1k v ∗ 1k = i and y2ky ∗′ 2k − y ′ 2k y ∗ 2k = i yields constraints |c ϕ 1 | 2 − |c ϕ 2 | 2 = 1 2k , (4.30a) |c h 1 | 2 − |c h 2 | 2 = 1 2k . (4.30b) Here, variations in the mode … view at source ↗

**Figure 2.** Spectra of k 2 2π2a4 ρk,(I) (solid) and k 2 2π2a4 ρ BD k,(I) (dashed) for ϕ (left) and h (right) as given in Eqs. (4.37) and (4.38) for BPa (upper row), BPb (middle row) and BPc (lower row). and h via Eqs. (4.38). In practice, we generate all spectra with wave number close to the unitarity cut-off kUV (which is MP in case of R2 -Higgs inflation [22, 24]). However, to find the quantum energy density from Eqs. (4.38),… view at source ↗

**Figure 3.** The energy densities ρ q (ϕ) (blue) and ρ q (h) (red) as defined in Eqs. (4.38) for the three benchmark points. The black line displays the background energy density ρinf. for BPb and BPc, m2 eff,(h) is large positive for N < 0 but oscillates for N ≳ 0. For BPb, after the end of inflation, m2 eff,(h) does not go below zero in the plotted range N ≲ 3 (see [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: Relative strength of different terms of (ω + Z ) 2 in Eq. (3.37) for BPa (left panel), BPb (middle panel) and BPc (right panel). The upper panel conforms k = a(tend)H(tend) while the lower panel is for k = 10 a(tend)H(tend). The blue, cyan, orange, and red lines in eac…

**Figure 5.** Figure 5: The energy densities ρ q Z (upper row) and ρ q W (lower row) for various values of Λ for the benchmark points BPa (left column), BPb (middle column) and BPc (right column). The black line displays the background energy density ρinf. 5.2.2 Goldstone bosons The Goldstone…

**Figure 6.** Figure 6: Different terms in ω 2 (ϕ2) (τ, k)/a2 . On the bottom panels, the magenta, green, red and black lines correspond to k 2 /a2 , m2 eff,(ϕ2) , m2 Z and E(ϕ2)Υ/a2 , see Eq. (3.55b), for all three BPs. Here as a reference we set k = a(tend)H(tend). We can see that the spike…

**Figure 7.** Figure 7: The energy densities ρ q (ϕ2) and ρ q (ϕ3) for the benchmark points BPa through c from left to right. We recall that ρ q (ϕ3) = ρ q (ϕ4) . The black line displays the background energy density ρinf. At sufficiently early times, when all modes of interest are deep insid…

**Figure 8.** Figure 8: The energy densities ρ q A for various values of Λ and all the benchmark points. The black line displays the background energy density ρinf. 0.0 0.5 1.0 1.5 2.0 1 × 10-7 5 × 10-7 1 × 10-6 5 × 10-6 1 × 10-5 5 × 10-5  0.0 0.5 1.0 1.5 2.0 1 × 10-6 5 × 10-5  0.0 0.5 1.0 …

**Figure 9.** Figure 9: Comparison of different masses for the three BPs after end of inflation. 7 Reheating temperature Our discussion in the preceding sections regarding preheating for different fields did not include effects such as decay and scattering of the produced particles. These int…

**Figure 10.** Figure 10: The ΓZ /H and ΓW /H shown, in the left panels, in blue and red respectively for BPb (top) and BPc (bottom). The energy densities of the Z (middle) and W (right) taking into account the particle decay for the BPb (top) and BPc (bottom), see Eq. (7.3). The black line di…

**Figure 11.** Figure 11: Representative conversion rates WW → ee, WW → hh, and hh → ee as a function of the Hubble constant and the W/h velocity β for a typical epoch at the end of inflation characterized by h0 ≃ 10−5MP , meff,(h) ∼ 2.5 × 10−5MP , mW ∼ 4 × 10−5MP , from [PITH_FULL_IMAGE:figu…

**Figure 12.** Figure 12: The helicity HA for various values of Λ and all the benchmark points BPa, BPb and BPc, from left to right. The parameter fθW encodes all the details on the EWPT dynamics with significant uncertainties fθW = − sin(2θW ) dθW d log T [PITH_FULL_IMAGE:figures/full_fig_p0…

**Figure 13.** Figure 13: Baryogenesis parameter space for BPb (top) and BPc (bottom). On the left panels we show the asymmetry parameter in function of Λ and fθW . The red line must be in between the light and dark blue curves to meet the observational constraint ηB = 8.7 × 10−11. On the righ…

**Figure 14.** Figure 14: Spectra of the transverse mode of the Z boson and the corresponding BD spectra for different values of N . 10-4 10-3 10-2 10-21 10-17 10-13 k [MP] k2 2 π a4 ρ k [M3 P] 10-4 10-3 10-2 10-20 10-18 10-16 10-14 10-12 k [MP] k2 2 π a4 ρ k [M3 P] 10-7 10-6 10-5 10-4 10-15 1…

**Figure 15.** Figure 15: Same figure as [PITH_FULL_IMAGE:figures/full_fig_p035_15.png]

**Figure 16.** Figure 16: Same figure as [PITH_FULL_IMAGE:figures/full_fig_p035_16.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.

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(3.32) can then be brought to the form ï eZ′′ + a2KZ eZ ò + ik ß eZ ′ 0 + a2 e− √ 2 3 φ MP gZ 2 h0 eϕ2 ™ + 8iM 2 P ξRΛ2 ∂τ Å F (φI )e √ 2 3 φ MP ã Ä k × eZ ä = 0

(3.33) The three-components of Eq. (3.32) can then be brought to the form ï eZ′′ + a2KZ eZ ò + ik ß eZ ′ 0 + a2 e− √ 2 3 φ MP gZ 2 h0 eϕ2 ™ + 8iM 2 P ξRΛ2 ∂τ Å F (φI )e √ 2 3 φ MP ã Ä k × eZ ä = 0. (3.34) This is a system of three differential equations. A linear combination of them is obtained by multiplying both sides of Eq. (3.34) by ik and utilizing E...

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completion

(3.47) Likewise, the three components of Eqs. (3.46) read as ï fW±′′ + a2KW fW± ò + ik ß fW ±′ 0 ± a2 e− √ 2 3 φ MP ie 2 √ 2sW h0( eϕ3 ± i eϕ4) ™ + 8iM 2 P ξRΛ2 ∂τ Å F (φI )e √ 2 3 φ MP ã Ä k × fW± ä = 0. (3.48) Applying the same procedure as for the Z boson, and using Eq. (3.48), we find fW ±′ 0 = ∓ a2 e− √ 2 3 φ MP ie 2 √ 2sW h0( eϕ3 ± i eϕ4). (3.49) Co...

2018

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