arxiv: 2601.04340 · v2 · submitted 2026-01-07 · 🌌 astro-ph.CO · gr-qc· hep-ph

Recognition: 1 theorem link

· Lean Theorem

Dark QCD Origin of the NANOGrav Signal and Self-Interacting Dark Matter

Zihan Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-16 15:57 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qchep-ph

keywords dark QCDNANOGravgravitational wavesself-interacting dark matterfirst-order phase transitionentropy dilutionpulsar timing arraysMeV scale

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The pith

A near-conformal dark SU(3) gauge theory produces a MeV-scale first-order phase transition that fits the NANOGrav gravitational wave signal while setting the dark matter abundance through entropy dilution.

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

The paper shows that an SU(3)_D gauge theory with 6-8 light flavors, staying close to the conformal window, undergoes a strong first-order phase transition at temperatures of 5-6 MeV. This transition generates a stochastic gravitational wave background whose amplitude matches the NANOGrav 15-year observations at a level comparable to supermassive black hole binary predictions. The same setup supplies a heavy dark baryon of mass around 40 GeV as self-interacting dark matter, mediated by a light pseudo-dilaton of 20-50 MeV, and ties the wave amplitude directly to the observed dark matter density via an entropy dilution factor D approximately equal to alpha to the power 3/4. Explicit calculations of the bounce action, bubble velocity, and extra relativistic degrees of freedom confirm that the benchmark parameters remain consistent with cosmology.

Core claim

For representative parameters T_n ≈ 5–6 MeV, α ∼ 500–1000, β/H_* ∼ 30–50, an SU(3)_D gauge theory with N_f ∼ 6–8 light flavors undergoing a first-order phase transition at the MeV scale produces gravitational waves in the pulsar timing array band that fit NANOGrav data, while the entropy dilution D ≈ α^{3/4} connects the wave amplitude to the relic density of a 40 GeV dark baryon whose self-interactions are mediated by a light pseudo-dilaton arising from walking dynamics near the conformal window.

What carries the argument

The central mechanism is the first-order phase transition in a near-conformal SU(3)_D gauge theory with walking dynamics, which generates the gravitational wave spectrum and the entropy dilution factor that fixes the dark matter abundance.

Load-bearing premise

The assumption that an SU(3)_D gauge theory with 6-8 light flavors stays near-conformal and produces a first-order phase transition at the MeV scale with transition strength alpha around 500-1000 and duration beta over H star around 30-50.

What would settle it

Future pulsar timing array data showing a gravitational wave spectrum that does not rise as frequency cubed at low frequencies and fall as frequency to the minus four at high frequencies, or a measurement of Delta N_eff larger than 0.1 without corresponding changes in the dark pion mass relative to the dilaton mass.

Figures

Figures reproduced from arXiv: 2601.04340 by Zihan Wang.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

The NANOGrav 15-year stochastic gravitational wave background (SGWB) amplitude $A_{\rm yr} \approx 2.4 \times 10^{-15}$ lies at the upper edge of population synthesis predictions for supermassive black hole binaries (SMBHBs), motivating exploration of additional cosmological sources. We present a phenomenological framework based on an $\text{SU}(3)_D$ gauge theory that can simultaneously accommodate the gravitational wave signal and resolve small-scale structure anomalies via Self-Interacting Dark Matter (SIDM). The dark matter candidate is a heavy dark baryon $\chi = QQQ$ with mass $m_\chi \approx 40$~GeV, which self-interacts through a light pseudo-dilaton $d$ $m_d \approx 20$--$50$~MeV as a pseudo-Goldstone boson of approximate scale invariance arising in near-conformal gauge theories with $N_f \sim 6$--$8$ light flavors. A first-order phase transition at the MeV scale, enabled by walking dynamics near the conformal window, produces gravitational waves in the PTA band. For representative parameters $T_n \approx 5$--$6$~MeV, $\alpha \sim 500$--$1000$, $\beta/H_* \sim 30$--$50$, the model provides a fit to NANOGrav data comparable to SMBHB while naturally connecting the gravitational wave amplitude to the dark matter relic density through entropy dilution $D \approx \alpha^{3/4}$. We present explicit calculations of the bounce action, bubble wall velocity, and $\Delta N_{\rm eff}$, demonstrating that the benchmark parameters are theoretically consistent and cosmologically safe ($\Delta N_{\rm eff} \lesssim 0.1$ for $m_\pi > 2m_d$). The distinctive spectral shape ($f^3 \to f^{-4}$) provides a robust prediction testable with future PTAs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A dark QCD model ties NANOGrav to SIDM via MeV-scale phase transition and entropy dilution, but the extreme transition strength looks hard to realize.

read the letter

The paper's core idea is a single SU(3)_D dark sector with near-conformal dynamics that produces a first-order phase transition at a few MeV. This generates a gravitational-wave background matching the NANOGrav amplitude while the same setup supplies self-interacting dark matter through heavy baryons mediated by a light pseudo-dilaton, with entropy dilution D scaling as alpha to the 3/4 linking the two observables. For the quoted benchmarks of T_n around 5-6 MeV, alpha 500-1000, and beta/H 30-50, the model claims a fit comparable to supermassive black hole binaries plus a distinctive spectral shape that future PTAs could test. It also reports explicit bounce-action, wall-velocity, and Delta N_eff calculations to argue the parameters stay safe. That combination of PTA signal and SIDM phenomenology in one framework is the main new element, and the concrete spectral prediction is a useful output. The calculations of the transition quantities are presented as done, which gives the reader something concrete to examine. The soft spot is the transition itself. Standard results for walking SU(3) theories with 6-8 flavors tend to produce milder transitions or crossovers, and alpha values above a few hundred at MeV temperatures would require supercooling that is not obviously supported by existing lattice or effective-potential work. The D scaling relation also appears tuned to close the relic-density loop rather than emerging independently. There are multiple free parameters (m_chi, m_d, T_n, alpha, beta/H) so the fit has flexibility. This is for people working on PTA interpretations or dark-sector models who want to see how the pieces might connect. A reader focused on gravitational-wave cosmology or self-interacting dark matter would get value from the specific linkage and the testable spectrum. It deserves a serious referee so that lattice and effective-theory experts can check whether the required transition strength is actually attainable.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes a phenomenological SU(3)_D dark gauge theory with N_f ∼ 6–8 near-conformal flavors that simultaneously accounts for the NANOGrav 15-year SGWB amplitude via a first-order phase transition at T_n ≈ 5–6 MeV and supplies a self-interacting dark matter candidate χ = QQQ whose interactions are mediated by a light pseudo-dilaton d. For benchmark values α ∼ 500–1000 and β/H_* ∼ 30–50 the model reproduces the observed A_yr while linking the GW amplitude to the χ relic density through entropy dilution D ≈ α^{3/4}; explicit calculations of the bounce action, wall velocity, and ΔN_eff are stated to confirm theoretical consistency and cosmological safety (ΔN_eff ≲ 0.1).

Significance. If the extreme transition parameters can be realized, the framework would provide a unified cosmological origin for the PTA signal and small-scale DM structure anomalies together with a distinctive f^3 → f^{-4} spectral shape. The claimed explicit computations of bounce action and ΔN_eff constitute a positive feature that could strengthen the result if supported by concrete derivations.

major comments (3)

[Abstract and benchmark parameters] Abstract and benchmark parameters: the values α ∼ 500–1000 and β/H_* ∼ 30–50 are selected to reproduce the NANOGrav amplitude and to enforce D ≈ α^{3/4}; no lattice result or effective-potential calculation is referenced showing that walking SU(3)_D dynamics with N_f ∼ 6–8 can generate vacuum-energy ratios α > 100 at T ∼ 5 MeV, contrary to standard expectations for near-conformal theories that typically yield milder transitions or crossovers.
[Entropy-dilution link] Entropy-dilution link: the claimed natural connection between GW amplitude and relic density is realized by setting D ≈ α^{3/4} where α itself is adjusted to fit the observed A_yr; this renders the link enforced by parameter choice rather than independently derived from the model dynamics.
[ΔN_eff and cosmological safety] ΔN_eff and cosmological safety: while the abstract asserts explicit calculations yielding ΔN_eff ≲ 0.1 for m_π > 2m_d, the manuscript does not provide the explicit range of m_d, the precise entropy-injection formula, or error propagation on the benchmark parameters, leaving the robustness of the cosmological constraint unverified.

minor comments (1)

[Notation] Notation for the pseudo-dilaton mass m_d and its relation to the approximate scale invariance should be defined more explicitly in the text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments on our manuscript. We address each major comment below, providing clarifications on the phenomenological framework, the origin of the parameter relations, and the explicit calculations. Where appropriate, we indicate revisions to strengthen the presentation.

read point-by-point responses

Referee: [Abstract and benchmark parameters] Abstract and benchmark parameters: the values α ∼ 500–1000 and β/H_* ∼ 30–50 are selected to reproduce the NANOGrav amplitude and to enforce D ≈ α^{3/4}; no lattice result or effective-potential calculation is referenced showing that walking SU(3)_D dynamics with N_f ∼ 6–8 can generate vacuum-energy ratios α > 100 at T ∼ 5 MeV, contrary to standard expectations for near-conformal theories that typically yield milder transitions or crossovers.

Authors: Our model is explicitly phenomenological, employing an effective potential for the walking SU(3)_D theory near the conformal window that permits strong first-order transitions with large α when the beta function is sufficiently flat. While we acknowledge that direct lattice simulations at MeV-scale temperatures for N_f = 6–8 are not yet available and standard expectations for near-conformal theories often favor milder transitions, the benchmark values are motivated by the expected enhancement of the vacuum energy in walking dynamics, as supported by effective-potential analyses in the literature on near-conformal gauge theories. We will add explicit references to such studies and a brief discussion of the effective-potential assumptions in the revised manuscript to address this concern. revision: partial
Referee: [Entropy-dilution link] Entropy-dilution link: the claimed natural connection between GW amplitude and relic density is realized by setting D ≈ α^{3/4} where α itself is adjusted to fit the observed A_yr; this renders the link enforced by parameter choice rather than independently derived from the model dynamics.

Authors: The relation D ≈ α^{3/4} is derived from the entropy release associated with the vacuum energy fraction α during the phase transition, which directly determines both the gravitational-wave amplitude (via the standard PT formulas) and the dilution factor affecting the χ relic density. Although α is tuned to reproduce A_yr, this single parameter choice simultaneously fixes the post-transition entropy and thus the DM abundance for the fixed m_χ, yielding a predictive connection rather than independent tuning. We will clarify this derivation and its implications in the revised text to emphasize that the link follows from the shared dynamics of the transition. revision: partial
Referee: [ΔN_eff and cosmological safety] ΔN_eff and cosmological safety: while the abstract asserts explicit calculations yielding ΔN_eff ≲ 0.1 for m_π > 2m_d, the manuscript does not provide the explicit range of m_d, the precise entropy-injection formula, or error propagation on the benchmark parameters, leaving the robustness of the cosmological constraint unverified.

Authors: We thank the referee for noting this omission. The explicit calculations of ΔN_eff, including the entropy-injection formula ΔN_eff ≈ (g_*/g_*^{SM}) (T_d/T_n)^4 with T_d determined by m_d decays, the benchmark range m_d = 20–50 MeV, and the condition m_π > 2 m_d, are presented in Section 4 of the manuscript. We will expand this section in the revision to include the full formula, tabulated values for the benchmark parameters, and a brief propagation of uncertainties on α and β/H_* to demonstrate the robustness of ΔN_eff ≲ 0.1. revision: yes

Circularity Check

1 steps flagged

Fitted α to NANOGrav determines D ≈ α^{3/4} for relic-density connection by construction

specific steps

fitted input called prediction [Abstract]
"For representative parameters T_n ≈ 5–6 MeV, α ∼ 500–1000, β/H_* ∼ 30–50, the model provides a fit to NANOGrav data comparable to SMBHB while naturally connecting the gravitational wave amplitude to the dark matter relic density through entropy dilution D ≈ α^{3/4}."

α is adjusted to match the observed SGWB amplitude; the same α then fixes D via the stated relation, which in turn fixes the relic density to the observed value. The connection is therefore enforced by the choice of the fitted parameter rather than predicted independently of the NANOGrav fit.

full rationale

The paper selects benchmark values of α (∼500–1000) to reproduce the NANOGrav amplitude for given T_n and β/H_*. The entropy dilution factor is then defined as D ≈ α^{3/4} and used to set the post-transition relic density of χ to the observed value. This makes the claimed 'natural connection' between the GW signal and the DM abundance a direct consequence of the parameter choice rather than an independent derivation. No other circular patterns (self-citation chains, uniqueness theorems, or ansatz smuggling) are present in the provided text. The bounce-action and ΔN_eff calculations support consistency of the chosen parameters but do not remove the fitting dependence for the central claim.

Axiom & Free-Parameter Ledger

5 free parameters · 2 axioms · 2 invented entities

The central claim rests on the existence of a near-conformal dark SU(3) theory, a strong first-order transition at MeV temperatures, and several parameters tuned to observations.

free parameters (5)

m_χ = 40 GeV
Mass of the dark baryon set near 40 GeV to produce the correct relic density after dilution
m_d = 20-50 MeV
Mass of the pseudo-dilaton chosen in 20-50 MeV to yield SIDM cross sections
T_n = 5-6 MeV
Nucleation temperature fixed at 5-6 MeV to place the GW peak in the PTA band
α = 500-1000
Transition strength parameter adjusted to 500-1000 to match observed GW amplitude
β/H_* = 30-50
Inverse duration parameter set to 30-50 to shape the spectrum

axioms (2)

domain assumption An SU(3)_D gauge theory with N_f ∼ 6-8 flavors lies near the conformal window and produces walking dynamics
Invoked to justify the existence of a light pseudo-dilaton and a first-order transition at MeV scale
standard math A first-order phase transition at T_n ∼ 5 MeV generates a stochastic GW background whose amplitude scales with α and β/H_*
Standard result from cosmological bubble nucleation calculations

invented entities (2)

Dark baryon χ = QQQ no independent evidence
purpose: Candidate for self-interacting dark matter
Postulated composite state whose mass and interactions are set by the dark gauge theory
Pseudo-dilaton d no independent evidence
purpose: Light mediator for SIDM and remnant of approximate scale invariance
Introduced as pseudo-Goldstone boson of the near-conformal dynamics

pith-pipeline@v0.9.0 · 5665 in / 2010 out tokens · 53370 ms · 2026-05-16T15:57:35.466838+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

For representative parameters T_n ≈ 5–6 MeV, α ∼ 500–1000, β/H_* ∼ 30–50, the model provides a fit to NANOGrav data comparable to SMBHB while naturally connecting the gravitational wave amplitude to the dark matter relic density through entropy dilution D ≈ α^{3/4}.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages · 21 internal anchors

[1]

This convergence suggests that the GW signal and the correct dark matter relic density are mutually consistent out- comes of a single dynamical mechanism

to the observed abundance ( Ωobsh2 ≈ 0.12). This convergence suggests that the GW signal and the correct dark matter relic density are mutually consistent out- comes of a single dynamical mechanism. IV. GRA VIT A TIONAL W A VE ANAL YSIS We evaluate the predicted GW spectrum against the NANOGrav 15-year dataset [ 7] to determine observa- tional viability. ...

work page
[2]

For our GW benchmark ( vS ≈ 18.8 MeV, gD ≈ 0.075), we obtain mZ′ ≈ 2.8 MeV

This breaks U (1)D → Z2, generating the mass spectrum: • Dark Gauge Boson ( Z ′): The gauge field ac- quires a mass mZ′ = gDqSvS = 2 gDvS. For our GW benchmark ( vS ≈ 18.8 MeV, gD ≈ 0.075), we obtain mZ′ ≈ 2.8 MeV. This massive vector is re- sponsible for the wall friction. • Pseudo-Dirac Dark Matter: The Yukawa term yD⟨S⟩ generates a Majorana mass splitt...

work page
[3]

We assume the scalar ϕ mixes with the Higgs or couples to vector-like leptons to generate an effective interaction with electrons: Lportal = geϕ¯ee

The Leptophilic Portal The dark sector communicates with the Standard Model primarily through a leptophilic scalar portal. We assume the scalar ϕ mixes with the Higgs or couples to vector-like leptons to generate an effective interaction with electrons: Lportal = geϕ¯ee. (A3) We fix ge ≈ 10−6. This choice is phenomenologically motivated to ensure the scal...

work page
[4]

prompt decay

Experimental Constraints The model benchmark ( mϕ ≈ 20 MeV, ge ≈ 10−6, ∆m ≈ 100 eV) satisfies all robust laboratory and astro- physical bounds: • Beam Dump Experiments (E137): The scalar decay length is cτϕ ≈ 18 cm[34]. This is suf- ficiently prompt that ϕ decays primarily within the thick shielding of fixed-target experiments (like E137), preventing the ...

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[5]

The T ree-Level Barrier Potential To simultaneously satisfy the small-scale structure con- straints of Self-Interacting Dark Matter (which requires a small gauge coupling gD ≈ 0.15) and the large vacuum energy required for the NANOGrav signal (which implies 6 α ≫ 1), we utilize a scalar potential with a tree-level bar- rier. We adopt a renormalizable fini...

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[6]

Derivation of T ransition Parameters The phase transition occurs when the bubble nucle- ation rate Γ ∼ e−S3/T becomes comparable to the Hub- ble expansion rate. Numerical minimization of the Eu- clidean bounce action S3 for our benchmark potential yields a nucleation temperature of: T∗ ≈ 1.24 MeV (B3) We now rigorously derive the transition strength param...

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We consider a co- annihilation scenario where the dark matter ground state χ1 and excited state χ2 are degenerate in mass ( m1 ≈ m2 ≈ mχ) during freeze-out

Thermal F reeze-out and Overproduction The evolution of the dark matter number density nχ is governed by the Boltzmann equation. We consider a co- annihilation scenario where the dark matter ground state χ1 and excited state χ2 are degenerate in mass ( m1 ≈ m2 ≈ mχ) during freeze-out. The relevant annihilation channel is the t-channel pro- cess χχ → ϕϕ. T...

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Assuming instantaneous reheating, the vacuum energy density is converted to radiation: ρvac = π2 30 g∗T 4 rh

Entropy Dilution The first-order phase transition releases vacuum energy ρvac, which is converted into radiation at the reheating temperature Trh. Assuming instantaneous reheating, the vacuum energy density is converted to radiation: ρvac = π2 30 g∗T 4 rh. (C7) Using the potential parameters derived in Appendix B , we find Trh ≈ 5.76 MeV. The dilution fac...

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Final Relic Density The diluted relic abundance is the thermal abundance divided by the dilution factor: Ωobsh2 = Ωthermalh2 D ≈ 12.0 100.2 ≈ 0.1198 ≈ 0.120. (C9) This result aligns perfectly with the Planck 2018 mea- surement Ωch2 = 0 .120 ± 0.001, demonstrating that the specific thermodynamics required for the GW signal nat- urally generate the exact di...

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[10]

The abundance today is given by: h2Ωsw(f ) = 2 .65×10−6Υ H∗ β κα 1 + α 2 100 g∗ 1/3 vwSsw(f )

The Sound W ave Amplitude The GW energy density spectrum from sound waves, Ωsw(f ), is calculated using the analytic fits derived from hydrodynamic simulations. The abundance today is given by: h2Ωsw(f ) = 2 .65×10−6Υ H∗ β κα 1 + α 2 100 g∗ 1/3 vwSsw(f ). (D1) We derive each component of this equation below

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For strong transitions ( α ≫ 1) where the bubble walls reach a terminal velocity, the energy transfer is highly eﬀicient

Eﬀiciency F actor ( κ): This factor represents the fraction of the released vacuum energy that is converted into the kinetic energy of the bulk fluid motion. For strong transitions ( α ≫ 1) where the bubble walls reach a terminal velocity, the energy transfer is highly eﬀicient. Using the fit from [ 35]: κ ≈ α 0.73 + 0.083√α + α . (D2) For our benchmark α...

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S2, the friction is suﬀicient to stop runaway ( γeq ∼ 104) but small enough that the wall remains ultra-relativistic

W all V elocity ( vw): As derived in Sec. S2, the friction is suﬀicient to stop runaway ( γeq ∼ 104) but small enough that the wall remains ultra-relativistic. We there- fore set the wall velocity to the speed of light, vw ≈ 1

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However, in strong transitions ( α ≫ 1), the acoustic period is truncated by the onset of non- linearities (shocks) and the expansion of the universe

RMS Fluid V elocity and Finite Lifetime Standard simulations assume the sound waves persist for a Hubble time. However, in strong transitions ( α ≫ 1), the acoustic period is truncated by the onset of non- linearities (shocks) and the expansion of the universe. The root-mean-square velocity of the plasma fluid is required to calculate the sound wave lifet...

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Red- shifting this frequency to today involves the ratio of scale factors a∗/a0, determined by entropy conservation (gsT 3a3 = const)

Peak F requency Derivation The peak frequency of the spectrum corresponds to the characteristic scale of the bubbles, f∗ ≈ 2/R∗. Red- shifting this frequency to today involves the ratio of scale factors a∗/a0, determined by entropy conservation (gsT 3a3 = const). The peak frequency today is: fp,0 ≈ 1.9 × 10−5 Hz g∗ 10 1/6 T∗ 100 GeV β H∗ 1 vw . (D11) Subs...

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It rises as f 3 due to causality at super-horizon scales and falls as f −4 at high frequencies: Ssw(f ) = f fp,0 3 7 4 + 3(f /fp,0)2 7/2

Spectral Shape F unction The frequency dependence Ssw(f ) follows a broken power law. It rises as f 3 due to causality at super-horizon scales and falls as f −4 at high frequencies: Ssw(f ) = f fp,0 3 7 4 + 3(f /fp,0)2 7/2 . (D13) This causal f 3 slope is a key discriminator against other sources . Appendix E: BUBBLE W ALL DYNAMICS AND FRICTION The accura...

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In the thin-wall limit, this is well- approximated by the vacuum energy density difference : Pdrive = ∆Veff ≈ ρvac = λeffv4 S 8

V acuum Driving Pressure The force driving the expansion of the bubble is the pressure difference between the false vacuum (symmet- ric phase) inside the bubble and the true vacuum (bro- ken phase) outside. In the thin-wall limit, this is well- approximated by the vacuum energy density difference : Pdrive = ∆Veff ≈ ρvac = λeffv4 S 8 . (E1) Using our bench...

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This interaction exerts a friction pressure Pfric on the wall

F riction Mechanisms As the bubble wall sweeps through the plasma, parti- cles transition from the symmetric phase to the broken phase. This interaction exerts a friction pressure Pfric on the wall. For a wall moving with an ultra-relativistic Lorentz factor γ ≫ 1, the dominant friction mechanism is not simple Boltzmann scattering ( P ∼ T 4), but rather t...

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(E4) We can rewrite this in terms of the transition strength parameter α = ρvac/ρrad(T∗), where ρrad ∼ π2 30 g∗T 4 ∗ : γeq ≈ ρvac g2 DT 4∗ = αρrad g2 DT 4∗ ≈ α g2 D π2g∗ 30

T erminal V elocity Derivation The bubble wall reaches a terminal steady state when the friction pressure counteracts the driving pressure: Pdrive = Pfric =⇒ ρvac ≈ γeqg2 DT 4 ∗ . (E4) We can rewrite this in terms of the transition strength parameter α = ρvac/ρrad(T∗), where ρrad ∼ π2 30 g∗T 4 ∗ : γeq ≈ ρvac g2 DT 4∗ = αρrad g2 DT 4∗ ≈ α g2 D π2g∗ 30 . (E...

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Runaway occurs only if the friction saturates or if the wall never reaches equilibrium before bubbles collide

V alidity of the Sound W ave T emplate To confirm that the energy is deposited into the plasma rather than accelerating the wall indefinitely, we must compare γeq to the runaway threshold. Runaway occurs only if the friction saturates or if the wall never reaches equilibrium before bubbles collide. The collision limit γcoll is determined by the size of th...

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Kinematic Blocking Mechanism A critical consistency requirement of the model is bal- ancing the wall friction with reheating eﬀiciency. The first is Friction Requirement: The Z ′ boson must couple strongly to the dark sector plasma ( gD ≈ 0.15) to gener- ate the transition radiation friction required to stop bub- ble runaway.There is also Reheating Requir...

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At the instant of re- heating, the vacuum energy ρvac is partitioned into dark radiation ( ρDR) and visible radiation ( ρSM): ρDR = Bρvac, ρ SM = (1 − B)ρvac

Derivation of Effective Neutrino Number We define the branching ratio to the dark radiation sector as B ≡ Γ(Z ′ → νD ¯νD)/Γtot. At the instant of re- heating, the vacuum energy ρvac is partitioned into dark radiation ( ρDR) and visible radiation ( ρSM): ρDR = Bρvac, ρ SM = (1 − B)ρvac. (F1) The deviation in the effective number of relativistic species, ∆N...

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The entropy dilution factor D scales with the reheating tem- perature as D ∝ T 3 rh

Impact on Entropy Dilution Finally, we verify that this energy leakage does not disrupt the relic density solution derived before. The entropy dilution factor D scales with the reheating tem- perature as D ∝ T 3 rh. Since a fraction B of the energy is diverted to dark radiation, the effective reheating tem- perature of the SM plasma is reduced: T ′ rh = T...

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Initial Conditions at Nucleation At the nucleation temperature T∗ ≈ 1.24 MeV, the collision of bubbles generates Magnetohydrodynamic (MHD) turbulence in the plasma

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Assuming equipartition between kinetic and magnetic energy, the initial magnetic energy density is: ρB,∗ ≈ ϵturbρvac

Magnetic Energy Density: A fraction ϵturb of the released vacuum energy ρvac is converted into tur- bulent kinetic energy. Assuming equipartition between kinetic and magnetic energy, the initial magnetic energy density is: ρB,∗ ≈ ϵturbρvac. (G1) However, the turbulence is generated at the bubble colli- sion scale R∗, which is smaller than the Hubble horiz...

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Helicity Generation: The dark sector contains chiral fermions Ψ charged under U (1)D. During the phase transition, the changing gauge fields and parity- violating wall interactions generate a non-zero Chern- Simons number, leading to maximal magnetic helicity: H = Z V A · B d3x ≈ λ∗ · EB, (G3) where λ∗ is the initial correlation length ( ∼ R∗) and EB is t...

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For non-helical fields, energy cascades to smaller scales and dissipates via viscosity, decaying rapidly as B ∝ a−2τ −1/2

The Helical Inverse Cascade The evolution of the magnetic field is governed by the decay of MHD turbulence. For non-helical fields, energy cascades to smaller scales and dissipates via viscosity, decaying rapidly as B ∝ a−2τ −1/2. However, for a maxi- mally helical field, the magnetic helicity H is a conserved quantity (up to resistivity, which is negligi...

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Present-Day Observables We evolve the field from T∗ to the present temperature T0. The redshift of the magnetic field strength, account- ing for the inverse cascade, leads to the present-day value B0: B0 ≈ 10−13 G T∗ 1 MeV 1/3 ϵturb 0.05 1/2 fH 1 1/3 , (G6) where fH ≈ 1 is the helicity fraction. Substituting our benchmark values ( T∗ ≈ 1.24 MeV): B0 ≈ 10−...

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