WIMP Freeze-Out in Diffusive Unimodular Gravity

Abstract: We study a non-standard cosmology (NSC) scenario within the Unimodular Gravity (UG) framework, sourced by a scalar field $φ$ that undergoes energy diffusion, parametrized by the diffusion parameter $x$, the initial energy densities rate $κ\equivρ<em>φ/ρ</em>γ|<em>{\text{ini}}$, and the end-of-domination temperature $T</em>{\text{end}}$. We compare this UG+NSC scenario with standard NSC and $Λ$CDM cosmologies for WIMP Dark Matter (DM) production via the freeze-out mechanism. We find that energy diffusion reshapes the allowed $(m_χ, \langleσv\rangle)$ parameter space, where $m_χ$ is the DM mass and $\langle σv \rangle$ is the thermally averaged annihilation cross section, opening regions otherwise excluded by DM overproduction in $Λ$CDM, and shifting the mass and cross-section ranges accessible to WIMP candidates depending on $x$, $κ$, $T_{\text{end}}$, and the barotropic index $ω$ of $φ$. As a concrete application, we implement this framework for the Real Singlet Scalar WIMP and test its $(m_χ, λ<em>{HS})$ parameter space against current direct detection bounds from the LZ experiment, showing that energy diffusion opens previously unconstrained regions in the Higgs-portal coupling $λ</em>{HS}$ and thereby alters the detectability prospects of this benchmark model in future searches.

• The paper proposes a new cosmological model using Unimodular Gravity with energy diffusion, which significantly enhances entropy production and alters WIMP freeze-out dynamics, shifting viable dark matter parameters by up to sixteen orders of magnitude.
• The diffusion mechanism in Unimodular Gravity modifies both the conservation equations and the Friedmann equation, leading to a more pronounced effect on cosmological expansion than bulk viscosity-based models.
• The research demonstrates that energy diffusion can relax constraints on dark matter direct detection experiments, particularly in the context of a singlet scalar model, providing new parameter spaces to explore.

This paper develops a non-standard cosmology (NSC) within Unimodular Gravity (UG), in which an early-universe field $\phi$ undergoes energy diffusion rather than decay alone, and examines its consequences for WIMP dark matter production via thermal freeze-out (2608.19482). The work extends a prior program in which dissipative NSCs based on bulk viscosity were applied to WIMP and FIMP candidates; the key distinction is that diffusion in UG modifies not only the conservation equations but also the first Friedmann equation, altering the expansion history more substantially than viscosity-based models.

Unimodular Gravity with energy diffusion

The authors adopt the modern formulation of UG based on volume-preserving diffeomorphisms ($\sqrt{-g}=\epsilon_0$), for which the Noether theorem yields a modified matter conservation law. The cosmological "constant" becomes a function $\lambda = \Lambda + Q(x)$, where $Q$ is the diffusion function encoding energy-momentum nonconservation, with $\Lambda$ recovered as an integration constant. For a spatially flat FLRW universe containing radiation and $\phi$, they assume diffusion sourced exclusively by $\phi$ with $Q = x\rho_\phi$, giving

$$3H^2 = \frac{\rho_\gamma + (1+x)\rho_\phi}{M_p^2}, \qquad \dot{\rho}_\phi + 3H\rho_\phi\left(\frac{\omega+1}{1+x}\right) = -\frac{\Gamma_\phi}{1+x}\rho_\phi.$$

Two effective quantities follow directly from this structure: an effective barotropic index $\omega_\text{eff}=(\omega - x)/(1+x)$ and an effective decay rate $\sqrt{-g}=\epsilon_0$0. For $\sqrt{-g}=\epsilon_0$1, both dilution and decay of $\sqrt{-g}=\epsilon_0$2 are slowed; notably, even a pressureless fluid ($\sqrt{-g}=\epsilon_0$3) acquires a negative $\sqrt{-g}=\epsilon_0$4, mimicking quintessence-like behavior without a fundamental scalar. Since $\sqrt{-g}=\epsilon_0$5 would accelerate depletion of $\sqrt{-g}=\epsilon_0$6, the analysis is restricted to $\sqrt{-g}=\epsilon_0$7. Because $\sqrt{-g}=\epsilon_0$8, the standard $\sqrt{-g}=\epsilon_0$9CDM limit is recovered once $\lambda = \Lambda + Q(x)$0 decays, ensuring consistency with post-BBN cosmology provided $\lambda = \Lambda + Q(x)$1 MeV.

Impact on entropy injection and the relic density

Compared to the classical NSC scenario at identical benchmark parameters, the diffusive UG evolution enhances entropy injection by roughly ten orders of magnitude in energy-density evolution. In the $\lambda = \Lambda + Q(x)$2–$\lambda = \Lambda + Q(x)$3 plane ($\lambda = \Lambda + Q(x)$4), the relic-density contour shifts toward smaller $\lambda = \Lambda + Q(x)$5: by about four orders of magnitude for $\lambda = \Lambda + Q(x)$6, seven orders for $\lambda = \Lambda + Q(x)$7, and up to sixteen orders at low $\lambda = \Lambda + Q(x)$8 for $\lambda = \Lambda + Q(x)$9, all at $Q$0. This scaling with $Q$1 follows directly from $Q$2: the larger the intrinsic barotropic index, the faster $Q$3 dilutes in standard NSC relative to its effectively slower UG behavior.

In the $Q$4 plane, the UG+NSC scenario opens regions excluded in $Q$5CDM by DM overproduction. Parameter scans show that increasing $Q$6 steepens the relic-density contours and permits smaller cross sections at lower masses; increasing $Q$7 shifts contours leftward toward lighter masses, while larger $Q$8 (earlier decay, less entropy injection) allows heavier candidates. Heavier WIMPs freeze out earlier (during radiation domination) and are therefore more sensitive to subsequent dilution, requiring less initial $Q$9 abundance; lighter candidates decouple later and are less affected. These results are stated to be model-independent, depending only on $\Lambda$0 and $\Lambda$1.

Application to Real Singlet Scalar DM

As a concrete benchmark, the authors consider the $\Lambda$2-symmetric real singlet scalar extension of the SM with Higgs-portal coupling $\Lambda$3, restricted to $\Lambda$4 GeV—above the top threshold and away from the Higgs resonance—where $\Lambda$5 can be treated as velocity- and temperature-independent and expressed analytically in $\Lambda$6. This approximation is what makes a full scan over the four-dimensional NSC parameter space tractable; near resonances and thresholds it breaks down, and the authors explicitly defer that treatment to future work.

Confronting the resulting parameter space with LZ spin-independent direct detection limits, the central result is that energy diffusion shifts the relic-density curve toward smaller $\Lambda$7, leaving a sizable region below the LZ exclusion line in mass ranges already ruled out under $\Lambda$8CDM and standard NSC assumptions. Varying $\Lambda$9 from $\phi$0 to $\phi$1 progressively enlarges this unconstrained region, with the relaxation reaching up to an order of magnitude in $\phi$2, and the effect is not saturated within the range of $\phi$3 studied—an implication being that next-generation direct detection experiments are required to probe the full parameter space opened by diffusion.

Limitations and open questions

Several caveats bear on these results. The diffusion function $\phi$4 is assumed proportional to the energy density of $\phi$5 only, and while this parametrization has been tested against late-time SNe Ia and OHD data elsewhere, no microphysical derivation of $\phi$6 or of the diffusion mechanism itself is provided. The singlet scalar analysis is confined to a single benchmark model, high masses ($\phi$7 GeV), and a temperature-independent cross-section approximation, so resonance and coannihilation regimes remain unexamined. Additionally, constraints beyond BBN consistency—such as $\phi$8 bounds from CMB measurements on extended radiation-like content—are not systematically imposed, and the interplay between diffusion and perturbation-level observables is left open.

Conclusion

The paper establishes that energy diffusion within UG constitutes a distinct class of NSC, distinguishable from viscous or purely decaying-field scenarios through its modification of the Friedmann equation via the $\phi$9 term. Quantitatively, it shifts viable WIMP parameters by up to sixteen orders of magnitude in the $\phi$0–$\phi$1 plane and relaxes LZ direct-detection tension by up to an order of magnitude in the Higgs-portal coupling for the singlet scalar benchmark. The framework's generality—depending only on $\phi$2 and $\phi$3—makes it applicable across WIMP realizations, though extending it to resonance regions, additional particle models, and precision cosmological constraints remains outstanding.