Post Matter–Radiation Equality Behavior

• Post matter–radiation equality behavior is the epoch when nonrelativistic matter dominates, driving gravitational collapse and cosmic structure formation.
• Comparative models like ΛCDM and f(R,L_m) gravity reveal distinct impacts on expansion rates, density perturbation growth, and CMB acoustic features.
• Alternative scenarios, including early dark energy, AdS vacua, and dark matter decay to dark radiation, offer testable predictions to address current cosmological tensions.

Post matter–radiation equality behavior refers to the physical, dynamical, and observational consequences of the transition and subsequent evolution of the Universe after the epoch at which nonrelativistic matter (dark and baryonic) first began to dominate the cosmic energy budget over radiation. This epoch, occurring at redshift $z_{\mathrm{eq}} \sim 3400$, marks a pivotal transition in structure formation, cosmological expansion, and the interplay of standard and nonstandard sectors. Contemporary research explores both ΛCDM and beyond–ΛCDM paradigms—including modifications from AdS vacua, early dark energy, dark sector interactions, alternative gravity, and nonstandard initial conditions—each with distinct implications for the cosmic expansion rate, growth of perturbations, and cosmological observables in the post–equality era.

1. Background: Dynamics and Expansion after Equality

Following matter–radiation equality, the Universe enters an era dominated by pressureless matter (cold dark matter and baryons), with residual radiation contributing subdominantly until recombination. In standard ΛCDM,

$$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$

where $a$ is the scale factor normalized to unity today. Matter domination promotes the linear growth of density perturbations, permitting the gravitational collapse of potential wells and the emergence of cosmic large-scale structure.

In modified gravity scenarios, such as $f(R,L_m)$ gravity, the expansion law and growth of structure acquire additional corrections. The effective dark energy–like term scales as $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$ with $w \simeq -0.005$, renormalizing the effective matter density and altering both the Hubble expansion rate and the corresponding growth equations (Goswami et al., 14 Jul 2025). This leads to small but testable deviations from the standard scaling, e.g., $a(t) \propto t^{2/3}$ in ΛCDM is replaced by $a(t) \propto t^{2/[3(1+w)]}$ in $f(R,L_m)$.

Non-gravitational processes, such as decay of dark matter into dark radiation, further perturb the background by draining cold dark matter density and injecting a component with radiation-like dilution that remains negligible at early times but grows in relative abundance post–equality (Bjaelde et al., 2012).

2. Growth of Structure and Evolution of Perturbations

The transition to matter domination is critical for the growth of cosmological perturbations. In ΛCDM, the matter density contrast $\delta(a)$ satisfies

$$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$0

which admits the familiar growing mode solution $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$1 in the matter-dominated regime ($$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$2).

In alternative models, such as $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$3 gravity, the effective gravitational coupling $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$4 can acquire scale or density dependence: $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$5 where for $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$6, $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$7 mildly increases at high redshift (Goswami et al., 14 Jul 2025). This results in a growth factor $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$8 and growth index $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$9 that deviate from their ΛCDM values ($a$0, $a$1).

Perturbative analyses employing the Meschersky equation incorporate the mass flow of baryon–radiation plasma from regions of concentrated cold dark matter and capture the collisionless nature of CDM during this epoch. In expanding backgrounds, the dominant mode remains $a$2, with c_s^2 terms negligible for collisionless matter (Kumar et al., 2012).

Gravity modifications and nontrivial dark sector couplings affect both the amplitude and rate of structure growth, producing observable signatures in the matter power spectrum, cluster and high-z galaxy abundance, and weak lensing signals (Goswami et al., 14 Jul 2025, Ghosh et al., 2024).

3. Early Dark Energy, Anti–de Sitter Episodes, and Beyond–ΛCDM Phenomenology

Several models posit the existence of non-negligible early dark energy (EDE) or additional vacuum episodes with anti–de Sitter (AdS) properties, both of which may alter post–equality dynamics.

• AdS–like Vacua and “Multiple–AdS” Scenarios: The presence of two AdS vacua—one at $a$3 (pre-recombination, dubbed AdS–EDE) and another at low redshift (negative cosmological constant today)—can punctuate the ΛCDM history, producing episodes of slowed or decelerated expansion (Wang et al., 4 Jun 2025). Field transitions are governed by a potential $a$4 with multiple AdS minima. The sequence of occupation—starting at a higher AdS minimum and transitioning through barriers via Hubble drag or quantum tunneling—produces observable effects on the CMB (reduced sound horizon $a$5, phase shifts), raises the best-fit $a$6 (bridging the $a$7 tension), and simultaneously lowers $a$8 to reconcile structure growth.
• Early Dark Energy from Spacetime Triggers: Models with curvature-coupled scalars generate EDE naturally and transiently at matter–radiation equality by exploiting geometric invariants that vanish in pure radiation or matter eras but become nonzero across the transition (Jing et al., 2024). The scalar field is “kicked,” injecting energy that decays rapidly (faster than radiation), so that $a$9 at peak, dropping sharply for $f(R,L_m)$0. This reduces $f(R,L_m)$1, briefly raises the effective equation of state $f(R,L_m)$2 to $f(R,L_m)$3–$f(R,L_m)$4, and decays away before interfering with late-time structure growth.

Both classes produce a range of observable signatures—most notably in the CMB sound horizon and low-$f(R,L_m)$5 supernova/BAO distances—and provide consistent fits to Planck, DESI, and SNe data when their EDE fractions are tightly controlled and dynamical consistency is maintained.

4. Observational Signatures and Constraints in the Post–Equality Era

The transition to matter domination, and the detailed properties of any additional dark sector effects or modified gravity, result in a suite of observable imprints:

• CMB Temperature and Polarization: The Sachs–Wolfe effect on the largest angular scales directly traces the time-constant CDM potential between equality and recombination, with $f(R,L_m)$6 (Kumar et al., 2012). Features such as EDE and AdS–EDE episodes reduce the sound horizon $f(R,L_m)$7 and shift acoustic peak phases, increasing the inferred $f(R,L_m)$8 (Wang et al., 4 Jun 2025, Jing et al., 2024).
• Gravitational Wave Background: Total equation of state oscillations in certain $f(R,L_m)$9 gravity models at $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$0 modulate $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$1 between $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$2 and $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$3 with a characteristic frequency $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$4–$\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$5. This induces band-limited enhancements of primordial gravitational wave energy density $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$6 by factors $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$7–$\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$8 at $\rho_{\rm de}(a) = \rho_{\rm de,0}\,a^{-3(1+w)}$9–$w \simeq -0.005$0 Hz (low-$w \simeq -0.005$1 modes), a signature potentially detectable by LiteBIRD through $w \simeq -0.005$2-mode polarization at large angular scales (Odintsov et al., 6 Jul 2025).
• Growth Rate of Structure: In $w \simeq -0.005$3 gravity and models with nonstandard dark matter–radiation interactions, the late-time growth factor $w \simeq -0.005$4 is enhanced by $w \simeq -0.005$5–$w \simeq -0.005$6\% over ΛCDM, accelerating structure formation and increasing the abundance of early halos and clusters. Changes in clustering amplitude $w \simeq -0.005$7, traced by redshift-space distortions and weak lensing, provide critical discriminants (Goswami et al., 14 Jul 2025).
• Matter Power Spectrum Suppression: Interactions between a nontrivial fraction $w \simeq -0.005$8 of dark matter and a dark radiation component around equality (so-called “DM-loading”) shift CMB acoustic peaks ($w \simeq -0.005$9 at high $a(t) \propto t^{2/3}$0), require a larger angular sound horizon $a(t) \propto t^{2/3}$1, and suppress small-scale matter fluctuations ($a(t) \propto t^{2/3}$2), both measurable signatures in the combination of Planck, BAO, SH0ES, and weak lensing data (Ghosh et al., 2024).
• Constraints on EDE and Dark Sector Parameters: Recent tomographic and fluid-plateau reconstructions cap post–equality EDE fractions at $a(t) \propto t^{2/3}$3 (95% C.L.), insufficient to solve the $a(t) \propto t^{2/3}$4 tension in standard forms (Gómez-Valent et al., 2021). Growth- and expansion-based probes strictly curtail deviations from canonical behavior.

5. Alternative Gravity and Mimetic Dark Matter Scenarios

Mimetic dark matter models, particularly those coupling to curvature invariants such as the Gauss–Bonnet term, provide a non-particle realization of cold dark matter whose post–equality behavior can be extremely close to dust: $a(t) \propto t^{2/3}$5 to within corrections of order $a(t) \propto t^{2/3}$6 (with $a(t) \propto t^{2/3}$7 the number of reheating e-folds), which are utterly negligible (Chamseddine et al., 9 Jan 2026). For certain nonlinear couplings, one can engineer a small “tilt” $a(t) \propto t^{2/3}$8, corresponding to an effective $a(t) \propto t^{2/3}$9 or smaller, which would slow structure growth only slightly. Such deviations are easily compatible with existing constraints on dark matter effective equation of state.

Models featuring dark matter decay into dark radiation ($a(t) \propto t^{2/[3(1+w)]}$0 at rate $a(t) \propto t^{2/[3(1+w)]}$1) induce a scale-dependent $a(t) \propto t^{2/[3(1+w)]}$2 that rises roughly linearly with $a(t) \propto t^{2/[3(1+w)]}$3 beyond equality. The resulting extra dark radiation fraction is limited by WMAP, ACT, and SPT to $a(t) \propto t^{2/[3(1+w)]}$4–$a(t) \propto t^{2/[3(1+w)]}$5, ensuring that any increase in $a(t) \propto t^{2/[3(1+w)]}$6 or CMB acoustic shifts remains within a few percent (Bjaelde et al., 2012). These models are distinguished by altered peak heights and positions and modest suppression in CMB temperature and polarization spectra.

6. Synthesis: Theoretical Consistency and Cosmological Tensions

The post–equality epoch remains an incisive regime for discriminating between cosmological models based on their predictions for expansion history, structure growth, and observable imprints on the CMB and large-scale structure. Theoretical consistency of models featuring metastable AdS vacua, rapid early dark energy decay, or dark sector interactions must guarantee vacuum stability, absence of instabilities (e.g., in $a(t) \propto t^{2/[3(1+w)]}$7 or $a(t) \propto t^{2/[3(1+w)]}$8 gravity or in mimetic/dark radiation models), and compatibility with structure formation and late-time acceleration.

A comparative overview of post–equality signatures is provided in the table below:

Model/Mechanism	Principal Effect Post-Equality	Key Observable Signature(s)
ΛCDM (standard)	Matter domination, $a(t) \propto t^{2/[3(1+w)]}$9	Canonical CMB, P(k), structure growth
$f(R,L_m)$0 gravity	Modified $f(R,L_m)$1, enhanced $f(R,L_m)$2	$f(R,L_m)$3 higher $f(R,L_m)$4, early collapse, enhanced $f(R,L_m)$5, lensing
EDE/AdS–EDE scenarios	Spike to $f(R,L_m)$6, rapid decay	Reduced $f(R,L_m)$7, higher $f(R,L_m)$8, CMB phase shift, $f(R,L_m)$9 lowered
DM-loaded dark radiation models	$\delta(a)$0, $\delta(a)$1, $\delta(a)$2	CMB peak shifts, power spectrum suppression
Mimetic/modified CDM	$\delta(a)$3	Tiny tilt in $\delta(a)$4, small change in structure growth
DM decay to DR	Extra $\delta(a)$5 post-equality	Expansion speed-up, CMB spectral effect

A central theme is that most viable nonstandard scenarios rapidly restore matter domination and standard growth after the equality-triggered episode, so as not to disrupt structure formation and CMB anisotropies. Tight combined constraints on EDE fractions (<1–2% post-equality), dark sector couplings ($\delta(a)$6few %), and modified expansion laws delimit the parameter space for models addressing cosmological tensions such as $\delta(a)$7 and $\delta(a)$8. Only those models in which energy injection decays sufficiently fast, and which do not introduce excessive clustering or negative pressure, can bring observables in alignment with current data.

7. Outlook and Prospects

Future CMB missions (e.g., LiteBIRD) targeting low-$\delta(a)$9 B-mode polarization, along with large-scale structure and lensing surveys, promise sensitivity to subtle post–equality features such as the predicted oscillatory $$\rho_m(a) = \rho_{m0}\,a^{-3}, \quad \rho_r(a) = \rho_{r0}\,a^{-4}, \quad H^2(a) = H_0^2 [\Omega_m a^{-3} + \Omega_r a^{-4} + \Omega_\Lambda],$$00 signatures, DM–loading induced CMB phase shifts, and mild departures from canonical growth. Confronting such models with the full suite of Planck, BAO, SNe Ia, weak lensing, and local measurements will further sharpen or exclude specific extensions to the standard cosmological model. Within current limits, the post–equality evolution remains remarkably robust to moderate deviations from ΛCDM, with only a narrow window available for exotic phenomena that both impact observables and remain theoretically consistent (Wang et al., 4 Jun 2025, Jing et al., 2024, Ghosh et al., 2024, Odintsov et al., 6 Jul 2025, Goswami et al., 14 Jul 2025, Gómez-Valent et al., 2021, Chamseddine et al., 9 Jan 2026, Bjaelde et al., 2012, Kumar et al., 2012).