Modified Friedmann Equations in Quantum Cosmology

• Modified Friedmann equations are generalizations of standard cosmological equations that incorporate quantum gravity, non-classical matter couplings, and entropic corrections.
• They employ methods like entropy-area modifications, generalized uncertainty principles, and gauge extensions to resolve singularities and trigger cosmic bounces.
• These models offer insights into early-universe behavior and late-time acceleration, providing testable predictions for cyclic and emergent-space cosmologies.

Modified Friedmann equations are generalizations of the standard Friedmann equations of cosmology that incorporate additional terms or structures motivated by quantum gravity, generalized entropy-area relations, non-classical matter couplings, gauge-theoretic extensions, noncommutative geometry, or other fundamental-physics considerations. These modifications reflect proposed new microphysics at short distances, corrections to gravitational dynamics, or phenomenological effects designed to resolve singularities, explain cosmic acceleration, or capture trans-Planckian phenomena. They have been developed within diverse frameworks, including entropy-corrected emergent gravity, loop quantum cosmology, generalized uncertainty principles, gauge extensions, non-additive entropies, and more.

1. Foundations: Standard and Emergent-Space Friedmann Equations

The classical Friedmann equations describe the evolution of a spatially homogeneous and isotropic universe (FLRW metric). In natural units ($c = \hbar = k_B = 1$):

$$H^2 + \frac{k}{a^2} = \frac{8\pi G}{3} \rho, \qquad \dot\rho + 3H(\rho+p) = 0$$

where $H\equiv\dot{a}/a$ is the Hubble rate, $a(t)$ the scale factor, $k$ the spatial curvature index, $\rho$ the total energy density, and $p$ the pressure.

Emergent-space scenarios (Padmanabhan, Cai et al.) reinterpret cosmic expansion as a thermodynamically driven process, with the expansion rate governed by the difference in degrees of freedom between the apparent-horizon surface and the bulk: $$\frac{dV}{dt} = L_p^2 \, H \, r_A \left(N^{\rm (eff)}_{\rm sur} - N_{\rm bulk}\right)$$ where $L_p^2=G$, $r_A$ is the apparent-horizon radius, and $N_{\rm sur}$, $N_{\rm bulk}$ count surface/bulk degrees of freedom. This approach provides a geometric mechanism for embedding quantum-gravitational and statistical-entropy corrections into the dynamical background equations (Yuan et al., 2013).

2. Entropic Corrections and Modified Dynamical Structure

A principal source of modified Friedmann equations is quantum-gravity-corrected entropy-area relations for horizons. General corrections take the form: $$S(A) = \frac{A}{4G} + \alpha \ln\frac{A}{4G} + \beta \frac{4G}{A} + \dots$$ with $(\alpha, \beta)$ encoding logarithmic and inverse-area quantum corrections, respectively.

The incorporation of such corrections into the horizon-thermodynamics yields modified Friedmann equations: $$H^2+\frac{k}{a^2} + \alpha L_p^2 \big( H^2+\frac{k}{a^2} \big)^2 - \beta L_p^4 \big( H^2+\frac{k}{a^2} \big)^3 = \frac{8\pi G}{3} \rho$$ (Yuan et al., 2013, Sheykhi, 2010, Salehi et al., 2017).

For entropy forms inspired by entanglement (power-law regime),

$$S(A) = \frac{A}{4G}[ 1 - K_\alpha A^{1 - \alpha/2} ]$$

which leads to

$$H^2+\frac{k}{a^2} - \frac{1}{r_c^2} \big( H^2+\frac{k}{a^2} \big)^{\alpha/2} = \frac{8\pi G}{3} \rho$$

(Yuan et al., 2013).

Quantum-gravity-motivated modifications also include generalized uncertainty principles (GUP, DSR-GUP), non-additive (Tsallis, Kaniadakis) or fractional entropies, exponential corrections, and alternative statistical frameworks:

• GUP-induced quadratic energy-density corrections: $H^2 = \frac{8\pi G}{3}\rho + c_1 \rho^2$ (Ökcü, 19 Nov 2025, Jusufi et al., 2022)
• Tsallis entropy ($S \sim A^\beta$): $(H^2+k/a^2)^{2-\beta} = \frac{8\pi G}{3} \rho$ (Sheykhi, 2018)
• Kaniadakis entropy yields: $$H^2+\frac{k}{a^2} - \alpha(H^2+\frac{k}{a^2})^{-1} = \frac{8\pi G}{3}(\rho+\rho_\Lambda)$$ (Sheykhi, 2023)

3. Physical Interpretation and Phenomenology

Early-Universe Behavior and Singularity Resolution

The higher-power $(H^2+k/a^2)^2$, $(H^2+k/a^2)^3$, and density-squared corrections are dominant at high curvature (small scale factor), yielding notable dynamical features:

• Repulsive quantum corrections generically trigger a cosmic bounce, replacing the initial singularity by a regular minimum of the scale factor for suitable parameter choices (typically for $\alpha > 0$) (Alonso-Serrano et al., 2022, Ökcü, 19 Nov 2025, Sheykhi, 2010).
• These corrections can support nonsingular cyclic or oscillating universes, especially for closed ($k=1$) or open ($k=-1$) spatial geometries (Salehi et al., 2017, Ökcü et al., 19 Jul 2024, Linsefors et al., 2013).
• In GUP, DSR-GUP, and zero-point-length scenarios, the modified Friedmann equations enforce maximum energy density scales and minimum horizon radii near the Planck regime (Ökcü, 19 Nov 2025, Ökcü et al., 2020, Jusufi et al., 2022), leading to universal avoidance of classical blow-up.

Late-Time Cosmological Effects

At late times, when $H^2 \ll 1$ (Planck units), higher-order curvature and density corrections become negligible, and all modified Friedmann equations reduce to their standard forms, ensuring compatibility with observational cosmology for appropriate parameter ranges (Yuan et al., 2013, Sheykhi, 2010, Sheykhi et al., 2011). Any significant late-time deviations require tuning of non-standard parameters (e.g., Tsallis parameter $\beta$) (Sheykhi, 2018).

The corrections may introduce effective dark-energy-like terms, modify the deceleration parameter, and impact cosmic acceleration or expansion history, but in generic cases (for standard matter EoS) they tend to be subdominant at low curvature (Sheykhi, 2023, Ökcü, 19 Nov 2025).

4. Classification by Underlying Microphysics and Methods

Correction Type	Example Entropy/Action	Modified Friedmann Structure
Logarithmic and $1/A$	$S = (A/4G) + \alpha \ln(A/4G) + \beta (4G/A)$	$(H^2+\frac{k}{a^2}) +\ldots = (8\pi G/3) \rho$
Power-law Entanglement	$S = (A/4G)[1 - K_\alpha A^{1-\alpha/2}]$	$$(H^2+\frac{k}{a^2}) - K(H^2+\frac{k}{a^2})^{\alpha/2} = \ldots$$
Nonadditive Tsallis	$S \sim A^\beta$	$(H^2+\frac{k}{a^2})^{2-\beta} = (8\pi G/3)\rho$
Kaniadakis Entropy	$S_\kappa = \kappa^{-1} \sinh(\kappa S_{BH})$	$$H^2+\frac{k}{a^2} - \alpha(H^2+\frac{k}{a^2})^{-1} = \ldots$$
Generalized Uncertainty	GUP, DSR-GUP, zero-point length	$H^2 \propto \rho(1 - \alpha \rho)$, maximum $\rho$
Emergent-Space Approaches	Holographic equipartition, Padmanabhan-type laws	$$dV/dt = L_p^2 H r_A(N_{\rm sur}^{(\rm eff)} - N_{\rm bulk})$$
Loop Quantum Cosmology	Holonomy corrections, effective Ashtekar variables	$$H^2 = \frac{8\pi G}{3} \rho (1 - \rho/\rho_{\rm crit})$$

Each approach modifies either the effective gravitational coupling, the structure of dominant Hubble/density terms, or both.

5. Cosmological and Observational Implications

• Resolution of Singularities: Most formulations predict a bounded energy density and/or Hubble parameter, generically resolving the big bang singularity by a regular bounce (Alonso-Serrano et al., 2022, Ökcü, 19 Nov 2025, Linsefors et al., 2013, Díaz-Barrón et al., 2019, Jusufi et al., 2022).
• Cyclic and Oscillatory Universes: Certain entropy-corrected equations allow stable periodic solutions, especially in closed or open universes, leading to cyclic cosmological histories with alternating expansion and contraction (Salehi et al., 2017, Ökcü et al., 19 Jul 2024).
• Acceleration and Dark-Energy Mimicry: Nonadditive and Kaniadakis-type entropies can effectively generate late-time acceleration or dark-energy-like effects for specific parameters, but most quantum corrections rapidly vanish in the infrared (Sheykhi, 2018, Sheykhi, 2023).
• Matter-Energy Interactions: Direct coupling between light and matter, as postulated in the opaque-universe scenario, yields additional negative-gravity terms altering the expansion dynamics, potentially explaining cosmic cyclicality and eliminating the need for $\Lambda$ (Vavryčuk, 2020).
• Gravitational Baryogenesis: GUP and DSR-GUP frameworks directly impact thermodynamic quantities such as $\dot R$, producing nonzero baryon asymmetry even during radiation domination and enabling new observational bounds on GUP parameters (Ökcü, 19 Nov 2025).
• Testing via Cosmological Observables: Any deviation in the expansion rate, deceleration parameter, or the evolution of density perturbations arising from modified Friedmann equations creates possible signatures accessible to cosmological probes (CMB, BAO, large-scale structure, 21-cm cosmology) (Ökcü et al., 19 Jul 2024, Vavryčuk, 2020, Sheykhi, 2018).

6. Model-Dependent Features and Extensions

Gauge Theory, Nonminimal Coupling, Noncommutativity

• Nonminimally Coupled Gravity: Theories with $$S = \int \sqrt{-g}(\kappa f_1(R) + f_2(R) \mathcal{L})d^4x$$ yield Friedmann equations with additional $f_2(R)\rho$ terms and introduce matter-curvature exchange, with the specific behavior depending on fluid Lagrangian prescription and $f_i(R)$ (Bertolami et al., 2013).
• Maxwell-Weyl Gauge Theory: Maxwell-Weyl extensions produce modified Friedmann equations via extra scalar modes (Dirac field, Maxwell gauge fields), yielding time-dependent effective cosmological constants, bounce and cyclic scenarios (Kibaroğlu, 2023).
• Noncommutative Effective LQC: Noncommutative deformations of loop quantum cosmology preserve the bounce but introduce new correction factors to $H^2$ and the Raychaudhuri equation. Effective scalar potentials and renormalized critical densities appear, but do not generate inflationary epochs or alter the qualitative bounce dynamics for generic initial data (Díaz-Barrón et al., 2019).

Deformed Gravity, Modified Statistics, Conformal Quantum Corrections

• Hořava-Lifshitz and Debye Gravity: HL-deformed entropy-area relations, especially with Debye-model equipartition, modify the Friedmann equations via temperature-dependent correction functions $D(x)$, suppressing gravity in the IR while introducing logarithmic corrections in the UV (Liu et al., 2010, Sheykhi et al., 2011).
• Fractional Entropy and Conformal Bohmian Gravity: Fractional black-hole entropy and Bohm–de Broglie quantum potentials yield modified Friedmann equations featuring fractional powers of curvature invariants or explicit quantum-potential derivatives, driving acceleration or modifying the $H^2$ structure (Çoker et al., 2023, Gregori et al., 2019).

7. Universality and Reduction to Standard Cosmology

All consistent formulations of modified Friedmann equations reduce to the standard Einstein-Friedmann system in the appropriate limit—i.e., as quantum-gravitational or non-standard parameters tend to zero, or in the late (low-curvature, large-horizon) universe. This property ensures that these scenarios recover classical cosmology and are not ruled out by current data unless correction parameters are fine-tuned to impact low-energy dynamics (Sheykhi, 2010, Sheykhi, 2023, Yuan et al., 2013).

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In summary, modified Friedmann equations systematically encode modifications to cosmological dynamics arising from quantum gravity, generalized thermodynamic/entropic structures, or new fundamental interactions. The resulting dynamical system can capture early-universe regularization (bounce), effective dark energy, cyclicity, and observable departures from ΛCDM, offering testable predictions in the domains of cosmological singularity-resolution, cosmic acceleration, and possibly large-scale structure formation. The precise phenomenology is highly sensitive to the form, order, and sign of the correction terms, as well as the underlying microphysical or statistical justification for the modified entropy or matter-gravity coupling employed.