Modified Friedmann Equation Overview

• Modified Friedmann Equation is a generalization of the classical Friedmann equation that incorporates quantum gravity, nonlocal, and thermodynamic corrections to resolve singularities.
• It introduces higher-curvature, logarithmic, and fractional corrections that become significant at Planck-scale densities and during cosmic bounce scenarios.
• These modifications yield observable effects such as nonsingular bounces, adjusted expansion histories, and effective changes to Newton’s constant in various cosmological models.

A modified Friedmann equation is any generalization of the standard Friedmann equations that incorporates additional theoretical corrections or interactions, typically motivated by quantum gravity, modified gravity, statistical mechanics, or nontrivial matter couplings. Such equations play a central role in modern cosmological model-building, particularly in efforts to resolve the Big Bang singularity, explain cosmic acceleration, or capture Planck-scale/new-physics effects in the early or late universe.

1. Definitions and Foundations

The standard (spatially flat) Friedmann equation in general relativity reads

$H^2 = \frac{8\pi G}{3} \rho,$

where $H=\dot a/a$ is the Hubble parameter, $a(t)$ the scale factor, $G$ Newton’s constant, and $\rho$ the total energy density. Modified Friedmann equations introduce quantum-corrected, nonlocal, higher-curvature, or thermodynamically induced terms, such as

$$H^2 = \frac{8\pi G}{3} \rho + \Delta_{\mathrm{mod}}(H, \rho, ...).$$

$\Delta_{\mathrm{mod}}$ terms typically scale with positive powers of $H^2$ or $\rho$, logarithmic or fractional powers, or explicit curvature invariants, and often vanish in the infrared ($H\to0$).

The need for such modifications arises from fundamental considerations:

• Quantum gravitational effects (including modified entropy–area relations and generalized uncertainty principles)
• Microscopic models of spacetime emergence
• Thermodynamic or statistical mechanics of horizons
• Topology-changing quantum gravity (e.g., baby universe production)
• Gauge-theoretic or nonminimally coupled gravity frameworks

2. Quantum-Gravity Corrections from Entropy–Area Modifications

Many derivations of modified Friedmann equations use non-standard entropy–area laws associated with the apparent/Hubble horizon. These arise as generic predictions of quantum gravity, including string-inspired models, loop quantum gravity, or deformed uncertainty principles.

Generalized Entropy-Area Laws:

$$S = \frac{A}{4G} - \alpha \ln \frac{A}{4G} + \beta \frac{4G}{A} + \cdots$$

or more generally, power-law or nonlocal expressions (e.g., Kaniadakis, Tsallis, or exponential entropies).

Generic Effects:

• Higher-order curvature corrections: $(H^2)^{n\geq2} \sim \rho^n$ or $H^4, H^6, ...$ terms (Sheykhi, 2010, Sheykhi, 2023, Sheykhi, 2018).
• Nonperturbative exponential or fractional power corrections (Çoker et al., 2023, Ökcü et al., 2024).
• GUP/DSR-motivated minimal-length effects: bounded $H^2$ and $\rho$ at high energy (Ökcü, 19 Nov 2025, Awad et al., 2014, Ökcü et al., 2020).
• Leading corrections most relevant near Planck-scale curvature; corrections generally negligible at late times.

Modified Entropy	Representative Correction to $H^2$	Reference
Logarithmic	$-\alpha(H^2)^2 - \beta(H^2)^3$	(Sheykhi, 2010)
Kaniadakis	$-\alpha(H^2)^2$	(Sheykhi, 2023)
Tsallis $(A^\beta)$	$(H^2)^{2-\beta}$	(Sheykhi, 2018)
Exponential	Nonperturbative, Eq. (58) in (Ökcü et al., 2024)	(Ökcü et al., 2024)
Fractional	$(H^2)^{(3\alpha-2)/(2\alpha)}$	(Çoker et al., 2023)
GUP/DSR-GUP	$H^4$, bounded $H^2,\rho$	(Ökcü, 19 Nov 2025, Ökcü et al., 2020)

3. Thermodynamic and Emergent Gravity Derivations

The modified Friedmann equations can systematically be derived by applying the first law of thermodynamics to the apparent horizon, or by using Padmanabhan’s emergent-space concept: $$dE = T_h dS_h + W dV \quad \longrightarrow \quad \text{Modified Raychaudhuri/Friedmann equations}$$ where $T_h$ is the horizon temperature, $S_h$ the modified entropy, and $W$ the work density. This approach accommodates both standard and quantum-corrected entropy.

Emergent-space frameworks postulate that spacetime expansion is governed by the mismatch in degrees of freedom between bulk and boundary: $$\frac{dV}{dt} = G (N_{\mathrm{sur}} - N_{\mathrm{bulk}})$$ A quantum-corrected $N_{\mathrm{sur}}$ from a non-standard entropy law leads directly to higher-curvature terms in the Friedmann dynamics (Zhang et al., 2017).

Approach	Correction Features	Reference
Thermodynamic/Horizon-based	$H^4$, $H^6$, log, power laws	(Sheykhi, 2010)
Emergent-space (Padmanabhan)	Quantum-bounce, $H^2(1-\rho/\rho_c)$, log running	(Zhang et al., 2017)
Entropic gravity (Verlinde)	Entropy-driven corrections	(Ökcü, 19 Nov 2025, Awad et al., 2014)

4. Dynamical and Cosmological Consequences

4.1 Quantum Bounce and Singularity Resolution

A notable feature of many modified Friedmann equations is the emergence of a critical density $\rho_c$ and an associated nonsingular bounce: $$H^2 = \frac{8\pi G}{3} \rho \left( 1 - \frac{\rho}{\rho_c} \right)$$ This structure is realized in various frameworks:

• Emergent-space models with MDR-corrected entropy (Zhang et al., 2017)
• Quantum-gravity inspired Raychaudhuri corrections (quadratic in $\rho$) (Alonso-Serrano et al., 2022)
• Effective loop quantum cosmology (LQC) with holonomy corrections (Linsefors et al., 2013)
• GUP/DSR-modified or equipartition-motivated frameworks (Ökcü, 19 Nov 2025, Ökcü et al., 2020)

Bounce solutions require $a(t)>0$ for all $t$, $H=0$, and $\ddot a>0$ at the bounce (Zhang et al., 2017, Alonso-Serrano et al., 2022). The effective negative sign in the higher-order corrections (e.g., $-\rho^2$) drives the repulsive behavior necessary to halt collapse.

4.2 Early- and Late-Time Cosmology

• Corrections become relevant when $H^2 \sim \ell_p^{-2}$ or $\rho \sim \ell_p^{-4}$. They are negligible at late times but can affect initial singularity and early inflationary dynamics.
• Modified equations can produce accelerated expansion without a cosmological constant via nontrivial coupling to baby universe/topology-changing processes (Ambjorn et al., 2017, Ambjorn et al., 2022).
• Fractional, Tsallis, or nonextensive entropic corrections can shift the threshold equation of state parameter $\omega$ for late-time acceleration, sometimes yielding acceleration without dark energy (Sheykhi, 2018, Çoker et al., 2023).

4.3 Effective Newton’s Constant and Running Couplings

Some corrections act as modifications to Newton’s constant: $G_{\mathrm{eff}} = \frac{G}{1 + 2\alpha H^2}$ leading to running gravitational coupling at high curvature (Sheykhi, 2023). Logarithmic corrections induce weak scale-dependence/running.

4.4 Structure Growth, Observational Consequences

Models with topology change (W₃ algebra) or extra surface terms can fit cosmic expansion history, resolve the Hubble tension, and reproduce large-scale structure observations without explicit $\Lambda$ (Ambjorn et al., 2022, Ambjorn et al., 2017).

Modified equations also predict bounded Hubble rates, finite Kretschmann scalar at bounce/maximum density, and sometimes inflationary-like behavior at high density (Ökcü, 19 Nov 2025, Linsefors et al., 2013).

5. Modified Friedmann Equations in Generalized Theories

5.1 Nonminimally Coupled Theories

Introducing nonminimal matter–gravity couplings via functions $f_1(R)$ and $f_2(R)$ (in the action $S = \int \sqrt{-g}[f_1(R) + f_2(R)\mathcal{L}_m]$) yields modified dynamics: $$H^2 = \frac{1}{6\kappa} \frac{\kappa f_2 \rho + 6H \dot\Phi + \Phi R - f_1}{\Phi}$$ where $$\Phi = F_1(R) + \frac{2}{\kappa} F_2(R) \mathcal{L}_m$$ and $F_i = df_i/dR$ (Bertolami et al., 2013). Such couplings alter both the form and matter-content dependence of the expansion, and can absorb or reinterpret the cosmological constant problem via a dynamical $f_2(R)$ function.

5.2 Gauge-Theoretic and Scalar-Tensor Extensions

Maxwell-Weyl gauge gravity introduces additional time-dependent scalars ($\psi(t), \phi(t)$) into the Friedmann equations, leading to extra friction/anti-friction and dynamical $\Lambda_{\rm eff}(t)$, enabling inflation, acceleration, bounces, or cyclic cosmologies in a unified framework (Kibaroğlu, 2023).

5.3 Conformal Bohm-de Broglie Gravity

Quantum potential-driven conformal rescalings yield modifications of the form: $$H^2 = \frac{8\pi G}{3} \rho - \dot Q H + \frac{1}{4} (\dot Q)^2$$ where $Q$ encodes the quantum potential. These terms act as a nonlocal, negative-pressure component and can drive late-time acceleration without a fundamental cosmological constant (Gregori et al., 2019).

6. Model-Dependent Features and Limitations

6.1 Bounce Realizability and Perturbative Validity

The bounce often appears only within a finite truncation (e.g., $O(\ell_P^2)$), whereas including all higher-order terms can affect or prevent the bounce unless full nonperturbative knowledge is available (Alonso-Serrano et al., 2022). The critical density at the bounce can lie outside the strict perturbativity regime, suggesting caution in interpretation.

6.2 Anisotropy and Quantum Shear Constraints

Anisotropic models (Bianchi I) show that most quantum-corrected solutions never return to classicality except for a special band in phase space, and may feature oscillatory or bounded states at Planckian scales (Linsefors et al., 2013).

6.3 Observational Viability

Some models can fit current cosmological data (e.g., W₃ algebra-based), reproduce late-time $w\lesssim -1$ without ghost fields, and resolve the $H_0$ tension with no violation of early-universe constraints (Ambjorn et al., 2022). The structure of modifications determines if these models are self-consistent and phenomenologically viable.

7. Schematic Overview: Selected Modified Friedmann Equations

Framework/Correction	Modified Friedmann Equation (flat, $k=0$)	Reference
Emergent-space, MDR entropy	$H^2 = \frac{8\pi G}{3}\rho + O(\eta^2\ell_p^2 H^4)$	(Zhang et al., 2017)
Quantum-gravity, thermodynamics	$$H^2 = \frac{8\pi G}{3}\rho - \frac{16\pi^2G^2D\ell_P^2}{3c^2}\rho^2$$	(Alonso-Serrano et al., 2022)
Entropy-log-correction	$$H^2 - \frac{\alpha G}{2\pi}H^4 = \frac{8\pi G}{3}\rho$$	(Sheykhi, 2010)
Kaniadakis entropy	$H^2 - \alpha H^4 = \frac{8\pi G}{3}\rho$	(Sheykhi, 2023)
Tsallis entropy	$(H^2)^{2-\beta} \propto \rho$	(Sheykhi, 2018)
GUP-modified equipartition	$$H^2 = \frac{8\pi G}{3}\rho\left(1-\frac{\rho}{\rho_c}\right)$$	(Ökcü, 19 Nov 2025)
Bohm–de Broglie, conformal factor	$$H^2 = \frac{8\pi G}{3}\rho - \dot Q H + \frac14 \dot Q^2$$	(Gregori et al., 2019)
W₃-algebra, baby universes	$$H^2 = \frac{8\pi G}{3}\rho + \frac{a}{\dot a} \frac{1+3F(x)}{F(x)^2}$$, $F^3-F^2+x=0$	(Ambjorn et al., 2017, Ambjorn et al., 2022)

• (Zhang et al., 2017) Wei Zhang and Xiao-Mei Kuang, "The quantum effect on Friedmann equation in FRW universe" (2017)
• (Alonso-Serrano et al., 2022) "Friedmann equations and cosmic bounce in a modified cosmological scenario" (2022)
• (Sheykhi, 2010) "Thermodynamics of apparent horizon and modified Friedman equations" (2010)
• (Sheykhi, 2023) "Corrections to Friedmann equations inspired by Kaniadakis entropy" (2023)
• (Sheykhi, 2018) "Modified Friedmann Equations from Tsallis Entropy" (2018)
• (Ökcü, 19 Nov 2025) "Friedmann equations from GUP-modified equipartition law" (2025)
• (Gregori et al., 2019) "Modified Friedmann equations via conformal Bohm -- De Broglie gravity" (2019)
• (Ambjorn et al., 2017, Ambjorn et al., 2022) Ambjørn, Watabiki et al., W₃ algebra cosmology (2017, 2022)
• (Linsefors et al., 2013) "Modified Friedmann equation ... in effective Bianchi-I loop quantum cosmology" (2013)
• (Awad et al., 2014) "Planck-Scale Corrections to Friedmann Equation" (2014)
• (Ökcü et al., 2020) "Modified Friedmann equations from DSR-GUP" (2020)
• (Çoker et al., 2023) "Modified Friedmann equations from fractional entropy" (2023)
• (Bertolami et al., 2013) "Modified Friedmann Equation from Nonminimally Coupled Theories of Gravity" (2013)
• (Ökcü et al., 2024) "Exponential correction to Friedmann equations" (2024)
• (Liu et al., 2010, Sheykhi et al., 2011) Entropic gravity and Debye corrections (2010, 2011)
• (Kibaroğlu, 2023) "Modified Friedmann Equations from Maxwell-Weyl Gauge Theory" (2023)

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This comprehensive landscape demonstrates that the modified Friedmann equation is a central tool in exploring gravitational phenomena beyond classical GR, quantum-corrected cosmology, and the statistical mechanics of horizons, with wide-ranging implications for singularity resolution, cosmic acceleration, dark energy alternatives, and observational cosmology.