Quantum Fluctuation Modified Gravity

• Quantum fluctuation modified gravity is a framework where quantum metric decompositions yield effective modifications to Einstein’s equations.
• The models incorporate nonminimal matter couplings, scalar-tensor extensions, and higher-order corrections to address cosmological and black-hole phenomena.
• Empirical tests in cosmology, baryogenesis, and black-hole thermodynamics constrain the fluctuation parameter and guide model extensions.

Searching arXiv for recent and foundational papers on quantum fluctuation modified gravity. Quantum fluctuation modified gravity denotes a class of effective gravitational theories obtained by decomposing the metric operator into a classical background and a quantum fluctuating part, taking expectation values in a quantum state, and truncating the resulting action at first or second order in the fluctuations. In the formulations most often studied, the expectation value of the quantum correction is modeled by a tensor $K_{\mu\nu}$ built from the metric, curvature, or matter variables, so that Einstein gravity is replaced by a modified theory with nonminimal geometry–matter couplings, trace couplings, and, in some variants, additional scalar or torsional degrees of freedom. The program was developed from general effective-action constructions to cosmology, black-hole physics, baryogenesis, teleparallel gravity, and braneworld settings (Dzhunushaliev et al., 2013, Dzhunushaliev et al., 2015, Yang, 2015, Liu et al., 2016, Yang et al., 2020).

1. Foundational construction

The central hypothesis is the decomposition

$\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$

with the quantum correction encoded either by a nonzero one-point function

$\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$

or, in the second-order treatment, by $\langle \delta g_{\mu\nu}\rangle=0$ together with nontrivial quadratic correlators and $\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$ (Dzhunushaliev et al., 2013, Dzhunushaliev et al., 2015). In the first-order framework, the effective Lagrangian acquires the generic structure

$$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$

so the quantum part of the metric reappears as a classical modified-gravity sector with explicit matter couplings (Dzhunushaliev et al., 2013, Liu et al., 2016).

Several ansätze recur in the literature. The simplest is

$K_{\mu\nu}=\alpha\,g_{\mu\nu},$

with $\alpha$ a dimensionless fluctuation parameter. In that case the effective action becomes

$$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$

or equivalent sign conventions thereof, depending on the source (Yang, 2015, Yang et al., 2024). A more general scalar–tensor realization takes

$K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$

with $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$0 dynamical and the total Lagrangian

$\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$1

which is the form used for static black-hole solutions induced by quantum metric fluctuations (Yang et al., 2020). In the more general 2016 construction, $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$2 may be taken as $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$3, $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$4, or more complicated combinations involving $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$5, $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$6, and related tensors (Liu et al., 2016).

The second-order treatment leads to a different effective sector. Under factorization assumptions for the two-point fluctuation correlator, the Einstein–Hilbert term becomes

$\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$7

which yields an $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$8-type modification together with nonminimal matter couplings (Dzhunushaliev et al., 2015). Taken together, these results indicate that “quantum fluctuation modified gravity” is not a single Lagrangian but a family of effective models generated by different prescriptions for $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$9 and by different truncations of the fluctuation expansion.

2. Effective field equations and exchange terms

For the one-parameter metric ansatz $\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$0, variation with respect to $\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$1 yields

$\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$2

where

$\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$3

For a perfect fluid with $\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$4, one has $\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$5 (Yang, 2015). The scalar–tensor extension modifies the Einstein equations by derivative terms in $\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$6, the scalar potential, and the coupling term $\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$7 (Yang et al., 2020).

A defining feature of the framework is the generic non-conservation of the matter stress tensor. In the one-parameter model,

$\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$8

while in the scalar–tensor cosmological realization the divergence is again nonzero for generic couplings (Yang, 2015, Liu et al., 2016). The 2016 cosmological study interprets this nonzero divergence as a process of matter creation, corresponding to an irreversible energy flow from the gravitational field to the matter fluid, with matter production and entropy growth as the thermodynamic interpretation (Liu et al., 2016).

The scalar–tensor construction adds a Klein–Gordon sector. In vacuum, the field equations reduce to

$\langle \delta\hat g_{\mu\nu}\rangle = K_{\mu\nu},$9

and

$\langle \delta g_{\mu\nu}\rangle=0$0

which govern the black-hole sector studied numerically in spherical symmetry (Yang et al., 2020).

A teleparallel analogue replaces the metric fluctuation by a tetrad fluctuation $\langle \delta g_{\mu\nu}\rangle=0$1, with $\langle \delta g_{\mu\nu}\rangle=0$2. The effective action then contains a correction term $\langle \delta g_{\mu\nu}\rangle=0$3, generating non-minimal torsion–matter couplings and, for a specific choice, a subclass of $\langle \delta g_{\mu\nu}\rangle=0$4 gravity (Chen et al., 2021). This preserves the core mechanism—quantum fluctuations producing classical backreaction—but transposes it into the torsional rather than Riemannian description.

3. Cosmological realizations

In a spatially flat FLRW spacetime with a perfect fluid $\langle \delta g_{\mu\nu}\rangle=0$5, the one-parameter model yields the modified Friedmann equations

$\langle \delta g_{\mu\nu}\rangle=0$6

together with the modified continuity equation

$\langle \delta g_{\mu\nu}\rangle=0$7

(Yang, 2015). A nonsingular bounce is possible for

$\langle \delta g_{\mu\nu}\rangle=0$8

which implies

$\langle \delta g_{\mu\nu}\rangle=0$9

The same model also allows decelerated expansion in a dark-energy regime when

$\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$0

(Yang, 2015).

For slow-roll inflation with $\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$1 and $\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$2, the quantum-corrected equations are

$\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$3

and the slow-roll parameters are rescaled as

$\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$4

Big-Bang Nucleosynthesis constrains the fluctuation parameter to

$\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$5

(Yang, 2015).

The broader scalar–tensor cosmological program exhibits a wider range of behaviors. In the Higgs-type potential model

$\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$6

late-time solutions generically approach de Sitter acceleration, while $\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$7, $\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$8, and $\langle \delta^2 g_{\mu\nu}\rangle=K_{\mu\nu}$9 oscillate at intermediate redshifts before settling into an accelerating phase (Liu et al., 2016). In the $$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$0 model with dust, varying $$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$1 produces cosmologies ranging from marginally decelerating to strongly accelerating (Liu et al., 2016). In the teleparallel realization with

$$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$2

one finds exact de Sitter solutions and Hybrid Expansion Law evolutions interpolating from matter domination to late-time de Sitter behavior (Chen et al., 2021).

Baryogenesis provides an additional cosmological application. Three interaction terms were analyzed: $$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$3

$$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$4

$$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$5

with $$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$6 studied in detail (Yang et al., 2024). In the radiation era, the observed asymmetry

$$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$7

can be reproduced, with example parameter values $$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$8, $$\mathcal{L}_{\rm eff}\sim -\frac{1}{2\kappa^2}\sqrt{-g}\,\bigl(R+G_{\mu\nu}K^{\mu\nu}\bigr) +\sqrt{-g}\,\mathcal{L}_m+\frac12\sqrt{-g}\,T_{\mu\nu}K^{\mu\nu},$$9, and $K_{\mu\nu}=\alpha\,g_{\mu\nu},$0 for the three couplings, respectively (Yang et al., 2024).

4. Black holes and horizon thermodynamics

The black-hole sector has been developed most explicitly in the scalar–tensor model with

$K_{\mu\nu}=\alpha\,g_{\mu\nu},$1

Using the mass function $K_{\mu\nu}=\alpha\,g_{\mu\nu},$2, the field equations reduce to a coupled ODE system for $K_{\mu\nu}=\alpha\,g_{\mu\nu},$3. After introducing the dimensionless radial variable

$K_{\mu\nu}=\alpha\,g_{\mu\nu},$4

and rescaled variables $K_{\mu\nu}=\alpha\,g_{\mu\nu},$5, $K_{\mu\nu}=\alpha\,g_{\mu\nu},$6, and $K_{\mu\nu}=\alpha\,g_{\mu\nu},$7, the equations become a closed first-order system that was solved numerically with a standard Runge–Kutta (4th-order) shooting method under asymptotic flatness conditions (Yang et al., 2020).

In both the zero-potential and Higgs-potential cases, the numerical integrations admit genuine black-hole solutions with a single event horizon $K_{\mu\nu}=\alpha\,g_{\mu\nu},$8, corresponding to $K_{\mu\nu}=\alpha\,g_{\mu\nu},$9 (Yang et al., 2020). The horizon is identified when $\alpha$0 and $\alpha$1, or equivalently by a singularity in the ODE system. The thermodynamic quantities were computed from the surface gravity

$\alpha$2

together with the specific heat, entropy, and evaporation time (Yang et al., 2020). The horizon radius, temperature, heat capacity, entropy, and evaporation time differ by $\alpha$3 from the Schwarzschild values, with temperature increases of up to $\alpha$4 in the $\alpha$5 case and $\alpha$6 changes in $\alpha$7, $\alpha$8, and $\alpha$9 for Higgs-type potentials in the range $$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$0 (Yang et al., 2020). The scalar field remains nontrivial outside the horizon, so the usual no-hair theorems are evaded once geometry–matter coupling is introduced (Yang et al., 2020).

A different exact black-hole family arises in the Kiselev-type fluid background with

$$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$1

where

$$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$2

This reproduces modified dust, radiation, quintessence, cosmological-constant, and phantom Kiselev solutions, and allows analytic discussion of the strong energy condition, horizon existence, and positivity of the Hawking temperature (Hua et al., 2024). For example, in the dust case with $$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$3,

$$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$4

while for radiation with $$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$5,

$$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$6

(Hua et al., 2024).

The observational program has extended these solutions to accretion-disk, shadow, quasinormal-mode, and greybody analyses. For thin disks in the Novikov–Thorne model, the ISCO radius increases with $$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$7 for $$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$8 and decreases for $$S=\int d^4x\,\sqrt{-g}\left[\frac{1-\alpha}{2\kappa^2}R+L_m-\frac{\alpha}{2}T\right],$$9; the shadow size follows the same sign pattern (Yin et al., 1 Mar 2025). In perturbation theory, the 6th-order WKB treatment of scalar and vector fields shows that the real part of the quasinormal frequencies decreases as the quantum-fluctuation parameter $K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$0 increases, while the damping time lengthens; the greybody factors are suppressed by larger $K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$1, larger $K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$2, and suitable changes in the fluid equation-of-state parameter (Sajjad et al., 2 Oct 2025).

5. Extensions and adjacent formulations

Several extensions preserve the same basic logic while changing the geometric setting. In the braneworld realization, the linear expectation-value ansatz $K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$3 induces the five-dimensional $K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$4 action

$K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$5

with a thick-brane solution

$K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$6

The vacua shift to

$K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$7

the brane tension depends on $K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$8, and the domain wall disappears as $K_{\mu\nu}=\alpha\,\phi(x)\,g_{\mu\nu},$9 (Almeida et al., 2023). Differential configurational entropy selects an absolute extremum at $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$00 and a secondary local extremum at $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$01 for $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$02 (Almeida et al., 2023).

A conceptually adjacent direction models quantum fluctuations of space-time by a random operational time whose increments belong to the domain of attraction of an $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$03-stable law. Averaging over the hitting process produces Caputo derivatives and fractional Hamilton equations,

$\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$04

leading in minisuperspace to early-time behavior $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$05 followed by late-time acceleration (Tajahmad, 2022). Another adjacent construction, based on precanonical quantum gravity in spin-connection variables, identifies

$\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$06

so that a MOND-like acceleration scale and the cosmological constant are both traced to quantum fluctuations of the spin connection (Kanatchikov et al., 2023).

These related formulations do not use an identical effective action, but they reinforce a common theme: quantum fluctuations can be re-expressed as classical modified-gravity terms, effective sources, or nonstandard propagation laws.

6. Phenomenology, constraints, and interpretive issues

The phenomenology is correspondingly broad. In cosmology, BBN restricts the one-parameter model sharply (Yang, 2015), while baryogenesis permits viable radiation-era asymmetry generation with small or moderate $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$07 (Yang et al., 2024). In black-hole physics, the 2020 scalar–tensor model suggests that shadows, accretion spectra, and gravitational-wave ringdowns may acquire $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$08 deviations from general relativity (Yang et al., 2020). The Kiselev family adds analytic bounds from the strong energy condition, horizon existence, and temperature positivity (Hua et al., 2024). Thin-disk and shadow calculations further indicate that current Event Horizon Telescope measurements, together with X-ray spectroscopy and continuum fitting, can already limit $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$09 in some ordinary-matter or radiation environments, while future improvements could push the range down to $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$10 (Yin et al., 1 Mar 2025).

A recurrent issue is model dependence. The effective theory depends on the ansatz for $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$11, on the matter Lagrangian, and, in scalar–tensor realizations, on the potential $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$12. In the black-hole solutions, thermodynamic quantities depend sensitively on the asymptotic data $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$13, $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$14, and on the potential parameters $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$15 (Yang et al., 2020). In cosmology, changing from $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$16 to $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$17, or to the teleparallel and braneworld analogues, changes both the field content and the interpretation of the non-conservation law (Liu et al., 2016, Chen et al., 2021, Almeida et al., 2023). This suggests that present applications probe a structured effective-field-theory landscape rather than a unique completed quantum-gravity theory.

A common misconception is that the framework merely rescales Newton’s constant. That statement is accurate only for the most elementary sector of the $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$18 ansatz. In the wider literature, quantum fluctuations generate trace couplings, $\hat g_{\mu\nu}=g_{\mu\nu}+\delta \hat g_{\mu\nu},$19-dependent source terms, scalar dynamics, torsion–matter couplings, higher-derivative curvature terms, matter creation, and nontrivial black-hole hair (Yang, 2015, Dzhunushaliev et al., 2015, Yang et al., 2020, Chen et al., 2021). Another misconception is that the constructions are confined to cosmology; in fact, they extend to black-hole thermodynamics, accretion observables, quasinormal modes, greybody factors, braneworld defects, and baryogenesis (Yang et al., 2020, Hua et al., 2024, Yang et al., 2024, Yin et al., 1 Mar 2025, Sajjad et al., 2 Oct 2025).

Within the scope of the cited works, quantum fluctuation modified gravity is therefore best understood as an effective semiclassical program in which quantum fluctuations of the metric, tetrad, or related geometric variables are encoded into modified classical dynamics. Its principal technical content lies in the choice of fluctuation tensor, the resulting field equations, and the observational sectors—cosmological, black-hole, or higher-dimensional—in which those effective modifications can be constrained.