Density perturbations in nonminimally coupled gravity: symptoms of Lagrangian density ambiguity

Technical Summary: Density Perturbations in Nonminimally Coupled Gravity and Lagrangian Ambiguity

Problem Statement
This work investigates the late-time evolution of cosmological density perturbations within a modified theory of gravity featuring a nonminimal coupling (NMC) between curvature and matter, specifically the $f(R, L_m)$ model. A central issue addressed is the "Lagrangian density ambiguity" inherent in perfect fluid descriptions. In General Relativity (GR), the choice between the matter Lagrangian densities $L_m = -\rho$ and $L_m = p$ (where $\rho$ is energy density and $p$ is pressure) yields identical gravitational dynamics because $L_m$ does not explicitly appear in the field equations. However, in NMC theories, the field equations and conservation laws depend explicitly on $L_m$, breaking this degeneracy. The paper aims to determine how these distinct choices affect the effective gravitational constant ($G_{\text{eff}}$) and the viability of NMC models intended to explain late-time cosmic acceleration.

Methodology
The authors analyze the scalar sector of perturbations on a Friedmann–Lemaître–Robertson–Walker (FLRW) background within the Newtonian gauge. The study proceeds through the following steps:

1. Derivation of Perturbed Equations: The authors derive the full set of perturbed field and conservation equations for the NMC $f(R)$ action, considering both $L_m = -\rho$ and $L_m = p$.
2. Quasistatic Approximation ($L_m = -\rho$): The paper first reviews the derivation of density perturbation equations under the quasistatic approximation (neglecting time derivatives of metric perturbations) for the $L_m = -\rho$ case.
3. Full Non-Quasistatic Evolution ($L_m = -\rho$): Recognizing potential pathologies in the quasistatic limit, the authors perform a full analysis without the quasistatic assumption. They introduce dimensionless parameters to decouple the system of differential equations, resulting in a fourth-order linear homogeneous differential equation for the density contrast $\delta$.
4. Alternative Lagrangian Analysis ($L_m = p$): The study repeats the quasistatic analysis for the $L_m = p$ choice, specifically focusing on the late-time dust-dominated era ($c_s^2 = 0$).
5. Numerical Application: The theoretical framework is applied to inverse power-law NMC models of the form $f_2(R) = 1 + (R_n/R)^n$. The authors numerically evolve the growth factor $D(z)$ and growth rate $f_g(z)$ using background parameters consistent with Pantheon+ data.

Key Contributions and Results

• Pathologies in $L_m = -\rho$ under Quasistatic Approximation:
  For the choice $L_m = -\rho$, the quasistatic approximation yields an effective gravitational constant $G_{\text{eff}}$ with a denominator that can approach zero for specific wavenumbers ($k$) and model parameters. This leads to unphysical singular behavior. The authors demonstrate that for late-time dominating models (where $f_2(R)$ decreases as $R$ increases), the condition $F_{,R} < 0$ (required for the model to recover GR at early times) combined with $L_m = -\rho < 0$ triggers this divergence. This suggests that the theory breaks down at the perturbative level even if the background evolution appears viable.

• Resolution via Full Non-Quasistatic Analysis ($L_m = -\rho$):
  When the full non-quasistatic equations are solved, the singular behavior is avoided. The leading-order terms in the denominator of the effective gravitational constant become proportional to higher powers of the wavenumber ($k^{14}$), which remain positive. However, the resulting $G_{\text{eff}}$ exhibits a strong scale dependence ($k^2$ relative to the denominator). To maintain a physically viable $G_{\text{eff}}$ consistent with large-scale structure formation, the magnitude of the NMC modifications ($f_2 - 1$) must be extremely small ($\sim 10^{-5}$). This constraint effectively prevents the NMC sector from driving the accelerated expansion of the Universe, rendering such models observationally unviable for background cosmology.

• Viability of $L_m = p$:
  In contrast, the choice $L_m = p$ yields significantly different results. In the late-time dust-dominated era ($p \approx 0$), the NMC terms in the background Friedmann equation vanish, restoring the standard $\Lambda$CDM background evolution. Crucially, the perturbative equations do not suffer from the singularities or severe scale-dependence found in the $L_m = -\rho$ case. The effective gravitational constant remains scale-independent and within observational bounds (deviating by at most $\sim 50\%$ from $G$). The growth rate $f_g$ shows a transient enhancement followed by a decrease, but the model remains physically consistent.

• Inverse Power-Law Models:
  Applying these findings to models with $f_2(R) = 1 + (R_n/R)^n$, the authors show that while the background remains $\Lambda$CDM, the perturbative dynamics differ markedly based on the Lagrangian choice. For $L_m = p$, the models predict a specific evolution of the growth factor and rate that could be tested against observational data, unlike the $L_m = -\rho$ case which is ruled out by the requirement of a non-singular, viable $G_{\text{eff}}$.

Significance and Claims
The paper claims to identify a critical, previously overlooked issue in the perturbative dynamics of NMC $f(R)$ gravity: the choice of matter Lagrangian density is not merely a formal ambiguity but has profound physical consequences for the theory's viability.

• Constraint on NMC Models: The authors argue that for $L_m = -\rho$, the requirement to avoid singularities and maintain a viable effective gravitational constant imposes such strict constraints on the NMC function that these models cannot account for late-time cosmic acceleration without fine-tuning or additional sectors (e.g., a cosmological constant in $f_1(R)$).
• Resolution of Ambiguity: The results suggest that $L_m = p$ is the physically preferred choice for late-time cosmology within this framework, as it avoids the pathological behavior of $G_{\text{eff}}$ and allows for non-pathological modifications to structure formation.
• Future Directions: The work highlights the need for a fundamental derivation of the perfect fluid Lagrangian in NMC contexts to resolve the ambiguity. It also proposes that the derived equations can be used to constrain NMC parameters via the $\sigma_8$ tension and growth rate data, similar to how the Hubble tension has been addressed in other modified gravity scenarios.

The authors conclude that while the $L_m = -\rho$ choice leads to unphysical singularities or triviality in the perturbative regime, the $L_m = p$ choice offers a consistent framework for studying late-time modifications to gravity, provided the background evolution is not solely driven by the NMC sector.