Perturbation Theory in the Minimally Extended Varying Speed of Light (meVSL) Model

TL;DR

The paper addresses how a time-varying speed of light, implemented through the meVSL framework with a time-dependent \\tilde{c} and a fixed parameter b, alters cosmological perturbations. It develops a full linear perturbation theory in meVSL by deriving perturbed Christoffel symbols, Ricci and Einstein tensors, and the energy-momentum tensor, then imposes gauge freedom and SVT decomposition with a normal-mode (Fourier) approach. The main result is that scalar perturbations acquire new terms proportional to d ln(\\tilde{c})/d ln a, causing ≈2% deviations in subhorizon growth of the matter density contrast and the Newtonian potential, with effects that depend on the sign of b and the perturbation scale; larger-scale modes show the strongest responses, including implications for the ISW effect and CMB lensing. The study highlights meVSL as a theoretically consistent modification to GR that can imprint observable signatures in large-scale structure and CMB observables, while noting that a full treatment beyond tight-coupling is needed to robustly confront data.

Abstract

Cosmological perturbation theory provides a fundamental framework for analyzing the evolution of density fluctuations and gravitational potentials in the Universe. It plays a crucial role in understanding large-scale structure formation and cosmic microwave background (CMB) anisotropies. In this study, we apply perturbation theory to the minimally extended varying speed of light (meVSL) model to investigate the effects of a varying speed of light on the matter density contrast and the Newtonian gravitational potential. Unlike conventional models with a constant speed of light, the meVSL model introduces modifications to the cosmological evolution equations, leading to potential deviations in structure formation and gravitational interactions. By deriving and analyzing the perturbed equations within this framework, we explore how a varying speed of light affects the growth of density perturbations and the evolution of gravitational potentials. Compared to the standard constant speed of the light model, we find deviations of approximately $2$\% in the subhorizon modes of both quantities. Although detecting these effects observationally remains a significant challenge, our results provide new theoretical insights into the meVSL model and its potential observational signatures, such as the integrated Sachs-Wolfe effect and gravitational lensing of the CMB.

Figures (3)

• Figure 1: The evolution of dark matter density contrast $\delta_c$ (left panel) and curvature potential $\phi$ (right panel) as a function of the e-folding number $n = \ln a$ for different wavenumbers: $k = 10^{-2}$ Mpc$^{-1}$ (dashed), $0.1$ Mpc$^{-1}$ (dotted), $0.6$ Mpc$^{-1}$ (solid), and $1$ Mpc$^{-1}$ (dot-dashed) for $b = 0$. a) The dark matter density contrasts for corresponding modes b) The curvature potentials for different modes.
• Figure 2: The figure shows the difference in the DM density contrast $\delta_c$ between the varying speed of light cases ($b = \pm 0.01$) and the standard case ($b = 0$). a) Comparison with $b = -0.01$ (faster past speed of light) case. b) Comparison with $b = +0.01$ (slower past speed of light) case.
• Figure 3: Difference in the curvature potential $\phi$ between the varying speed of light cases ($b = \pm 0.01$) and the standard case ($b = 0$). a) The case of $b = -0.01$. b) $b = +0.01$ case.