Diffeomorphism Invariance Breaking in Gravity and Cosmological Evolution

TL;DR

The paper investigates explicit diffeomorphism invariance breaking in gravity within an effective two-derivative framework, asking whether small symmetry violations can yield a consistent cosmological evolution that remains close to General Relativity (GR). It introduces a diffeomorphism-violating Lagrangian with five leading two-derivative densities and coefficients alpha_a, and derives a modified FLRW cosmology with a clock function f and scale factor a, examining single- and multi-fluid universes. Analytically solved cases include radiation and vacuum sectors, while matter requires numerics; multi-fluid evolutions (Radiation+Matter, Matter+Vacuum, Radiation+Matter+Vacuum) test stability and continuity with GR. The main finding is that all explored configurations yield scale-factor evolution that smoothly converges to GR as alpha_a → 0, with no instabilities observed up to two derivative order, though vacuum-dominated cases can show time-dependent deviations for large symmetry-breaking that become negligible for realistic small coefficients. The results support the viability of mild diffeomorphism breaking in low-energy effective gravity and its compatibility with cosmological evolution.

Abstract

Diffeomorphism invariance breaking has been investigated in the literature in several contexts, including emergent General Relativity (GR). If GR emerges from an underlying theory without diffeomorphism invariance, there may be small violations of this symmetry at low energies. Since such small violations should not cause instabilities in cosmological evolution, it is a suitable framework for examining such symmetry-breaking effects. In this paper, the cosmological evolution with broken diffeomorphism invariance is investigated in the (modified) FLRW spacetime in the effective theory framework. The GR Lagrangian is augmented with all diffeomorphism-breaking but Lorentz-invariant terms in the leading order, namely, those involving two derivatives. The magnitudes of (minor) violations are kept general modulo the conditions arising in the linearized theory. The analytic solutions of the scale factor in the full non-linear theory for the single-component universes are attempted; the radiation and vacuum solutions are found analytically, whereas the matter solution is worked out numerically since an analytic solution does not exist in the required form. It is observed that the solutions smoothly connect to those of GR in the limit of vanishing symmetry-breaking. The more realistic, two-component, and three-component universes are numerically studied, and no signs of singular behavior are observed: minor diffeomorphism-violating modifications to GR at the level of two derivatives do not cause instabilities in the basic cosmological evolution.

Figures (12)

• Figure 1: Logarithmic plots of the numerical solutions of $a(\tau)$ and $f(\tau)$ in the theory with broken diffeomorphism for the matter-only case for three benchmark points (BPs), given in Table \ref{['BPs']}. The dimensionless time parameter $\tau$ is chosen to run from 1 to $1000$.
• Figure 2: In the top and mid panels, the comparison of the $a(\tau)$ solutions for the matter-only fluid in the theory with broken diffeomorphism (referred to as broken GR in the plots) to the GR case, $a_{\tiny{\hbox{GR}}}(\tau)=(\frac{3}{2}\tau)^{2/3}$, is displayed for each BP, given in Table \ref{['BPs']}. In the bottom panel, we compare the Hubble parameters, which are the same for all three cases.
• Figure 3: Comparison of $a(\tau)$, $f(\tau)$, and the Hubble parameter $H(\tau)$ for the matter-only case for different initial conditions, denoted by pink, orange, and dashed lines, for three benchmark points (BPs), given in Table \ref{['BPs']}. Note that the orange lines correspond to the solutions in the previous figure.
• Figure 4: Logarithmic plots of the $a(\tau)$ and $f(\tau)$ solutions in theory with broken diffeomorphism (i.e. broken GR) for the case of a two-component fluid; a radiation-dominated system with small amount of matter. The plots are displayed for BP1, given in Table \ref{['BPs']}. Figs. \ref{['PlotRadMatBP1a']} and \ref{['PlotRadMatBP1b']} are the same plots in different ranges, given to demonstrate the evolution from the radiation solution through a matter-like realm. The $f(\tau)$ solution is displayed in Fig. \ref{['PlotRadMatBP1c']}. In Fig. \ref{['PlotRadMatBP1d']}, we display the $a(\tau)$-$f(\tau)$ parametric plot, for the reader's convenience.
• Figure 5: Comparison of $a(\tau)$ solution in the broken theory to that of GR for different BPs (given in Table \ref{['BPs']}) for the radiation-matter case in the top and mid panels. The comparison of the Hubble parameters is displayed at the bottom, which is the same for all three cases.