[Paper Review] Cosmology and gravitational waves in consistent $D o 4$ Einstein-Gauss-Bonnet gravity

Katsuki Aoki, Mohammad Ali Gorji+1 more|arXiv (Cornell University)|May 18, 2020

Cosmology and Gravitation Theories37 references20 citations

TL;DR

This paper proposes a consistent 4D Einstein-Gauss-Bonnet gravity theory by breaking part of diffeomorphism invariance, enabling two dynamical gravitational degrees of freedom without violating the Lovelock theorem. It shows that tensor gravitational waves acquire a $k^4$ dispersion correction, leading to a bound $ ilde{ ho} \lesssim (10\,{\rm meV})^{-2}$ from gravitational wave observations, distinguishing it from both GR and inconsistent $D\to 4$ limits.

ABSTRACT

In a very recent paper [1], we have proposed a novel $4$-dimensional gravitational theory with two dynamical degrees of freedom, which serves as a consistent realization of $D o4$ Einstein-Gauss-Bonnet gravity with the rescaled Gauss-Bonnet coupling constant $ ildeα$. This has been made possible by breaking a part of diffeomorphism invariance, and thus is consistent with the Lovelock theorem. In the present paper, we study cosmological implications of the theory in the presence of a perfect fluid and clarify the similarities and differences between the results obtained from the consistent $4$-dimensional theory and those from the previously considered, naive (and inconsistent) $D ightarrow 4$ limit. Studying the linear perturbations, we explicitly show that the theory only has tensorial gravitational degrees of freedom (besides the matter degree) and that for $ ildeα>0$ and $\dot{H}<0$, perturbations are free of any pathologies so that we can implement the setup to construct early and/or late time cosmological models. Interestingly, a $k^4$ term appears in the dispersion relation of tensor modes which plays significant roles at small scales and makes the theory different than not only general relativity but also many other modified gravity theories as well as the naive (and inconsistent) $D o 4$ limit. Taking into account the $k^4$ term, the observational constraint on the propagation of gravitational waves yields the bound $ ildeα \lesssim (10\,{ m meV})^{-2}$. This is the first bound on the only parameter (besides the Newton's constant and the choice of a constraint that stems from a temporal gauge fixing) in the consistent theory of $D o 4$ Einstein-Gauss-Bonnet gravity.

Motivation & Objective

• To resolve the inconsistency of the naive $D\to 4$ limit of Einstein-Gauss-Bonnet gravity, which violates the Lovelock theorem and leads to strong coupling problems.
• To construct a consistent 4D gravitational theory with two physical degrees of freedom by breaking part of diffeomorphism invariance, avoiding the infinite strong coupling issue of previous approaches.
• To study cosmological implications of the consistent theory, particularly in the presence of a perfect fluid, and compare results with the naive $D\to 4$ limit.
• To analyze linear gravitational perturbations and confirm the absence of scalar modes and pathologies in the tensor sector for $\tilde{\alpha} > 0$ and $\dot{H} < 0$.
• To derive observational constraints on the Gauss-Bonnet coupling $\tilde{\alpha}$ using gravitational wave dispersion relations and cosmological data.

Proposed method

• Construct a consistent 4D gravity theory by modifying the action to break part of diffeomorphism invariance, ensuring the theory avoids the Lovelock theorem's constraints while preserving second-order equations of motion.
• Use a rescaled Gauss-Bonnet coupling $\tilde{\alpha} = (D-4)\alpha$ in the $D\to 4$ limit, ensuring finite corrections to the Einstein equations without introducing extra scalar degrees of freedom.
• Derive the Hamiltonian formulation with constraints, identifying $N$ and $\lambda_{\rm GF}$ as Lagrange multipliers, and enforce second-class constraints via gauge fixing to maintain two physical degrees of freedom.
• Analyze linear perturbations around a Friedmann-Lemaître-Robertson-Walker (FLRW) background, focusing on tensor and scalar modes to confirm the absence of scalar gravitons and pathologies.
• Compute the dispersion relation for tensor modes, showing a $k^4$ correction term $\beta k^4 / M_*^2$ that distinguishes the theory from GR and other modified gravity models.
• Use the $k^4$-modified dispersion relation to derive observational bounds on $\tilde{\alpha}$ from gravitational wave propagation, yielding $\tilde{\alpha} \lesssim (10\,{\rm meV})^{-2}$.

Experimental results

Research questions

• RQ1 Can a consistent 4D Einstein-Gauss-Bonnet gravity theory be formulated that avoids the strong coupling problem and violates the Lovelock theorem only partially?
• RQ2 How do cosmological solutions in the consistent 4D theory compare with those from the naive $D\to 4$ limit, particularly in the presence of a perfect fluid?
• RQ3 What is the structure of linear gravitational perturbations in the consistent 4D theory, and do they contain scalar modes or pathologies?
• RQ4 How does the $k^4$ term in the tensor mode dispersion relation affect gravitational wave propagation and observational constraints?
• RQ5 Can the consistent 4D theory provide a viable alternative to the inconsistent $D\to 4$ limit for modeling early and late-time cosmology?

Key findings

• The consistent 4D Einstein-Gauss-Bonnet theory features only tensor gravitational modes and no scalar degrees of freedom, ensuring stability and absence of pathologies for $\tilde{\alpha} > 0$ and $\dot{H} < 0$.
• The dispersion relation for tensor modes includes a $k^4$ correction term $\beta k^4 / M_*^2$, which distinguishes the theory from general relativity and other modified gravity models at small scales.
• The $k^4$ term leads to a stringent observational bound: $\tilde{\alpha} \lesssim (10\,{\rm meV})^{-2}$, derived from gravitational wave propagation constraints.
• The consistent theory reproduces the background equations and scalar perturbations of the naive $D\to 4$ limit but differs fundamentally in the tensor sector, especially in the UV regime.
• The black hole solutions from the naive $D\to 4$ limit are also solutions in the consistent theory, but quasinormal mode analysis is expected to differ due to the modified dispersion relation.
• The theory opens a new framework for minimally modified gravity, generalizable to actions with higher-order spatial derivatives, such as $k^6$ terms, by relaxing certain constraints in the action.