[Paper Review] Modified Theories of Gravity

Fulvio Sbisà|arXiv (Cornell University)|Jun 12, 2014

Cosmology and Gravitation Theories4 citations

TL;DR

This PhD thesis investigates modified gravity theories—specifically the Cascading DGP and dRGT massive gravity—as alternatives to dark energy for explaining cosmic acceleration. It analyzes perturbations in nested brane setups and the Vainshtein screening mechanism, finding a new critical tension value in the 6D Cascading DGP model and precisely characterizing parameter regions where global solutions and Vainshtein screening exist in massive gravity.

ABSTRACT

The recent observational data in cosmology seem to indicate that the universe is currently expanding in an accelerated way. An intriguing interpretation of these data is that they may just be signalling that Einstein's General Relativity is not the correct description of gravity when we consider distances of the order of the present horizon of the universe. In this thesis we consider two models which modify General Relativity at very large distances, the Cascading DGP and the dRGT massive gravity, and investigate their phenomenological viability. We start with a general introduction to standard cosmology and we introduce the late time acceleration problem and the cosmological constant problem. We then provide a pedagogical introduction to the DGP model, of which the Cascading DGP is an extension, and to the dRGT massive gravity. Concerning the Cascading DGP, we show that the thin limit of the 4D brane inside the (already thin) 5D brane is well defined, at least for the class of configurations that we consider, and confirm that gravity is regularized in these set-ups. We give a geometrical interpretation of the presence of the critical tension, and comment on the difference between the results in the literature and our results, which we support with a numerical calculation. Regarding the dRGT massive gravity, we focus on the branch of solutions in which the Vainshtein mechanism can occur. We determine analytically the number and properties of local solutions which exist asymptotically on large scales (but still below the gravitational Compton wavelength), and of local (inner) solutions which exist on small scales. We characterize exactly the properties of global solutions in every point of the phase space, and characterize precisely in which regions the Vainshtein mechanism takes place. We also provide numerical solutions which confirm our analysis.

Motivation & Objective

• To assess whether modified gravity theories like Cascading DGP and dRGT massive gravity can explain the observed late-time acceleration of the universe without invoking dark energy.
• To investigate the viability of these models by analyzing ghost instabilities and screening mechanisms on cosmological and astrophysical scales.
• To determine the conditions under which the Vainshtein mechanism effectively screens new gravitational degrees of freedom in massive gravity.
• To compute the critical tension in the 6D Cascading DGP model that separates ghost-free from ghost-containing configurations.
• To characterize the existence and matching of inner (small-scale) and asymptotic (large-scale) solutions in massive gravity, particularly in the context of Vainshtein screening.

Proposed method

• Performs a first-order perturbative analysis of gravitational fields and brane positions in the minimal 6D Cascading DGP model with induced gravity on nested branes.
• Applies the thin limit approximation to the 4D brane within a pre-existing thin 5D brane, confirming the finiteness of the gravitational field on the 4D brane.
• Uses gauge-invariant variables and master equations to analyze perturbations, with regularization of gravity via induced gravity terms.
• Derives and solves quintic equations for gravitational potentials in the weak-field limit of dRGT massive gravity, focusing on nonlinearities in the Vainshtein mode.
• Analyzes asymptotic and inner solution behaviors using power-series expansions and matching conditions in the intermediate regime.
• Employs numerical checks in known exact solution cases to validate analytical results, particularly for the critical tension in Cascading DGP.

Experimental results

Research questions

• RQ1 What is the precise value of the critical tension in the 6D Cascading DGP model that separates ghost-free from ghost-instability configurations?
• RQ2 How does the Vainshtein mechanism operate in dRGT massive gravity, and in which parameter regions do global solutions with screening exist?
• RQ3 Under what conditions can asymptotically flat and non-flat solutions match with inner configurations exhibiting Vainshtein screening?
• RQ4 How do the leading-order behaviors of inner and asymptotic solutions in massive gravity influence the finiteness of gravitational potentials at the origin?
• RQ5 What is the role of nonlinear terms in the Vainshtein mode in enabling effective screening of the extra gravitational degrees of freedom?

Key findings

• A new value for the critical tension in the 6D Cascading DGP model is found, differing from previous literature results, with numerical validation in a known exact solution case.
• The thin limit of the 4D brane inside a thin 5D brane is well-defined for the considered configurations, and the gravitational field on the 4D brane remains finite.
• In dRGT massive gravity, asymptotically flat solutions only match with inner configurations that exhibit the Vainshtein mechanism.
• Non-asymptotically flat solutions can match with either Vainshtein-screens or self-shielding inner solutions, depending on the parameter regime.
• Global solutions fail to exist in certain regions of the parameter space, and these regions are precisely characterized in terms of the Vainshtein mechanism's applicability.
• The inner diverging solution D behaves as $ h(\rho) \sim -\sqrt[3]{2/\beta} \cdot \rho_v / \rho $ at small radii, with the next-order term scaling as $ \rho $, ensuring finiteness of the potential in the limit $ \rho \to 0^+ $.