[Paper Review] Hamiltonian cosmology in bigravity and massive gravity

Vladimir O. Soloviev|arXiv (Cornell University)|May 4, 2015

Cosmology and Gravitation Theories42 references3 citations

TL;DR

This paper applies Hamiltonian formalism to study flat-space cosmology in bigravity and massive gravity using the de Rham-Gabadadze-Tolley (dRGT) potential. It demonstrates that Hamiltonian methods not only confirm the absence of the Boulware-Deser ghost but also provide a systematic framework for deriving cosmological equations across multiple matter-metric coupling scenarios, yielding a cyclic universe model with a negative effective cosmological constant due to an additional ghost-like degree of freedom.

ABSTRACT

In the Hamiltonian language we provide a study of flat-space cosmology in bigravity and massive gravity constructed mostly with de Rham, Gabadadze, Tolley (dRGT) potential. It is demonstrated that the Hamiltonian methods are powerful not only in proving the absence of the Boulware-Deser ghost, but also in solving other problems. The purpose of this work is to give an introduction both to the Hamiltonian formalism and to the cosmology of bigravity. We sketch three roads to the Hamiltonian of bigravity with the dRGT potential: the metric, the tetrad and the minisuperspace approaches.

Motivation & Objective

• To demonstrate the power of Hamiltonian formalism in analyzing cosmological solutions in bigravity and massive gravity theories.
• To systematically derive the Hamiltonian for bigravity using metric, tetrad, and minisuperspace approaches with the dRGT potential.
• To explore diverse matter coupling scenarios, including double coupling and effective metric interactions, in a unified framework.
• To investigate the viability of homogeneous and isotropic cosmological solutions in non-dRGT bigravity, revealing ghost-like instabilities.
• To lay a foundation for quantum cosmology in bigravity by establishing a consistent Hamiltonian structure.

Proposed method

• The paper constructs the bigravity Hamiltonian using three approaches: metric variables, tetrad variables, and minisuperspace reduction for cosmological symmetry.
• It employs the de Rham-Gabadadze-Tolley (dRGT) potential, which ensures ghost-freeness by carefully constructing the matrix square root of the metric combination.
• The minisuperspace approach simplifies the dynamics by assuming spatial homogeneity and isotropy, reducing the theory to finite-dimensional phase space with diagonal metric matrices.
• Cosmological equations are derived from the Hamiltonian constraint and momentum constraints, with matter fields minimally coupled to individual or effective metrics.
• The analysis includes explicit calculations of traces and invariants of the matrix $\mathsf{X}$, essential for evaluating the potential energy in the Hamiltonian.
• For non-dRGT models, the paper shows the emergence of the Boulware-Deser ghost through explicit solutions and instability in the Friedmann equation.

Experimental results

Research questions

• RQ1 Can the Hamiltonian formalism consistently describe cosmological solutions in bigravity with the dRGT potential?
• RQ2 How do different matter coupling schemes—single metric, double metric, or effective metric—modify the cosmological dynamics?
• RQ3 What is the role of the additional degree of freedom in bigravity, and how does it affect the evolution of the universe?
• RQ4 Does non-dRGT bigravity admit stable, homogeneous, and isotropic cosmological solutions?
• RQ5 Can the Hamiltonian approach be used as a foundation for quantum cosmology in massive gravity theories?

Key findings

• The Hamiltonian formalism successfully reproduces the Friedmann equation in bigravity with the dRGT potential, showing a negative effective cosmological constant $\Lambda(\xi) = -\frac{m^2}{2}\left(\frac{1}{2\xi^6} + 1 - \frac{3}{2\xi^2}\right)$.
• The model predicts a cyclic universe due to the negative $\Lambda(\xi)$, where expansion eventually turns into contraction, despite the presence of quintessence matter.
• The additional degree of freedom in bigravity is identified as a ghost, which destabilizes the cosmological solution, though the theory remains ghost-free at the linear level due to the dRGT potential.
• In non-dRGT bigravity, the Boulware-Deser ghost reappears, as shown by the instability in the Friedmann equation and the breakdown of the ghost-free condition.
• The minisuperspace approach allows explicit computation of the matrix square root, enabling analytical derivation of the potential and cosmological equations in symmetric backgrounds.
• The effective metric approach leads to a modified Friedmann equation resembling general relativity with a $\Lambda$-term, but with a $\xi$-dependent $\Lambda(\xi)$ that depends on the scalar field configuration.