---
title: Cosmological Solutions in dRGT
url: https://www.emergentmind.com/topics/cosmological-solutions-in-drgt
type: topic
---

# Cosmological Solutions in dRGT

The de Rham-Gabadadze-Tolley (dRGT) theory of massive gravity supplies a ghost-free nonlinear completion of Fierz-Pauli massive gravity, with the graviton acquiring a strictly nonzero mass $m$. A central avenue of investigation in this framework is its cosmological sector: can the theory accommodate spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) solutions that fit the universe’s observed expansion and cosmic acceleration? Addressing this requires detailed scrutiny of background solutions, their stability, phenomenology, and data concordance, both in the pure dRGT theory and in its extended incarnations. The following sections systematically survey the technical developments and key results in cosmological solutions within dRGT massive gravity and its extensions.

## 1. Foundation: dRGT Action and FLRW No-Go Results

The core dRGT action in four dimensions is
\[
S = \frac{M_{\mathrm{Pl}}^2}{2} \int d^4x\, \sqrt{-g}\left[ R[g] - m^2\sum_{n=0}^4\beta_n\,e_n(\sqrt{g^{-1}f}) \right] + S_{\text{matter}}[g, \Psi],
\]
where $g_{\mu\nu}$ is the physical metric, $f_{\mu\nu}$ the fixed fiducial metric, $\beta_n$ are dimensionless parameters, and $e_n$ the elementary symmetric polynomials constructed from the square root matrix $\sqrt{g^{-1}f}$ [1701.02873, 1901.09331, 1301.4993]. The theory’s ghost freedom is preserved by the specific structure of the interaction terms.

For standard FLRW ansätze—$g_{\mu\nu}$ and $f_{\mu\nu}$ both homogeneous and isotropic—the pure dRGT theory does not admit nontrivial flat or closed FLRW solutions (i.e., with $k=0$ or $k=+1$): the system imposes strong constraints that force the physical scale factor $a(t)$ to be static, except for Minkowski space [1701.02873, 1301.4993, 2411.19873]. Open ($k=-1$) FLRW models do exist, with two solution branches:

- **Normal branch**: Corresponds to generalized Milne universes (hyperbolic slicing of Minkowski), with no genuine cosmic acceleration.
- **Self-accelerating branch**: An algebraic constraint fixes the ratio $f/a=\xi$ (with $f(t)$ the Stüeckelberg temporal field), producing a genuine de Sitter expansion $H^2 = \frac{m^2}{3}\sum_{n=0}^3\beta_n\,\xi^n$, even in the absence of a bare cosmological constant [1701.02873, 1301.4993].

Nevertheless, these FLRW backgrounds in pure dRGT confront severe obstacles in their linear perturbation theory. Specifically, vector and scalar modes can acquire vanishing kinetic terms (helicity-1 and helicity-0 become infinitely strongly coupled), rendering the theory unstable or non-predictive at cosmological scales [1701.02873].

## 2. Evading the No-Go: Singular Reference Metrics and Stückelberg Inhomogeneity

A crucial development for constructing isotropic and homogeneous cosmologies invoked either singular reference metrics or inhomogeneous, anisotropic Stüeckelberg profiles. These prescriptions relax the rigid structure in $f_{\mu\nu}$ and the Stückelberg sector, opening nontrivial dynamical FLRW solutions [2411.19873, 1901.09331].

- **Singular Reference Metrics**: Making $f_{00}=0$ (or more generally, dropping rank in the reference metric) allows the lapse term in the massive potential to vanish, evading the constraint that would otherwise freeze $a(t)$ [1901.09331]. For example, $f_{\mu\nu} = \text{diag}(0,1,1,1)$ yields a consistent Hamiltonian constraint and viable cosmological evolution.

- **Inhomogeneous Stückelbergs**: Enforcing FLRW symmetry only at the level of the square-root tensor $X^{\mu}_{\nu}$ (not pointwise in $\phi^a$) leads to two branches of flat FLRW solutions [2411.19873]:
  - The $\Lambda$-branch (“self-accelerating”) has $G = \text{const}$, manifesting as a cosmological constant-like graviton stress-energy.
  - The mixed branch (“bi-fluid”): The graviton sector acts as a mixture of several perfect fluids, producing more general behavior.

In both cases, the dynamical solutions acquire dark energy and dark matter-like effective fluids, with energy densities scaling as $1/a$, $1/a^2$, and $1/a^3$ respectively [1901.09331].

## 3. Friedmann Equations and Effective Cosmic Components

Upon varying the dRGT action in mini-superspace ansatz, the modified Einstein equations yield generalized Friedmann and Raychaudhuri equations. For singular reference metrics as in [1901.09331]:
\[
H^2 + \frac{k}{a^2} = \frac{\kappa^2}{3} \rho - m^2\left( \frac{c_1}{2a} + \frac{c_2}{a^2} + \frac{c_3}{a^3} \right),
\]
where $c_i$ are the coefficients of the ghost-free interaction terms. The graviton “fluid” energy and pressure are
\[
\rho_g = -\frac{3m^2}{\kappa^2}\left(\frac{c_1}{2a} + \frac{c_2}{a^2} + \frac{c_3}{a^3}\right), \quad p_g = \frac{m^2}{\kappa^2}\left( \frac{c_1}{a} + \frac{c_2}{a^2} \right).
\]
The emergent terms provide:
- $1/a$ scaling (“domain wall-like”): acts as dynamical dark energy.
- $1/a^3$ scaling: mimics dark matter.
- $1/a^2$ scaling: shifts effective curvature.

At late times, the $1/a$ term (with $c_1$) dominates and can drive cosmic acceleration without an explicit cosmological constant. For certain parameter choices, the effective equation of state $w_g(a)$ can cross $-1$ [1901.09331].

## 4. Extensions with Additional Scalar Fields and Higher Dimensions

Several model extensions introduce new scalar DoFs or higher-dimensional origins to ameliorate pure dRGT’s pathologies:

- **Brans-Dicke–dRGT (BD-dRGT)**: Augments the theory with a Brans-Dicke–type scalar $\varphi$ (with coupling function $\omega(\varphi)$) and a rescaling in the graviton mass term. The full action in the Jordan and Einstein frames admits flat FLRW, self-accelerating de Sitter vacua, with all perturbations (tensor, vector, scalar) free of ghosts or gradient instabilities [2204.05595]. Stability of vector and scalar modes is set by explicit no-ghost and sound-speed conditions on the kinetic matrices (e.g., $\det\mathcal{F}>0$ for scalars).

- **DBI-dRGT**: Inclusion of a Dirac-Born-Infeld scalar with noncanonical kinetic structure, coupled to the graviton mass potential, yields self-accelerating de Sitter branches. Stabilizing vector and scalar perturbations requires constraints both on the graviton mass and on the DBI tension $T(\sigma)$; phenomenological viability is determined by satisfying $M_{\rm GW}^2 > 0$ and sound-speed bounds [2205.10863].

- **Dimensional Reduction**: Reducing higher-dimensional dRGT yields 4D effective actions combining quasi-dilaton or mass-varying massive gravity. Late-time de Sitter attractors arise with stabilization of the radion, and the onset of acceleration is controlled by fixed points in the autonomous dynamical system, characterized by the parameters $(\alpha_3, \alpha_4)$, curvature of the extra dimensions, and scalar potential [1712.09349].

## 5. Observational Concordance and Parameter Constraints

Phenomenological viability necessitates consistency with supernovae (SNe Ia), cosmic microwave background (CMB), and baryon acoustic oscillation (BAO) data sets. For theories with singular reference metrics [1901.09331], the Friedmann equation is rewritten in terms of observable energy densities:
\[
\frac{H^2}{H_0^2} = \Omega_{k0}/a^2 + \Omega_{m0}/a^3 + \Omega_\Lambda(a) + \cdots,
\]
with $\Omega_\Lambda(a)$ and other scale-dependent terms determined by $(m^2/H_0^2) \times c_i$.

Markov chain Monte Carlo fits (e.g., via CosmoMC) yield allowed regions for the graviton mass and interaction coefficients, such as
\[
\Omega_{m0} = 0.287^{+0.049}_{-0.072},\;\Omega_{10}=0.077,\;\Omega_{1p0}=0.357,\;\Omega_{40}=-0.079,\;\frac{3c_3}{c_2}=0.952,
\]
with $1\sigma$ and $2\sigma$ contours leaving significant parameter space for dRGT contributions [1901.09331]. In the BD-dRGT and DBI-dRGT extensions, additional constraints from gravitational wave propagation (via $M_{\rm GW} \lesssim 10^{-22}$ eV from GW170817) and linear-perturbation stability further restrict the viable parameter regions [2204.05595, 2205.10863].

## 6. Pathologies: Lapse Function, Big-Brake Singularities, and Strong Coupling

Certain parameter regions in the pure dRGT FLRW sector exhibit unphysical lapse function behavior—specifically, $N^2(a)<0$, rendering the metric signature ill-defined. Addition of a cosmological background energy density with negative pressure ($w_\mathrm{bg}=-1$) rectifies this, restoring a positive-definite lapse and guaranteeing acceleration at late times [1901.03967].

“Big-brake” singularities—a divergence in $H$ at finite $a$—can arise generically in these cosmologies due to the structure of the modified Friedmann equation. Avoiding such future singularities demands parameter tuning such that the denominator of $H^2$ does not vanish for finite $a$ [1301.4993, 1712.09349].

Strong coupling in scalar/vector perturbations is a generic feature in self-accelerating branches of both pure dRGT and certain mixed branches with inhomogeneous Stüeckelbergs: kinetic terms vanish (for example, $K^2_E \propto -\dot H$ on $\Lambda$-branch), and only tensor modes propagate at quadratic order [2411.19873, 1701.02873]. Extensions with extra scalar degrees of freedom are engineered to alleviate or eliminate this pathology [2204.05595, 2205.10863].

## 7. Comparative Overview of Solution Branches and Extensions

| Model/Branch         | FLRW Solutions   | Stability                     | Late-Time Acceleration | Observational Fit            |
|----------------------|-----------------|-------------------------------|------------------------|------------------------------|
| Pure dRGT (Minkowski $f_{\mu\nu}$, homogeneous Stückelbergs) | Flat/Closed: Only static; Open: two branches [1701.02873, 1301.4993] | Self-acc. branch: strong coupling in scalar/vector [1701.02873, 2411.19873] | Self-acceleration in open branch | Open parameter space allowed [1901.09331] (extensions preferred) |
| Singular $f_{\mu\nu}$ or inhomogeneous Stückelbergs | Flat FLRW now viable [2411.19873, 1901.09331] | $\Lambda$-branch: tensor only; mixed: partial restoration, but with strong-coupling issues [2411.19873] | Yes, via effective $\Lambda$ | Yes, fits SNe, CMB, BAO [1901.09331]           |
| BD-dRGT, DBI-dRGT, higher-dimensional, quasi-dilaton, etc. | Flat FLRW with self-acceleration [2204.05595, 2205.10863, 1712.09349] | Ghost-/gradient-free for suitable parameters | Robust, dynamical in scalar sector | Consistent with late-time expansion, GW bounds   |

These results establish that, while pure dRGT admits only limited cosmological solutions compatible with observation and often suffers from severe strong-coupling pathologies, allowing singular reference metrics or leveraging scalar-tensor and higher-dimensional generalizations yields technically natural models admitting robust, healthy cosmic acceleration sourced by the graviton mass term, with phenomenology that can be tuned to current data.

Source: https://www.emergentmind.com/topics/cosmological-solutions-in-drgt