Cosmology of $F(R)$ nonlinear massive gravity

The theory of nonlinear massive gravity can be extended into the $F(R)$ form as developed by Cai et al. [Phys. Rev. D 90, 064051 (2014)]. Being free of the Boulware-Deser ghost, such a construction has the additional advantage of exhibiting no linear instabilities around a cosmological background. We investigate various cosmological evolutions of a universe governed by this generalized massive gravitational theory. Specifically, under the Starobinsky Ansatz, this model provides a unified description of the cosmological history, from early-time inflation to late-time self-acceleration. Moreover, under viable $F(R)$ forms, the scenario leads to a very interesting dark energy phenomenology, including the realization of the quintom scenario without any pathology. Finally, we provide a detailed analysis of the cosmological perturbations at linear order, as well as the Hamiltonian constraint analysis, in order to examine the physical degrees of freedom.

Figure 1

Four inflationary solutions for the Starobinsky Ansatz (55), corresponding to (a) $m_g=10^{-50}$, ${\alpha }_3=2$, ${\alpha }_4=-1$, $a_k=5\times 10^{-41}$, $\xi =10^{10}$ (black solid), (b) $m_g=10^{-50}$, ${\alpha }_3=10$, ${\alpha }_4=10$, $a_k=10^{-41}$, $\xi =10^9$ (red dashed), (c) $m_g=10^{-50}$, ${\alpha }_3=1$, ${\alpha }_4=1$, $a_k=10^{-40}$, $\xi =10^{10}$ (blue-dash dotted), (d) $m_g=10^{-50}$, ${\alpha }_3=10$, ${\alpha }_4=1$, $a_k=2\times 10^{-41}$, $\xi =10^{10}$ (green dotted). All parameters are in Planck units. The two horizontal lines mark the $N=50$ and $N=60$ e-folding regimes.

Figure 2

Four late-time accelerating solutions for the Starobinsky Ansatz (55), corresponding to (a) $m_g=2$, ${\alpha }_3=1$, ${\alpha }_4=1$, $a_k=0.05$, $\xi =0.2$ (black solid), (b) $m_g=1.5$, ${\alpha }_3=1$, ${\alpha }_4=-2$, $a_k=0.01$, $\xi =0.5$ (red dashed), (c) $m_g=3$, ${\alpha }_3=3$, ${\alpha }_4=-5$, $a_k=0.05$, $\xi =0.5$ (blue-dash dotted), (d) $m_g=3$, ${\alpha }_3=10$, ${\alpha }_4=1$, $a_k=0.05$, $\xi =0.5$ (green dotted). All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.31$, ${\mathrm{\Omega }}_{DE0}\approx 0.69$, ${\mathrm{\Omega }}_{k0}\approx 0.01$ at the present scale factor $a_0=1$ [1].

Figure 3

Upper graph: The dark energy equation-of-state parameter $w_{DE}$ as a function of the redshift $z$, in the case of the exponential $F(R)$ Ansatz (56), for $m_g=1$, ${\alpha }_3=3$, ${\alpha }_4=-2$, $a_k=0.1$, $\beta =1.5$, $R_S=0.25$. All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.31$, ${\mathrm{\Omega }}_{DE0}\approx 0.69$ and ${\mathrm{\Omega }}_{k0}\approx 0.01$ at present [1]. Lower graph: The corresponding evolution of the dark energy density parameter ${\mathrm{\Omega }}_{DE}$ (back solid), as well as its two constituents ${\mathrm{\Omega }}_{F_R}$ (red dashed) and ${\mathrm{\Omega }}_{MG}$ (green dotted), of the $F(R)$ and massive-gravity sectors respectively.

Figure 4

Upper graph: The dark energy equation-of-state parameter $w_{DE}$ as a function of the redshift $z$, in the case of the exponential $F(R)$ Ansatz (56), for $m_g=0.75$, ${\alpha }_3=1$, ${\alpha }_4=-2$, $a_k=0.1$, $\beta =1.1$, $R_S=0.3$. All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.31$, ${\mathrm{\Omega }}_{DE0}\approx 0.69$ and ${\mathrm{\Omega }}_{k0}\approx 0.01$ at present [1]. Lower graph: The corresponding evolution of the dark energy density parameter ${\mathrm{\Omega }}_{DE}$ (back solid), as well as its two constituents ${\mathrm{\Omega }}_{F_R}$ (red dashed) and ${\mathrm{\Omega }}_{MG}$ (green dotted), of the $F(R)$ and massive-gravity sectors respectively.

Figure 5

Upper graph: The dark energy equation-of-state parameter $w_{DE}$ as a function of the redshift $z$, in the case of the power-law-like Starobinsky $F(R)$ Ansatz (57), for $m_g=0.5$, ${\alpha }_3=2$, ${\alpha }_4=1$, $a_k=0.1$, $\lambda =0.3$, $n=2$, $R_c=2$. All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.31$, ${\mathrm{\Omega }}_{DE0}\approx 0.69$ and ${\mathrm{\Omega }}_{k0}\approx 0.01$ at present [1]. Lower graph: The corresponding evolution of the dark energy density parameter ${\mathrm{\Omega }}_{DE}$ (back solid), as well as its two constituents ${\mathrm{\Omega }}_{F_R}$ (red dashed) and ${\mathrm{\Omega }}_{MG}$ (green dotted), of the $F(R)$ and massive-gravity sectors respectively.

Figure 6

Upper graph: The dark energy equation-of-state parameter $w_{DE}$ as a function of the redshift $z$, in the case of the power-law-like Starobinsky $F(R)$ Ansatz (57), for $m_g=2$, ${\alpha }_3=2$, ${\alpha }_4=1$, $a_k=0.1$, $\lambda =25$, $n=2$, $R_c=10$. All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.31$, ${\mathrm{\Omega }}_{DE0}\approx 0.69$ and ${\mathrm{\Omega }}_{k0}\approx 0.01$ at present [1]. Lower graph: The corresponding evolution of the dark energy density parameter ${\mathrm{\Omega }}_{DE}$ (back solid), as well as its two constituents ${\mathrm{\Omega }}_{F_R}$ (red dashed) and ${\mathrm{\Omega }}_{MG}$ (green dotted), of the $F(R)$ and massive gravity sectors, respectively.