$F(R)$ nonlinear massive theories of gravity and their cosmological implications

We propose a nonlinear massive gravitational theory which includes $F(R)$ modifications. This construction inherits the benefits of the de Rham-Gabadadze-Tolley model and is free of the Boulware-Deser ghost due to the existence of a Hamiltonian constraint accompanied by a nontrivial secondary one. The scalar perturbations in a cosmological background can be stabilized at the linear level for a wide class of the $F(R)$ models. The linear scalar mode arisen from the $F(R)$ sector can absorb the nonlinear longitudinal graviton, and hence, our scenario demonstrates the possibility of a gravitational Goldstone theorem. Finally, due to the combined contribution of the $F(R)$ and graviton-mass sectors, the proposed theory allows for a large class of cosmological evolutions, such as the simultaneous and unified description of inflation and late-time acceleration.

Figure 1

The left panel presents three inflationary solutions corresponding to (a) $m_g=10^{-50}$, ${\alpha }_3=2$, ${\alpha }_4=-1$, $a_0=5\times 10^{-41}$, and $\xi =10^{10}$ (solid black line), (b) $m_g=10^{-50}$, ${\alpha }_3=10$, ${\alpha }_4=10$, $a_0=10^{-41}$, and $\xi =10^9$ (dashed red line), and (c) $m_g=10^{-50}$, ${\alpha }_3=1$, ${\alpha }_4=1$, $a_0=10^{-40}$, and $\xi =10^{10}$ (dash-dotted blue line). The two horizontal lines mark the $N=50$ and $N=60$ $e$-folding regimes. All parameters are in Planck units. The right panel depicts three late-time accelerating solutions corresponding to (a) $m_g=3$, ${\alpha }_3=3$, ${\alpha }_4=-5$, $a_0=0.05$, and $\xi =0.5$ (solid black line), (b) $m_g=1.5$, ${\alpha }_3=1$, ${\alpha }_4=-2$, $a_0=0.01$, and $\xi =0.5$ (dashed red line), and (c) $m_g=3$, ${\alpha }_3=10$, ${\alpha }_4=1$, $a_0=0.05$, and $\xi =0.5$ (dash-dotted blue line). All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.31$, ${\mathrm{\Omega }}_{\mathrm{DE}0}\approx 0.69$, and ${\mathrm{\Omega }}_{k0}\approx 0.01$ at the present scale factor $a_0=1$.