Cosmology with three interacting spin-2 fields

Theories of massive gravity with one or two dynamical metrics generically lack stable and observationally viable cosmological solutions that are distinguishable from $\mathrm{\Lambda }$ cold dark matter (CDM). We consider an extension to trimetric gravity, with three interacting spin-2 fields which are not plagued by the Boulware-Deser ghost. We systematically explore every combination with two free parameters in search of background cosmologies that are competitive with $\mathrm{\Lambda }\mathrm{CDM}$. For each case we determine whether the expansion history satisfies viability criteria, and whether or not it contains beyond-$\mathrm{\Lambda }\mathrm{CDM}$ phenomenology. Among the many models we consider, there are only three cases that seem to be both viable and distinguishable from standard cosmology. One of the models has only one free parameter and displays a crossing from above to below the phantom divide. The other two provide scaling behavior, although they contain future singularities that need to be studied in more detail. These models possess interesting features that make them compelling targets for a full comparison to observations of both cosmological expansion history and structure formation.

Figure 1

Visual depiction of star trigravity (left panel) and path trigravity (right panel). The white circles and the single lines between them represent, respectively, the three metrics and the interaction terms mixing them. The shaded circles and the double lines represent the matter fields and their couplings to the physical metric, $g$, respectively.

Figure 2

Evolution of $r_1^{\prime }$, the matter density parameter ${\mathrm{\Omega }}_{\mathrm{m}}$, and the equation of state $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,1}{\beta }_{2,1}$ model of star trigravity with $B_{11}=1$. In this and subsequent figures, we remind the reader that $r_i$ effectively stands for time, as it monotonically increases or decreases throughout cosmological evolution. Of course, whether $r_i$ increases or decreases with time can be determined from the sign of $r_i^{\prime }$.

Figure 3

(Left panel) Evolution of $r_1^{\prime }$, the matter density parameter ${\mathrm{\Omega }}_{\mathrm{m}}$, and the effective equation of state $w_{\mathrm{eff}}$ in terms of $r_1$ for the ${\beta }_{1,1}{\beta }_{2,3}$ model of star trigravity with $B_{13}=\{2,20\}$. (Right panel) Time evolution of $r_1$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ for the finite branch $[r_1^{\mathrm{sing}},r_1^{\mathrm{fix}+}]$ of the ${\beta }_{1,1}{\beta }_{2,3}$ model with $B_{13}=2$, together with the time evolution of ${\mathrm{\Omega }}_{\mathrm{m}}$ and $w_{\mathrm{eff}}$ for standard $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. The right vertical line shows $N=0$, i.e., today, while the left vertical line represents the would-be initial value of $N$ for $B_{13}=2$, i.e., the would-be initial size of the Universe.

Figure 4

Evolution of $r_1^{\prime }$, the matter density parameter ${\mathrm{\Omega }}_{\mathrm{m}}$, and the effective equation of state $w_{\mathrm{eff}}$ in terms of $r_1$ for the ${\beta }_{1,3}{\beta }_{2,3}$ model of star trigravity with an interaction parameter ratio of $B_{33}=2$.

Figure 5

Evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,1}{\beta }_{2,1}$ model of path trigravity with ratio of the interaction parameters $B_{11}=1.8$ representing case (a).

Figure 6

Evolution of $r_1$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $N=\ln a$ in the ${\beta }_{1,1}{\beta }_{2,1}$ model of path trigravity with $B_{11}=1.8$ representing case (a). In order to compare it to standard cosmology, the evolution of ${\mathrm{\Omega }}_{\mathrm{m}}$ and $w_{\mathrm{eff}}$ is plotted also for $\mathrm{\Lambda }\mathrm{CDM}$ with ${\mathrm{\Omega }}_{\mathrm{m},0}=0.3$. The left vertical line represents $N=0$ (today). (Left panel) Evolution for the finite branch $[0,r_1^{\mathrm{sing}}]$, where the right vertical line, at $N=0.28$ (in the future), represents the time at which $r_1$ takes its final value $r_1^{\mathrm{sing}}$. (Right panel) Evolution for the infinite branch $[r_1^{\mathrm{sing}},\infty ]$, where the right vertical line, at $N=0.57$, represents the time at which $r_1$ takes its final value $r_1^{\mathrm{sing}}$.

Figure 7

The ${\beta }_{1,1}{\beta }_{2,1}$ model of path trigravity with the ratio of the interaction parameters $B_{11}=1.5$ representing case (b). (Left panel) Evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$. (Right panel) Evolution of $r_1$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $N\equiv \ln a$ for the finite branch of the model. The vertical line represents $N=0$ (today). In order to compare it to standard cosmology, the evolution of ${\mathrm{\Omega }}_{\mathrm{m}}$ and $w_{\mathrm{eff}}$ is plotted also for $\mathrm{\Lambda }\mathrm{CDM}$ with ${\mathrm{\Omega }}_{\mathrm{m},0}=0.3$.

Figure 8

(Left panel) The evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,1}{\beta }_{2,1}$ model of path trigravity with $B_{11}=1$ representing case (c). (Right panel) The same for the ${\beta }_{1,1}{\beta }_{2,2}$ model of path trigravity with the interaction parameter ratio of $B_{12}=1$.

Figure 9

(Left panel) Evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,1}{\beta }_{2,3}$ model of path trigravity with the interaction parameter ratios of $B_{13}=2$ (solid lines) representing case (Ia) and $B_{13}=0.5$ (dotted lines) representing case (Ib). (Right panel) The same for the ${\beta }_{1,2}{\beta }_{2,1}$ model of path trigravity with the interaction parameter ratio of $B_{21}=1$ representing case (a).

Figure 10

Evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,2}{\beta }_{2,1}$ model of path trigravity with the interaction parameter ratios of $B_{21}=\sqrt{3}$ (left panel) representing case (b) and $B_{21}=2$ (right panel) representing case (c).

Figure 11

Evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,2}{\beta }_{2,3}$ model of path trigravity with the interaction parameter ratios of $B_{23}=2$ (solid lines) representing case (Ia) and $B_{23}=0.7$ (dotted lines) representing case (Ib).

Figure 12

Evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,3}{\beta }_{2,1}$ model of path trigravity with an interaction parameter ratio of $B_{31}=1$.

Figure 13

Evolution of $r_1^{\prime }$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and $w_{\mathrm{eff}}$ as functions of $r_1$ for the ${\beta }_{1,3}{\beta }_{2,2}$ model of path trigravity with an interaction parameter ratio of $B_{32}=2$ (left panel) representing case (a) and $B_{32}=0.7$ (right panel) representing case (b).