Signatures of a graviton mass in the cosmic microwave background

There exist consistent low energy effective field theories describing gravity in the Higgs phase that allow the coexistence of massive gravitons and the conventional $1/r$ potential of gravity. In an effort to constrain the value of the graviton mass in these theories, we study the tensor contribution to the CMB temperature anisotropy and polarization spectra in the presence of a nonvanishing graviton mass. We find that the observation of a $B$-mode signal consistent with the spectrum predicted by inflationary models would provide the strongest limit yet on the mass of an elementary particle—a graviton—at a level of $m\lesssim {10}^{-30}\mathrm{eV}\approx (10\mathrm{Mpc}{)}^{-1}$. We also find that a graviton mass in the range between $(10\mathrm{Mpc}{)}^{-1}$ and $(10\mathrm{kpc}{)}^{-1}$ leads to interesting modifications of the polarization spectrum. The characteristic signature of a graviton mass in this range would be a plateau in the $B$-mode spectrum up to angular multipoles of $\ell \sim 100$. For even larger values of the graviton mass the tensor contribution to the CMB spectra becomes strongly suppressed.

Figure 1

This plot summarizes the different behaviors of modes and which range of multipole coefficients they contribute to. Class I corresponds to modes that are relativistic at recombination. Class II corresponds to modes that are nonrelativistic as they enter the horizon and during their subsequent evolution. Class III corresponds to modes that enter the horizon when they are relativistic but become nonrelativistic before recombination.

Figure 2

This plot shows the quantity ${\mathcal{I}}^2$ in $(\mu K{)}^2$ as a function of mass for a scalar amplitude of ${\Delta }_{\mathcal{R}}^2=2.41\times {10}^{-9}$, a tensor-to-scalar ratio of $r=1$, and a tensor spectral index $n_T=0$.

Figure 3

These plots show the $C_{BB,\ell }^T$ multipole coefficients for the range of masses that lead to the most interesting signal in the CMB. The masses are given by $m_g=\mu \times 3000H_0$, where $\mu$ is given in the legend. Longer dashes correspond to larger mass. All plots are for a scalar amplitude ${\Delta }_{\mathcal{R}}^2=2.41\times {10}^{-9}$, a tensor-to-scalar ratio, $r=1$, and a tensor spectral index $n_T=0$. For the remaining cosmological parameters parametrizing the background, we use the five-year WMAP values [27].

Figure 4

This plot shows $T$ (upper left panel), $E$ (lower left), $TE$ (upper right) and $B$ (lower right) spectra for the massive case with $\mu =10$ (solid line) and for the massless case (dashed line).

Figure 5

These plots show marginalized likelihood plots obtained from a Markov chain Monte Carlo study of a massive gravity model with a mass $m_g=3\times {10}^4{H}_0$, or equivalently $\mu =10$ using the five-year WMAP data. The dark and light blue contours correspond to 68% and 95% confidence level, respectively.