Massive photon and dark energy

We investigate the cosmology of massive electrodynamics and explore the possibility whether the massive photon could provide an explanation of dark energy. The action is given by the scalar-vector-tensor theory of gravity, which is obtained by nonminimal coupling of the massive Stueckelberg QED with gravity; its cosmological consequences are studied by paying particular attention to the role of photon mass. We find that the theory allows for cosmological evolution where the radiation- and matter-dominated epochs are followed by a long period of virtually constant dark energy that closely mimics a $\mathrm{\Lambda }\mathrm{CDM}$ model. We also find that the main source of the current acceleration is provided by the nonvanishing photon mass governed by the relation $\mathrm{\Lambda }\sim m^2$. A detailed numerical analysis shows that the nonvanishing photon mass on the order of $\sim 10^{-34}\mathrm{eV}$ is consistent with current observations. This magnitude is far less than the most stringent limit on the photon mass available so far, which is on the order of $m\le 10^{-27}\mathrm{eV}$.

Figure 1

Evolution of $\widehat{f}$ (left) and $\widehat{\phi }$ (right) as a function of logarithmic scale factor $N=\ln a(t)$. In both panels, we have used $\widehat{\eta }=0.9$, $\widehat{\omega }=-0.35$, $\chi =10^{-7}$, and $\widehat{m}=10^{-3}$. We have also set the initial values ${\widehat{\phi }}_i=8$ and ${\widehat{f}}_i\simeq a_i^{(1+\sqrt{25-24\widehat{\eta }})/2}\simeq 2\times {10}^{-4}$ at the initial epoch $\ln a_i=-20$.

Figure 2

(Left) Evolution of energy density of the vector field (red), radiation (blue), and matter (green curve). (Right) Evolution of the equation-of-state parameter. The same model parameters have been used as in Fig. 1.

Figure 3

Marginalized one-dimensional probability distributions of Hubble constant ($H_0$) and five model parameters [${\log }_{10}\widehat{m}$, $-{\log }_{10}(-\chi )$, ${\mathrm{\Omega }}_m{h}^2$, ${\mathrm{\Omega }}_b{h}^2$, ${\log }_{10}{\widehat{\phi }}_i$], favored by the current observations: $H(z)+\mathrm{SN}+\mathrm{BAO}+\mathrm{PLANCK}$ (blue) and $H(z)+\mathrm{SN}+\mathrm{BAO}+\mathrm{WMAP}9$ (red histograms), respectively.

Figure 4

(Left) Observed Hubble parameters versus redshift (grey and black dots with error bars; see text). (Right) The Hubble diagram for Union 2.1 compilation of SN type Ia. In both figures, the red curve represents the best-fit prediction of our model constrained with $H(z)$, SN, BAO, and CMB data sets.

Figure 5

Marginalized likelihood distributions for $(H_0,{\log }_{10}\widehat{m})$ (left) and $({\log }_{10}{\widehat{\phi }}_i,{\log }_{10}\widehat{m})$ (right) with 68.3% and 95.4% confidence limits, obtained by the joint parameter estimation with $H(z)+\mathrm{SN}+\mathrm{BAO}+\mathrm{PLANCK}$ (blue) and $H(z)+\mathrm{SN}+\mathrm{BAO}+\mathrm{WMAP}9$ (red) data sets, respectively.