Detecting dark energy in orbit: The cosmological chameleon

We show that the chameleon scalar field can drive the current phase of cosmic acceleration for a large class of scalar potentials that are also consistent with local tests of gravity. These provide explicit realizations of a quintessence model where the quintessence scalar field couples directly to baryons and dark matter with gravitational strength. We analyze the cosmological evolution of the chameleon field and show the existence of an attractor solution with the chameleon following the minimum of its effective potential. For a wide range of initial conditions, spanning many orders of magnitude in initial chameleon energy density, the attractor is reached before nucleosynthesis. Surprisingly, the range of allowed initial conditions leading to a successful cosmology is wider than in normal quintessence. We discuss applications to the cyclic model of the universe and show how the chameleon mechanism weakens some of the constraints on cyclic potentials.

Figure 1

A typical scalar potential for the chameleon field.

Figure 2

Scalar potential for cyclic models of the universe. In this paper, we will only consider Region a).

Figure 3

Chameleon effective potential for large and small matter density $\rho$. This illustrates that, as $\rho$ decreases, the minimum shifts to larger values of $\varphi$ and the mass of small fluctuations decreases. The solid curve is the sum of the contribution from the actual potential $V(\varphi )$ (dashed curve) and the contribution due to the matter density (dotted curve).

Figure 4

Evolution of the chameleon field $\varphi$ as a function of $1+z$. In this example the potential is $V(\varphi )=M^6/{\varphi }^2$, $\beta =1$ and $M=10\mathrm{M}\mathrm{e}\mathrm{V}$, the latter being larger than the required $M={10}^{-3}\mathrm{e}\mathrm{V}$ due to numerical limitations. The initial conditions for the field are ${\varphi }_i=2M_{\mathrm{P}\mathrm{l}}$ and ${\dot{\varphi }}_i=0$ at $z={10}^{20}$. The dashed line presents the case in which the contribution of particles which become nonrelativistic is neglected. The solid line includes the contribution of the particles in Table  to $T_{\mu }^{\mu }$. Since ${\varphi }_i\gtrsim \beta M_{\mathrm{P}\mathrm{l}}$ in this case, the field does not reach the minimum by the onset of BBN ($z\approx {10}^9$). As explained in the text, this solution is therefore ruled out by BBN constraints on time-variation of particle masses (see Sec. 5).

Figure 5

The root mean square quantity $<(\varphi -{\varphi }_{\min }{)}^2{>}^{1/2}$ plotted as a function of redshift for the undershoot solution with ${\varphi }_i={10}^{-2}{M}_{\mathrm{P}\mathrm{l}}$ and neglecting kicks. The potential is as in the previous two figures. The dotted line is the slope predicted by Eq. (59).

Figure 6

The chameleon starts near the minimum and is subjected to the contribution to $T_{\mu }^{\mu }$ from some particle species becoming nonrelativistic. In this example, the potential is $V(\varphi )=M^6/{\varphi }^2$, $\beta =1$ and $M=10\mathrm{M}\mathrm{e}\mathrm{V}$, the latter once again chosen to be much larger than the required $M={10}^{-3}\mathrm{e}\mathrm{V}$ due to numerical limitations. We see that the field oscillates and eventually settles back to the minimum.