Symmetron cosmology

The symmetron is a scalar field associated with the dark sector whose coupling to matter depends on the ambient matter density. The symmetron is decoupled and screened in regions of high density, thereby satisfying local constraints from tests of gravity, but couples with gravitational strength in regions of low density, such as the cosmos. In this paper we derive the cosmological expansion history in the presence of a symmetron field, tracking the evolution through the inflationary, radiation- and matter-dominated epochs, using a combination of analytical approximations and numerical integration. For a broad range of initial conditions at the onset of inflation, the scalar field reaches its symmetry-breaking vacuum by the present epoch, as assumed in the local analysis of spherically symmetric solutions and tests of gravity. For the simplest form of the potential, the energy scale is too small for the symmetron to act as dark energy, hence we must add a cosmological constant to drive late-time cosmic acceleration. We briefly discuss a class of generalized, nonrenormalizable potentials that can have a greater impact on the late-time cosmology, though cosmic acceleration requires a delicate tuning of parameters in this case.

Figure 1

Schematic plots of the symmetron effective potential, illustrating the symmetry-breaking phase transition.

Figure 2

Numerical solution of the symmetron evolution as a function of redshift through the radiation- and matter-dominated eras, starting from ${\varphi }_{\mathrm{rad}\mathrm{\text{−}}\mathrm{i}}={10}^{-2}{M}_{\mathrm{Pl}}$, with $M={10}^{-4}{M}_{\mathrm{Pl}}$. Figure 2a confirms that the field is initially frozen because of Hubble friction, but eventually undergoes damped oscillations. See main text for details. Figure 2b is a zoom-in on the boxed region, highlighting that the oscillations are smooth.

Figure 3

Evolution of the symmetron around the phase transition. (a) At the onset of the transition ($z_{\mathrm{tran}}=1$), the symmetron rapidly evolves towards ${\varphi }_0$, the minimum of the potential, and undergoes damped oscillations around this point. The dashed line indicates ${\varphi }_{\min }$, where we have chosen $M={10}^{-4}{M}_{\mathrm{Pl}}$. (b) Zoom-in on the boxed region, which shows that the oscillations are smooth. The solid line indicates ${\varphi }_{\min }(z)$, the minimum of the effective potential given by (62).

Figure 4

The fractional energy density in the symmetron for the quartic potential.

Figure 5

The generalized potential studied in Sec. 4b, shown here for illustrative purposes with $M=0.1M_{\mathrm{Pl}}$. The minima occur at $\varphi \sim \pm M$. By construction, the height of the potential is $\sim H_0^2{M}_{\mathrm{Pl}}^2$, and the mass around the minima is $\sim H_0$.

Figure 6

The behavior of $\varphi$ after the transition for the generalized potential (63) and coupling function (67). The dashed line denotes the minimum ${\varphi }_0$, given by (65), where we have taken $M={10}^{-4}{M}_{\mathrm{Pl}}$. We have chosen parameters such that $z_{\mathrm{tran}}=1$. Note that the plot extends to future redshifts ($z<0$). Because of the asymmetric form of the potential, in this case the symmetron has not yet settled to the minimum by the present time.

Figure 7

Hubble parameter as a function of redshift for the symmetron with generalized potential (63) and coupling function (67) [solid line], where $H_0=72\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ denotes the measured value. For comparison, the thick dashed line is a $\Lambda \mathrm{CDM}$ cosmology with ${\Omega }_{\mathrm{m}}^{(0)}=0.25$, while the thin dashed line is an Einstein-de Sitter (${\Omega }_{\mathrm{m}}^{(0)}=1$) universe. In all three cases, we have fixed the expansion history deep in the matter-dominated era.

Figure 8

Symmetron evolution during reheating for different parameter values. In each case, the thick line indicates the numerical solution, whereas the thin line is the Mathieu cosine function (80). All plots have $\omega ={10}^4{H}_{\inf }$ and $M={10}^{-4}{M}_{\mathrm{Pl}}$. (a) In this case, ${\Gamma }_{\inf }\gg m_{\inf }$, hence reheating has essentially completed before the tachyonic amplification can kick in. (b) In the opposite regime, ${\Gamma }_{\inf }\ll m_{\inf }$, reheating is slow. Since $m_{\inf }=\omega$ lies in the unstable region of the Mathieu cosine solution, the field is exponentially amplified. The amplification is eventually curtailed once the inflaton decays and reheating takes place. The dashed line indicates the maximum allowed amplification, corresponding to ${\varphi }_{\mathrm{rad}\mathrm{\text{−}}\mathrm{i}}\simeq M$. (c) This case also has ${\Gamma }_{\inf }\ll m_{\inf }$, as in (b), but $m_{\inf }={10}^3\omega$ lies in the stability region of the Mathieu cosine. Note the logarithmic scale on the time axis in this case.