Measuring the speed of dark: Detecting dark energy perturbations

The nature of dark energy can be probed not only through its equation of state but also through its microphysics, characterized by the sound speed of perturbations to the dark energy density and pressure. As the sound speed drops below the speed of light, dark energy inhomogeneities increase, affecting both cosmic microwave background and matter power spectra. We show that current data can put no significant constraints on the value of the sound speed when dark energy is purely a recent phenomenon, but can begin to show more interesting results for early dark energy models. For example, the best fit model for current data has a slight preference for dynamics [$w(a)\ne -1$], degrees of freedom distinct from quintessence ($c_s\ne 1$), and early presence of dark energy [${\Omega }_{\mathrm{de}}(a\ll 1)\ne 0$]. Future data may open a new window on dark energy by measuring its spatial as well as time variation.

Figure 1

The deviation of the power spectrum of the matter density perturbations (Newtonian gauge) from the $c_s=1$ case is plotted vs wave number $k$. Three regions—above the Hubble scale (small $k$), below the sound horizon (large $k$), and the transition in between—can clearly be seen. The models have $w=-0.8$ (deviations will be smaller for $w$ closer to $-1$) and constant sound speed as labeled. For the $c_s=0.1$ case, we also show the result (dashed curve) in terms of the gauge invariant variable $D_g$ as defined in 49 (in that work $\Phi$ is equal to minus our $\varphi$). This illustrates that the low $k$ behavior is strongly gauge dependent.

Figure 2

The equation of state (lower three curves) and sound speed (upper three curves) as a function of scale factor are illustrated for two models. The aether model takes $s=3$ (solid curves) or $s=1$ (dashed curves) and $w_0=-0.99$; the early dark energy density ${\Omega }_e$ is determined from these parameters. Note that the cEDE model (dotted curves, also taking $w_0=-0.99$, and here setting $c_s=0$) is a close match to the aether model.

Figure 3

CMB temperature power spectrum for $w=-0.8$ and $c_s=1$, explicitly showing the contribution of the late-time ($z<10$) ISW effect.

Figure 4

Left panel: CMB temperature power spectrum for $c_s=0$, and its difference from the $c_s=1$ case, are plotted for $w=-0.8$, along with the cosmic variance. Right panel: The signal relative to the noise (here just cosmic variance) is low, with the total summed over all multipoles $S/N\simeq 1.0$. Compensating the difference between the models by varying the other cosmological parameters would make the $S/N$ even smaller.

Figure 5

The ratio of the dark energy to dark matter density power spectra (Newtonian gauge) is plotted for various values of constant $w$ and $c_s$. Although $c_s=0$ gives dramatically more power on subhorizon scales than $c_s=1$, the direct ratio of the dark energy power to the matter power is negligible.

Figure 6

Left panel: dark energy (lower four, thin curves) and dark matter (upper, thick curves) density power spectra for different choices of the dark energy equation of state and sound speed. Right panel: relative differences in the potential ($\varphi$) and matter density (${\delta }_m$) power spectra between $c_s=0$ and $c_s=1$ (matter and dark energy perturbations in Newtonian gauge).

Figure 7

Left panel: CMB temperature spectra for the early dark energy cEDE model with ${\Omega }_e=0.03$, $w_0=-0.8$ are plotted for $c_s=0$ and 1. The effect of changing the sound speed on the late ISW effect is a little stronger than in the case of ordinary $w=-0.8$ dark energy (also shown), but the major difference comes from higher $\ell$, where the early dark energy exhibits significant differences between $c_s=0$ and $c_s=1$, while ordinary dark energy does not. Right panel: Signal to noise squared per mode for distinguishing $c_s=1$ from the $c_s=0$ fiducial is plotted vs multipole. The late ISW (treated as $\ell <21$) contributes only $(S/N{)}^2=1.8$; including higher $\ell$, say all $\ell \le 2000$, gives $(S/N{)}^2=8.8\times {10}^3$. However, the differences at high $\ell$ can at least partly be compensated by varying other cosmological parameters.

Figure 8

Constant equation of state case, plotting the marginalized one-dimensional probability distributions using data from supernovae (Union2), CMB (WMAP5), galaxy autocorrelation (SDSS LRG), and the cross correlation between large scale structure tracers (see text) and CMB temperature anisotropies. Solid lines are for the model with $\log (c_s)$ a free parameter (with a flat prior), whereas the dotted lines correspond to fixed $c_s=1$.

Figure 9

Early dark energy case, plotting the marginalized one-dimensional probability distributions using data from supernovae (Union2), CMB (WMAP5), galaxy autocorrelation (SDSS LRG), and the cross correlation between large scale structure tracers (see text) and CMB temperature anisotropies. Solid lines are for the model with $\log (c_s)$ a free parameter (with a flat prior), whereas the dotted lines correspond to fixed $c_s=1$.

Figure 10

Confidence level contours of 68.3%, 95.4%, and 99.7% in the dark energy model with constant equation of state. The constraints are based on current data including CMB, supernovae, LRG power spectrum, and cross correlation of CMB with matter tracers. The small likelihood variations at $w=-1$ are not physical (the sound speed has no observable effect when $w=-1$), but are due to finite chain length.

Figure 11

Confidence level contours of 68.3%, 95.4%, and 99.7% in the cEDE early dark energy model in the $w_0\mathrm{\text{−}}\log c_s$ (left), ${\Omega }_e\mathrm{\text{−}}\log c_s$ (middle), and ${\Omega }_e\mathrm{\text{−}}{w}_0$ (right) planes. The constraints are based on current data including CMB, supernovae, LRG power spectrum, and cross correlation of CMB with matter tracers.