Fingerprinting dark energy. II. Weak lensing and galaxy clustering tests

The characterization of dark energy is a central task of cosmology. To go beyond a cosmological constant, we need to introduce at least an equation of state and a sound speed and consider observational tests that involve perturbations. If dark energy is not completely homogeneous on observable scales, then the Poisson equation is modified and dark matter clustering is directly affected. One can then search for observational effects of dark energy clustering using dark matter as a probe. In this paper we exploit an analytical approximate solution of the perturbation equations in a general dark energy cosmology to analyze the performance of next-decade large-scale surveys in constraining equation of state and sound speed. We find that tomographic weak lensing and galaxy redshift surveys can constrain the sound speed of the dark energy only if the latter is small, of the order of $c_s\lesssim 0.01$ (in units of $c$). For larger sound speeds the error grows to 100% and more. We conclude that large-scale structure observations contain very little information about the perturbations in canonical scalar field models with a sound speed of unity. Nevertheless, they are able to detect the presence of cold dark energy, i.e. a dark energy with nonrelativistic speed of sound.

Figure 1

The derivatives with respect to $\log c_s^2$ of the $Q$ parameter Eq. (18) (red solid line), the growth factor Eq. (30) (blue dashed line), the redshift distortion parameter Eq. (32) (green dotted line), and the matter power spectrum $P_0(k)$ (black dot-dashed line). The derivatives are evaluated at a $z=0.5$ except the one for $P_0(k)$ which is at $z=0$; for all the derivatives the sound speed is $c_s^2={10}^{-4}$. The vertical dotted line gives the scale of the sound horizon for $c_s^2={10}^{-4}$ at $z=0.5$.

Figure 2

The derivatives of the matter power spectrum with respect to $\log c_s^2$ for two different values of the sound speed: $c_s^2={10}^{-2}$ and $c_s^2={10}^{-4}$, red solid and blue dashed lines, respectively. The vertical dotted lines give the scale of the sound horizon for two different sound speeds, $c_s^2={10}^{-4}$ (left) and $c_s^2={10}^{-2}$ (right).

Figure 3

Confidence region at 68% for three different values of $z_{\max }=2,3,4$, blue dashed line, green long-dashed line, and solid contour, respectively. The upper panel shows the confidence region when the sound speed is $c_s^2=1$, and the bottom panel with the sound speed $c_s^2={10}^{-5}$. The parameter equation of state is for both cases $w_0=-0.8$.

Figure 4

The relative errors on the sound speed and their corresponding fits for our benchmark surveys, assuming only $Q_{\gamma }$ in the red (upper) line and assuming only $Q_{\Phi }$ in the blue (bottom) line.

Figure 5

Confidence region at 68% for three different values of $z_{\max }=2,3,4$, blue dashed line, green long-dashed line, and solid countour, respectively. The upper panel shows the confidence region when the sound speed is $c_s^2=1$, and the bottom panel with the sound speed $c_s^2={10}^{-5}$. The parameter equation of state is for both cases $w_0=-0.8$.

Figure 6

The Fisher matrix element $G-G$, $\beta -\beta$, $P_0-P_0$, solid blue (middle), red (bottom), and green (upper) lines, respectively. For $c_s^2=1$.

Figure 7

The upper panel shows $G-G$, $\beta -\beta$, $P_0-P_0$, solid blue (upper), red (bottom), and green (middle) lines, respectively. For $c_s^2={10}^{-5}$.

Figure 8

The relative errors on the sound speed for two different experiments and their fits: WL in red (upper) and $P(k)$ in black (bottom), both for our benchmark survey.