Can a galaxy redshift survey measure dark energy clustering?

A wide-field galaxy redshift survey allows one to probe galaxy clustering at largest spatial scales, which carries invaluable information on horizon-scale physics complementarily to the cosmic microwave background (CMB). Assuming the planned survey consisting of $z\sim 1$ and $z\sim 3$ surveys with areas of 2000 and $300{\mathrm{deg}}^2$, respectively, we study the prospects for probing dark energy clustering from the measured galaxy power spectrum, assuming the dynamical properties of dark energy are specified in terms of the equation of state and the effective sound speed $c_{\mathrm{e}}$ in the context of an adiabatic cold dark dominated matter model. The dark energy clustering adds a power to the galaxy power spectrum amplitude at spatial scales greater than the sound horizon, and the enhancement is sensitive to redshift evolution of the net dark energy density, i.e. the equation of state. We find that the galaxy survey, when combined with CMB expected from the Planck satellite mission, can distinguish dark energy clustering from a smooth dark energy model such as the quintessence model ($c_{\mathrm{e}}=1$), when $c_{\mathrm{e}}\lesssim 0.04$ (0.02) in the case of the constant equation of state $w_0=-0.9$ ($-0.95$). An ultimate full-sky survey of $z\sim 1$ galaxies allows the detection when $c_{\mathrm{e}}\lesssim 0.08$ (0.04) for $w_0=0.9$ ($-0.95$). These forecasts show a compatible power with an all-sky CMB and galaxy cross correlation that probes the integrated Sachs-Wolfe effect. We also investigate a degeneracy between the dark energy clustering and the nonrelativistic neutrinos implied from the neutrino oscillation experiments, because the two effects both induce a scale-dependent modification in the galaxy power spectrum shape at largest spatial scales accessible from the galaxy survey. It is shown that a wider redshift coverage can efficiently separate the two effects by utilizing the different redshift dependences, where dark energy clustering is apparent only at low redshifts $z\lesssim 1$.

Figure 1

Upper panel: Shown is how the dark energy clustering effect amplifies the power of linear power spectrum $P(k)$ at $z=0.6$ on spatial scales greater than the dark energy sound horizon, relative to the smooth dark energy model ($P_{\mathrm{sm}}$). Note that nonrelativistic neutrinos are ignored in this plot. The dark energy equation of state is taken to be $w_0=-0.8$, and the sound speed is varied to show how the scale-dependent modification in $P(k)$ changes with $c_{\mathrm{e}}$: with decreasing $c_{\mathrm{e}}$, the transition in $P(k)$ amplitude appears at smaller length scales, i.e. larger $k$. Shaded boxes around the curve of $c_{\mathrm{e}}=0.1$ show the expected $1\mathrm{\text{−}}\sigma$ measurement errors for spherically averaged $P(k)$ at each $k$ bin for a survey covering $2000{\mathrm{deg}}^2$ and $0.5<z<1.3$ (see text for the details), but note that we have multiplied the error bars by a factor $1/2$ for illustrative purpose. Comparing the solid curves with the error bars leads to a naive expectation that the dark energy clustering can be detected from this type of galaxy survey only when the sound speed is smaller than $c_{\mathrm{e}}\sim 0.1$. Middle panel: Redshift dependence of the power spectrum shape modification due to the dark energy clustering, where $c_{\mathrm{e}}=0.1$ and $w_0=-0.8$ are fixed. Lower panel: As the fiducial value of $w_0$ is away from $w_0=-1$, the amplification in $P(k)$ at small $k$ is more enhanced, for a fixed $c_{\mathrm{e}}=0.1$. In all of the panels, the bold solid curve shows the same result for $c_{\mathrm{e}}=0.1$, $w_0=-0.8$, and $z=0.6$.

Figure 2

A significance of discriminating the dark energy clustering model from a smooth dark energy model ($c_{\mathrm{e}}=1$) as a function of the fiducial value of $c_{\mathrm{e}}$, expected from the galaxy survey combined with Planck. The fiducial value of the dark energy equation of state is kept fixed to $w_0=-0.8$. The bold solid curve shows the forecast for the nominal survey design of WFMOS (see Table ), while the dot-dashed curve shows the result when the Gz1 survey alone is used. The dashed curve shows the result for an ultimate all-sky survey of $z\sim 1$ galaxies.

Figure 3

A maximum dark energy sound speed with $(S/N){|}_{c_{\mathrm{e}}}\ge 1$ against the fiducial value of $w_0$. The solid and dashed curves are the results for WFMOS (Gz1 plus Gz3) and the full-sky $z\sim 1$ survey, respectively. As the underlying true cosmology approaches to the cosmological constant model, one can detect dark energy clustering from a galaxy survey only when the sound speed $c_{\mathrm{e}}$ is sufficiently small.

Figure 4

Shown is how each redshift slice of the WFMOS survey contributes to the dark energy constraint shown in Fig. 2. The number in the parentheses for the label of each curve denotes the comoving survey volume for each redshift slice (see Table ). The lowest redshift slice ($0.5\le z\le 0.7$) yields the most contribution even though the slice has the smallest survey volume and the smallest $k_{\max }$, as dark energy is more prominent at lower redshifts. Also interesting is a jagged feature appearing around $c_{\mathrm{e}}\approx 0.05$ when one redshift slice alone is used, while the jagged feature is significantly weakened when all the slices are combined. The reasons for this to happen are explained in the text as well as the subsequent figures.

Figure 5

Effects of the dark energy clustering (solid curves) and the massive neutrinos (dashed) on the linear power spectrum shape, relative to the smooth dark energy model. The five curves from top to bottom are the results for $z=0.6$, 0.8, 1.0, 1.2, and 3, respectively. We assume $c_{\mathrm{e}}=0.05$, corresponding to the jagged feature in the curve of Fig. 4, and $w_0=-0.8$ for the dark energy fiducial model and assume $f_{\nu }=0.01$ and $N_{\nu }^{\mathrm{nr}}=2$ for the neutrino mass and the number of nonrelativistic neutrino species. Note that the dashed curves for the neutrinos are multiplied by an arbitrary factor so that $P_{\nu }(k)/P_{\mathrm{de},\mathrm{sm}}\to 1$ at large $k$, taking into account the fact that the galaxy power spectrum amplitude involves uncertainty related to galaxy bias. It is clear that the redshift dependences of the two effects are quite different. The two arrows in the horizontal axis denote $k_{\max }$ for the $z\sim 1$ and $z\sim 3$ surveys as given in Table .

Figure 6

Projected 68% ellipses in $f_{\nu }$ and $c_{\mathrm{e}}$ subspace. The error ellipses shrink more as more redshift slices are added. In particular, for the nominal WFMOS survey, a $z\sim 3$ survey helps break the degeneracy between the two parameters efficiently, even though the slice is insensitive to dark energy. Note that the prior $\sigma (w_0)=0.05$ is employed.

Figure 7

Difference in ${\sigma }_8$ when a generalized dark energy contribution is included and the power spectrum is normalized by the primordial curvature perturbation. When $w_0>-1$ and the sound speed is large enough, ${\sigma }_8$ is reduced compared to the $\Lambda \mathrm{CDM}$ prediction, because the structure formation slows down due to the more accelerated cosmic expansion at intermediate redshifts. On the other hand, when the sound speed is sufficiently small, the dark clustering enhances the power spectrum amplitude, yielding greater ${\sigma }_8$. The plot shows $c_{\mathrm{e}}\approx 0.005$ is the critical value of $c_{\mathrm{e}}$, where the two effects above cancel out so that the resulting ${\sigma }_8$ is independent of $c_{\mathrm{e}}$ and $w_0$, and similar to the $\Lambda \mathrm{CDM}$ prediction.