Counting voids to probe dark energy

We show that the number of observed voids in galaxy redshift surveys is a sensitive function of the equation of state of dark energy. Using the Fisher matrix formalism, we find the error ellipses in the $w_0\text{−}{w}_a$ plane when the equation of state of dark energy is assumed to be of the form $w_{\mathrm{CPL}}(z)=w_0+w_{a}z/(1+z)$. We forecast the number of voids to be observed with the ESA Euclid satellite and the NASA WFIRST mission, taking into account updated details of the surveys to reach accurate estimates of their power. The theoretical model for the forecast of the number of voids is based on matches between abundances in simulations and the analytical prediction. To take into account the uncertainties within the model, we marginalize over its free parameters when calculating the Fisher matrices. The addition of the void abundance constraints to the data from Planck, HST and supernova survey data noticeably tighten the $w_0\text{−}{w}_a$ parameter space. We, thus, quantify the improvement in the constraints due to the use of voids and demonstrate that the void abundance is a sensitive new probe for the dark energy equation of state.

Figure 1

We represent the prediction from the Sheth and Van de Weygaert model matched to the simulation. The shaded areas show the effects of the variation of the value of ${\delta }_v$ (varied by 0.05 and 0.1 in the figure) for the $\mathrm{\Lambda }\mathrm{CDM}$ case. The marginalization over ${\delta }_v$ allows us to take into account the fact that the Sheth and Van de Weygaert fit is qualitative, and only roughly matches simulations due to the discreteness of the galaxy distribution as opposed to a smooth density field.

Figure 2

Matching of the abundance of voids in simulations for models with different $w_0$. The lines show the abundances from simulations for models with different $w_0$ (and fixed $w_a$). The prediction from the Sheth and Van de Weygaert model are represented with dashed lines.

Figure 3

Matching of the abundance in simulations for models with different $w_a$. The thick lines show the abundances from simulations for models with different $w_a$ (and fixed $w_0$). The prediction from the Sheth and Van de Weygaert model are represented with dashed lines.

Figure 4

$1\text{−}\sigma$ error ellipses for several combinations of data sets. From outer ellipse to inner: $\text{Planck}+\mathrm{HST}$, $\text{Planck}+\mathrm{HST}+\mathrm{SN}$, $\text{Planck}+\mathrm{HST}+\text{Euclid}$ Voids. The light-blue filled contour shows the result of the marginalization over the parameter ${\delta }_v$. For comparison, we also show in the striped-filled contour the BAO constraints from Euclid (also added to $\text{Planck}+\mathrm{HST}$), from [18].

Figure 5

$1\text{−}\sigma$ error ellipses for several combination of data sets. From outer ellipse to inner: $\text{Planck}+\mathrm{HST}$, $\text{Planck}+\mathrm{HST}+\mathrm{SN}$, $\text{Planck}+\mathrm{HST}+\mathrm{WFIRST}$ Voids. The light-blue filled contour shows the result of the marginalization over the parameter ${\delta }_v$. For comparison, we also show in the striped-filled contour the BAO constraints from Euclid (also added to $\text{Planck}+\mathrm{HST}$), from [18].

Figure 6

Comparison of $1\text{−}\sigma$ error ellipses for WFIRST and Euclid voids when the supernovae data are also considered (in each case the void abundance constraints are added to $\text{Planck}+\mathrm{HST}+\mathrm{SN}$ data). We also show the constraints obtained by considering an estimate of the combination of voids from both surveys (see Sec. 4). (It should be noted that the axis range is smaller in this figure.)