CLUSTER–VOID DEGENERACY BREAKING: DARK ENERGY, PLANCK, AND THE LARGEST CLUSTER AND VOID

We assume a flat ΛCDM model with a power-law power spectrum of primordial density perturbation, and massive neutrinos. The model is specified by today's values of mean matter density ${{\rm{\Omega }}}_{{\rm{m}}}$, mean baryonic matter density ${{\rm{\Omega }}}_{{\rm{b}}}$, sum of neutrino masses ${\rm{\Sigma }}{m}_{\nu }$, statistical spread of the matter field at quasi-linear scales ${\sigma }_{8}$, and scalar spectral index ${n}_{{\rm{s}}}$. We assume ${\rm{\Sigma }}{m}_{\nu }=0.06\;{\rm{eV}}$, one neutrino mass eigenstate and three neutrino species so that the effective relativistic degrees of freedom ${N}_{{\rm{eff}}}=3.046$.

The model for number counts is

$$\begin{eqnarray}&&{N}_{{\rm{obs}}}=\int \int \int p(O| {O}_{{\rm{t}}})n[M({O}_{{\rm{t}}}),z]\displaystyle \frac{{dM}}{{{dO}}_{{\rm{t}}}}\displaystyle \frac{{dV}}{{dz}}{{dzdO}}_{{\rm{t}}}{dO},\end{eqnarray} \tag{ 1 }$$

where O is the size observable (mass, radius) for a type of object (here cluster or void), ${O}_{{\rm{t}}}$ is the true physical value of the observable O, and $M({O}_{{\rm{t}}})$ is the mass of the object. The differential number density is given by $n(M,z)$, $p(O| {O}_{{\rm{t}}})$ is the measurement pdf for the observable O, and dV/dz is the cosmic volume element. For integrating Equation (1), we use ${M}_{{\rm{void}}}=\frac{4}{3}\pi {R}^{3}{\rho }_{{\rm{m}}}(1+{\delta }_{{\rm{v}}}^{{\rm{m}}})$. We highlight the fact that while we write the expression for voids in terms of mass, they are observationally defined by radius and density contrast. Redshift integration is performed according to survey specifications (Table 1), or "observable universe" specification.

The differential number density of objects in a mass interval ${dM}$ about M at redshift z is

$$\begin{eqnarray}&&n(M,z){dM}=-F(\sigma ,z)\displaystyle \frac{{\rho }_{{\rm{m}}}(z)}{M\sigma (M,z)}\;\displaystyle \frac{d\sigma (M,z)}{{dM}}\;{dM},\end{eqnarray} \tag{ 2 }$$

where $\sigma (M,z)$ is the dispersion of the density field at some comoving scale ${R}_{{\rm{L}}}={(3M/4\pi {\rho }_{{\rm{m}}})}^{1/3}$, and ${\rho }_{{\rm{m}}}(z)={{\rho }_{{\rm{m}}}(z=0)(1+z)}^{3}$ is the matter density. The expression can be written in terms of linear-theory radius R_L for voids. The multiplicity function (MF) denoted as $F(\sigma ,z)$ is described in the following for clusters and voids.

The cluster (halo) MF ${F}_{h}(\sigma )$ encodes the halo collapse statistics. We use the MF of Watson et al. (2013), their Equations (12)–(15), accurate to within 10% for the regime we consider. Since our measured mass is defined at overdensity 200 (rather than 178), we convert to a ${\rm{\Delta }}=200$ overdensity using a scaling relation, their Equations (17)–(19) (Watson et al. 2013).

Neutrinos are treated according to Brandbyge et al. (2010). Neutrinos free-stream on cluster scales, and do not participate in gravitational collapse, so cluster masses should be rescaled: $M=4\pi {R}_{{\rm{L}}}^{3}[(1-{f}_{\nu }){\rho }_{{\rm{m}}}+{f}_{\nu }{\rho }_{{\rm{b}}}]/3$, where ${f}_{\nu }=[{\rm{\Sigma }}{m}_{\nu }/93\;{\rm{eV}}]/{{\rm{\Omega }}}_{{\rm{dm}}}{h}^{2}$ is the fraction of dark-matter abundance ${{\rm{\Omega }}}_{{\rm{dm}}}$ in neutrinos, and ${\rho }_{{\rm{b}}}$ is the baryon density. This gives a good first-order approximation, used with massless-neutrino MFs. Massive neutrinos also affect cluster and void distributions by shifting the turn-over scale in the matter power spectrum, and suppressing the power on the neutrino free-streaming scale.

The theoretical description of the void MF is not robustly known, mainly due to ambiguities in void definition, dynamics, and selection (e.g., Chongchitnan 2015). We use the Sheth–van de Weygaert (SvdW) MF (Sheth & van de Weygaert 2004) based on excursion set theory, but adapt it to supervoids, to describe a first-crossing distribution for voids with a particular density contrast ${\delta }_{{\rm{v}}}$. Void-in-cloud statistics are unimportant for our size of void. We pick the SvdW MF as representative also of alternative void MFs for our range of parameter values.

Achitouv et al. (2015) show that several spherical-expansion + excursion set theory predictions of the void MF consistently describe the dark-matter void MF from N-body simulations within simulation uncertainties across at least $R=1\mbox{--}10\;{h}^{-1}$ Mpc. The SvdW MF is defined for a top-hat filter in momentum space. The observation is defined in real space, which introduces non-Markovian corrections. Achitouv et al. (2015) derive a "DDB" void MF including such corrections, and stochasticity in the void formation barrier. We have computed that these corrections give an ∼10% increase in the void MF compared to SvdW for $R\gt 100\;{h}^{-1}$ Mpc and ${\delta }_{{\rm{v}}}\gt -0.3$. The SvdW MF predicts ${ \mathcal O }(1)$ voids with $R\sim 200\;{h}^{-1}$ Mpc in the WISE–2MASS survey patch, consistent with the peaks-theory prediction by Nadathur et al. (2014) rescaled to the survey patch, and extreme-void statistics in the largest N-body simulations (e.g., Hotchkiss et al. 2015). Large, small-underdensity voids should be close to linear and trace initial underdensity peaks, and hence are not so sensitive to expansion dynamics and interactions. Indeed, the excursion-set-peaks prediction (Paranjape & Sheth 2012) falls in between the SvdW and DDB MFs. Thus, SvdW should be a good approximation to the dark-matter void MF to within $\lesssim 50\%$. This uncertainty is significantly smaller than the Poisson uncertainty and propagated uncertainty in radius. These considerations motivate interest in a supervoid, rather than a fully nonlinear void.

While there is no consensus on the normalization of the dark-matter void MF, its shape is relatively well-understood for large supervoids. The same applies to voids defined by biased tracers, such as galaxies, provided that ${\delta }_{{\rm{v}}}$ is calibrated to survey specifications (Pisani et al. 2015). The radius and density contrast defined from tracers are larger than corresponding dark-matter quantities. Since we use a dark-matter density contrast, but a galaxy-density radius, we choose a converse approach. We set ${\delta }_{{\rm{v}}}$ to the measured dark-matter value (within uncertainties), and allow the radius to be self-calibrated (Hu 2003) to a dark-matter radius by the data. This procedure is successful in that only void radius, but not density contrast, is re-calibrated in parameter estimation. This is analogous to the scaling-relation strategy used for galaxy clusters (Allen et al. 2011).

The void MF for (nonlinear) radius R should be evaluated at corresponding linear radius ${R}_{{\rm{L}}}$, related as $R/{R}_{{\rm{L}}}={(1+{\delta }_{{\rm{v}}}^{{\rm{m}}})}^{-1/3}$, with ${\delta }_{{\rm{v}}}^{{\rm{m}}}$ nonlinear. The linear density contrast ${\delta }_{{\rm{L}}}$ defines the MF density threshold. We approximate the spherical-expansion relationship by ${\delta }_{{\rm{v,L}}}^{{\rm{m}}}=c[1-{(1+{\delta }_{{\rm{v}}}^{{\rm{m}}})}^{-1/c}]$, c = 1.594 (Jennings et al. 2013).

Neutrinos are treated according to Massara et al. (2015). Neutrino density contributes to the dynamical evolution of the void, but lacks significant density contrast itself. We use ${\delta }_{{\rm{v}}}^{{\rm{m}}}=[1-{f}_{\nu }(1+{{\rm{\Omega }}}_{{\rm{b}}}/{{\rm{\Omega }}}_{{\rm{m}}})]{\delta }_{{\rm{v}}}^{{\rm{cdm}}}$, where ${\delta }_{{\rm{v}}}^{{\rm{cdm}}}$ is the cold-dark-matter density contrast (which is our observational quantity).

We use a log-normal pdf $p(O| {O}_{{\rm{t}}})$ in Equation (1) for the observable O (i.e., M₂₀₀ or R) given its true value ${O}_{{\rm{t}}}$, with the mean and variance matched to the mean and variance of the observational cluster mass or void radius determinations, ${\mu }_{\mathrm{ln}O}=\mathrm{ln}{O}_{{\rm{t}}}/\sqrt{1+{({\sigma }_{O}/{O}_{{\rm{t}}})}^{2}}$ and ${\sigma }_{\mathrm{ln}O}^{2}=\mathrm{ln}[1+{({\sigma }_{O}/{O}_{{\rm{t}}})}^{2}]$. This approximation does not bias predicted abundances. Including this probability distribution in predictions corrects for the otherwise-present Eddington bias. We also perform an Eddington-biased analysis for which $p(O| {O}_{{\rm{t}}})=\delta (O-{O}_{{\rm{t}}})$. Redshift uncertainties are neglected because they are contained in the survey patches.

Number counts follow Poisson statistics, corrected for object-to-object clustering. The fractional correction to Poisson counts due to mild $[{N}_{{\rm{c}}}(\gt M)\ll 1]$ clustering is $-{N}_{{\rm{c}}}(\gt M)/2$ (Colombi et al. 2011), where ${N}_{{\rm{c}}}(\gt M)\equiv \bar{\xi }(\gt M)N(\gt M)$ and $\bar{\xi }(\gt M)$ is the average auto- or cross-correlation for objects above the mass threshold(s) within the patch. For all of our cluster and void cases, we estimate ${N}_{{\rm{c}}}(\gt M)\lesssim 0.01$, based on Davis et al. (2011), and correlation function and bias results in Sheth & van de Weygaert (2004), Hamaus et al. (2014), and Clampitt et al. (2016). Hence, our number counts follow non-clustered Poisson statistics to sub-percent-level accuracy.

Extreme value (or Gumbel) statistics describe the pdf of extrema of samples drawn from random distributions (Gumbel 1958). Consider a large patch of the universe, populated with clusters and voids. Let ${M}_{{\rm{max}}}$ be the mass of the most extreme cluster (or void), and ${p}_{{\rm{G}}}({M}_{{\rm{max}}})$ be the pdf of ${M}_{{\rm{max}}}$. For non-clustered Poisson statistics, we can write (Colombi et al. 2011; Davis et al. 2011)

$$\begin{eqnarray}&&{p}_{{\rm{G}}}(M)=\displaystyle \frac{{dN}(\gt M)}{{dM}}{e}^{-N(\gt M)},\end{eqnarray} \tag{ 3 }$$

where $N(\gt M)$ is the mean number of clusters (or voids) above the threshold M within the patch, according to Equation (1). The total Gumbel likelihood (under non-clustering) is then

$$\begin{eqnarray}&&{p}_{{\rm{G}}}({M}_{{\rm{halo}}},{M}_{{\rm{void}}})={p}_{{\rm{G}}}^{{\rm{halo}}}({M}_{{\rm{halo}}}){p}_{{\rm{G}}}^{{\rm{void}}}({M}_{{\rm{void}}}).\end{eqnarray} \tag{ 4 }$$

This likelihood is multiplied with measurement pdfs and external priors, and evaluated as described in Section 3.3.

We include measurement uncertainties in cluster and void properties as normally distributed with means and variances as detailed in Section 2.

In line with recent Planck analyses (Planck collaboration 2015), we use the following external priors.

$$\begin{eqnarray}&&h=0.706\pm 0.033\;{\rm{(maser+Cepheids}},{\rm{Efstathiou}}\ 2014),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{b}}}{h}^{2}=0.023\pm 0.002\;({\mathrm{BBN}}^{3}),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{n}_{{\rm{s}}}=0.9677\pm 0.006\;({\mathrm{CMB}}^{4}),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\rm{\Sigma }}{m}_{\nu }=0.06\;\mathrm{eV}(1\;\mathrm{mass}\;\mathrm{eigenstate})({\mathrm{NO}}^{5}),\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{N}_{{\rm{eff}}}=3.046\;(\mathrm{Std}.\mathrm{Model},3\;\mathrm{neutrino}\;\mathrm{species}).\end{eqnarray} \tag{ 9 }$$

We specify ${T}_{{\rm{CMB}}}=2.7255$ K (Fixsen 2009) for the radiation energy density. With these priors, our analysis is directly comparable to the Planck results. For an equivalent analysis independent of CMB anisotropies, we can set ${n}_{{\rm{s}}}\approx 1$, assuming an inflationary origin of primordial perturbations, or based on large-scale-structure surveys.