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On the Abundance of Extreme Voids

Cosmic voids have been shown to be an effective probe of cosmology, complementary to galaxy clusters. In this work, we present a simple theoretical framework for predicting of the size of the largest voids expected within a given redshift and volume. Our model is based on the exact extreme-value statistics which has previously been successfully applied to massive galaxy clusters. We implement our formalism using the void-abundance models and compare the extreme-void predictions to simulations and observations. We find that the simplest void models can only explain the extreme-void abundance with ad hoc parameter adjustments. We argue that extreme-void distributions should be used as an additional test on theories of void abundance.

At the core of our work is the exact extreme-value statistics previously used for predicting the distribution of the most massive clusters [28,29]. We will use this framework, together with the SVdW theory, to produce a probability distribution of the largest voids expected to be observed at a given redshift. Our framework is then tested against simulation and observational data. We then discuss how this approach can be improved and extended to other void models.
Even though simulations give us access to the full distribution of void number counts, the results are highly dependent on the choice of void finder (e.g. ZOBOV [30], VIDE [31], amongst others, see [32] for a recent discussion). Our goal is in this work is to investigate whether the extreme-void predictions from current theory is consistent with the extreme voids seen in simulations and observations. And if not, how could we address the inconsistency?
Throughout this paper, we will use the Planck+WMAP cosmological parameters as given in [33].

Void abundance
We begin with an overview of the SVdW theory, which gives the differential abundance of voids in the radius interval [R, R + dR]. The model is based on the so-called 'excursion set' theory, in which the local linear overdensity, δ(x), performs a random walk in the (δ, σ 2 ) plane, where σ 2 is the variance of the overdensity, smoothed on comoving scale R: (2.1) D(z) is the amplitude of the growing mode (normalised to D(0) = 1), P (k) is the linear matter power spectrum and W (k, R) is the Fourier-space window function. A void of radius R is deemed to form when δ first makes an 'excursion' below the barrier δ v . In the original paper, the value δ v = −2.7 was recommended (this corresponds to a nonlinear overdensity ∆ v ≈ −0.8). This value of δ v is derived from the 'shell-crossing' condition, analogous to the Einstein-de Sitter critical overdensity, δ c = 1.686, commonly used to define an object formed by spherical collapse. The SVdW model gives the differential abundance of voids with radius R as [18] dn where V s (r) = 4πr 3 /3. The oscillatory behaviour of f at high R can be avoided with the approximation given by Jennings, Li and Hu [19]: The SVdW theory has been shown to be consistent with dark-matter simulations where only spherical voids are considered [19]. However, void finders based on the watershed algorithm typically pick out voids that are non-spherical and irregular. This led many authors to take δ v as a free parameter in the model, and simply adjust δ v to match simulation results. Whilst this is a rather ad hoc approach, such an adjustment appears to do the trick, for instance, in accurately matching the void abundances in the simulations of [34], with the parameter δ v ≈ −1 for z 1. We will return to these simulation results in §6.

Void number counts
The probability density function (pdf) for voids with radius in the interval [R, R + dR], in the redshift bin centred z, width ∆z, is given by where f sky is the fraction of the sky observed and dV /dz is the volume element given by In (3.1), the total number of voids, N tot , with radius above R min in the redshift bin [z − ∆z/2, z + ∆z/2] is where R max,V is determined by the volume of the redshift bin. Fig. 1 shows total number of voids N tot as a function of z (top panel), with ∆z = 0.05, R min = 1 and f sky = 1. Since we will be concerned with the largest voids, our results are insensitive to changes in R min . In the same plot, we consider what happens when δ v varies from −0.9 to −1.1, and the fractional difference compared to the δ v = −1 model is plotted in the lower panel.
We see that when the void barrier is decreased (voids become emptier), the number count increases (in this case by about 10 − 15%), and vice versa. This can be interpreted as the fragmentation within the void population -smaller voids are typically emptier, and they can combine to form shallower, larger voids, which are therefore less numerous. This simple picture is consistent with detailed studies of void profiles in previous simulations [7,13].
Having obtained the pdf (3.1), we can construct the cumulative probability distribution (cdf), F (R), which gives the probability that an observed void has radius ≤ R. It is given by: (3.5)

Distribution of extreme voids
Harrison and Coles [35] showed that the cdf naturally leads to the prediction of the most massive clusters expected at a given redshift. This section follows their treatment, but adapted to cosmic voids. Consider N observations of voids drawn from the cdf, F (R). We ask: what is the probability that the largest void observed is of radius R * ? The cdf required is where the last step assumed that void radii are independent, identically distributed variables (we will come back to this point). Differentiating Φ with respect to R * gives the exact extreme-value pdf Figure 2 shows an example of the exact extreme-value pdf, φ(R), at z = 0.225. The pdf is non-Gaussian but can be well-approximated as a sum of a few Gaussian terms. On the figure we also show the contour in dashed lines representing the 5th and 95th percentile of the distribution. Assume that all available voids are observed (N = N tot ), the peak of the pdf occurs where It follows that the peak of the extreme-value pdf occur at the zero of the function  These zeroes can be used to roughly locate the most likely extreme radius (although when comparing with data in the next section, we will again use the percentiles). Fig. 3 shows R max , the peak of the extreme-value distribution (i.e. the root of X(R) = 0), as a function of redshift. With δ v = −1, we see that the largest void predicted by the SVdW theory is most likely ∼ 50 h −1 Mpc in radius, at z ∼ 0.2 − 0.3, with R max very gradually decreasing at higher redshifts. This slow redshift evolution agrees with the simulation of [36], in which the maximum void radius appears to be almost independent of redshift.

Sensitivity to δ v
In Fig 3 , we also plotted the extreme-void predictions for varying δ v . We observe that decreasing |δ v | results in larger extreme-voids across all redshifts, in accordance with the previous section where we deduced that larger voids are shallower.
In addition, we also included the case δ v = −2.7, the shell-crossing value originally recommended by SVdW. However, such a large magnitude of |δ v | predicts the peak of R max at only ∼ 20h −1 Mpc, which is effectively ruled out by recent simulations in which much larger voids have been seen (we will elaborate on this in the next section). Our finding adds weight to the poor fit of δ v = −2.7 to the void abundance dn/d ln R as seen in the simulations of [34].

Sensitivity to dark energy
The radius of extreme voids is linked to the properties of dark energy. Here, we consider the effects of varying the dark-energy density parameter, Ω Λ , and equation of state, w, from the fiducial values (Ω Λ = 0.73, w = −1).
If Ω Λ were to increase whilst flatness is maintained, then the matter density parameter must decrease. This gives rise to a reduced σ 8 and, intuitively, we expect structures to be generally less pronounced; clusters are less massive, and voids are smaller.
If the dark-energy equation of state, w, is larger than −1, we expect that void radius will not grow as fast at late times, in accordance with Friedmann equation which predicts R ∼ t 2/(3(w+1)) . Hence, we expect smaller voids if w > −1. Figure 4 shows the peak of the extreme-void pdf when Ω Λ and w are changed from their fiducial values (Ω Λ is varied whilst flatness is maintained). These behaviours are in accord with our intuitive discussion above. Note that the effect of varying w seen here qualitatively agrees with the results in [37], in which the effects of w on voids are further investigated.
In both cases, the variations shown are degenerate with small changes in δ v in the fiducial model. However, current observational constraints on Ω Λ and w are so tight that we do not expect these degeneracies to be significant.

Comparison with simulations
We now compare our extreme-void model with simulation results. We study the void distributions in two simulations: 1) Chan, Hamaus and Desjacques [34], using a 1.5 h −1 Gpc box, containing 1024 3 particles, at redshift snapshots z = 0, 0.5 and 1, 2) Sutter et al., using a 1 h −1 Gpc box, containing 1024 3 particles, at snapshot z = 0. The maximum void were extracted from these distributions, giving 4 data point in the z − R max plane.
On top of these data points, we superimposed the theoretical extreme-value distribution, calculated as in §4, except that the pdf and the total number of voids are calculated without  Figure 5. The radius, R max , of the largest void computed using the SVdW theory without volume factors. The diagonal stripe comprises the peak of the extreme-value pdf (centre -solid line), and the 95% and 5% percentiles of the distribution (top and bottom dashed lines). Data points (x) are from the simulation of [34] whilst the point ( * ) is from Sutter et al. [39].
This discrepancy from their recommended value of δ v ∼ −1 can be explained by examining their Fig. 2 carefully. Even though the bulk of the void distribution can be described by the SVdW model with δ v ∼ −1, the mismatch at high values of R appears to be significant.
In conclusion, our results favour a redshift-dependent barrier δ v (z) -similar to the modifications of the SVdW model advocated by [21,38]. Although SVdW theory with δ v ≈ −1 appears to fit dn/d ln R well across the redshifts, we have shown that the data on extreme voids are not consistent (i.e. outside the central 90%) with theoretical expectations at any of the redshift snapshots. This illustrates the use of our extreme-value formalism as an additional consistency check for void theories.
The reader may be aware of much larger void radii that have been reported in other simulations and observations. For instance, [26,36] reported voids as large as 350 h −1 Mpc, which, according to the SVdW, should be unobservably rare. However, these voids are not defined using the same criterion as the SVdW definition (in which voids are region with mean density ρ v ≤ 0.2ρ). It is beyond the scope of this work to try to reconcile these two definitions. We simply note here that such enormous voids cannot yet be theoretically described using the SVdW framework without further ad hoc adjustments.

Comparison with observation
We perform a similar comparison with voids identified from the SDSS10 catalogue as presented in [40]. Voids in the catalogue were binned with ∆z = 0.05, and the maximum in each bin is picked out. Changing the bin size has no significant effect on neither the theory nor the observed extreme-void distributions. The result is shown in Fig 6. The theory curve with δ v = −1 (solid line) is calculated using the formalism in §4. The 5th and 95th percentiles are shown in dashed lines. As in the previous section, it appears that no single value of δ v is a suitable fit to the observed extreme voids. In contrast with the simulated voids, real voids appear to favour δ(z) which increases with redshift.
The discrepancy between theory and observation is not surprising: it is unclear how the SVdW predictions, which are relevant only for dark matter voids, can be extrapolated to galaxy voids. Furlanetto and Piran [20] previously tackled this question by modifying the SVdW theory to fit voids in the 2dF survey. Perhaps a formalism such as theirs is could be used our extreme-value formalism to ameliorate the discrepancy seen in Fig. 6.

Conclusion and discussion
In summary, we have given a simple theoretical framework for the prediction of the size of the largest voids expected within a given redshift bin. Our model is based on the exact extreme-value calculations previously applied to galaxy clusters. We implement our formalism using the void-abundance model of Sheth and Van de Weygaert, with δ v taken as a free parameter. Our results show that the simplest SVdW with δ v = a constant is inconsistent with the extreme sizes of voids seen in the simulations and observations. For instance, δ v which decreases with redshift would produce a better fit to simulations.  Figure 7. Comparison between the SVdW model and the volume-conserving 'V dn' model of [19].
The central message in this work is that any theory which predicts void abundance must also give a sensible prediction for the distribution of extreme voids. Our work provides a framework to perform this check.
There are several scopes for improvement of our extreme-value model.
• Void correlation. Of course, voids are not completely uncorrelated, as previously shown in the theoretical work of [21,22,34,41]. Preliminary works suggest that introducing a void correlation will reduce the SVdW abundance for spherical voids, although the effects on the largest voids are likely to be small and predictable by linear theory [42,43]. Our work on including void correlation into the extreme-void statistics is in progress.
• Alternative void abundances. There is increasing evidence that the SVdW model cannot accurately explain simulation data without ad hoc adjustments, simply because real voids are unlikely to be spherical, nor are they all likely to form out of the shellcrossing condition. Apart from simple modification of the SVdW model, alternative void abundances include those directly derived from the Press-Schechter cluster abundance [16,44], or one derived using de-Sitter geometry [45]. The extreme-value formalism presented here remains unchanged and can be readily applied using these alternative models.
A crucial shortfall of the SVdW model is the fact that it assumes that voids are isolated spheres, and as voids evolve from the initial Lagrangian space to the late-time Eulerian space, their volume is not conserved (instead, their comoving number density is conserved -and only their sizes change). Jennings, Li and Hu [19] showed that this assumption can yield volume fraction in voids exceeding unity. They suggested a modified volume-conserving SVdW model -so-called 'V dn' model, and showed that this formalism produces a good fit to their simulation, but only for spherical voids with R ≤ 20h −1 Mpc (using δ v =-2.7).
It is curious to see if the V dn model performs any better than the SVdW model when incorporated into our extreme-value formalism. Figure 7 shows the comparison (with δ v =-1 in both cases). Unfortunately the V dn suppresses the radius of extreme-voids, and exascerbates the underprediction of extreme void radii.
• Alternative void definitions. Voids are defined in the SVdW theory as spherical regions in which the density satisfies ρ v ≤ 0.2ρ (where ρ is the mean matter density). We have found that decreasing the ratio ρ v /ρ to below 0.2 increases the extreme sizes of voids. However, such an adjustment is unsatisfactory as there are consistency relations between this ratio and δ v (see the Appendix of e.g. [19]). Even with the consistency relation in place, we were unable to reproduce a significant number of extreme voids with radius beyond ∼ 100 Mpc unless model parameters describe extremely shallow voids. We anticipate that this problem requires a fresh approach to the modelling of void abundance, which must include the shape and profile of voids.
Perhaps the greatest challenge for void theory is the prediction for the abundances of real galaxy voids seen, for instance, in the SDSS data [46,47]. A solution to this problem will have a significant implication, in particular, on explaining large-scale observables associated with supervoids. Given the highly nonlinear and complex nature of real voids, perhaps it is more productive to strive towards a semi-analytic modelling of voids led by accurate simulations, similar to what has been achieved with the universal-mass-function approach for galaxy clusters.