Massive Neutrinos Leave Fingerprints on Cosmic Voids

Massive neutrinos uniquely affect cosmic voids. We explore their impact on void clustering using both the DEMNUni and MassiveNuS simulations. For voids, neutrino effects depend on the observed void tracers. As the neutrino mass increases, the number of small voids traced by cold dark matter particles increases and the number of large voids decreases. Surprisingly, when massive, highly biased, halos are used as tracers, we find the opposite effect. How neutrinos impact the scale at which voids cluster and the void correlation is similarly sensitive to the tracers. This scale dependent trend is not due to simulation volume or halo density. The interplay of these signatures in the void abundance and clustering leaves a distinct fingerprint that could be detected with observations and potentially help break degeneracies between different cosmological parameters. This paper paves the way to exploit cosmic voids in future surveys to constrain the mass of neutrinos.


INTRODUCTION
The cosmic web (Bond et al. 1996) is a powerful tool to constrain neutrino properties. Cosmic voids are large (typically 10 − 100 h −1 Mpc) underdense regions of the cosmic web that have undergone minimal virialization and are dominated by inward or outward bulk flows (Shandarin 2011;Falck & Neyrinck 2015;Ramachandra & Shandarin 2017). In contrast to halos, which have undergone non-linear growth that can wash out primordial information, voids offer a pristine environment to study cosmology. As such, voids are a complementary probe to measurements of the cosmic microwave background and galaxy clustering and can help break existing degeneracies between cosmological parameters, thus becoming increasingly popular to study with both simulations and observations (see e.g. Ryden 1995;Goldberg & Vogeley 2004;Colberg et al. 2008;Viel et al. 2008;Van De Weygaert & Platen 2011;Paranjape et al. 2012;Chan et al. 2014;Hamaus et al. 2014b;Sutter et al. 2014b,c;Hamaus E-mail: ckreisch@astro.princeton.edu † E-mail: apisani@astro.princeton.edu Szapudi et al. 2015;Qin et al. 2017;Alonso et al. 2018;Pollina et al. 2018, and references therein).
The discovery of neutrino oscillations demonstrate that at least two neutrino families must have a nonzero mass (Becker-Szendy et al. 1992;Fukuda et al. 1998;Ahmed et al. 2004), evidence for beyond the standard model physics. Cosmological observables provide stringent upper bounds on the sum of neutrino masses, Σm ν (see e.g. Planck Collaboration et al. 2018), and may soon determine the last missing parameter in the standard model.
At linear order, neutrinos do not cluster on scales smaller than their free-streaming length, which is a function of the mass m ν of the single neutrino species (Lesgourgues & Pastor 2006). For example, neutrinos have free-streaming lengths of 130 h −1 Mpc and 39 h −1 Mpc for Σm ν = 0.06 eV and Σm ν = 0.6 eV (assuming 3 degenerate neutrino species), respectively. Neutrino free-streaming scales for Σm ν of interest thus fall within the range of typical void sizes, making voids an interesting tool for studying neutrinos.
Since voids are under-dense in matter, they are particularly sensitive to the effects of diffuse components in the universe like radiation and dark energy. For this reason, voids offer an appealing, new avenue to constrain neutrino properties. Villaescusa-Navarro et al. (2013) studied how massive neutrinos affect voids at high redshifts with Lyα forest analyses using hydrodynamical simulations (see also Krolewski et al. 2017). Massara et al. (2015) focused on how neutrinos affect void abundance, density profiles, ellipticities, the correlation function, and velocity profiles with N-body simulations that included massive neutrinos as an additional collisionless particle component. Banerjee & Dalal (2016) observed that neutrinos affect the scale-dependent void bias for voids traced by the CDM particle field. They use a spherical void finder and a small volume simulation (700 h −1 Mpc box length). In recent data analyses voids have been found using finders that do not assume spherical voids (e.g. Hamaus et al. 2017;Pollina et al. 2017). It is interesting to analyze the effects of neutrinos on voids with non-spherical shapes, such as in Massara et al. (2015), which have advantage of closely following the cosmic web pattern. Work such as Hamaus et al. (2014a) analyzed void power spectra without discussion of neutrinos. Thus far, the effect of neutrinos on voids has not been considered in depth without assuming spherical voids, and their effect on voids traced by halos is especially unexplored. Previous simulations with massive neutrinos did not have the volume and resolution to explore the effect of neutrinos on voids derived from the halo distribution and Halo Occupation Distribution (HOD) mocks (see e.g. Massara et al. 2015).
For the first time, we use N-body simulations with densities and volumes large enough to distinguish the effects neutrinos have on voids derived from the halo distribution and on voids derived from the particle distribution. The paper is organized as follows. In §2 we describe the two sets of massive neutrino simulations used in this work, the Dark Energy and Massive Neutrino Universe Project (DEMNUni) and the Cosmological Massive Neutrino Simulations (MassiveNuS), as well as the void finder used to build our void catalog. We show how neutrinos impact voids in §3 and discuss these results in §4. We conclude and discuss application to future surveys in §5.

SIMULATIONS AND VOID FINDER
In this work, we use two sets of massive neutrino simulations: the Dark Energy and Massive Neutrino Universe (DEMNUni, Carbone et al. 2016;Castorina et al. 2015), and the Cosmological Massive Neutrino Simulations (MassiveNuS 1 , Liu et al. 2018). We isolate the effects of Σm ν by comparing the large volume DEMNUni simulations (2 h −1 Gpc box length, 2048 3 CDM particles plus 2048 3 ν particles) with the smaller but more highly resolved MassiveNuS simulations (512 h −1 Mpc box length, 1024 3 CDM particles-i.e. eight times higher resolution than DEMNUni but 60 times smaller in volume). We focus our analysis on the simulation snapshots at z = 0.
Comparing how neutrinos affect voids for different tracers is imperative when looking towards constraining the sum of neutrino masses with upcoming surveys. Surveys observe galaxies, which are biased tracers of the CDM fluctuations Castorina et al. 2014), and void properties are sensitive to the tracer used to build the void catalog (Pollina et al. 2016(Pollina et al. , 2017. We rely on the optimal features of both simulations to be sensitive to neutrino effects at different scales, show consistency, check that our results are physical, and robustly test the sensitivity of our results to simulation design (see Appendix B for volume and resolution tests). The small volume and high resolution of MassiveNuS causes these simulations to be dominated by small voids, capturing the small scale impacts of Σm ν , whereas the large volume of the DEMNUni simulations captures large scale effects. MassiveNuS's high resolution enables the use of halos above a minimum mass M min = 3 × 10 11 h −1 M whereas DEMNUni's minimum halo mass is M min = 2.5 × 10 12 h −1 M , making MassiveNuS halos less biased than DEMNUni. The two simulations also use different methods to capture the effect of massive neutrinos-DEMNUni neutrinos are treated as particles and MassiveNuS neutrinos use a fast linear response algorithm (Ali-Haïmoud & Bird 2013).
The sum of neutrino masses Σm ν is varied in each simulation suite with other cosmological parameters kept fixed. The DEMNUni simulations assume a baseline cosmology according to the Planck results (Planck Collaboration et al. 2013), with h = 0.67, n s = 0.96, A s = 2.1265 × 10 −9 , Ω m = 0.32, and Ω b = 0.05. The relative energy densities of cold dark matter Ω c (and neutrinos, Ω ν ) vary for each model as Ω c = 0.27, 0.2659, 0.2628 and 0.2573, for Σm ν = 0, 0.17, 0.30 and 0.53 eV, respectively. In the considered cases, since A s is fixed while varying the neutrino mass, the simulations with massive neutrinos have a lower value of σ 8 with respect to the massless neutrino ΛCDM case. We use the three fiducial models of MassiveNuS in this work, where Σm ν = 0, 0.1, 0.6 eV and all other parameters are held constant at A s =2.1×10 −9 , Ω m =0.3, h=0.7, n s =0.97, w=−1, and Ω b =0.05.
We use the public void finder VIDE 2 to locate voids in the simulations . Because the void finder  . Void abundance in the sub-sampled cold dark matter field of the DEMNUni simulation. Colors denote the sum of neutrino masses used in each simulation. The bottom panel shows the ratio between void number densities (with uncertainties) for different Σm ν values and the number density in the massless neutrino case. Increasing Σm ν increases the number of small voids and decreases the number of large voids derived from the particle field. All abundance plots are cut at ∼ 2 times the mean particle separation in the simulation and where voids are so large that there are too few voids for informative uncertainties. All figures are for z = 0.
runs on a tracer distribution and uses the position of these objects, we can find voids from both the halo distribution (in this work we use the friends-of-friends (FoF) catalogs) and the CDM particle distribution. For the latter, running the void finding procedure on a large number of CDM particles (e.g. directly on the 2048 3 particles) is computationally expensive. We thus subsampled the CDM particle field to 1.5% of the original particle number for both DEMNUni and Mas-siveNuS. See Sutter et al. (2014a) for a discussion on how subsampling the tracer distribution affects voids. We note that for the DEMNUni subsampling this corresponds roughly to 505 3 particles, which is comparable to the CDM particle number density in the work done by Massara et al. (2015). Throughout the paper we refer to the subsampled CDM particle field simply as "CDM particles". We do not subsample the halo field unless specified. See Appendix A for more information on the simulations and void finder.

RESULTS
The sum of neutrino masses affects both the number of voids and the void bias. As the sum of neutrino masses increases, there are fewer large voids and more small voids seen in the CDM field. However, if we use halos as tracers there are more large voids and fewer small voids. The total number of voids changes, as well (see Section 3.1). Neutrinos affect how voids cluster and produce a strong scale dependent trendthis is a distinctive feature (see Section 3.2).
We note that we have also analyzed void catalogs  built from the mock HOD 3 galaxy catalog obtained from the DEMNUni simulations. The HOD's are built using the model described in Zheng et al. (2005), and the luminosity dependence is described in de la Torre et al. (2013). Results for the HOD catalogs are consistent with those obtained for the halo field. From now on we focus our analysis only on void catalogs extracted from the CDM and halo fields.

Void abundance
The impact of Σm ν on the void abundance, i.e. the void size function, depends upon the tracer. In Figure 1 and Figure 2 we show the void abundances derived from the subsampled CDM distribution and the halo distribution, respectively, for the DEMNUni simulation. All abundance plots have Poisson uncertainties.
The trend with Σm ν for the void abundance derived from the halo distribution is inverted relative to that derived from the CDM particle field. The void abundance derived from the CDM field shows that increasing Σm ν increases the number of small voids and decreases the number of large voids. Our findings are consistent with Massara et al. (2015)'s results based on a simulation with lower volume and mass resolution than DEMNUni. Conversely, for the void abundance derived from the halo distribution (see Figure 2) increasing Σm ν decreases the number of small voids and increases the number of large voids, although the magnitude of the effect is lower in absolute value than in the CDM case. As explained in Appendix B, although the number density of the tracers changes when changing Σm ν , the number density is not the origin of the opposite trends observed in the different void abundance plots.
Previous simulations lacked a sufficient combination of volume and mass resolution to investigate the void abundance derived from the halo field in detail and so were unable to discriminate between these two different trends in the void statistics (see e.g. Section 5 of Massara et al. 2015, whose simulations had 512 3 CDM particles, 512 3 neutrinos, and a 500 h −1 Mpc box length).
Varying Σm ν not only impacts the void abundance but also the total number of voids, as expected. In Figure 3 we show the total number of voids in the DEMNUni and Mas-siveNuS simulations derived from both the halo distribution and CDM particle distribution as a function of Σm ν . For voids derived from the CDM distribution, the total number of voids increases as Σm ν increases. There are more small voids and less large voids for the CDM case as Σm ν increases. The simulation volume is kept fixed, so overall there is a larger total number of voids that fill the volume.
For the halo case, the DEMNUni and MassiveNuS simulations show opposite behavior for the total number of voids as a function of Σm ν . Increasing Σm ν decreases the total number of DEMNUni voids derived from the halo field. This occurs because increasing Σm ν decreases the number of small voids and increases the number of large voids in the DEMNUni halo case, so there must be a lower (with respect to the massless neutrino case) total number of voids to fill the simulation volume. For the MassiveNuS simulations, the number of voids increases with Σm ν in both the halo and CDM cases. The MassiveNuS simulations have a smaller volume than the DEMNUni simulation, yielding a smaller total number of voids and, thus, larger uncertainties in the void abundance than the DEMNUni simulation. Nonetheless, the MassiveNuS void abundances for both the halo case and CDM case appear to be consistent with the trends seen in the DEMNUni CDM case. Thus, since the MassiveNuS simulation has more small voids for both the halo case and CDM case for nonzero Σm ν relative to the massless case, the total number of voids must also increase with Σm ν in both cases. We include the Mas-siveNuS void abundances in Appendix C. MassiveNuS has halos with smaller masses than DEMNUni, and thresholding the halo mass in MassiveNuS to match that of DEMNUni gives concordance between the two simulations for the total number of voids traced by halos. We also test different halo mass cuts in the halo catalogs and discuss the number of voids for the mass cut MassiveNuS simulation in Section 3.2 and comment further on these trends in Section 4.

Power Spectra & Correlation Functions
The void distribution is sensitive to Σm ν and how Σm ν impacts the underlying tracer distribution. Increasing the sum of neutrino masses damps the CDM power spectrum, P cc , on small scales in the DEMNUni simulation, as expected since neutrinos do not cluster on scales smaller than their freestreaming length (Lesgourgues & Pastor 2006). As Σm ν increases, the effect becomes more significant.
The halo-halo power spectrum, P hh , for DEMNUni shows an overall boost in power as Σm ν increases and biases the halo distribution (see Figure 4, all power spectra have a k bin size ∆k ≈ 0.008 hMpc −1 unless otherwise noted and have uncertainties computed by VIDE and estimated from scatter in the bin average). Neutrinos reduce the growth of CDM perturbations. Therefore, at a fixed redshift, virialized halos have a smaller mass than in the massless neutrino case at the same redshift. The densest initial fluctuations in the matter density field will still form halos large enough to be detected in our simulations, but, depending on the value of Σm ν , fluctuations with sufficiently low densities will no longer form halos with masses above the simulation mass threshold. Because only halos at the densest overdensities can be detected in simulations, halos at all scales are more highly correlated (with respect to the massless neutrino case), leading to a larger halo bias b h . The larger halo bias tends to compensate the suppression of the matter power spectrum due to free-streaming neutrinos, and the cumulative effect depends on Σm ν . The halo power spectrum is given by:  Raccanelli et al. 2017;Vagnozzi et al. 2018). We note that a similar inversion in the effect of the sum of neutrino masses on the matter power spectrum and the halo power spectrum has been seen We find that increasing Σm ν boosts the correlation between voids derived from the halo distribution while it damps the correlation between voids derived from the CDM particle field for the DEMNUni simulation (see Figure 5).
To understand the effects of halo mass on the power spectra in the presence of neutrinos, we analyze the void distribution in the MassiveNuS simulations, which have a lower halo mass threshold. We plot the halo-halo power spectra and the void-void power spectra, as a function of Σm ν in Figure 6 and Figure 7, respectively. The MassiveNuS simulations do not show the overall boost in the halo power for increasing Σm ν that we see in the DEMNUni halo distribution. The void-void power spectra show a similar trend: for the MassiveNuS simulations, the power spectra of voids found in the halo distribution behave as the power spectra of voids derived from the CDM particle field, even if the differences due to neutrinos effects are much less pronounced in the former than in the latter. In other words, the MassiveNuS void spectra do not show the same inversion between the halo and CDM cases as that observed in the DEMNUni simulations. We discuss the physical explanation behind this apparent contradiction in Section 4. Colors denote the sum of neutrino masses used in each simulation, dashed denotes voids traced by the CDM particle field, and solid denotes voids traced by the halo field. The bottom panel shows the ratio between the different Σm ν cases and the respective massless case. Increasing Σm ν boosts the power spectrum for voids derived from the halos but damps the power spectrum for voids derived from the particle distribution. The power spectrum spans the scales accessible to the DEMNUni simulation.

The Effects of Tracer Bias
While on the one hand neutrinos have a physical impact on the total number of voids (see Section 3.1), on the other hand the number of voids directly maps to the void shot noise, which can be approximated at small scales as 1/n v , where n v = (Number of Voids)/Volume is the void density.
To disentangle the impacts on the void power spectra of the void number and halo bias as they change with Σm ν , we remove shot noise and subsample the DEMNUni simulation in two different manners: (i) we bias the halo distribution by making two mass cuts such that each of them contains only halos with M ≥ 5 × 10 12 h −1 M or M ≥ 1 × 10 14 h −1 M ; (ii) we randomly subsample the halo distribution so that the number of halos matches that of the two subsamples defined in (i). In this way we produce sub-sets of halos with the same bias as the full halo distribution of the DEMNUni simulations, but with the same halo number density as the highly biased subsamples in (i) (see e.g. Figure B2 in Appendix B for a similar application to the MassiveNuS simulations).
To remove the effects of void number density we model the shot noise for the void-void power spectrum as scaledependent following the prescription by Hamaus et al. (2014a), which is well approximated by 1/n v for small scales: where P vv is the void-void power spectrum and P vc is the void-CDM cross-correlation power spectrum. Thus, we can write the void power spectrum with shot noise removed as The sum of neutrino masses affects the amplitude and phase of the void-void power spectrum. In Figure 8 we plot the void power spectra (with shot noise removed) for the two highly biased catalogs of (i), and compare them with the void spectra of the corresponding randomly subsampled catalogs of (ii) (see Figure B3 in Appendix B for analogous void power spectra including shot noise for multiple halo mass thresholds from MassiveNuS). At large scales, the void power spectrum tracks the tracer power spectrum: the power at large scales for the voids traced by halos with a higher mass threshold is larger, as expected for a more biased sample. Nonetheless, the large scale power is of the same order of magnitude for both the mass thresholds at large scales (compare top and bottom panels of Figure 8). For the highly biased tracers (M ≥ 1 × 10 14 h −1 M , bottom panel), the power at large scales is dominated by uncertainties because there are less small voids that correlate at large scales. For the less highly biased tracers (M ≥ 5 × 10 12 h −1 M , top panel) there is a discernible difference for the two neutrino masses at large scales because there is a large number of small voids traced by smaller halos, improving the uncertainties.
The power at small scales dramatically increases with Σm ν when increasing the halo bias (compare top and bottom panels of Figure 8). The small voids that remain when increasing Σm ν have highly biased halos forming their walls. These highly biased halos sit near overdensities, forming a concentrated cosmic web with voids that are, thus, tightly packed, boosting their correlation. The minimum at scales just larger than k = 10 −1 hMpc −1 corresponds to the scale at which voids are uncorrelated (see e.g. Hamaus et al. 2014a). The scale of the local maximum to the right of this minimum corresponds to the void exclusion scale, k exc ≈ π/R v , wherē R v is the average void radius. This is the smallest scale at Increasing Σm ν shifts the power from small scales to large scales for the DEMNUni voids found in the halo distribution. Σm ν may create a scale-dependent bias in voids, but this effect must be more thoroughly investigated to determine if the scale dependence is due to neutrino properties, non-linearities, or other effects. Increasing the halo bias increases the scale-dependent impact Σm ν has on the void power spectra. This is seen most clearly near the void exclusion scale. This shift in power from small voids to large voids is consistent with Σm ν decreasing the number of small voids and increasing the number of large voids for voids derived from the halo distribution in the DEMNUni simulations, thus causing the average void radius to increase and k exc to decrease (see Section 3.1).
On the other hand we find that, for the MassiveNuS simulations, increasing Σm ν shifts the power in the void power spectra (with shot noise removed) from large to small scales for both the CDM voids and halo voids (see near the exclusion scale k ≈ 0.5 hMpc −1 in Figure 9, which has bin size ∆ log k ≈ 0.08 hMpc −1 ). This is in contrast to the shift in power from small to large scales seen for the DEMNUni simulation in Figure 8. We note that the DEMNUni void power spectra (with shot noise removed) for CDM voids is consistent with that of MassiveNuS. Tracer bias influences how different kinds of voids respond to Σm ν : a low mass threshold, and so a low tracer bias, does not produce an inversion between the CDM case and halo case for the void abundance and power spectra. We have verified that sampling the MassiveNuS halo distribution so it has the same minimum halo mass as the DEMNUni simulation, M ≥ 2.5 × 10 12 h −1 M and thus increasing the tracer bias, leads to the inverted behavior between the biased halo case and the CDM case for the abundances, total number of voids (see Figure 3), and the power spectra, like seen for the DEMNUni simulation. The exclusion scale in the biased Mas-siveNuS distribution also shifts from small scales to match the DEMNUni exclusion scale.
The correlation functions are a useful tool to view the Σm ν inversion effects in real space. In Figure 10 we plot the void auto-correlation function for voids derived from the CDM particle field and the halo field. All correlation functions are computed by VIDE via an inverse Fourier transform  Figure 10. The void auto-correlation function for DEMNUni voids, including uncertainties. We scale the correlation functions by r 2 to emphasize the effects at large r. Colors denote the sum of neutrino masses used in each simulation. The top panel corresponds to voids found in the CDM particle field, while the bottom panel corresponds to voids found in the halo field. Increasing Σm ν diminishes void clustering for voids traced by CDM particles while it enhances void clustering for voids traced by halos. All correlation functions are cut at 2 times the mean particle separation in the simulation and where scales are so large that noise dominates. Voids traced by the CDM particles are so small that the correlation function does not become negative for scales larger than 2 times the particle separation due to the simulation resolution.
of the power spectra, have an r bin size ∆ log r ≈ 0.04 h −1 Mpc, and have uncertainties computed by VIDE and estimated from scatter in the bin average. ξ vv peaks at the void exclusion scale 2R v because this is the average distance at which voids are most tightly packed, i.e. the walls of neighboring spherical voids with a radius equal to the average void radius meet. ξ vv decreases for smaller scales, i.e. scales smaller than 2R v , since voids do not overlap. As explained in Massara et al. (2015), this decline is gradual because voids are not perfect spheres and they have different sizes. For scales larger than the exclusion scale, voids do not cluster as much and so ξ vv falls. We note that the void auto-correlation function becomes negative at scales larger than the Baryon Acoustic Oscillations (BAO) before approaching zero since voids trace the matter distribution at large scales. Voids are not likely to be separated by this distance. Increasing Σm ν suppresses void clustering for the CDM case at scales smaller than the BAO peak position, and reduces the anticorrelation at large scales since there are more voids spread throughout the field. Voids derived from the halos cluster more near the exclusion scale, showing opposite behavior to the CDM case just like the power spectra.
In the upper panel of Figure 11 we compare, for two different values of Σm ν , voids derived from the less highly biased halo catalog defined in (i) to the corresponding catalog, defined in (ii), derived from the original halo catalog with the same halo density for two different Σm ν . Increasing Σm ν boosts the correlation of voids derived from the biased halo sample, analogous to the effect on halos with large bias. Increasing the neutrino mass reduces the number of small voids traced by halos in the field, so the remaining voids are more highly correlated, resulting in a higher correlation peak. Since there are less small voids and more large voids, there is more void clustering for scales larger than the exclusion scale.
Σm ν 's impacts on the amplitude and scale are most prominent for voids traced by highly biased tracers. In the lower panel of Figure 11 we show the void auto-correlation function for voids derived from the highly biased halo sample and the original catalog with the same halo density. Decreasing the tracer density and increasing the halo bias both shift the average void radius to larger scales, causing the corre-lation function to peak at larger scales (compare upper and lower panels). For the voids traced by the less dense and highly biased halos, increasing Σm ν strongly shifts the entire correlation function to larger scales, similarly to the impact on the power spectra in Figure 8.
The impact of Σm ν on the correlation functions is not simply explained by the effects of void abundance. In the upper panel of Figure 11 we see that increasing Σm ν boosts the correlation for the voids traced by the less highly biased halos without significantly changing the peak location relative to the massless case. On the contrary, for the highly biased case in the lower panel of Figure 11, the amplitude at the correlation peak does not change between the massless and Σm ν = 0.53 eV cases. If void abundance solely drove Σm ν 's impacts on the correlation functions, the correlation peak's amplitude would decrease as the average void radius increases (see e.g. the void auto-correlation functions in Massara et al. 2015, for different void sizes). Neutrinos impact the clustering of voids -Σm ν influences void bias. We further explore the impact of Σm ν in our upcoming paper.

DISCUSSION
Our work indicates that voids respond to Σm ν in two distinct manners, determined by if they are derived from the halo distribution or the cold dark matter particle field. Both the halo and CDM distributions should be utilized to properly study voids and the impact neutrinos have on them. For forecasting constraints on Σm ν , the void catalog ideally should be built from the survey mock or HOD populated simulation rather than the CDM distribution.
Increasing Σm ν slows down the growth of the CDM perturbations, reducing the CDM overdensities present today. Since the evolution of the overdensities has slowed, fewer mergers of the small overdensities have occurred, resulting in a larger number of small CDM overdensities and fewer large CDM overdensities relative to the massless neutrino case. The numerous smaller CDM overdensities yield smaller voids since the small overdensities fragment what would be large voids. Hence, increasing Σm ν increases the number of small voids and decreases the number of large voids derived from the CDM particle field. Since there are more small overdensities in the field as Σm ν increases, voids become less biased near the correlation peak since they are not as localized and less antibiased for scales larger than the BAO peak position, as it is more likely to find voids separated by larger distances.
We note that our void finding procedure in the CDM case only uses CDM particles and does not include the neutrino particles. A different approach is to locate voids in the total matter field, such as in the work of Banerjee & Dalal (2016) that included neutrino particles and CDM particles. In our work, we have established that the inversion is unique to voids derived from halos because halo bias drives the inversion. Therefore, our results are particularly relevant to interpreting void observations.
For the halo case, increasing Σm ν makes halos less massive, leaving only the halos that sit at large density perturbations detectable in our simulations. Thus, these halos are more highly correlated and we see a bias effect in the halohalo power spectra. For the DEMNUni simulation, only mas-sive halos remain due to the limited mass resolution of the simulations, so there are no longer small halos that could segment a larger void into separate voids. For this reason and since larger voids are defined by larger overdensities, increasing Σm ν increases the number of large voids derived from the halo catalog and decreases the number of small voids.
The high resolution of the MassiveNuS simulation produces a lower minimum halo mass and, thus, halos that are less biased tracers of the CDM particle field than the DEM-NUni simulation. MassiveNuS can identify halos at smaller CDM overdensities than DEMNUni, and, consequently, these halos have masses and bias lower than the DEMNUni mass resolution. However, MassiveNuS has a finite resolution and cannot identify halos at the smallest CDM overdensities, so its halo catalog is still biased (even if its effective bias is smaller than the DEMNUni halo catalogs), and its halos have a higher correlation than the CDM overdensities.
Since increasing Σm ν leads to more small CDM overdensities and MassiveNuS has a low effective halo bias, Mas-siveNuS halos trace these small CDM overdensities more than the DEMNUni halos. Halos in MassiveNuS are less biased tracers of the matter density field; therefore, the increased correlation due to the halo's bias from the simulation resolution and Σm ν is not substantial enough to overpower the damping effects from the neutrino free-streaming. Thus, the MassiveNuS void power spectra for voids found in the CDM field and for voids found in the halo field damp as Σm ν increases.

CONCLUSIONS & FUTURE PROSPECTS
We have explored the impact of the sum of neutrino masses Σm ν on void properties with the N-body simulations DEMNUni and MassiveNuS. For the first time we have shown that: (i) the effect Σm ν has on void properties depends on the type of tracer the void catalog was built from, (ii) using voids only derived from the cold dark matter particle field to study neutrinos, as has been assumed in the literature, is not sufficient to capture the effects of neutrinos on voids. Voids are not always smaller and denser in the presence of neutrinos, and tracer properties can actually lead to larger voids, a smaller number of voids, and enhanced void clustering, (iii) the impact of Σm ν on the void abundance and voidvoid power spectrum for the DEMNUni void catalog derived from the halo distribution is opposite to that for the void catalog derived from the CDM particle field. For voids derived from the cold dark matter field, increasing Σm ν increases the number of small voids, decreases the number of large voids, and damps the void-void power spectrum. The opposite is true for voids derived from the biased halo distribution due to the effects of halo bias, (iv) halo bias influences how Σm ν affects voids -this will have interesting impacts on future surveys aiming to constrain the sum of neutrino masses, and (v) void power spectra and auto-correlation functions are powerful tools for distinguishing neutrino masses. Neutrinos leave a distinct fingerprint on voids, which can potentially help break the degeneracy between cosmological parameters in halo measurements. We plan to thoroughly explore breaking degeneracies, such as σ 8 , in upcoming work.
By comparing observations of the number of voids, void abundance, and void clustering to ΛCDM simulations with volume and resolution matching the survey volume and galaxy number density, surveys have a new avenue to place constraints on Σm ν . Upcoming surveys like PFS, DESI, Euclid, and WFIRST have halo densities near that of DEMNUni, and the densest may even exceed the density of DEMNUni. For these upcoming observations, simulations such as DEMNUni and MassiveNuS are the best tools for evaluating the impact of neutrinos on the observed voids. In the final stages reliable mocks will also be necessary to correctly evaluate the mask and survey boundary effects.
The opposite behavior of the DEMNUni and MassiveNuS simulation to Σm ν indicates there exists a threshold halo bias for which the void power spectra, correlation functions, and abundances for voids derived from the halo distribution will be less sensitive to Σm ν . It would be interesting to compare surveys with halo biases above and below the threshold at which Σm ν induces the inversion effect in the void abundances, number, power spectra, and correlation functions, since lower densities increase the minimum halo mass, and so halo bias, of the survey. In this sense one could imagine an extraordinarily dense low-z survey to be particularly interesting. Within the same survey, it will be interesting to compare void properties for tracers with different luminosity or mass thresholds, i.e. with different biases. The use of multi-tracer techniques is another promising tool for constraining Σm ν and its impact on voids. Utilizing the redshift dependence of these effects and redshift coverage of these surveys could further yield unique constraints on neutrino properties. We explore this interdependence in our upcoming paper. The DEMNUni simulations have been performed using the tree particle mesh-smoothed particle hydrodynamics (TreePM-SPH) code GADGET-3 Springel et al. (2001), specifically modified by Viel et al. (2010) to account for the presence of massive neutrinos. They are characterized by a softening length ε = 20kpc, start at z in = 99, and are performed in a cubic box of side L = 2000 h −1 Mpc, containing N p = 2048 3 CDM particles, and an equal number of neutrino particles when Σm ν 0 eV. These features make the DEMNUni set suitable for the analysis of different cosmological probes, from galaxy-clustering, to weak-lensing, to CMB secondary anisotropies.
Halos and sub-halo catalogs have been produced for each of the 62 simulation particle snapshots, via the friendsof-friends (FoF) and SUBFIND algorithms included in Gadget III Springel et al. (2001);Dolag et al. (2010). The linking length was set to be 1/5 of the mean inter-particle distance (Davis et al. 1985) and the minimum number of particles to identify a parent halo was set to 32, thus fixing the minimum halo mass to M FoF 2.5 × 10 12 h −1 M .

A2 The MassiveNuS simulation suite
The MassiveNuS simulations consists a large suite of 101 Nbody simulations, with three varying parameters Σm ν , A s , and Ω m . In order to avoid shot noise and high computational costs typically associated with particle neutrino simulations, MassiveNuS adopts a linear response algorithm (Ali-Haïmoud & Bird 2013), where neutrinos are described using linear perturbation theory and their clustering is sourced by the full non-linear matter density. This method has been tested robustly against CDM particle simulations and agreements are found to be within 0.2% for Σm ν ≤ 0.6 eV.
The simulations use the public code Gadget-2, patched with the public code kspace-neutrinos to include neutrinos 4 . The MassiveNuS halo catalogues are computed using the public halo finder code Rockstar 5 (Behroozi et al. 2013), also a friends-of-friends-based algorithm.  2017)) and embeds the ZOBOV code (Neyrinck 2008).
With VIDE we define the void radius as: where the volume V is the total volume of all the Voronoi cells composing the void (following VIDE's convention). It is important to notice that VIDE is able to find voids regardless of the shape, so it is particularly adapted to correctly capture the non-spherical feature of voids.

APPENDIX B: ROBUSTNESS TO VOLUME AND RESOLUTION EFFECTS
To further investigate the inversion described in the main text, we compare results we find with the DEMNUni simulations to the smaller but highly resolved MassiveNuS simulations described in §2.
The main differences between the two simulations are their volume and resolution. Thus, comparing the void behavior in these simulations allows us to check if the inversion in the void abundance and power spectra is a volume and/or resolution artifact or physical in nature.

B1 Testing the effect of volume
Simulation volume can affect the number and size of voids: a simulation with an insufficiently large volume could miss large voids, and if the tracer density is held constant, reducing the simulation volume will decrease the number of voids found, eventually increasing the uncertainties so much that trends become indiscernible. It is therefore important to probe if the volumes of the simulations we use have an effect on our results.
In Figure B1 we plot the DEMNUni void-void power spectra after cutting the volume of the simulation to match that of the MassiveNuS simulation. We included voids with x, y, and z positions 0 − 512 h −1 Mpc of the origin and removed all others to produce the volume cut catalog.
Cutting the simulation volume maintains the overall shape of the void auto-correlation power spectrum. The elbow near k ≈ 10 −1 hMpc −1 is still present, as is the rise to the left of the elbow. The scales probed by the volume cut simulation are smaller, so the power spectrum spans from only k = 10 −2 hMpc −1 to higher k for which the DEMNUni mass resolution becomes less reliable. For this reason, bins and uncertainties are larger for k 10 −1 hMpc −1 in the volume cut simulation than in the original version.
Since increasing Σm ν still boosts the overall power in the volume cut DEMNUni simulation, we conclude that the size of the DEMNUni and MassiveNuS simulations does not influence the inversion behavior we observe.

B2 Testing the effect of halo density
To probe how halo density affects the inversion, we randomly subsample the MassiveNuS simulation. We plot the voidvoid power spectra for different halo densities in Figure B2. Decreasing the halo density shifts the elbow towards large scales because the average void radius increases, and so the exclusion scale increases. Small scales increase in power due to the dependence of shot noise on tracer density (Hamaus et al. 2014a).
While decreasing the tracer density in the MassiveNuS simulation boosts the power, especially at small scales, it does not induce the Σm ν inversion effect. This is in stark contrast to changing the minimum halo mass (see Figure B3), which induces an inversion effect as the threshold halo mass increases, increasing the halo bias, decreasing the total number of voids, and increasing the average void radius. This suggests physical characteristics of halos induce the inversion effect, justifying the paper's focus on halo bias.

APPENDIX C: MASSIVENUS VOID ABUNDANCE
In Figure C1 and Figure C2 we show the MassiveNuS abundances for the voids seen in the CDM field and the voids seen in the halo distribution, respectively. Uncertainties are large in Figure C2 due to the number of voids, making it difficult to definitively see clear trends for the different Σm ν . However, for all Σm ν , there are more small voids and less large voids relative to the massless case for voids seen in the halo field. Thus, it appears that abundances for voids seen in both the CDM field and the halo field are consistent with an increased number of small voids and decreased number of large voids as Σm ν increases. This is in contrast to the DEMNUni abundance plots, which show clear opposite trends for the 2 tracer fields.  Figure B3. Void-void power spectra for the MassiveNuS simulations with different halo mass thresholds to illustrate the effects of halo bias. Colors denote the mass threshold of the simulation, where black is the original mass resolution. Dashed and solid lines denote the sum of neutrino masses used. The bottom panel shows the power spectra ratio between different Σm ν for each halo mass threshold. As the mass threshold increases there is an inversion effect due to a larger halo bias and a smaller total number of voids.  Figure C2. Void abundance in the halo field of the MassiveNuS simulation. Colors denote the sum of neutrino masses used in each simulation. The bottom panel shows the ratio between void number densities (with uncertainties) for different Σm ν values and the number density in the massless neutrino case. Nonzero Σm ν appears to increase the number of small voids and decrease the number of large voids relative to the massless case, in contrast to the DEMNUni abundance for voids traced by halos.