title: The impact of massive neutrinos on halo assembly bias
authors: Titouan Lazeyras Francisco Villaescusa-Navarro Matteo Viel
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      Ω_rρ_ cr,0 h  Mpc^-1Ø𝒪 h^-1 MpcPhys. Rev. DPhys. Rep.MNRASJCAPAstrophys. J.Astrophys. J. Lett.BAASρ_ cr,0JHEPa,b,c]Titouan Lazeyras,d]Francisco Villaescusa-Navarro,a,b,c,e]Matteo Viel[a]SISSA, Via Bonomea 265, 34136 Trieste, Italy[b]INFN, Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy[c]IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2, 34151, Trieste, Italy[d]Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ, 08544, USA[e]INAF-OATS, via Tiepolo 11, 34131 Trieste, Italytlazeyra@sissa.it, fvillaescusa@princeton.edu, viel@sissa.itUsing the publicly available Quijote simulations, we present the first measurements of the assembly bias of dark matter halos in N-body simulations which include massive neutrinos. We focus on the dependence of the linear bias b_1 on three halo properties: 1) concentration c, 2) spin λ, and 3) ellipticity s. Although these simulations cover a large volume, superior to any future surveys, we do not detect any effect of neutrinos on the relations b_1(c), b_1(λ) and b_1(s) at fixed halo mass. We further study the dependence of halo properties and environment on neutrinos, finding these quantities to be impacted by neutrino masses at the same level as assembly bias. We find that the effect of neutrinos on spin and shape can be largely attributed to the change in the cold dark matter σ_8 in neutrinos simulations, which is not the case for concentration.The impact of massive neutrinos on halo assembly bias
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      § INTRODUCTION
      
      
      It is now well established that most of the observed tracers of the large-scale structure (LSS) of the Universe, such as galaxies, reside in dark matter halos. Hence the statistics of halos determine those of galaxies on large scales, making their distribution one of the key ingredients of the theoretical description of LSS. In the context of perturbation theory, the statistics of halos are written in terms of bias parameters multiplying operators constructed out of the matter density field (see <cit.> for a recent review). On sufficiently large scales, a linear relation between the halo density field _̣h and the matter one _̣m is enough to describe the clustering pattern of halos
      _̣h(,τ) = b_1(τ) _̣m(,τ) + … ,
      
      where δ_a is the density contrast of the considered field, defined as δ_a=ρ_a/ρ̅_a-1, b_1 is the linear bias of dark matter halos and the dots indicate that we only wrote the first term of the expansion.
      
      This bias parameter was commonly thought to depend only on the redshift and mass of the considered halo population, implying that the clustering of dark matter halos is unaffected by the halo environment. In the last fifteen years however, several studies showed that such a model of halo biasing is too simplistic, and that halo bias depends on halo assembly history and other halo properties (see for instance <cit.> and references therein). This phenomenon is now known as assembly bias. Note that some papers refer to the dependence of halo bias on halo properties as secondary bias, reserving the term assembly bias for the dependence of halo bias on the assembly history (i.e. mass accretion rate or formation redshift). We do not make this difference here, and from now on, when we refer to assembly bias, we mean that the clustering of halos depends on properties other than halo mass and redshift.
      
      Ref. <cit.> were the first to show that halo formation depends on the halo environment, which provided the first indirect evidence of assembly bias. Shortly after, <cit.> presented the first direct measurements of this effect, showing that halos with a lower half-mass assembly redshift are less clustered than average, and vice versa. Since then, numerous works, using numerical simulations, have studied and found assembly bias as a function of various halo properties such as formation time, concentration, spin, shape, substructure content or mass accretion rate. One thing that quickly became clear is that possible correlations between two halo properties are not sufficient to explain the change in clustering observed with respect to these quantities <cit.>. Since then, many papers studied the dependence of halo properties and linear bias as a function of their local environment and formation time (see <cit.> for recent works). While it is now clear that the halo environment, characterized by its position in the cosmic web and surrounding tidal field, plays a crucial role in assembly bias, a fully consistent picture explaining the physical mechanisms behind assembly bias at all mass, and which halo properties dominate it is still lacking to this day.
      
      Assembly bias is not only studied in simulations. Indeed, several studies claimed to have observed galaxy assembly bias (see for instance <cit.>). However, some of these claims were re-investigated by further studies without clear evidence for assembly bias <cit.>. More recently, <cit.> presented the first evidence for assembly bias on galaxy cluster scales. This signal was however re-investigated by <cit.> who concluded that the signal found in <cit.> was due to projection effects. Their results for assembly bias on cluster scales are consistent with zero. Excitement was revived very recently with <cit.> who claimed to have detected anisotropic assembly bias <cit.> in the BOSS DR12 data.
      
      Since halo assembly bias has been shown to be a significant effect (up to 30%), it is crucial to incorporate it correctly in the modeling of LSS for the analysis of current and future survey data, since a given galaxy sample could preferentially reside in halos with particular properties. This will be even more relevant if galaxy assembly bias is confirmed to be detected convincingly in observations. Notice that in theory the complete perturbative bias expansion automatically takes into account assembly bias if all necessary terms are included (see section 9.2 of <cit.>). Nevertheless, several efforts have recently been made to incorporate it in semi-analytical models such as (decorated) HOD  and abundance matching techniques (see <cit.> and references therein). These methods have the advantage to allow to extract information from the non-perturbative regime, where perturbation theory fails.
      
      LSS observables are also affected by the presence of massive neutrinos <cit.>, 
      and future surveys such as Euclid or WFIRST are expected to put stringent constraint on their total mass. Present constraints set an upper limit at the 2σ C.L. of 0.12 eV using either Planck data (including lensing) in combination with BAO <cit.> or Planck, BAO and the Lyman-α forest <cit.>. However, we stress that these limits are model dependent and strictly valid in a one-parameter extension (the neutrino mass) of a vanilla ΛCDM model, obtained by using Bayesian statistics.
      Due to their large thermal velocities, neutrinos suppress the clustering of matter on small scales and leave a scale dependent imprint on the clustering on large-scales, as was recently measured in b_1 by <cit.>. Furthermore, the effect of neutrinos is degenerated with other ones (such as σ_8<cit.>) and hence it is important to model their impact correctly in order to achieve percent level precision cosmology in an unbiased way with future surveys. In this paper we investigate the impact of massive neutrinos on halo assembly bias in the Quijote simulations <cit.> which include massive neutrinos as light particles. These simulations are state-of-the-art ones and cover very large volumes which should allow us to detect any effect of neutrinos on assembly bias relevant for future surveys. 
      
      This paper is organized as follows. In sims we present the simulation suite used for our study and the halo finding procedure. assemblybias describes our procedure to measure assembly bias, while our results are presented in results. We first focus on the impact of neutrinos on assembly bias in assemblynu before investigating their impact on halo property and environment in haloprop. We conclude in concl. The appendices present the importance of including neutrinos in the halo definition (nuhalo), additional results for other neutrino masses than those presented in the main text (addnumass), and the effect of varying σ_8 on b_1 (sigma8), which is relevant for interpreting our results as we will discuss later. 
      
      
      
      
      
      § SIMULATIONS AND HALO FINDING
       
      
      In this section, we briefly describe the characteristics of our simulation suite and the procedure used to identify halos and determine their properties.
      
      Our results are based on the suite of Quijote simulations described in 
      <cit.>. These gravity-only simulations have a very large combined total volume, and cover a wide range of different cosmologies (see table 1 in <cit.>). We use a subset of five cosmologies dubbed “fiducial”, “M_ν^+”,“M_ν^++”,“M_ν^+++”, and “σ_8^+” in the nomenclature of <cit.>. The fiducial cosmology is a flat ΛCDM one with the following parameters: =0.3175, =0.049, h=0.6711, n_s=0.9624, and σ_8=0.834. The volume of each simulation is 1  Gpc^3/h^3 sampled with 512^3 particles, yielding a mass resolution m_p=6.56×10^11 M_⊙/h. We use 500 realizations, probing a combined volume of 500 ( Gpc/h)^3; simulations were initialized with second-order perturbation theory (2LPT) at z_i=127.  
      
      The neutrino simulations adopt the same parameters but further implement massive neutrinos as light particles, using 512^3 mass elements again. Crucially, the total matter density  and σ_8 are kept fixed between the fiducial and neutrinos simulations, while the more relevant quantity might have been the CDM+baryon fluid ones as we discuss later. The three neutrino simulations model a sum of the neutrino masses equal to 0.1, 0.2, and 0.4 eV for the M_ν^+,M_ν^++, and M_ν^+++ respectively. An additional difference between the neutrino simulations and the fiducial one is that neutrino ones are initialized with the Zel'dovich approximation. 
      
      Finally, in order to disentangle[It is well known that the effect of massive neutrinos can be mimicked by a change in σ_8. Besides, when considering halos or galaxies, the relevant quantity is not the r.m.s. of the matter field on a sphere of 8  Mpc/h, σ_8,m, but of the CDM+baryons field σ_8,c<cit.>.] the effect of neutrinos to that of varying σ_8 in the CDM+baryons field (σ_8,c), we make use of the set of massless neutrino simulations with enhanced σ_8 which has a value σ_8=0.849. This value matches pretty well the value of σ_8 for the baryon-CDM fluid in the M_ν^++ cosmology. These simulations are again initialized with 2LPT. We refer the reader to <cit.> for further detail on these simulations.
      
      The dark matter halos were identified using the Amiga Halo Finder (hereafter AHF) <cit.> at z=0, which identifies halos with a spherical overdensity (SO) algorithm. We set the value of the overdensity to 200 times the background density to define halos. By default, we take the cold dark matter-baryon fluid only to define halo (i.e. we do not include neutrinos in halos), but we do not expect the inclusion or not of neutrinos to affect our results in a significant way since they are much lighter than the rest. In addition, AHF removes unbound particles found within a halo, which we expect to be the case for a majority of neutrinos since, on these small scales, they are expected to have a large thermal velocity <cit.>. However, we investigate the impact of including neutrinos in halos in nuhalo. 
      
      The bound particles are then used to calculate canonical properties of halos like the density profile, rotation curve, mass, spin, and ellipticity. As this algorithm is based on particles, it natively supports multi-mass simulations. In this paper, we only use distinct halos and do not consider their sub-halos. Further, we will restrict our analysis to halos with at least 200 particles within r_200 (the radius corresponding to an overdensity of 200 with respect to the background density) to ensure convergence of the halo properties considered, such as concentration. With these parameters, the smallest halos have a mass of 10^14.12  h^-1 M_⊙, well above M_⋆ (the mass of halos with present day peak significance equal to 1). We have made use of 100 realizations of a simulation of the fiducial cosmology with a mass resolution eight times higher to conduct a convergence study of halo properties. We found the mean properties measured in the high resolution simulation to be within 3% of the ones measured in the low resolution one at all mass, and already within 2% for all but the lower mass bin we consider. Furthermore, in this paper we will study ratios between quantities measured in simulations including or not neutrinos, where convergence should be very robust. All of this justifies considering halos with down to 200 particles.
      
      
      
      
      § MEASURING ASSEMBLY BIAS
      
      
      We now describe the procedure we use to measure assembly bias. Even though neutrinos travel at high velocities, they are nonrelativistic at low redshift, and therefore they contribute to the total matter budget in the Universe, i.e.
      ρ_m = ρ_ cdm + ρ_b + ρ_ν,
      
      where the subscript b stands for baryons. Equivalently
      δ_m = f_ cdmδ_ cdm + f_b_̣b + f_v_̣ν = (1-f_ν)δ_c + f_νδ_ν,
      
      where f_i=Ω_i/Ω_m are the fractional background energy densities of each species, and in the last equality we assumed δ_b=δ_ cdm≡δ_c (which is correct for the simulations used in this paper) and used f_b+f_ cdm=1-f_ν. 
      
      The linear bias of dark matter halos can be obtained by taking the ratio of the halo-matter cross-power spectrum to the matter-matter auto-power spectrum, which tends to a constant, b_1, at low k. However, due to their large thermal velocities, that lead to large free streaming lengths λ_ fs≳ 100, neutrinos are expected to create a scale-dependence in galaxy clustering on linear scales, as was measured in b_1 by <cit.>. It was however shown in those works, that if halo bias is defined with respect to CDM+baryons only (subscript c in the following), the bias becomes almost scale-independent <cit.>. This restores universality of the mass function and limits the scale dependence of halo bias on large-scales to a small effect (of the order f_ν≲ 1%). While this effect could also be present in assembly bias, we do not attempt to measure it here by lack of signal-to-noise ratio[Notice that <cit.> used a very large set of N-body simulations, with a carefully set up for the neutrino component, to measure this effect. Given the realistic neutrino masses we are considering here, it is unlikely that we can detect this effect with our simulations.], and we assume that the ratio of the power spectra indeed tends to a constant on the largest scales.
      Hence, for a given halo sample of mass M, we define the large-scale linear bias as
      
      b_1(M)=lim_k→ 0P_ hc(k,M)/P_ cc(k),
      
      where the subscript c stands for CDM+baryons. The extension of b1 to other halo samples than halos selected by mass only is straightforward. In particular, if we further select halos in a given mass bin by a secondary property p, we have
      
      b_1(p | M)=lim_k→ 0P_ hc(k,p | M)/P_ cc(k).
      
      In practice, and in order to maximize the signal-to-noise ratio, we make use of modes up to k∼ 0.12 by fitting a second order polynomial in order to take care of nonlinear effects appearing on smaller scales. We have checked the stability and robustness of our results under a change in this value and found it to be the best compromise in term of signal-to-noise ratio while still ensuring that our linear treatment is valid. This means that we measure the linear bias of a given halo sample as
      P_ hc(k)/P_ cc(k)=b_1+A k^2,
      
      for k<0.12. The amplitude A includes higher-order terms such as b_2 or b_K^2 in the bias expansion, as well as higher-derivative terms like b_∇ ^2 that we do not take into account in our modeling. Notice as well that because we use the cross-power spectrum in the numerator of b1 we do not need to include a shot-noise term since it is negligible for dark matter particles.
      
      We divide the total halo catalog into 5 tophat mass bins in logarithmic scale of equal size log M=0.26 (where log is the base 10 logarithm) from log M = 14.12 to log M = 15.43. For each secondary property considered, we then divide the subcatalogs at each mass in 4 quartiles, and we measure the linear bias in each quartile using asblyb1 to obtain the b_1(p | M) relation.
      
      In order to facilitate the comparison of our results with those of, e.g. <cit.>, we follow <cit.> and, for each halo property that we consider, we define
      
      p'=  ln(p/p̅)/σ( lnp),
      
      where p is the mean value of the property in a given quartile, p̅ is the mean value in a given mass bin, and σ is the square root of the variance at fixed mass. We then plot the relations b_1(p' | M).
      
      We investigate the dependence of the linear halo bias on three halo properties: the concentration c, spin parameter λ, and shape s. The halo concentration is quantified using the usual Navarro-Frenk-White (NFW) <cit.> concentration parameter c measured as in <cit.>. More specifically, AHF computes the ratio between the maximum of the circular velocity V_ max and V_200, the circular velocity at r_200. For the case of the NFW halo profile <cit.>, this ratio is given by
      V_ max/V_200=(0.216   c/f(c))^1/2,
      
      where f(c) is given by
      
      f(c)=ln (1+c)-c/1+c.
      
      Computing c from the circular velocity at two different radii is hence straightforward. However, this way of inferring the concentration is not as robust as a proper fit of the halo density profile. For the halo spin, we use the spin parameter as defined in <cit.>λ = |J̌|/√(2)MVr_200,
      
      where the angular momentum J̌, the mass M and the circular velocity V are evaluated at position r_200.
      Finally, following the works of e.g. <cit.>, we also measure the bias as function of halo shape given by
      
      s=r_3/r_1,
      
      where r_1 > r_2 > r_3 are the axes of the moment-of-inertia tensor of the halo particles. This ratio is part of the standard output of AHF, which computes the moment-of-inertia tensor in the standard way from the distribution of particles in the sphere of radius r_200.
      
      
      
      
      
      § RESULTS
      
      
      We now turn to our results. We start by measuring the impact of massive neutrinos on assembly bias, before turning to the dependence of halo properties and environment on neutrinos. We assess the importance of including neutrinos in the halo finding procedure in nuhalo. In this section we present results for the M_ν^+++ cosmology (i.e. a neutrino mass of 0.4 eV) since we expect the potential effects to be maximal for this case. Results for neutrino masses of 0.1 and 0.2 eV are presented in addnumass for completeness.
      
      
      
       §.§ Impact of neutrinos on assembly bias
      bofp presents results for b_1  as a function of concentration c, spin parameter λ, and ellipticity s. The points linked by solid lines show the relative linear bias 1=b_1(p |M )/b_1(M), and the errorbars show the 1σ error. We divide the bias as function of a property at fixed mass by the average one at the same mass to remove the vertical mass dependence that is of no interest in our context since it has already been extensively studied before. The color coding, indicating the mass, is the same on each figure and for each set of curves. Each time, the left panel presents results in the fiducial cosmology, while the right one shows results for the massive neutrinos simulations M_ν^+++.
      
      Before getting into the analysis of these figures, we would like to stress again that, although we will refer to low and high masses, one should keep in mind that all our results are for halo masses much above M_⋆, and hence technically very massive halos (cluster-sized halos). 
      Furthermore, as explained in sims, we used a spherical overdensity algorithm to identify halos. This most likely has an impact on our findings <cit.>, but we do not investigate how they would change if we used, e.g., a friends-of-friends (FoF) algorithm.
      
      The results presented on bofp are in agreement with previous results from the literature (e.g. <cit.>). The top left panel of bofp shows the bias as a function of concentration (more concentrated halos are on the right) for the fiducial cosmology. We get a monotonically decreasing relation, and the effect is more important for the lower mass bins. The trends are very similar in all simulations involving massive neutrinos. On the other hand, the middle row of bofp shows that halos with a higher angular momentum are more clustered, and again the effect is stronger in the lower mass bins. Again, the effect of neutrinos is not visible by eye from these figures. Finally, the bottom row of bofp presents the linear bias as a function of halo shape at fixed mass. More spherical halos (with positive s') are more biased, while more elliptical ones have a lower bias. The effect is larger for lower masses, and we cannot see any effect induced by neutrinos by eye.
      
      We do not go in more details discussing these figures since all these results are already known and investigated in the case of massless neutrinos (e.g. <cit.>), and that our goal here is to detect the effect of neutrinos on assembly bias.
      
      Since bofp does not allow to see any clear effect of the massive neutrinos on assembly bias, we further plot the ratios b^ rel,  ν_1(p|M)/b^ rel,   fiducial_1(p|M) (i.e. the ratio of the ratios), where the superscript “ν” indicates that we measure this quantity in simulations including massive neutrinos, while the denominator is the one measured in the fiducial cosmology. The results are presented on ratios for the M_ν^+++ simulations, and in addnumass for the two smaller neutrino masses. This figure is the main results of this paper. The color coding, indicating halo mass, is the same as on previous figures. The errorbars show the 1σ error on the mean obtained by error propagation from the errors showed on bofp.
      
       
      
      We start again by discussing assembly bias as a function of concentration shown on the top left panel of ratios. More concentrated halos have a higher c' value. We do not detect any measurable effect of neutrinos on the b_1(c) relation at M_ν=0.4 eV, and the effect is even smaller for masses of 0.1 and 0.2 eV. We could argue that there seems to be a mass-dependent, percent-level trend with more concentrated halos having a lower bias in neutrinos cosmologies. This is however not statistically significant given the size of the errorbars. We did not attempt to increase the signal-to-noise ratio by adding more simulation volume for two reasons: first, we would only be able to investigate the effect for the highest neutrino mass of 0.4 eV which is already an unreasonably high mass ruled out by observations <cit.>. Secondly we already use a total volume of 500 ( Gpc/h)^3 at a single redshift, which is much larger than any current or future survey volume.
      
      Turning to b_1(λ), the top right panel of ratios shows no clear effect due to neutrinos at any halo mass. The conclusions are the same at lower neutrino masses. One could argue a very small decrease in the overall amplitude of the bias (0.5%) at all values of λ in the simulation with total neutrino mass of 0.4 eV but this is again not statistically significant. Overall, we can safely conclude that neutrinos have no effect on assembly bias as a function of halo spin, at least for the volumes considered in this work.
      
      Finally, the bottom panel of ratios shows the impact of neutrinos on the bias as a function of ellipticity in the M_ν^+++ cosmology. More spherical halos are on the right with s' closer to 1, while halos with a negative s' are more elongated. As in the case of concentration and spin parameter, neutrinos do not seem to impact the relation b_1(s). The only effect that can be seen is a percent effect on the overall amplitude of the bias at the three lowest mass considered in this paper. In this case there is a constant decrease of b_1 at all values of s. The two higher mass bins do not show this trend and have more scatter around the mean value, probably due to the smaller number of halos in these bins. Results presented on ratios2 in addnumass show similar trends while less statistically significant with all curved consistent with 1.
      
      The results presented on this figure imply that it will be very challenging to constraint neutrino masses from data using their impact on assembly bias[Notice that assembly bias in galaxies or galaxy clusters should in any case first be established.]. On the other hand, this also means that this is not an additional effect to take into account and model when analyzing future survey data. 
      
      Before concluding this section we make a few important remarks. First, we note that we use only the baryon-CDM fluid in the halo finding procedure (“bc halos”), as explained before. While we do not expect that including neutrinos in halos will have a large impact, we inspect this in nuhalo. We indeed find the same results as in this section (ratiownu). Furthermore, we also check the impact of σ_8 on assembly bias in sigma8. This is important since, as we said before, the total matter σ_8 is kept fixed with respect to the fiducial cosmology in neutrino simulations. This means that the baryon-CDM fluid has a modified (enhanced) σ_8 which impacts halo clustering and can impact assembly bias. As shown in ratiosigma8 we do not find any significant impact of σ_8 on assembly bias as a function of any properties. However we can observe some small trends by eye which go in the same way as the ones due to neutrinos discussed in this section. This minimizes the impact of neutrinos on assembly bias even further. Finally, using the 100 realizations of the high resolution simulation, we checked the impact of mass resolution on our results. We did not find any significant impact, and conclude that mass resolution does not induce any spurious assembly bias signal. Finally, we emphasis that choosing a more conservative minimum number of particles per halo such as 400 would only remove the lowest mass bin, leaving our conclusions that massive neutrinos do not impact the assembly bias of cluster-sized halos unchanged. 
      
      
      
       §.§ Dependence of halo properties and environment on neutrinos
      
      
      In this section we study the impact of neutrinos on halo properties, as well as their large-scale environment, defined as the overdensity (both of total matter and of baryon-CDM only) in a sphere of radius R=20.
      
      
      
      We start with the concentration, spin, and ellipticity - mass relations in order to investigate how neutrinos impact them. Since assembly bias is defined as the dependence of halo bias on properties other than mass, it is logical to assess the level at which neutrinos affect dark matter halo properties. We note that we do not expect any of the studied quantities to be impacted by neutrinos significantly, (i.e. more than assembly bias itself) since otherwise this would lead to a spurious signal in assembly bias due to an Eddington-like bias in the case of mass. To be clearer, if neutrinos affected strongly a given property p without affecting the clustering itself (i.e halos would be assigned a “wrong” value for p), assembly bias as a function of p would then be different in the fiducial and neutrino cosmologies, which in turn would show on ratios. This is however not what we observed, which is why we do not expect any strong impact of neutrinos on the three halo properties studied in this paper.
      
      The results are shown in propsofM for the fiducial case (orange) and in simulations with neutrinos of mass 0.2 (cyan) and 0.4 eV (blue) for halos built from the baryon-CDM+neutrino fluid. We further show the same relation for halos built only with the baryon-CDM fluid in the M_ν^+++ cosmology in magenta to assess the importance of including neutrinos in halos. Finally we also show the effect of enhancing σ_8 in red (σ_8^+) in order to disentangle the effect of σ_8 from the one of neutrinos. The relations showed are mean relations but of course the scatter (not shown) around these means is big. We do not go into a detailed analysis of the general trends common to all cosmologies in these relations but simply note that we recover all the already known ones (e.g. <cit.> and references therein). In particular more massive halos tend to be less concentrated, have a smaller angular momentum and are more elliptical (this last relation can seem a little surprising but can be understood, for very massive objects, when realizing that more spherical halos are also more concentrated and hence correspond to less massive objects, e.g. <cit.>). We see that neutrinos have a small effect on all three properties, and that including them or not in the halo definition has no effect on the inference of properties. We develop further on this last point and its impact on assembly bias in nuhalo. A final point to note is that, while the impact of enhancing σ_8 is the same as the one due to neutrinos in the case of M_ν=0.2 eV for the spin and shape of halos, it is not the case for concentration. Indeed enhancing σ_8 increases halo concentration but neutrinos tend to slightly make it smaller, as expected. We hence conclude that the effect of neutrinos on spin and shape can be largely attributed to the change in σ_8 but not for the case of concentration.
      
      
      
      In order to be more quantitative, ratiopropsofM shows the ratio of each property in the M_ν^+++ cosmology to the one in the fiducial simulation (left panel), as well as for the σ_8^+ case (right panel). Clearly neutrinos affect all quantities in a mass dependent way, the largest effect being seen at large masses, and that effect is of a couple percents maximum. We see again that the change in σ_8 seems to explain the effect of neutrinos on λ and s (with smaller amplitude since the change in σ_8 presented here corresponds to a neutrino mass of 0.2 eV), but not on c. These results go in the same direction as what we observed for assembly bias and are hence expected for reasons that we explained before. More specifically, halos in massive neutrinos cosmologies tend to be slightly more spherical and have a larger spin parameter, and their concentration tends to decrease at very high mass. While the effect on concentration has been partially studied before <cit.>, it is to our knowledge the first time that  the dependence of halo spin and ellipticity on neutrino mass is measured (see however <cit.> for the effect of massive neutrinos on halo spin flip). These results can be easily understood when considering the large velocity dispersion of neutrinos (in particular this directly explains the larger angular momentum observed) but, as we explained before, they can be largely attributed to the change in the baryon-CDM σ_8. A maybe more surprising result is the fact that the concentration is only very weakly affected by neutrinos, since one would expect the large free streaming scale of neutrinos to diminish the clustering of matter and hence also the concentration of halos by making their profile more shallow. This might still be true. However, since σ_8 for the baryon+CDM fluid also varies in neutrino cosmology and tends to increase the concentration, the two effects probably oppose and cancel each other leading to no net total difference between halo concentration in the fiducial and neutrino cosmologies.
      
      
      
      We now turn to the impact of neutrinos on the large scale density of halos, defined as the overdensity in a sphere of radius R=20 around the halo center, as a function of halo properties. This is motivated by the fact that halo assembly bias has now been shown to be related to halo environment and position in the cosmic web <cit.>. For example, high mass halos (that dominate their environment) with a lower concentration are expected to have a higher large-scale density on average since they have accreted less material. The average (or stacked) large-scale density of halos in turn corresponds to the cross correlation of the matter with the halo field, and is hence linked to the linear bias (higher environmental density gives rise to a higher bias). In this picture it is then clear why less concentrated objects are more biased, as can be seen on the top row of bofp. Since we do not see any strong difference between assembly bias in the fiducial and neutrino cosmologies, we do not expect them to affect the large scale density of halos in a property-dependent way. More precisely, while neutrinos might affect the overall overdensity around halos, we expect this effect to be constant over the range of value taken by a given property.
  