Cosmology with massive neutrinos III: the halo mass function and an application to galaxy clusters
Abstract
We use a suite of Nbody simulations that incorporate massive neutrinos as an extraset of particles to investigate their effect on the halo mass function. We show that for cosmologies with massive neutrinos the mass function of dark matter haloes selected using the spherical overdensity (SO) criterion is well reproduced by the fitting formula of Tinker et al. (2008) once the cold dark matter power spectrum is considered instead of the total matter power, as it is usually done. The differences between the two implementations, i.e. using instead of , are more pronounced for large values of the neutrino masses and in the high end of the halo mass function: in particular, the number of massive haloes is higher when is considered rather than . As a quantitative application of our findings we consider a Plancklike SZclusters survey and show that the differences in predicted number counts can be as large as for eV. Finally, we use the PlanckSZ clusters sample, with an approximate likelihood calculation, to derive Plancklike constraints on cosmological parameters. We find that, in a massive neutrino cosmology, our correction to the halo mass function produces a shift in the relation which can be quantified as and assuming one () or three () degenerate massive neutrino, respectively. The shift results in a lower mean value of with for and for , respectively. Such difference, in a cosmology with massive neutrinos, would increase the tension between cluster abundance and Planck CMB measurements.
a,b]Matteo Costanzi, c]Francisco VillaescusaNavarro, c,b]Matteo Viel, d]JunQing Xia, a,b,c]Stefano Borgani, e]Emanuele Castorina, f,g]Emiliano Sefusatti
Prepared for submission to JCAP
Cosmology with massive neutrinos III: the halo mass function and an application to galaxy clusters

Università di Trieste, Dipartimento di Fisica,
via Valerio, 2, 34127 Trieste, Italy 
INFNNational Institute for Nuclear Physics,
via Valerio 2, 34127 Trieste, Italy 
INAFOsservatorio Astronomico di Trieste,
via Tiepolo 11, 34133 Trieste, Italy 
Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science,
P.O. Box 9183, Beijing 100049, Peopleâs Republic of China 
SISSA  International School For Advanced Studies, Via Bonomea, 265 34136 Trieste, Italy

The Abdus Salam International Center for Theoretical Physics,
Strada Costiera 11, 34151, Trieste, Italy 
INAF, Osservatorio Astronomico di Brera,
Via Bianchi 46, I23807 Merate (LC) Italy
Keywords: cosmology: largescale structure of Universe; neutrinos; galaxies: clusters.
Contents
1 Introduction
Neutrinos are spin onehalf leptons carrying no electric charge. Within the particle standard model they are described as elementary massless particles. Measurements of the boson lifetime have pointed out that the number of active neutrinos is 3 (, [1]). On the other hand, the neutrino oscillation phenomenon indicates that at least two of the three neutrino families have to be massive. Unfortunately, measurements involving neutrino flavour changing only provide us with information about the mass square differences between the different mass eigenstates, i.e. they can not be used to determine the absolute neutrino mass scale. Recent experiments using solar, atmospheric and reactor neutrinos quantified these differences as: and (see e.g. [2, 3]), where , and are the masses of the different neutrino mass eigenstates. Since we are not capable to measure the sign of , two different mass ordering (hierarchies) are possible: a normal hierarchy () and an inverted hierarchy (). Therefore, the sum of the neutrino masses is constrained from below as 0.056, 0.095 eV depending on whether neutrinos follow the normal or inverted hierarchy, respectively. Knowing the absolute neutrinos mass scale is of great importance, since it is related to physics beyond the particle standard model. For this reason, a huge effort from both the theoretical and the experimental side is currently ongoing with the purpose to weight neutrinos.
From the cosmological point of view, the Big Bang theory predicts the existence of a cosmic neutrino background (see, e.g. [4, 5] for a review). In the very early Universe, cosmic neutrinos contributed to the total radiation energy density, affecting the nucleosynthesis process and therefore the primordial abundance of light elements. At the linear order, massive neutrinos impact cosmology in different ways, depending on which parameters are fixed: they shift the matterradiation equality time, at fixed , and they slow down the growth of matter perturbations during the matter and Dark Energy dominated era. The combination of the above two effects produces a suppression in the amplitude of the matter power spectrum on small scales (see for instance [5]).
The imprints left by massive neutrinos on the CMB and on the Large Scale Structucture (LSS) of the Universe have been used to set upper limits on their masses. Numerous recent works point towards neutrino masses, , below 0.3 eV at [e.g. 6, 7, 8, 9, 10], with the notable exceptions of [11] in which authors used Lyman data to set an upper limit of and [12] who found eV () by combining data from BAO, CMB and the WiggleZ galaxy power spectrum^{1}^{1}1Private communication.
Great attention has been recently drawn to the tension between the Planck measurements of the primary CMB temperature anisotropies [10] and measurements of the current expansion rate [13], the galaxy shear power spectrum [14] and galaxy cluster counts [15, 16, 17]. Besides unresolved systematic effects, it has been suggested by many authors [17, 18, 19, 20] that the discrepancy can be alleviated by extending the standard CDM model to massive neutrinos, either active or sterile. A common finding of those works is that a neutrino mass of provides a better fit to the combination of CMB data and low redshift Universe measurements than the vanilla CDM model.
Among the different probes of the LSS, galaxy clusters have played a significant role in the definition of the “concordance” CDM model [e.g. 21, 22], and many ongoing (Planck, SPT, DES), upcoming and future (eROSITA, LSST, Euclid) surveys will aim to use their abundances and spatial distribution to strongly constrain cosmological parameters. In order to fully exploit for cosmology the ever growing number of clusters detected, it is mandatory to have a reliable theoretical predictions for the cluster abundance (the halo mass function, HMF) [23, 24, 25], together with an accurate calibration of the observablemass relation. As for the former, since the pioneering work of Press & Schechter [26] many forms for the HMF have been proposed in literature [e.g. 27, 28, 29, 30, 31, 24], often calibrated against large suites of cosmological simulations. Despite the great improvement of the numerical results over the past decade many sources of systematic error still affect the HMF, including finite simulation volume, mass and force resolution, baryonic physics and massive neutrino effects. Here we focus on the consequences of nonvanishing neutrino masses.
The effects of neutrino masses on the halo mass function has already been studied in different works [32, 33, 34, 35]. In the work of [32], the authors measured the halo mass function from Nbody simulations incorporating massive neutrinos using a hybrid scheme to simulate neutrino particles. They showed that the halo mass function in models with massive neutrinos can be well reproduced by the Sheth and Tormen (ST) [27] mass function by using , instead of , when establishing the relation between the halo mass and the tophat window function radius (, see section 3 for details). Those results were later independently verified in [33, 35] using a different set of Nbody simulations. More recently, the authors of [34] investigated the gravitational collapse of a spherical region in a massive neutrino cosmology, showing that neutrinos play a negligible role in the process. This led to the conclusion that the cold dark matter power spectrum should be used to compute the r.m.s. of the matter perturbations, , required predict the halo mass function. In [36] this was tested againts Nbody simulations, resulting in an excellent agreement.
This paper is the last of a series of three papers. Paper I [37] introduces a large set of numerical simulations incorporating massive neutrinos as particles. It then studies the effect of neutrino masses on the spatial distribution of dark matter haloes, finding that halo bias, as typically defined w.r.t. the underlying total matter distribution, exhibits a scaledependence on large scales for models with massive neutrinos. In addition, Paper I investigates as well massive neutrinos effects on the spatial distribution of galaxies by constructing mock galaxy catalogues using a simple halo occupation distribution (HOD) model.
In Paper II [36] the universality of the HMF and of linear bias in massive neutrino cosmologies is discussed in terms of halo catalogues determined with the FriendsofFriends algorithm on the simulations introduced in Paper I. It is shown that the proper variable to describe the HMF of a massive neutrino model is the variance of cold dark matter perturbations, rather than the total ones (i.e. including neutrinos) typically assumed in previous analyses [32, 33, 35]. If the correct prescription is used then the HMF becomes nearly universal with respect to the neutrino mass. The paper discusses also similar results for the bias of haloes at large scales, which is found to be almost scale independent and universal when expressed in terms of CDM quantities alone.
In this paper we explore how the results of Paper II affect the determination of cosmological parameters from galaxy clusters data. Here we study the HMF of dark matter haloes identified using the Spherical Overdensity (SO) algorithm. The reason to use SO haloes is that the mass proxy in Xray and SZ measurements is calibrated with spherically defined objects rather than with the FriendsofFriend (FoF) haloes considered in Paper II. We show that the abundance of SO haloes is well reproduced by the Tinker fitting formula once the cold dark matter mean density and linear power spectrum are used, in agreement with Paper II and the work of [34]. Then, we show that our findings have interesting implications for cosmology using cluster number counts, especially due to the recently highlighted tension between cosmological parameter constraints inferred from CMB temperature data and the SZ clusters datasets [38]. As a case study, we choose the Planck SZselected sample of clusters [38], for which we perform a Monte Carlo Markov Chain analysis in order to compare constraints obtained using different prescriptions for the halo mass function. We find that using the CDM linear matter power spectrum when computing the r.m.s. of the smoothed linear density field, (i.e. when using a better description for the HMF in massive neutrinos cosmologies) changes the degeneracy direction between the parameters and and decreases the mean value. These changes increase the tensions between cosmological parameters constraints from CMB data and from SZ cluster counts.
The paper is organized as follows. In Section 2 we describe the numerical simulations we have used to calibrate the HMF of dark matter haloes identified using the SO criterion. The halo mass functions for the different cosmological models and the procedure used to compute them are shown in section 3. The implications of our results, in terms of cluster number counts, are presented in section 4, while the likelihood analysis is shown in section 5. Finally, we draw the main conclusions of this work in section 6.
2 Nbody simulations
For this paper we have used a subset of the large suite of Nbody simulations presented in Paper I. We summarize the main features of these simulations here and refer the reader to [37] for further details.
Name 
Box  
[eV]  []  
H6  0.2708  0.050  0.7292  0.0131  0.7  1.0  
H3  0.2708  0.050  0.7292  0.0066  0.7  1.0  
H0  0.2708  0.050  0.7292  0  0.7  1.0  
H6s8  0.2708  0.050  0.7292  0.0131  0.7  1.0 
The Nbody simulations have been run using the TreePM code GADGET3, which is an improved version of the code GADGET2 [39]. The neutrinos have been simulated using the socalled particlebased implementation (see [40, 41, 42, 43, 44] for the different methods used to simulate the cosmic neutrino background).
The starting redshift of the simulations was set to . The initial conditions were generated at that redshift by displacing the particles positions from a regular cubic grid, using the Zel’dovich approximation. We incorporate the effects of baryons into the CDM particles by using a transfer function that is a weighted average of the transfer functions of the CDM and the baryons, obtained directly from the CAMB code [45]. The Plummer equivalent gravitational softening of each particle type is set to of their mean interparticle linear spacing. For each simulation we saved snapshots at redshifts 0, 0.5, 1 and 2.
The different cosmological models used for this paper are shown in Table 1, together with the values of their cosmological parameters. Each simulation consists of eight independent realizations obtained by generating the initial conditions using different random seeds. The size of the cosmological boxes are 1 Gpc for all the simulations. The cosmological models span from a massless neutrino model (H0) to cosmologies with = 0.3 eV (H3) and = 0.6 eV (H6 and H6s8). The simulations H6, H3 and H0 share the value of the large–scale power spectrum normalisation , whereas the value of this parameter has been tuned in the simulation H6s8 to obtain the same value of of the simulation H0. The values of the other cosmological parameters are common to all the simulations: , , , and . In all the simulations the value of the parameter is given by , i.e. is fixed by requiring that the total matter density of the Universe is the same for all the cosmological models. The number of CDM particles is , and for the models with massive neutrinos the number of neutrinos is also . The masses of the CDM particles are M for the model H0, while for the others model the masses are slightly different since the value of varies from model to model.
3 The halo mass function
We start this section by explaining how we identify the dark matter haloes from the snapshots of the Nbody simulations. We then investigate whether the Tinker fitting formula [31] along with the socalled cold dark matter prescription reproduces the HMF of Nbody simulations for cosmological models with massive neutrinos.
The dark matter haloes have been identified using the SUBFIND algorithm [46]. Even though SUBFIND is capable of identifying all the haloes and subhaloes from a given particle distribution, we have used it to identify spherical overdensity (SO) haloes, which correspond to the groups identified by SUBFIND. The virial radius of a given dark matter halo corresponds to the radius within which the mean density is 200 times the mean density of the Universe. We restrict our analysis to SO haloes containing at least 32 particles.
SUBFIND has only been run on top of the CDM particle distribution. This is equivalent to neglect the contribution of neutrinos to the masses of the dark matter haloes. Such assumption is supported by different studies [47, 32, 48, 35] which have shown that the contribution of massive neutrinos to the total mass of dark matter haloes is below the percent level for the neutrino mass range relevant for this paper. We have explicitly checked that the contribution of neutrinos with = 0.6 eV to the total masses of dark matter haloes ranges from for haloes with M to for the most massive haloes with M. To make sure that our results are not affected by selecting the haloes on top of the CDM particle distribution we have run SUBFIND on top of the total matter (i.e. CDM plus neutrinos) density field. We find that the HMF of SO haloes changes by less than on a very wide range of masses. However, the masses of some low mass haloes are slightly changed when including neutrinos. This is because some of these low mass haloes contain many unbound neutrino particles, which bias the estimate of their masses by an unreasonable amount. This effect is less important for more massive haloes and/or for simulations in which the number of neutrino particles is much larger than the number of CDM particles. In order to avoid this spurious contamination in the masses of some dark matter haloes we decided to rely on the halo catalogues obtained by running SUBFIND just on top of the CDM particle distribution.
Now we turn our attention to the halo mass function for cosmologies with massless and massive neutrinos. It is a common practice to parametrize the abundance of dark matter haloes in the following way:
(3.1) 
where is the comoving number density of dark matter haloes per unit mass at redshift , is the comoving mean density of the Universe and is defined as:
(3.2) 
with being the linear matter power spectrum at redshift , while is the Fourier transform of the tophat window function of radius . The relationship between the halo mass, , and the radius in the tophat window function is given by .
Our aim here is to compare the results of the left and righthand side of eq. (3.1). The lefthand side can be directly measured from the Nbody simulations, whereas the righthand side can be computed using a fitting formula for the function together with some prescriptions for cosmological models with massive neutrinos. We calculate the lefthand side of eq. (3.1) by approximating the quantity by , where the width of the mass intervals has been chosen to be . The comoving number density of dark matter haloes in a given mass interval has been directly obtained from the Nbody halo catalogue. In order to compute the righthand side of eq. (3.1) we need the following three ingredients: 1) the function ; 2) the value of to establish the relation between the halo mass and the radius in the tophat window function and 3) the linear matter power spectrum .
Since we are considering SO haloes, we compare our Nbody results to the fitting formula of Tinker et al. [31], also defined in terms of SO haloes. The Tinker fit assumes the functional form proposed by [49] with
(3.3) 
where , , and are the bestfit parameters^{2}^{2}2Notice that the Tinker best fit parameters have an explicit redshift dependence, i.e. the Tinker halo mass function is not universal in redshift. for the overdensity presented in [31].
In a standard CDM cosmology the quantities and appearing in eqs. (3.1, 3.2) are evaluated for the total dark matter field. However, it is not obvious which quantities have to be used for a model with massive neutrinos. The work of [32] demonstrated that the abundance of dark matter haloes in massive neutrino cosmologies cannot be reproduced by the ST fit if the total matter density and linear power spectrum were used when calculating the r.h.s. of eq. (3.1). The authors proposed to use, instead, the mean cold dark matter density , computing, however, the variance still in terms of the total matter power spectrum. Such prescription, that we will refer to as the matter prescription, was later corroborated by several works [33, 35].
More recently, the authors of [34] studied the gravitational collapse of a spherical region in a massive neutrino cosmology, showing that neutrinos play a negligible role in the process and leading to the conclusion that the cold dark matter power spectrum should be used to predict the halo mass function. Indeed, in Paper II we show that, for FriendsofFriends (FoF) haloes, a good agreement between the MICE fitting formula [24] and our Nbody simulations is obtained if both and are computed in terms of CDM quantities alone. We call this the cold dark matter prescription for massive neutrino cosmologies, and in Paper II we show that it is the only way of obtaining a mass function that is nearly universal with respect to changes in the background cosmology.
We now compare the abundance of dark matter haloes from the Nbody simulations with the Tinker prediction evaluated with both the matter and cold dark matter prescriptions. We emphasize that for cosmologies with massless neutrinos the above two prescriptions become the same. We show the results of this comparison in Fig. 1 where the data points correspond to the mean of the mass function, , measured from eight realizations while the error bars represent the error on the mean. Predictions using the Tinker fitting formula along with the matter and cold dark matter prescriptions are shown by the dashed and solid lines, respectively. We show the results at redshifts 0, 0.5 and 1 for the simulations H0, H3 and H6 (results at are noisy). For clarity we do not display the results of the simulation H6s8 since they are very close to those of the simulation H0.
We find that the cold dark matter prescription reproduces much better the abundance of dark matter haloes extracted from the Nbody simulations. The agreement between the Tinker fitting formula (plus the cold dark matter prescription for massive neutrinos) and our results is pretty good at , while at higher redshift is a bit poorer. We note that the differences between the results from the Nbody simulations and the Tinker fitting formula along with the cold dark matter prescription are almost independent of the cosmological model, likely arising from the different method used to identify the SO halos with respect to Tinker et al. [31]. In addition, Paper II shows that the halo mass function for FoF haloes () in our Nbody simulations is very well reproduced (within a ) by the fitting formula of Crocce et al. [24] at all redshifts. We emphasize that the use of a different halo mass function will not change the main conclusions of this paper.
In Fig. 2 we show the ratio of the halo mass function for cosmologies with massive neutrinos to the halo mass function for the cosmology with massless neutrinos. We find that the abundance of SO haloes is very well reproduced by the Tinker fitting formula once the cold dark matter prescription is used for cosmologies with massive neutrinos.
4 An application to cluster number counts
A different prescription for the HMF can affect the constraints on cosmological parameters provided by cluster number counts by changing the number of clusters predicted for a given cosmology and survey.
The number of cluster expected to be detected within a survey with sky coverage in a redshift bin can be expressed as:
(4.1) 
where is the comoving volume element per unit redshift and solid angle, is the survey completeness and is the halo mass function. In what follows we adopt the Tinker functional form for the mass function defined in eq. (3.3) with the bestfit parameters for the overdensity as provided by [31].
The completeness function depends on the strategy and specifics of the survey. For the purpose of this work we can simply express this function as
(4.2) 
where the lower limit in the mass integral, , represents the minimum value of the observed mass for a cluster to be included in the survey, and it is determined by the survey selection function and the fiducial signaltonoise level adopted.
The function gives the probability that a cluster of true mass has a measured mass given by and takes into account the uncertainties that a scaling relation introduces in the knowledge of the cluster mass. Under the assumption of a lognormaldistributed intrinsic scatter around the nominal scaling relation with variance , the probability of assigning to a cluster of true mass an observed mass can be written as [50]:
(4.3) 
where the parameter represents the fractional value of the systematic bias in the mass estimate.
We now turn to the implications of the prescription choice on the HMF prediction. By replacing with one neglects the suppression of the total DM density fluctuations on scales smaller than their freestreaming length, the scale below which neutrinos cannot cluster due to their high thermal velocity (see, e.g. [5]). The magnitude of the suppression depends on the sum of the neutrino masses, while the scale below which it takes place depends on the individual neutrino mass and on redshift. This in turn affects the halo mass function by shifting the maximum cluster mass (i.e. the scale beyond which the halo mass function is exponentially suppressed) to larger values, thus increasing the predicted number of rare massive clusters. The effect is larger for larger total neutrino mass, larger number of massive neutrinos and higher redshift.
In Fig. 3 we show the ratio of the cluster counts predicted using the prescription over the one predicted using (colour coded) for different combinations of () values and for two neutrino mass split schemes: a single massive neutrino (left panel) and three degenerate massive neutrinos (right panel). In the former case the total neutrino mass is assigned to one neutrino species ( and ), in the latter one it is equally split among three massive species ( with ). The plots have been obtained by varying and and keeping fixed , , , and to the Planck mean value ([10]; Table 2, Planck+WP). In order to mimic a Planck SZcluster survey, we computed the number counts integrating eq. (4.1) between with a sky coverage and we approximated the Planck SZcluster completeness function using as lower limit in eq. (4.2) the limiting mass ^{3}^{3}3Following the recipe given in [51], the limiting mass has been converted to – the limiting mass within a radius encompassing an overdensity equal to 200 times the mean density of the Universe – consistently with the chosen halo mass function. provided by the Planck Collaboration (dashed black line in Fig. 3 of [17]). Moreover, since we are simply interested in quantify the relative effect of using an improved HMF calibration we assumed no uncertainties in the estimation of the true mass () and we set and in eq. (4.3). Power spectra have been computed using CAMB [45], where has been obtained exploiting the relation
(4.4) 
with , and being the CDM, baryon and total matter transfer functions, respectively.
Assuming one massive neutrino, changing the matter power spectrum to the cold dark matter one in the HMF prediction increases the expected number of clusters by in the minimal normal hierarchy scenario (eV), reaching differences of for masses of . Considering instead three degenerate massive neutrinos, the CDM prescription gives even a larger correction to the cluster counts: the splitting of the total neutrino mass between three species causes the freestreaming length to increase, therefore widening the range in which is suppressed with respect to . As a result, the difference in cluster counts computed with the two prescriptions reaches for neutrino mass of the order of . For a given cosmology the magnitude of the ratio slightly depends also on the specifics of the survey: a lower would entail a larger difference between the expected number of clusters computed with the two different calibrations.
The difference in the predictions in turn affects the degeneracy between cosmological parameters. An example of this effect is shown in figure 4, in the () plane (left panel) and the corresponding () plane (right panel). The curves correspond to constant number counts obtained using (black) or (red) in the halo mass function definition, following the same procedure of figure 3 to compute the expected number of clusters and keeping the other cosmological parameters (, , , , ) fixed to the Planck mean value. Solid and dashed curves are for models with one massive neutrino and three degenerate massive neutrinos, respectively. The different slope of the curves indicates a different degeneracy direction between parameters. Consistently with the results shown in figure 3 the change in the slope is more pronounced in the case of three massive neutrinos.
As illustrated in the next section both these effects can contribute to modify the information on cosmological parameters inferred from cluster data in models with massive neutrinos.
5 Implications for cosmological constraints
The ultimate aim of an analytic expression for the halo mass function is to link the observed abundance of galaxy clusters to the underlying cosmology. The recently released Planck data indicate some tension between the cosmological parameters preferred by the primary CMB temperature measurements and those obtained from cluster number counts using Xray [15], optical richness [16] and SZselected clusters [52, 53, 17]. In particular, the values of and inferred from cluster analyses are found to be lower than the values derived from CMB data. It has been argued that this discrepancy could be due to a wrong calibration of cluster mass (see e.g. [54, 55]) or alternatively it may indicate the need to extend the minimal CDM to massive neutrinos [17, 18, 19, 20]. In the latter case, the results presented in this paper could in principle affect derived cosmological constraints which relies on an incorrect calibration of the HMF in cosmological models with massive neutrinos. In fact, in all previous cluster studies, the variance of the total dark matter field has been used to put constraints on neutrino masses. In section 4 we have shown that, given a background cosmology, using the “wrong”prescription for the HMF could lead to differences up to in the expected number of cluster. However that is not the the only reason to use the variance of the CDM field. Indeed, a key assumption in previous cosmological analyses of clusters counts is that the shape of the HMF is independent of the underlying cosmology, and the same functional form can be used through all the parameter space. In Paper II we show that universality of the HMF with respect to neutrino masses, and more in general cosmology, is recovered only if the cold dark matter prescription is adopted. This is another important effect that should be taken into account by future studies.
In order to assess the effects of the cold dark matter prescription on the parameter estimation we choose as a case study the sample of 188 SZselected clusters with measured redshift published in the Planck SZ Catalogue [38]. The cosmological constraints have been obtained using the likelihood function for Poisson statistics [56]:
(5.1) 
where , represent respectively the number of clusters observed and theoretically predicted in the th redshift bin. The redshift range has been divided in bins of width between and , also including in the analysis redshift bins with no observed clusters. We computed the predicted number of clusters , , for a Plancklike SZcluster survey following the procedure described on section 4. The parameter space has been explored by means of the Monte Carlo Markov Chain technique using the publicly available code CosmoMC^{4}^{4}4http://cosmologist.info/cosmomc/ [57], where we included a module for the computation of the likelihoods function described above. Since we are interested in the effects that a different prescriptions for the HMF has on parameter constraints rather than the constraints themselves, we kept fixed , and , allowing only , , and to vary.
For the same reason we neglect errors on nuisance parameters, which have been kept fixed to and in order to roughly reproduce the mean values of and obtained by the Planck Collaboration with PlanckSZ+BAO+BBN data [17]. We also checked that our results in the plane fixing to were in good agreement with those obtained by the Planck Collaboration. Finally, due to the weak sensitivity of the clusters sample to some cosmological parameters, we set a Gaussian prior on the total neutrino mass, , and one on the expansion rate, km/s/Mpc (from BAO measurements [58]). Note that the actual Planck cluster likelihood is not publicly available. Therefore, a quantitative comparison with the SZ Planck cluster results is not possible. However, since we are presenting results in terms of relative effects between different HMF calibrations, we expect that our findings will be robust and could be quantitatively similar to those to be derived with a more accurate likelihood analysis.
The joint constraints on the  plane resulting from this analysis are shown in figure 5 with green contours for the CDM prescription and blue contours for the matter prescription. The left panel is for a model with one massive neutrino while the right one for a model with three degenerate massive neutrinos. The dashed lines show the relation with and parameters obtained by fitting a powerlaw relation to the likelihood contours. Also shown in the right panel with orange contours are the constraints from Planck+WP+BAO datasets for a CDM cosmology with free ^{5}^{5}5Chains publicly available at http://www.sciops.esa.int/.. While the constraining power of clusters is almost unaffected by different HMF prescriptions the degeneracy direction become steeper in the CDM case. For one massive neutrino the shift can be quantified as , or . The effect is even larger when considering three massive neutrino, for which we obtain a shift of and . The different degeneracy of the CDMcase contours can be understood as follows: for our set of free parameters moving toward large values in order to keep constant the number of clusters one has to compensate with lower and larger values. Using the CDM prescription, however, for a given matter density and neutrino mass the value of which reproduces the right number of cluster is smaller than the one recovered using the matter prescription; moreover, the difference between values inferred from the two HMF prescriptions increases with the total neutrino mass and it is more pronounced when assuming three massive neutrinos (see Fig. 4).
Using the orange contours as a reference one can see that the shift of the contours caused by the CDM prescription goes in the direction of increasing the tension with the Planck+BAO results. This means that when using the CDM prescription in trying to reconcile the Planck CMB measurements with cluster number counts, when extending the CDM model to massive neutrinos, a larger value will result from the combination of the two datasets.
The effects of the usage of the CDM prescription on parameter estimation are clearly visible but with low statistical significance for the cluster sample chosen for this work. However, owing to the much stronger constraining power expected from upcoming and future cluster surveys, corrections to the  degeneracy direction of the order of would offsets the resulting constraints by a statistically significant amount [59, 60].
6 Summary and perspectives
By using a set of large boxsize Nbody simulations containing CDM and neutrinos particles we have studied the abundance of dark matter haloes, identified using the SO criterion, in cosmological models with massive neutrinos. The SO haloes have been extracted from the Nbody simulations by running the SUBFIND algorithm on top of the CDM particle distribution to avoid spurious mass contamination in the low mass haloes from unbounded neutrino particles. We have however explicitly checked that our results do not change if SUBFIND is run on top of the total matter density field. We have compared the abundance of dark matter haloes in cosmologies with massless and massive neutrinos with the Tinker fitting formula along with the matter prescription and the cold dark matter prescription. In both prescriptions we use instead of when setting the relation between the halo mass and the radius in the tophat window function: . However, in the cold dark matter prescription we use the CDM linear power spectrum, , when computing the value of , whereas in the matter prescription we use the total matter linear power spectrum, .
We find that the abundance of SO haloes is much better reproduced by the Tinker fitting formula once the cold dark matter prescription is used, in agreement with the claims of Ichiki & Takada [34] and the results of Paper II [36]. The agreement is very good at while it worsens a bit at higher redshift. Once we present the results as ratios of the halo mass functions for cosmologies with massive neutrinos to the halo mass function for cosmologies with massless neutrinos the agreement with theoretical predictions improves significantly at all redshifts. We stress that the conclusions of this paper are not affected if a different halo mass function fitting formula was used.
We have investigated the effects that the cold dark matter prescription has on theoretically predictions of number counts and on the estimation of cosmological parameter from cluster samples. By using the Tinker fitting formula for the HMF we computed the expected number of clusters for a Plancklike SZcluster survey. We found that for a cosmology with massive neutrinos the predicted number of clusters is higher when using the cold dark matter prescription with respect to the results obtained by using the matter prescription. For a given value of the effect is more pronounced for large neutrino masses and in the case of a splitting of the total neutrino mass between three degenerate species. Assuming one massive neutrino family (and two massless neutrino families) the difference in the predicted number counts between the two prescriptions is nearly for , while it reaches in models with three degenerate massive neutrinos.
The different prediction for the HMF in turn affects the degeneracy direction between cosmological parameters and the mean values inferred from the cluster sample. To quantify these effects we use as a case study the Planck sample of 188 SZselected clusters with measured redshifts. We performed a Monte Carlo Markov Chains analysis for the parameters , , and , both splitting the sum of the neutrino masses between one and three massive species. Looking at the combination , the cold dark matter prescription provides a steeper degeneracy direction (higher ) which causes the mean value to lower. The shift can be quantified as and for one and three massive neutrino respectively, or in terms of the mean value as and . The offset has a low statistical significance for the cluster sample used in this work but it could entail a significant correction when the sample is combined with other probes or for large cluster samples that will be provided by future cluster surveys. Furthermore, taking into account such an effect has the consequences of exacerbating the tension between the cosmological parameters derived from CMB data and those of cluster number counts [17].
Acknowledgments
MC thanks Barbara Sartoris for the useful discussions. FVN thanks Weiguang Cui for helpful discussions about the identification of dark matter haloes. The calculations for this paper were performed on SOM2 and SOM3 at IFIC and on the COSMOS Consortium supercomputer within the DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of BIS and the University of Cambridge, as well as the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http:// www.hpc.cam.ac.uk/), provided by Dell Inc. using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England. This work has been supported by the PRININAF09 project “Towards an Italian Network for Computational Cosmology”, by the PRINMIUR09 “Tracing the growth of structures in the Universe”, by the PD51 INFN grant and by the Marie Curie Initial Training Network CosmoComp (PITNGA2009238356) founded by the European Commission through the Framework Programme 7. FVN and MV are supported by the ERC Starting Grant CosmoIGM. J.Q. X. is supported by the National Youth Thousand Talents Program and the Grants No. Y25155E0U1 and No. Y3291740S3. ES was supported in part by NSFAST 0908241.
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