L Carbone

Guzzo

M Zennaro

J Bel

J Dossett

C Carbone

L Guzzo

Cosmological constraints from galaxy clustering in the presence of massive neutrinos

Keywords: bias, clustering, neutrinos

Cosmological constraints from galaxy clustering in the presence of massive neutrinos

INTRODUCTION

Present-time as well as forthcoming galaxy surveys, while on the one hand will allow us to reach unprecedented precision on the measurement of the galaxy clustering in the universe, on the other hand will challenge us to produce more accurate and reliable predictions. The effect of massive neutrinos on the clustering properties of galaxies, that in the past has been either neglected or considered as a nuisance parameter, is nowadays regarded as one of the key points to be included in the cosmological model in order for it to reach the required accuracy. At the same time, while allowing for more realistic predictions of cosmological observables, this process also helps in shedding light on some open issues of fundamental physics, such as the neutrino total mass or the hierarchy of their mass splitting.

From the experiments measuring neutrino flavour oscillations, particle physics has been able to draw a constraint on the mass splitting of the massive eigenstates of neut-rinos, and set a lower bound to the total neutrino mass, Mν = mν,i 0.06 eV at 95% level (Gonzalez-Garcia et al. 2012, 2014;Forero et al. 2014;Esteban et al. 2017).

On the other hand, the absolute scale of magnitude of neutrino masses is still an open issue. Beta decay experiments such as the ones carried out in Mainz and Troitsk have set as an upper limit at 95% level on the electron neutrino mass of m(νe) < 2.2 eV (Kraus et al. 2005). While future experiments like Katrin prospect much higher sensitivities, of order 0.2 eV (Bonn et al. 2011), present day cosmology can already intervene in the debate about neutrino mass.

Since neutrinos are light and weakly interacting, they decouple from the background when still relativistic. Therefore, even at late times they are characterised by large random velocities that prevent them from clustering on small scales. As a consequence, neutrinos introduce a characteristic scale-dependent and redshift-dependent suppression of the clustering, whose amplitude depends on the value of their mass. In fact, the presence of massive neutrinos influences the evolution of matter overdensities in the universe, depending on their mass. Last years have seen a large number of works extensively studying the interplay between cosmology and neutrino physics (see, for instance, Lesgourgues & Pastor 2006, 2012, 2014, and references therein).

As we aim to describe the clustering of galaxies in cosmologies with massive neutrinos, we have to cope with the description of the galaxy-matter bias. As a matter of fact, galaxies do not directly probe the matter distribution in the universe, being in fact a discrete sampling of its highest density peaks. We choose to describe galaxy clustering through a recently introduced observable that, on sufficiently large scales, does not depend on the galaxy-matter bias, the clustering ratio (Bel & Marinoni 2014).

In standard ΛCDM cosmologies, this observable has already been proved to be a reliable cosmological probe for constraining cosmological parameters, being particularly sensitive to the amount of matter in the universe, as shown in Bel et al. (2014).

In this work we aim at studying how the clustering properties of galaxies are modified by the presence of neutrinos, and in particular we want to extend the clustering ratio approach to cosmologies including massive neutrinos. By proving that this observable maintains its properties, we want to exploit it to constrain the total neutrino mass. This paper will be organized as follows. In Sec. 2 we will introduce the statistical observable we are going to use, the clustering ratio, and its properties. We will show why this observable can be considered unaffected either by the galaxy-matter bias on linear scales and redshift-space distortions, and we will introduce its estimators.

In Sec. 3 we will describe the effects of massive neutrinos on the matter and galaxy clustering. We will introduce the DEMNUni simulations, the set of cosmological simulations we use to test the properties of the clustering ratio in a cosmology with massive neutrinos. Finally we will show that the properties of the clustering ratio hold as well in cosmologies that include massive neutrinos, in particular confirming the independence of the clustering ratio from bias and redshift-space distortions on linear scales in the DEMNUni simulations.

Sec. 4 is devoted to presenting our results. We use a measurements of the clustering ratio in the Sloan Digital Sky Survey Data Release 7 and 12 to draw a constraint on the total neutrino mass and on the equation of state of dark energy. In particular we study the joint posterior distribution of the parameters of the model, including Mν and w, obtained from the clustering ratio measurement and the latest cosmic microwave temperature and polarization anisotropy data from the Planck satellite.

CLUSTERING RATIO

In order to describe the statistical properties of the matter distribution in the universe, we introduce the overdensity field δ(x, t) = ρ(x, t)

ρ(t) -1, (1)
where ρ(x, t) is the value of the matter density at each spatial position, while ρ(t) represents the mean density of the universe. This is assumed to be a random field with null mean. Information on the distribution must therefore be sought in its higher order statistics, such as the variance σ 2 = δ 2 (x) c and the 2-point autocorrelation function ξ(r) = δ(x)δ(x + r) c of the field. Here • c denotes the cumulant moment, or connected expectation value (Fry 1984).

In this work we will always consider the matter distribution smoothed on a certain scale R by evaluating the density contrast in spherical cells, i.e.

δR(x) = δ(x )W |x -x | R d 3 x , ( 2
)
where W is the spherical top-hat window function. As a consequence, the variance and correlation function will be smoothed on the same scale, and will be denoted σ 2 R and ξR(r).

An equivalent description of the statistical properties of the matter field can be obtained in Fourier space in terms of the matter power spectrum. Starting from the Fourier transform of the matter overdensity field,

δ(k) = d 3 x (2π) 3 e -ik•x δ(x) , (3)
the matter density power spectrum is defined according to

δ(k1) δ(k2) = δ D (k1 + k2)P (k1) , (4)
while the adimensional power spectrum can be written as ∆ 2 (k) = 4πP (k)k 3 . The variance and correlation function of the matter field are linked to its power spectrum, representing in fact different ways of filtering it. The variance is the integral over all the modes, modulated by the Fourier transform of the filtering function Ŵ ,

σ 2 R = ∞ 0 ∆ 2 (k) Ŵ 2 (kR) d ln k, (5)
and the correlation function is, in addition, modulated by the zero-th order spherical Bessel function j0(x) = sin(x)/x,

ξR(r) = ∞ 0 ∆ 2 (k) Ŵ 2 (kR)j0(kr) d ln k. (6)
The explicit expression of Ŵ (kR) is

Ŵ (kR) = 3 kR j1(kR) = 3 sin(kR) -kR cos(kR) (kR) 3 . (7)
However, in practice, we are not able to directly access the matter power spectrum. The reason is that the galaxies we observe do not directly probe the distribution of matter in the universe. In fact, they represent a discrete biased sampling of the underlying matter density field and the biasing function is, a priori, not known. A way to overcome this problem is to refine the independent measurements of the bias function through weak lensing surveys. Otherwise, one can parametrise the bias adding additional nuisance parameters to the model and, consequently, marginalize over them.

On the other hand, a completely different approach is to seek new statistical observables, which can be considered to be unbiased by construction. This is the path followed by Bel & Marinoni (2014) by introducing the clustering ratio,

ηR(r) ≡ ξR(r) σ 2 R , (8)
that is the ratio of the correlation function over the variance of the smoothed field.

We assume that the relation between the matter density contrast and the galaxy (or halo) density field is a local and deterministic mapping, which is enough regular to allow a Taylor expansion (Fry & Gaztanaga 1993) as

δg,R = F (δR) N i=1 bi i! δ i R . (9)
Moreover we assume the growth of fluctuations to occur hierarchically (see Bernardeau et al. 2002;Bel & Marinoni 2012), so that each higher order cumulant moment can be expressed according to powers of the variance and 2-point correlation funtion,

δ n R c = Snσ 2(n-1) R , δ n i,R δ m j,R c = CnmξR(r)σ 2(n+m-2) R , (10)
the former applying to the 1-point statistics and the latter to the 2-point ones.

It has been shown by Bel & Marinoni (2012) that the bias function only modifies the clustering ratio of galaxies at the next order beyond the leading one and that it is not sensitive to third order bias

ηg,R(r) ηR(r) + 1 2 c 2 2 ηR(r)ξR(r) -(S3,R -C12,R)c2 + 1 2 c 2 2 ξR(r) (11)
where c2 ≡ b2/b1. By choosing a sufficiently large smoothing scale, the higher order contribution in Eq. ( 11) becomes negligible and we obtain

ηg,R(r) = ηR(r), (12)
meaning that in this case the clustering ratio of galaxies can be directly compared to the clustering ratio predicted for the matter distribution.

The local biasing model is not the best way of describing the bias function between matter and haloes/galaxies (Mo & White 1996;Sheth & Lemson 1999;Somerville et al. 2001;Casas-Miranda et al. 2002). It can be improved by introducing a non-local component depending on the tidal field. However, it has been shown by Chan et al. (2012) and by Bel et al. (2015) that, when dealing with statistical quantities which are averaged over all possible orientations, then the non-local component is degenerate with the second order bias c2, thus expression (11) remains valid and we do not consider non-local bias in our analysis.

On linear scales, the clustering ratio is expected to be independent from redshift. Since, in a ΛCDM universe, we can write (normalizing all quantities to the present time variance on the scale r8 = 8 h -1 Mpc, σ 2 8 (z = 0)) the evolution of the variance and the correlation function as

σ 2 R (z) = σ 2 8 (z = 0)D 2 (z)FR ξR(r, z) = σ 2 8 (z = 0)D 2 (z)GR(r), (13)
where D(z) is the linear growth factor of matter density fluctuations and

FR = ∞ 0 ∆ 2 (k) Ŵ 2 (kR)d ln k ∞ 0 ∆ 2 (k) Ŵ 2 (kr8)d ln k GR(r) = ∞ 0 ∆ 2 (k) Ŵ 2 (kR)j0(kr)d ln k ∞ 0 ∆ 2 (k) Ŵ 2 (kr8)d ln k
, depend only on the shape of the power spectrum. Hence, the clustering ratio ξR(r)/σ 2 R = GR(r)/FR, does not depend on D(z), which cancels outs.

In practice, we include weak nonlinearities which introduce a small, but nevertheless detectable, redshift dependence.

Not only measures of the clustering of galaxies are biased with respect to predictions for the matter field, but they also are affected by the peculiar motion of galaxies. This motion introduces a spurious velocity component (along the line-of-sight) that distorts the redshift assigned to galaxies. Since, for the clustering ratio, we are interested in large smoothing scales and separations, we can focus on the linear scales, where the only effect is due to the coherent motion of infall of galaxies towards the overdense regions in the universe.

We can link the position of a galaxy (or dark matter halo) in real space to its apparent position in redshift-space. Let us denote r as the true comoving distance along the line-of-sight; in redshift space it becomes

s = r + v p (1 + z) H(z) êr, (14)
where v p is the line-of-sight component of the peculiar velocity and êr is the line-of-sight versor. Considering the Fourier space decomposition of the density contrast, the relation linking its value in redshift space to the one in real space (Kaiser 1987) is

δ s (k) = (1 + f µ 2 )δ(k), (15)
where quantities in redshift space are expressed with the superscript s and µ is the cosine of the angle between the wavemode k and the line-of-sight. Here f is the so called growth rate, defined as the logarithmic derivative of the growth factor of structures with respect to the scale factor, f ≡ d ln D/d ln a. Averaging over all angles ϑ, the variance and the correlation function in redshift space result modified by the same multiplicative factor

σ s 2 R = Kσ 2 R ξ s R (r) = KξR(r) (16)
where K = 1 + 2f /3 + f 2 /5 is the Kaiser factor. As a consequence, the clustering ratio is unaffected by redshift-space distortions on linear scales. This argument allows us to rewrite the identity (12) as

η s g,R (r) ≡ ηR(r), (17)
meaning that, by properly choosing the smoothing scale R and the correlation length r, measures of the clustering ratio from galaxies in redshift space can be directly compared to predictions for the clustering ratio of matter in real space.

Estimators

The clustering ratio can be estimated from count-in-cells, where, under the assumption of ergodicity, all ensemble averages become spatial averages. We follow the counting process set up by Bel & Marinoni (2012), we define the discrete density contrast as

δN,i = Ni N -1, ( 18
)
where Ni is the number of objects in the i-th cell and N is the mean number of objects per cell. The estimator of the variance is therefore

σ2 R = 1 p p i=1 δ 2 i ( 19
)
and the one of the correlation function is

ξR(r) = 1 pq p i=1 q j=1 δiδj (20)
leading to the definition of the estimator of the clustering ratio as

ηR(r) = ξR(r) σ2 R . ( 21
)
Throughout this work we will often express the correlation length r as a multiple of the smoothing scale, i.e. r = n R.

Since we are dealing with a discrete counting process, the shot noise needs to be properly accounted for. We follow the approach of Bel & Marinoni (2012) and correct the estimator of the variance according to

σ2 R = δ 2 n (x) - 1 N = 1 p p i=1 δ 2 i - 1 N , ( 22
)
where N is the mean number of objects per cell. On the other hand, the correlation function needs no correction, as long as the spheres do not overlap.

Effects of massive neutrinos

We introduce massive neutrinos as a subdominant dark matter component. For simplicity, we consider three degenerate massive neutrinos, with total mass Mν = i mν,i and present-day neutrino energy density in units of the critical density of the universe Ων,0h 2 = Mν /(93.14 eV). The neutrino fraction is usually expressed with respect to the total matter as fν = Ων /Ωm. For a more complete treatment of neutrinos in cosmology, we refer the reader to Lesgourgues & Pastor (2006, 2012, 2014). Neutrinos of sub-eV mass, which seem to be the most likely candidates both from particle physics experiments and cosmology, decouple from the primeval plasma when the weak interaction rate drops below the expansion rate of the universe, at a time when the background temperature is around T 1 MeV. This corresponds to a redshift 1 + z dec ∼ 10 9 . Since the redshift of their non-relativistic transition, obtained equating their rest-mass energy and their thermal energy, is given by

1 + znr 1890 mν,i 1 eV , (23)
when neutrinos decouple, they are still relativistic. As a consequence, since the momentum distribution of any species is frozen at the time of decoupling, neutrino momenta keep following a Fermi-Dirac distribution even after their nonrelativistic transition, and neutrinos end up being characterised by a large velocity dispersion. An effective description of the evolution of neutrinos can be achieved employing a fluid approximation (Shoji & Komatsu 2010). In this framework we can define a neutrino pressure, pν = wν ρν c 2 , computed integrating the momentum distribution. Such pressure is characterised by an effective adiabatic speed of sound (Blas et al. 2014)

cs,i = 134.423 (1 + z) 1eV mν,i km s -1 , (24)
that represents the speed of propagation of neutrino density perturbations. Such speed of sound defines the minimum scale under which neutrino perturbations cannot grow, called the free streaming scale. It corresponds to a wavenumber

kF S (z) = 4πGρa 2 c 2 s 1/2 = 3 2 H 2 Ωm(z) (1 + z) 2 c 2 s 1/2 , ( 25
)
or a proper wavelength

λF S = 2πa/kF S . (26)
At each redshift, neutrino density fluctuations of wavelength smaller than the free streaming scale are suppressed, their gravitational collapse being contrasted by the fluid pressure support. As a consequence, neutrinos do not cluster on small scales and remain more diffuse compared to the cold matter component.

Neutrino free streaming does not only affect the evolution of neutrino perturbations, in fact it affects the evolution of all matter density fluctuations. We can model the growth of matter fluctuations employing a two-fluid approach (Blas et al. 2014;Zennaro et al. 2017). In this case, the solution of the equations of growth for the neutrino and cold matter fluids are coupled,

     δcb + H δcb - 3 2 H 2 Ωm {fν δν + (1 -fν )δ cb } = 0 δν + H δν - 3 2 H 2 Ωm fν - k 2 k 2 F S δν + (1 -fν )δ cb = 0, (27)
where derivatives are taken with respect to conformal time, dτ = dt/a, and both the Hubble function and the matter density parameter are functions of time, H = H(τ ) and Ωm = Ωm(τ ).

The coupling of these equations requires the evolution of the CDM density contrast to be scale dependent, unlike in standard ΛCDM cosmologies. We therefore expect to find an observable suppression even in the CDM+baryon power spectrum, starting from the mode corresponding to the size of the free-streaming scale at the time of the neutrino nonrelativistic transition, knr = kF S (znr), and affecting all the scales smaller than this one.

CLUSTERING RATIO WITH MASSIVE NEUTRINOS

In order to investigate the behaviour of the clustering ratio in cosmologies with massive neutrinos, i.e. whether it maintains all the properties described in Sec. 2, we analyse the cosmological simulations "Dark Energy and Massive Neutrino Universe" (DEMNUni), presented in Castorina et al. (2015) and Carbone et al. (2016). These simulations have been performed using the Gadget-III code by Viel et al. (2010) based on the Gadget simulation suite (Springel et al. 2001;Springel 2005). This version includes three active neutrinos as an additional particle species1 .

The DEMNUni project comprises two set of simulations. The first one, which is the one considered in the present work, includes 4 simulations, each implementing a different neutrino mass. Besides the reference ΛCDM simulation, which has Mν = 0 eV, the other ones are characterised by Mν = {0.17, 0.30, 0.53} eV. The second set includes 10 simulations, exploring different combinations of neutrino masses and dynamical dark energy parameters.

All simulations share the same Planck-like cosmology, with Hubble parameter H0 = 67 km s -1 Mpc -1 , baryon density parameter Ω b = 0.05, primordial spectral index ns = 0.96, primordial amplitude of scalar perturbations As = 2.1265 × 10 9 (at a pivotal scale kp = 0.05 Mpc -1 ) and optical depth at the time of recombination τ = 0.0925. The density parameter of the cold dark matter, Ω cdm , is adjusted in each simulation, depending on the neutrino mass, so that all simulations share the same total matter density parameter Ωm = 0.32, see Tab. 1. Each simulation follows the evolution of 2048 3 CDM particles and, when present, 2048 3 neutrino particles, in a comoving cube of 2 h -1 Gpc side. The mass of the CDM particle is ∼ 8 × 10 10 h -1 M , and changes slightly depending on the value of Ω cdm . All simulations start at an initial redshift zin = 99 and reach z = 0 with 62 comoving outputs at different redshifts. In this work we focus on the snapshots at redshift z = 0.48551 and z = 1.05352.

Dark matter haloes have been identified through a Friend-of-Friends (FoF) algorithm with linking length b = 0.2 and setting the minimum number of particles needed to form a halo to 32. Thus, the least massive haloes have mass of about 2.6 × 10 12 h -1 M . In order to check the stability of our results regarding the choice of the definition of a halo we also have access to halo catalogues where haloes have been identified using spherical over-densities. For the purpose of the present work we constructed halo catalogues in redshift space by modifying the positions along the z direction according to the projected velocity (properly converted in length) in that direction (see Eq. 14).

Regarding error estimation, being these simulations very large, a jackknife method has been implemented by subdividing the box in 64 sub-cubes. The standard error on the measured value of ηR(r) is then taken to be the dispersion obtained from the jackknife process

σ 2 η R = Nj -1 Nj N j i=1 [ηR,i(r) -ηR(r)] 2 , ( 28
)
where Nj is the number of jackknife resamplings, in our case Nj = 64.

In the following, we first check the reliability of the clustering ratio in the presence of massive neutrinos. In particular we are interested in proving that the identity η z R,g (r) ≡ ηR(r) still holds. To this end, we must prove that the clustering ratio at the scales of interest does not depend on the galaxy-matter bias (so that we can compare predictions for matter and measures from galaxies) and that it is not affected by redshift-space distortions (to be safe when comparing the real space predictions to measures obtained in a galaxy redshift survey).

Bias sensitivity

In order to test the independence of the clustering ratio from the bias on linear scales, we divide the dark matter haloes in 9 mass bins, reported in Tab. 2. The various halo populations evolve in a different way, therefore they present different biasing functions with respect to the dark matter field. Thus, we will use the linear bias bL to characterise each halo sample. In Tab. 3 we show how both the FoFs and the spherical overdensities from the simulations populate these mass bins in the simulations. Due to the minimum number of particle required to identify a halo, the first two mass bins do not contain any. On the other hand, the only mass limit to the spherical over-densities is given by the mass resolution of the simulation, hence all the mass bins are populated.

In Fig. 1 we show the estimated correlation functions of each FoF sample in the two extreme cases of Mν = 0 eV and Mν = 0.53 eV (the same holds for the SOs as well).

We also represent the corresponding correlation function of the dark matter field, which is used to estimate the linear bias bL characterizing each halo sample:

bL ≡ ξ F oF R (nR) ξR(nR) . ( 29
)
We find that our cut in mass does indeed correspond to different tracers, with higher bias for higher-mass objects. However, such different halo populations still show a constant bias with respect to scale, which allows us to fit the measured bias in Fig. 1 with flat lines.

The independence of the FoF-matter bias from scale is confirmed also in the massive neutrino case (Fig. 1, right). In particular, we note here that the bias is generally higher when considering massive neutrinos. This is due to the fact that, since they smooth the matter distribution, neutrinos make haloes of a given mass rarer then in a standard ΛCDM cosmology.

Secondly we compute the clustering ratio for both the FoFs and the spherical overdensities at redshift z = 0.48551 and z = 1.05352. In each of these cases, we analyse the ΛCDM simulation, which does not include massive neutrinos, and the ΛCDMν simulations with Mν = {0.17, 0.30, 0.53} eV. In the two plots in Fig. 2 we show the clustering ratio for fixed smoothing radius R = 16 h -1 Mpc and correlation length r = 2R in the same mass bins shown in Tab. 2 for the FoFs and spherical overdensities respectively. Points are measurements in the simulations, while lines are the predictions obtained from the matter power spectrum. The ratio between measures and prediction is in the bottom panel. At z = 0.48551, the measured clustering ratio in the bins with masses < 30.43h -1 M agrees with the predictions for the matter at 3% level. Below 100 h -1 M the agreement is within 5%. For objects with mass greater than this, we observe a more scattered trend. We blame a lower statistical robustness, due to fewer objects falling in these mass bins. In any case, we do not observe any peculiar dependence of the clustering ratio on the mass of the objects, confirming up to a few percent accuracy its independence on the bias at these scales.

Conversely, at z = 1.055 we do see a dependence of the FoF clustering ratio on the mass. The reason is likely to be a lack of mass resolution in the simulations, as moving towards higher redshifts not enough haloes have formed in the high-mass end of the mass function. For this reason, we expect this situation to be even more sever in the cases with masses neutrinos, which is confirmed in the right power of Fig. 2.

As we find similar results for both the FoFs and the spherical overdensities, we conclude that such results do not depend on the tracer we choose to observe.

Finally, we claim that the clustering ratio is insensitive to the bias on linear scales in a cosmology including massive neutrinos irrespective of either the mass of the tracer or the nature of the tracer itself or the total mass of neutrinos considered. This allows us to directly compare real-space predictions of the matter clustering ratio with real-space measures of the clustering ratio of any biased matter tracer, i.e. ηg,R(r) ≡ ηR(r).

Redshift space

In redshift space, as described in Sec. 2, the apparent position of galaxies is modified according to the projection of their peculiar velocity along the line-of-sight. This effect distorts the clustering properties of the distribution and we thus expect its correlation function and variance to be affected. However, we expect redshift-space distortions not to affect the clustering ratio on linear scales (Eq. 17) as the effect is cancelling out into the ratio. In order to verify the accuracy of this approximation we created the redshift space catalogues of the FoFs and spherical overdensities in the simulations, moving the positions of the tracers along an arbitrary direction, chosen as the line-of-sight direction.

Having shown that the clustering ratio does not depend on the way we define the haloes nor on their mass tracer, from now on we focus on the dark matter halo catalogues identified with the Friend-of-Friends algorithm and we compare the two extreme cases of Mν = 0 eV and Mν = 0.53 eV. Note that we use the Mν = 0 simulation as a reference for comparisons, since it has already been shown that these c 0000 RAS, MNRAS 000, 000-000

Constraints from clustering ratio with neutrinos 7 all the neutrino masses. The smoothing radius is R = 16 h -1 Mpc and the correlation length is twice the smoothing radius. Points are measures in the simulations, while lines are the theoretical predictions. At z = 0.48551 the agreement between measures and predictions is better than 3% in the bins with masses < 30.43h -1 M (i.e. where the bias is < 2), while in the bins with larger masses, corresponding to larger values of bias, the agreement is at 5% level. Errors are larger in the highest mass bin because it is more sparsely populated. At z = 1.05352 the discrepancy between measures and predictions shows a specific dependence on the mass of the tracer. We consider this effect to be due a mass resolution effect in the simulation. Finally we point out that we find similar results for FoF and SO, meaning that the clustering ratio at these scales is insensitive to the linear bias irrespective of the mass tracer we choose. c 0000 RAS, MNRAS 000, 000-000 properties are valid when neutrino are massless (see Bel & Marinoni 2014;Bel et al. 2014).

Fig. 3 shows, at the scales of interest, the independence of the clustering ratio from redshift-space distortions. The ratio between measurements of the clustering ratio in redshift and in real space is of order 1, either with and without massive neutrino, both at z = 0.48551 and z = 1.05352.

In particular we show that we recover the results already obtained in other works (Bel & Marinoni 2012) in the ΛCDM case. In the case including massive neutrinos we find an agreement between redshift and real space measurements at 3% level at redshift z = 0.48551 and at better than 1% on scales R 16 h -1 Mpc at z = 1.05352. In this case the accuracy is higher at higher redshift as the growth of structures is more linear.

Moreover we note that the agreement with predictions is better for the simulation with massive neutrinos with respect to the ΛCDM one. This is due to the fact that massive neutrinos lower the matter fluctuations (see the values of σ8,cc in Tab. 1) and therefore tend to reduce the velocity dispersion, resulting in redshift-space distortions that are more into the linear regime.

We therefore propose to use the clustering ratio as a cosmological probe to constrain the parameters of the cosmological model. As a matter of fact, the analysis of the simulations has shown that the clustering ratio, besides being independent from the matter tracer and the bias, is not affected by redshift-space distortions on linear scales. This implies that Eq. ( 17), η s g,R (r) ≡ ηR, still holds in the presence of massive neutrinos, allowing us to directly compare clustering ratio measurements in redshift-survey galaxy catalogues to the theoretical matter clustering ratio predictions.

RESULTS

Optimisation

In order to estimate and predict the clustering ratio of galaxies, we need to choose two different scales: the smoothing scale R, i.e. the radius of the spheres we use for counting objects, and the correlation length r, that for simplicity we assume to be some multiple of the smoothing scale, r = nR. Choosing the best combinations of R and r is seminal to maximise the information we can extract from this statistical tool.

The smoothing scale R controls the scale under which we make our observable blind to perturbations. A sufficiently large value of R allows us to screen undesired nonlinear effects, that would compromise the effectiveness of the clustering ratio. On the other hand, an excessively large smoothing scale can lead to more noisy measurements, since in the same volume we can accommodate fewer spheres. Moreover, if R is too large, the entire signal would be screened and the measurement would become of little interest.

Also for the correlation length, choosing small values of R and n implies coping with small-scale nonlinearities, which risk to invalidate the identity expressed in Eq. ( 17). Large values of correlation distances, however, would make it difficult to accommodate enough couples of spheres in the volume to guarantee statistical robustness. An additional constraint comes from the strategy we adopt to fill

η s R (r)/η r R (r) 10 15 20 R [h -1 Mpc]
Figure 3. The clustering ratio smoothed on the scale R and at correlation length r = nR, n = 2 as a function of the smoothing scale. We show in red the measures in the ΛCDM simulation and in blue the ones in the simulation with the highest neutrino mass, Mν = 0.53 eV, which are the two extreme cases. Filled dots are measures in real space, while empty dots in redshift space. In the bottom panel the ratio between the clustering ratio in redhift space over the real space case is shown. Since on linear scales the monopole contribution coming from redshift-space distortions enhances the correlation function and the variance by the same multiplicative factor, we expect the clustering ratio to be unaffected. The ratio between redshift and real space measurements is, in fact, of order 1 with an accuracy better then 3%. This is even better confirmed in the case at redshift z = 1.055 (right) because at higher redshift the matter growth is more linear.

the volume with spheres and perform the count-in-cells. In this framework, if the correlation length is below twice the smoothing scales, r < 2R, the spheres of our motif of cells would overlap, resulting in an additional shot-noise contribution. For this reason, we only allow values of n 2.

The main information we want to extract is the total neutrino mass. The sensitivity of the clustering ratio to this parameter can be quantified as an effective signal-to-noise ratio, defined as

S/N = η ν R (r) -η Λ R (r) σ Λ R , (30)
where η ν R (r) is the clustering ratio measured in a simulation with neutrino mass Mν , η Λ R (r) is measured in the reference ΛCDM simulation and σ Λ R is the uncertainty on the clustering ratio measured in the ΛCDM simulation. This quantity measures how much a massive neutrino cosmology is distinguishable from a ΛCDM one, given the typical errors on the measures of the clustering ratio for the specific volume and number density of tracers, as a function of R and r. Fig. 4 shows the (n, R) plane, constructed as a grid with correlation lengths n ∈ [2, 2.75] with step ∆n = 0.05 and smoothing scales R ∈ [15, 30] with step ∆R = 1, all distances being expressed in units of h -1 Mpc. At each point on the grid a color is associated, representing the value of this effective signal-to-noise ratio. The effect of massive neutrinos is, as expected, appreciable on small scales (both small n and small R), and eventually becomes negligible moving towards large scales. c 0000 RAS, MNRAS 000, 000-000

Constraints from clustering ratio with neutrinos 9 M ν = 0.17 eV, z = 0.48551 0.5000 1.0000

1.4518 [η ν R (r) -η Λ R (r)]/σ Λ η Figure 4
. The effect of neutrinos on the clustering ratio compared to a ΛCDM cosmology. In the (n, R) plane, we plot colour contours corresponding to (η R (r, ν) -η R (r, ΛCDM)/ση(ΛCDM).

As expected, the sensitivity to the neutrino total mass increases at small smoothing scales and correlation lengths (red regions).

While we want to maximise the effects of neutrinos, we want to minimise errors. In particular, we can define a theoretical error, that accounts for the combinations of smoothing radii R and correlation lengths r where the assumptions under which we can apply the identity expressed in Eq. ( 17) break down. Such theoretical error can be quantified as

δ th = ηR(r) -η th R (r) ση , (31)
where ηR(r) is the clustering ratio measured in the simulation with a given cosmology, η th R (r) is the prediction obtained with a Boltzmann code, and ση the uncertainty on the measurement. In Fig. 5 we show as a color map the values that we obtain for this theoretical error in the same (n, R) plan introduced above. We can see that on very small scales the effect of nonlinearities is not negligible, and we cannot use the clustering ratio as an unbiased observable.

From Fig.s 4 and 5 we see that we need to balance between the requirement coming from the signal-to-noise ratio (that is maximum on small scales), and those from the theoretical errors (that is minimum on large scales). We introduce, therefore, a way of combining these pieces of information into a single colour map, which we use to seek the sweet-spots, in this parameter space, where both conditions are satisfied.

First, we define a combined percentage error (that accounts both for statistical errors and discrepancies from the model) as

δ combined = η ν R (r) -η ν,th R (r) η ν R (r) -η Λ R (r) σ Λ η η ν R (r) -η Λ R (r) . ( 32
)
The quantity in the first parenthesis is related to how much the statistical error is important with respect to the effects of neutrinos, while the second parenthesis is a weight that accounts for the typical uncertainty on the measurement in each bin of n and R. Finally, we define the neutrino contrast M ν = 0.17 eV, z = 0.48551 0.500 1.000 1.500 2.000 3.508 as

[η ν R (r) -η th R (r)]/σ η ν
C = S/N max(S/N) - δ combined max(δ combined ) . ( 33
)
Here we have normalised the signal/noise defined in Eq. ( 30) and the combined error defined in Eq. ( 32) to their respective maxima (on the considered grid) and we are interested in finding the regions where this contrast is dominated by the signal/noise, i.e. where C(n, R) ∼ 1. In Fig. 6 we show the neutrino contrast on the (n, R) grid, for the simulation with Mν = 0.17 eV at redshift z = 0.48551. We have repeated this analysis for the three massive neutrino simulations (with Mν = 0.17, 0.30, 0.53 eV) and for different redshifts, spanning the range from z = 0.48551 to z = 2.05053. Our conclusion is that the combination R = 22 h -1 Mpc, n = 2.1 is the most viable candidate for all these cosmologies and redshifts.

Likelihood

We aim at comparing measurements and predictions of the clustering ratio, in order to find the set of parameters of the model that maximizes the likelihood. We exploit the different dependence of measurements and predictions on the cosmological model. In particular, the measured value of the clustering ratio depends on the way we convert redshifts and angles into comoving distances, that depends on the total matter density Ωm, on the dark energy fraction ΩΛ and on the background expansion rate H(z).

On the other hand, the theoretical prediction for the clustering ratio depends on the entire cosmological model and is therefore sensitive also to the value of the total neutrino mass Mν .

We choose six baseline free parameters in our analysis, M ν = 0.17 eV, z = 0.48551 -0.526 0.000 0.500 0.786 Neutrino Contrast Figure 6. In this colour plot we subtract to the neutrino signalto-noise normalised to 1 a combined error normalised in the same fashion. Details on the definition are in the text. We are interested in the regions in the (n, R) plane where the neutrino signal-to-noise dominates (∼ 1) on the error (∼ 0), graphically visible as hot spots. The region with smoothing scales 20 < R < 23 h -1 Mpc seems to be the most promising. In particular, by repeating this test for different redshifts and neutrino masses, we chose as our candidate scales R = 22 h -1 Mpc, n = 2.1.

namely the baryon and cold dark matter density parameters Ω b h 2 and Ω cdm h 2 , the Hubble parameter H0, the optical depth at the recombination epoch τ , the amplitude of the scalar power spectrum at the pivotal scale As and the scalar spectral index ns. Moreover, we extend this parametrization with two additional free parameters, the total neutrino mass Mν and the equation of state of the dark energy fluid w. The most general vector of parameters therefore is p = {Ω b h 2 , Ω cdm h 2 , H0, τ, As, ns, Mν , w}.

We follow Bel & Marinoni (2014), who showed that the likelihood function of the clustering ratio (given a fixed set of parameters) is compatible with being a Gaussian. Therefore we will compute the logarithmic likelihood as ln L = -χ 2 /2 (apart from a normalization term) where

χ 2 (p) = i (ηR,i(r) -η th R,i (r)) 2 σ 2 η i , ( 34
)
where we neglect the covariance between the different redshift bins.

We account for the dependence of the measurements on the cosmological model assumed, induced by the cosmologydependant conversion of redshifts into distances, whenever we compare measurements of the clustering ratio (obtained in the fiducial cosmology) to its predictions (in a generic cosmology). That is, when computing the likelihood for the set of parameters ϑ, we must keep in mind that the measured value has been computed in a different cosmology, the one with the fiducial set of parameters ϑ F .

We keep the measurements fixed in the fiducial cosmology and rescale the predictions accordingly. We consider that, due to Alcock-Paczyński effect, at same redshift and angular apertures we can associate different lengths depending on cosmology (Alcock & Paczyński 1979).

Our measurements depend on distances only through the smoothing scale R. This is because the correlation length is always expressed as a multiple of the smoothing scale, r = nR. This means that, since the measure has been obtained in the fiducial cosmology using spheres of radius R F = 22 h -1 Mpc, they need to be compared to predictions obtained in a generic cosmology using a smoothing length R = αR F , where α is our Alcock-Paczyński correction.

We write the Alcock-Paczyński correction α as (Eisenstein et al. 2005)

α = E F (z) E(z) DA D F A 2 1/3 , ( 35
)
where E(z) ≡ H(z)/H0 is the normalised Hubble function and DA the angular diameter distance. Therefore, we are going to compare

η F,s g,R (nR) ≡ ηαR(nαR), (36)
the left hand side of Eq. ( 36) being the clustering ratio of galaxies measured in redshift space assuming the fiducial cosmology, while the right hand side is the predicted clustering ratio for matter in real space, rescaled to the fiducial cosmology to make it comparable with observations. In order to efficiently explore the parameter space we have modified the public code CosmoMC (Lewis & Bridle 2002), adding a likelihood function that implements this procedure.

Constraints using SDSS data

We measure the clustering ratio in the 7 th (Abazajian et al. 2009) and 12 th (Alam et al. 2015) data release of the Sloan Digital Sky Survey (SDSS) by smoothing the galaxy distribution with spherical cells of radius R and counting the objects falling in each cell. We divide the sample into three redshift bins that have mean redshifts z = {0.29, 0.42, 0.60}. The first redshift bin is extracted from the DR7 catalogue, while the two bins at higher redshift come from the DR12 catalogue, after removing the objects already present in the other bin.

To perform the count-in-cell procedure, we convert redshifts into distances, assuming a cosmology with H0 = 67 km s -1 Mpc -1 , Ωm = 0.32 and in which we fix the geometry of the universe to be flat, Ω k = 0, forcing ΩΛ = 1 -Ωr -Ωm. Therefore, this is to be considered our fiducial cosmology. We compute the clustering ratio using the estimators presented in Sec. 2.1, employing our optimised smoothing size and correlation length, R = 22 h -1 Mpc and r = 2.1 R. Our measures of the clustering ratio in each redshift bin are at 0.15 z 0.43, ηg,R(r) = 0.0945 ± 0.0067, at 0.30 z 0.53, ηg,R(r) = 0.0914 ± 0.0055, at 0.53 z 0.67, ηg,R(r) = 0.1070 ± 0.0110. Details on the computation of the clustering ratio and its errors in the SDSS catalogue can be found in Bel et al. (2015), where, though, measures are performed assuming a different fiducial cosmology. In Fig. 7 we show, for some relevant parameters, the joint posterior distribution obtained fitting at the same time the Planck temperature and polarization data and the clustering ratio measurements in SDSS DR7 and DR12, leaving free to vary the six baseline parameters and the total neutrino mass Mν . Already by eye, adding the clustering ratio to the CMB information does not seem to improve much the upper bound of the total neutrino mass parameter. In general, the most significant improvement seems to occur on the constraint of the cold dark matter density parameter.

Moreover, we have also checked how constraints change when we leave the equation of state of dark energy, w, as an additional free parameter. As a matter of fact, w is known to be strongly degenerate with the other parameters of the model, when only CMB data are used. In general, we need information from a geometrical probe sensitive to the late time universe in order not to find non-physical solutions. Fig. 8 shows that the clustering ratio is indeed able to break such degeneracy.

Also the combination of other cosmological probes can help breaking degeneracies and tightening constraints. For this reason we compare the constraining power of the clustering ratio to that of two other observables, the fit of the BAO peak in the correlation function measured by the BOSS collaboration in the DR11 CMASS and LOWZ datasets (Anderson et al. 2014) and the lensing of the CMB signal due to the intervening matter distribution between the last scattering surface and us, where the amplitude of the lensing potential, AL, has been kept fixed to 1 (Planck Collaboration et al. 2016).

In the first part of Tab. 4 we show the mean, 68% and 95% levels obtained for the different parameters combining the likelihoods presented above. To better show the beha-viour of the clustering ratio with respect to the other probes considered, in Fig. s 9-10, we focus particularly on the parameters w, Mν and H0.

In general, adding the clustering ratio considerably improves on the parameter constraints obtained with CMB data alone, especially when also w is free to vary. In particular, the clustering ratio is able to break the degeneracy between w and the other cosmological parameters, that affects the constraints drawn with the sole CMB data. On the other hand, the clustering ratio does not seem to improve much the constraint on the Mν parameter, especially when compared to probes such as the CMB lensing and the BAO peak position.

The clustering ratio proves to be extremely sensitive to the cold dark matter fraction Ω cdm h 2 , as adding the clustering ratio to the CMB analysis results in a 12% improvement on the 95% confidence level.

To improve our understanding of the results presented in the previous section, we investigate how well the clustering ratio allows us to recover a certain known cosmology.

To this purpose, we use the measurements of the clustering ratio in one of the DEMNUni simulations, the one with Mν = 0.17 eV, which represents the closest value to the current available constraints on the neutrino total mass. The clustering ratio is measured in the simulation at the same redshifts, and with the same binning, as in the SDSS data. The error on each measurement in the simulation is taken to be the one obtained from the SDSS measurements.

The likelihood using the CMB data is computed in this case fixing the bestfits to the values of the parameters in the cosmology of the simulation, and employing the covariance matrix contained in the publicly available Planck data release.

The posterior distribution obtained with this procedure is shown in Fig. 11, while the second part of Tab. 4 summarizes the improvements on the constraints on the parameters that we obtain adding the clustering ratio. We correctly recover the bestfits of our known cosmology, with errors comparable with the true ones. We conclude that the reason why we did not achieve a significant improvement on the constraint on the total neutrino mass using the SDSS data resides in the fact that the clustering ratio requires smaller error bars to be effective in constraining such parameter. We can therefore expect that, with upcoming, large galaxy redshift surveys, the clustering ratio will reach a larger constraining power.

We test such an hypothesis in the next section, analysing the clustering ratio expected for a Euclid-like galaxy redshift survey, in combination with CMB data.

Forecasts for a Euclid-like galaxy redshift survey

In order to forecast the constraining power of the clustering ratio, expected from a future, Euclid-like galaxy redshift survey, we construct the synthetic clustering ratio data in the following way:

• We imagine to have 14 redshift bins, from z = 0.7 to z = 2, with ∆z = 0.1

• In each redshift bin, the synthetic measurement of the clustering ratio is given by the predicted clustering ratio (computed using a Boltzmann code), to which we add a small random noise (within 1 standard deviation).

• We measure the errors (at the same redshifts) in the DEMNUni simulations; the errors in the simulations are then rescaled, according to the operative formula presented below, to match the volume and number density of our Euclid-like survey.

The relative error on the clustering ratio depends on the volume and number density of the sample, and can be parametrised, following Bel et al. (2015), as δη η = AV -1/2 exp 0.14 ln ρ -ln 2 ρ 2 ln(0.02) (37) where V is the volume expressed in h -3 Mpc 3 , ρ is the object number density in h 3 Mpc -3 and A is a normalization factor computed with the reference volume and number density.

We use these data to explore the posterior distribution of the parameters of the model. The results are shown in Fig. 12, and the constraints are shown in the last part of Tab. 4. c 0000 RAS, MNRAS 000, 000-000

Constraints from clustering ratio with neutrinos 13 In this case there is a much larger improvement on the constraints of all the parameters. The neutrino total mass parameter improves by 14% on the 95% limit with respect to using Planck alone. Most notably, the constraint on the cold dark matter density parameter, Ω cdm h 2 , improves by over 40%. Also the spectral index ns shows a 10% improvement and the constraint on the Hubble constant H0 improves by 20%.

This means that, when new data, covering a larger volume, will be available, clustering ratio measurements are expected to contribute with a significant improvement on the constraints on the parameters of the cosmological model.

We also note that, as more different observations are carried out, it becomes very interesting to enhance the constraining power of the clustering ratio also combining its measurements in different datasets. This can be easily done since the clustering ratio is a single measurement, thus scarcely dependent on the survey geometry.

SUMMARY AND CONCLUSION

Neutrino effects are being increasingly included in cosmological investigations, becoming in fact part of the standard cosmological model. Thanks to these investigations, the description of the statistical properties of the universe is gaining the precision required by forthcoming experiments and, at the same time, neutrino physics gains tighter constraints.

In this work we have considered the clustering ratio, an observable defined as the ratio between the smoothed correlation function and variance of a distribution, and extended its range of applicability to cosmologies that include a massive neutrino component. As a matter of fact, the clustering ratio, which has already been tested in ΛCDM cosmologies including only massless neutrinos, is unbiased and independent from redshift-space distortions on linear scales. As massive neutrinos introduce characteristic scale dependencies in the clustering of galaxies (and matter), such peculiar properties of the clustering ratio needed to be confirmed (or denied) in this cosmological framework.

We divided our analysis into two steps: first, we studied the properties of the clustering ratio in simulations that include massive neutrinos; afterwards, we used the clustering ratio to compute the likelihood of the parameters of the cosmological model, using both real data and forecasts of future data.

In the first part of this work, we employed the DEM-NUni simulations to test the clustering ratio in the presence of massive neutrinos. These are the largest available simulations that include massive neutrinos as a separate particle species along with cold dark matter. We computed the clustering ratio using different tracers (dark matter FoF haloes and spherical overdensities), divided into different mass bins (spanning the interval from ∼ 6 × 10 11 to 10 14 h -1 M ), and we explore different choices of neutrino mass (Mν = {0, 0.17, 0.3, 0.53} eV) in real and redshift space.

From such analysis we conclude that the properties of the clustering ratio hold also in cosmologies with massive neutrinos. In particular its main property, the fact that the galaxy clustering ratio in redshift space is directly comparable to the clustering ratio predicted for matter in real space on a range of linear scales, is proven valid.

We have therefore moved to employing the clustering ratio as a cosmological probe to find the set of parameters of the model that maximizes the likelihood function, given a set of data. We have used the data from the SDSS DR7+DR12 catalogue. We have computed the clustering ratio in three redshift bins and used these measures in combination with the temperature and polarization anisotropies of the CMB measured by the Planck satellite to explore the likelihood in parameter space with an MCMC approach. We find that the clustering ratio is able to break the degeneracy, present in the CMB data alone, between the equation of state of dark matter, w, and the other parameters. Moreover it improves the 95% limit on the CDM density parameter by ∼ 12%. However, we do not find an appreciable improvement in the constraint on the neutrino total mass.

By analysing simulations we conclude that we blame such lack of improvement on the statistical errors, which, with current data, are not yet competitive enough. We have therefore tested the constraining power of the clustering ratio using the error bars expected from a Euclid-like galaxy survey.

In this case we find that not only does the clustering ratio greatly improve (with respect to using CMB data alone) the constraint on the CDM density parameter (shrinking the 95% limit by ∼ 40%) and on the Hubble parameter H0 (whose 95% limit improves by 20%), but it is also able to improve by ∼ 14% the 95% upper bound on the total neutrino mass.

In conclusion, the clustering ratio appears to be a valuable probe to constrain the parameters of the cosmological model, especially with upcoming large galaxy redshift surveys. Being easy to model and to measure, it provides us with a powerful tool to complement other approaches to galaxy clustering analysis, such as the measurement of the galaxy correlation function or power spectrum. Moreover, we note that, given the simplicity of combining the clustering ratio measured in different surveys, we expect its true constraining power to emerge when it will be measured in a number of different datasets. 4. Mean, 68% and 95% levels of the marginalised posterior distributions. In the first part of the table the datasets of Planck's CMB temperature and polarization anisotropies, BOSS measurement of the BAO peak, and Planck's CMB lensing signal are used in combination with the clustering ratio measured in the SDSS DR7+12 sample. In the second part of the table bestfits have been fixed to the ones of the fiducial cosmology, errors for Planck are obtained from the publicly available Planck parameter covariance matrix and error for the clustering ratio are the ones measured in SDSS DR7 and DR12. Finally, in the last part of the table, bestfits have been fixed to the ones of the fiducial cosmology, errors for Planck are obtained from the publicly available Planck parameter covariance matrix and errors for the clustering ratio are the ones predicted for a Euclid-like galaxy survey, following the procedure described in the text. c 0000 RAS, MNRAS 000, 000-000

Figure 1 .Figure 2 .

 12 Figure1. Correlation function of the Friend-of-Friends in the simulations with two different cosmologies. In the plot on the left there are the measurements in the ΛCDM cosmology, while in the plot on the right the ΛCDMν simulation with Mν = 0.53 eV. In both cases, left panel is at redshift z = 0.48551 and right panel at z = 1.05352. In the top panel we show the correlation function measured in the mass bins presented in Tab. 2 (points) compared to the theoretical smoothed matter correlation function (black solid line). In the bottom panel there is the FoF-matter bias, computed as b = ξ F oF R (r)/ξ R (r). The linear bias in the simulation with massive neutrinos is larger than in the standard ΛCDM case, because, as neutrinos reduce clustering, massive haloes become rarer. We the bias values with a straight line between R = 16 and 22 h -1 Mpc. As the fit shows, the linear bias is compatible with being constant with scale.

Figure 5 .

 5 Figure 5. Discrepancy between the clustering ratio measured in the simulation and the theoretical prediction in the (n, R) plane. Colours represent the quantity (η R (r) -η th R (r))/σ ( R r), the blue regions being the ones with the best agreement with the predictions. Sufficiently large smoothing scales screen the effects of the nonlinear growth of perturbations, allowing us to exploit the clustering ratio as a cosmological probe. By smoothing our distribution on scales R > 19 h -1 Mpc we ensure an agreement with the model better then ∼ 1.5 standard deviations.

Figure 7 .

 7 Figure7. Joint posterior distribution obtained using Planck temperature and polarization data and the clustering ratio measured in SDSS DR7 and 12. We fit a cosmological model with seven free parameters, the six baseline parameters of Planck and the neutrino total mass, but here only four of them are shown.

Figure 8 .

 8 Figure 8. Joint posterior distribution obtained employing CMB temperature and polarization data from Planck and the clustering ratio measurements from SDSS DR7 and DR12 catalogues. Besides the six standard parameters of the model, also the equation of state of dark energy is left free.

Figure 9 .Figure 10 .

 910 Figure9. Degeneracy between the equation of state of dark energy, w, and, on the left, the Hubble parameter today H 0 , and on the right the total neutrino mass Mν . The considered likelihoods are the one computed using Planck data alone, and its combinations with the clustering ratio measured in SDSS DR7 and 12, the BAO position from SDSS DR11 or both.

Figure 11 .

 11 Figure11. Joint posterior distribution obtained using Planck temperature and polarization data and the clustering ratio. Bestfits here are fixed, errors for Planck come from the publicly available covariance matrix, errors on the clustering ratio have been compute in SDSS DR7 and DR12. Four of the seven free parameters are shown.

Table 1 .

 1 The cosmological parameters that vary among the 4 DEMNUni simulations considered in the present work, depending on the assumed neutrino total mass.

		ΛCDM NU0.17 NU0.30 NU0.53
	Mν [eV]	0	0.17	0.30	0.53
	Ωc	0.27	0.2659	0.2628	0.2573
	σ 8,cc	0.846	0.813	0.786	0.740

Table 2 .

 2 Subdivision of the halo catalogues in mass bins.

	Bin Mass range [10 12 h -1 M ]
	0	0.58 M < 1.16
	1	1.16 M < 2.32
	2	2.32 M < 3.28
	3	3.28 M < 4.64
	4	4.64 M < 6.55
	5	6.55 M < 9.26
	6	9.26 M < 30
	7	30 M < 100
	8	M 100

Table 3 .

 3 Population of the 9 mass bins for the Friend-of-Friends (FoF) and spherical overdenities with respect to the critical density (SO) at redshift z = 0.48551 and z = 1.05352.

			bin 0	bin 1	bin 2	bin 3	bin 4	bin 5	bin 6	bin 7	bin 8
		Mν = 0.00 eV	0	0 2902221 3509393 2402274 1708375 2878557 758008 145410
	z = 0.48551	Mν = 0.17 eV Mν = 0.30 eV	0 0	0 3152025 3178910 2430866 1667315 2712374 690356 122241 0 3116471 3269324 2263001 1642694 2589654 634844 104539
	FoF	Mν = 0.53 eV Mν = 0.00 eV	0 0	0 3273517 3026718 2144285 1501769 2334332 532432 0 2571902 3039264 2003119 1358767 2044706 389064	76127 38299
	z = 1.05352	Mν = 0.17 eV Mν = 0.30 eV	0 0	0 2713196 2674546 1958721 1277479 1836860 328973 0 2620295 2674912 1767015 1218810 1679768 283298	28852 22244
		Mν = 0.53 eV	0	0 2619539 2335248 1570202 1036782 1386909 206588	13059
		Mν = 0.00 eV 453415 2973241 2783664 2361431 1669494 1210036 2064503 527389	88178
	z = 0.48551	Mν = 0.17 eV 471602 2993986 2904400 2118437 1668026 1166042 1917029 470522 Mν = 0.30 eV 486007 2997527 2842491 2159522 1559049 1112575 1806821 424202	72097 60046
	SO	Mν = 0.53 eV 508713 3259563 2582711 1956593 1424500 1014066 1587591 343288 Mν = 0.00 eV 363148 2449897 2433901 2030689 1373779 952365 1446087 263959	41497 22482
	z = 1.05352	Mν = 0.17 eV 359806 2376103 2479355 1766898 1330746 Mν = 0.30 eV 354938 2304946 2374992 1750620 1206312	885172 1280841 218832 819053 1157522 184575	16462 12287
		Mν = 0.53 eV 341645 2359735 2069025 1500656 1039892	694588	932357 130319	6834

  Figure12. Joint posterior distribution obtained combining Planck temperature and polarization data and the clustering ratio measured in a Euclid-like galaxy survey. For CMB data, errors come from the publicly available covariance matrix, for the clustering ratio errors have been measured in the DEMNUni simulations and rescaled to match the volume and number density of our mock survey.

							Ω b h 2				Ωch 2						τ	CMB CMB + CR	Mν
				Pl (fixed w)		0.02222	±0.00017 ±0.00033 0.11978 ±0.00147 ±0.00291 0.07851 ±0.01713 ±0.03355	0.16722	< 0.19150	< 0.49402
				Pl + CR (fixed w)	0.02222	±0.00016 ±0.00031 0.11972 ±0.00128 ±0.00255 0.07801 ±0.01744 ±0.03330	0.15795	< 0.18088	< 0.47835
				Pl			0.02222	±0.00016 ±0.00034 0.11971 ±0.00142 ±0.00281 0.07737 ±0.01793 ±0.03483	0.22153	< 0.26698	< 0.60851
				Pl + CR			0.02216	±0.00017 ±0.00033 0.12042 ±0.00149 ±0.00290 0.07469 ±0.01754 ±0.03403	0.20304	< 0.24510	< 0.53081
				Pl + CLens		0.02217	±0.00017 ±0.00035 0.11967 ±0.00153 ±0.00299 0.06927 ±0.01749 ±0.03420	0.32882	±0.19711	< 0.67219
				Pl + BAO		0.02225	±0.00015 ±0.00030 0.11949 ±0.00134 ±0.00263 0.07728 ±0.01705 ±0.03304	0.11571	< 0.14235	< 0.30423
		0.0228	Pl + CLens + CR	0.02213	±0.00016 ±0.00032 0.12026 ±0.00145 ±0.00288 0.06909 ±0.01681 ±0.03237	0.29440	±0.17524	< 0.59471
				Pl + BAO +CR	0.02223	±0.00015 ±0.00030 0.11965 ±0.00132 ±0.00261 0.07731 ±0.01666 ±0.03147	0.10844	< 0.13143	< 0.28339
	Ωbh 2	0.0224				w				ln(10 10 As)				ns	H 0
		0.0220 0.123	Pl (fixed w) Pl + CR (fixed w) -1.00000 -1.00000 Pl -1.68615 Pl + CR -1.25376 ±0.24254 ±0.52000 3.08537 ±0.03368 ±0.06550 0.96389 ±0.00487 ±0.00967 73.04263 --3.09167 ±0.03332 ±0.06510 0.96531 ±0.00478 ±0.00951 66.36205 --3.09042 ±0.03375 ±0.06497 0.96550 ±0.00456 ±0.00906 66.47015 ±0.29543 ±0.59285 3.08881 ±0.03490 ±0.06751 0.96500 ±0.00473 ±0.00953 86.91446	+1.93320 -0.79827 +1.72753 -0.68157 +12.41268 -4.70920 ±6.83317	±3.14533 ±2.93440 ±15.59971 ±14.51362
	Ωch 2	0.120 0.117	Pl + CLens Pl + BAO Pl + CLens + CR	-1.67628 -1.05867 ±0.07959 ±0.16354 3.08846 ±0.03305 ±0.06428 0.96612 ±0.00451 ±0.00886 68.60048 ±0.35984 ±0.66750 3.07148 ±0.03387 ±0.06578 0.96452 ±0.00502 ±0.00978 84.62450 -1.22298 ±0.23981 ±0.51234 3.07259 ±0.03189 ±0.06176 0.96385 ±0.00493 ±0.00946 71.08454	+14.64438 -5.54025 ±1.67361 ±6.31034	±16.85512 ±3.36935 ±13.48706
		0.114	Pl + BAO + CR	-1.05267	±0.07749 ±0.15731 3.08903 ±0.03209 ±0.06098 0.96595 ±0.00453 ±0.00888 68.42143	±1.63637	±3.26416
			0.60				Ω b h2				Ωch 2						τ	Mν
	Σmν	0.30 0.45	Pl (fixed w) Pl + CR (fixed w)	0.02244 0.02242	±0.00016 ±0.00034 0.11948 ±0.00157 ±0.00314 0.09332 ±0.01644 ±0.03291 ±0.00015 ±0.00030 0.11950 ±0.00136 ±0.00265 0.09199 ±0.01659 ±0.03229	0.20477 0.20843	< 0.26441 ±0.11909	< 0.42993 < 0.41013
			0.15				w				ln(10 10 As)				ns	H 0
				Pl (fixed w)		-1.00000	-	-	3.05893 ±0.03169 ±0.06381 0.95937 ±0.00518 ±0.01015 66.65175	±1.43657	±2.67394
			0.16	Pl + CR (fixed w) -1.00000	-	-	3.05620 ±0.03264 ±0.06422 0.95969 ±0.00458 ±0.00918 66.60347	±1.23252	±2.41851
	τ	0.12				Ω b h 2				Ωch 2						τ	Mν
			0.08	Pl (fixed w)		0.02243	±0.00016 ±0.00031 0.11942 ±0.00146 ±0.00290 0.09335 ±0.01747 ±0.03445	0.20513	±0.12382	< 0.43157
			0.04	Pl + CR (fixed w)	0.02245	±0.00013 ±0.00026 0.11926 ±0.00084 ±0.00167 0.09422 ±0.01655 ±0.03298	0.17749	±0.11133	< 0.37693
							w				ln(10 10 As)				ns	H 0
	ln(10 10 As)	3.04 3.12	Pl (fixed w) Pl + CR (fixed w) -1.00000 -1.00000	--	--	3.05859 ±0.03378 ±0.06670 0.95983 ±0.00491 ±0.00964 66.66588 3.06008 ±0.03262 ±0.06476 0.96046 ±0.00443 ±0.00854 66.98261	±1.38464 ±1.12263	±2.62998 ±2.18365
			Table
			2.96
			0.97
	ns	0.96
			0.95
			62.5 65.0 67.5 70.0	0.0220	0.0224	0.0228	0.114 0.117 0.120 0.123	0.15 0.30 0.45 0.60	0.04	0.08	0.12	0.16	2.96	3.04	3.12	0.95	0.96	0.97
				H0		Ωbh 2			Ωch 2	Σmν			τ				ln(10 10 As)	ns

c 0000 RAS, MNRAS 000, 000-000

The simulations do not account for an effective neutrino number N eff > 3, as possible neutrino isocurvature perturbations which could produce larger N eff (therefore affecting galaxy and CMB statisticsCarbone et al.

2011) are currently excluded by present data (see, eg,Di Valentino & Melchiorri 2014)

ACKNOWLEDGEMENTS

LG, JB, CC, and JD acknowledge financial support from the European Research Council through the Darklight Advanced Research Grant (n. 291521). CC acknowledges the support from the grant MIUR PRIN 2015 "Cosmology and Fundamental Physics: illuminating the Dark Universe with Euclid". The DEMNUni simulations were carried out at the Tier-0 IBM BG/Q machine, Fermi, of the Centro Interuniversitario del Nord-Est per il Calcolo Elettronico (CINECA, Bologna, Italy), via the five million cpu-hrs budget provided by the Italian SuperComputing Resource Allocation (ISCRA) to the class-A proposal entitled "The Dark Energy and Massive-Neutrino Universe".