## Abstract

Cosmic neutrinos have been playing a key role in cosmology since the discovery of their mass. They can affect cosmological observables and have several implications being the only hot dark matter candidates that we currently know to exist. The combination of massive neutrinos and an adequate theory of gravity provide a perfect scenario to address questions on the dark sector that have remained unanswered for years. In particular, in the era of precision cosmology, galaxy clustering and redshift-space distortions afford one of the most powerful tools to characterise the spatial distribution of cosmic tracers and to extract robust constraints on neutrino masses. In this chapter, we study how massive neutrinos affect the galaxy clustering and investigate whether the cosmological effects of massive neutrinos might be degenerate with f(R) gravity cosmologies, which would severely affect the constraints.

### Keywords

- massive neutrinos
- gravity theories
- structure formation
- galaxy clustering

## 1. Introduction

From first principles, it is well known that a theory of gravity is needed to describe the spatial properties and dynamics of the large-scale structures (LSS) of the universe. The observational data collected for several decades provide strong support to the concordance model Lambda cold dark matter (ΛCDM), which yields a consistent description of the main properties of the LSS [1, 2, 3, 4]. However, since cosmological observations have entered in an unprecedented precision era, one of the current aims is to test some of the most fundamental assumptions of the concordance model of the universe. In this sense, the ΛCDM model assumes (i) the general theory of relativity (GR) as the theory describing gravitational interactions at large scales, (ii) the standard model of particles and (iii) the cosmological principle. Moreover, in this framework, the universe is currently dominated by dark energy (DE) in the form of a cosmological constant, responsible for the late-time cosmic acceleration [5, 6, 7] and by a cold dark matter (CDM) component that drives the formation and evolution of cosmic structures.

Recently, several shortcomings have been found in the ΛCDM scenario, like a possible tension in the parameter constraint of *H* _{0} and *f*(*R*) gravity are the favourite ones because of their generality and rich phenomenology [11, 12]. Moreover, modified gravity (MG) models represent one of the most viable alternatives to explain cosmic acceleration [13] that require satisfying simultaneously solar system constraints and to be consistent with the measured accelerated cosmic expansion and large-scale constraints [14, 15, 16, 17]. An extra motivation to study MG models is given by the fact that massive neutrinos, the only (hot) dark matter candidates we actually know to exist, can affect these observables and have several cosmological implications [18]. However, the degeneracy between some MG models and the total neutrino mass [19, 20, 21] give rise to a limitation of many standard cosmological statistics [22, 23]. In this context, the clustering analysis and redshift-space galaxy clustering have been proven to be a powerful cosmological probe to discriminate among MG scenarios with massive neutrinos, as will be discussed in the following sections.

Regarding the dynamics of background universe in the standard framework, it is well described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, whose line element in natural units

where *a*(*t*) is the so-called scale factor, (*k* defines the geometry of the universe under consideration to be flat (*a*(*t*) and the dynamic growth of the universe are called *Friedmann equations* and are given by

that can be re-expressed as

after eliminating the curvature term. These equations lead to the definition of the Hubble parameter as *h*, defined by the expression H_{0} = 100 h km s^{−1} Mp c^{−1} at the present epoch. In order to have a full description of the background universe, it is necessary an equation of state (EoS) of the cosmic fluid, considering that it has three principal components: baryonic matter, dark matter and radiation (see Figure 1a). A first approximation consists in assuming a linear relationship between *w* is a parameter that in principle can be time-dependent *i*, related to radiation *cosmic sum rule*. Therefore, a Friedmann universe can be described by the cosmological parameters

This equation is usually written as a dimensionless function defined by *a*. For further details concerning the background universe, see Refs. [25, 26, 27].

## 2. Modified gravity and massive neutrinos

### 2.1 Modified gravity models

One of the most interesting modifications of GR is that which modifies the Einstein-Hilbert action by introducing a scalar function, *f*(*R*), as follows:

where *R* is the Ricci scalar, *G* is the Newton’s gravitational constant, *g* is the determinant of the metric tensor *f*(*R*) theories of gravity and the assumptions needed to arrive to the various versions of *f*(*R*) gravity and GR, see, e.g., [28]. Thus, for a general *f*(*R*) model, one can consider a spatially flat FLRW universe with metric

where *t*. In general, the background evolution of a viable *f*(*R*) is not simple as it has been shown by [29, 30, 31]. However it is possible to get an approximation in a way that is analogous to the DE models, by neglecting the higher derivative and the non-linear terms. By defining the growth rate as *f*(*R*) model is approximated by [29, 32]

where *Geff *is the effective gravitational constant that can be written as [33]. A plausible *f*(*R*) function able to satisfy the solar system constraints, to mimic the ΛCDM model at high-redshift regime where it is well tested by the CMB and, at the same time, to accelerate the expansion of the universe at low redshift but without a cosmological constant [34], suggests that

which can be satisfied by a broken power law function such that

where the mass scale *m* is defined as *c* _{1}, *c* _{2} and *n* are non-negative free parameters of the model [34]. For this *f*(*R*) model, the background expansion history is consistent with the ΛCDM case by choosing

Nowadays it is generally accepted that some MG theories such as the [34] *f*(*R*) are strongly degenerated in a wide range of their observables with the effects of massive neutrinos; see, e.g., [19, 20, 21, 35]. This represents a serious challenge constraining cosmological models from current and future galaxy surveys requiring robust and reliable methods to disentangle both phenomena. Furthermore, for some specific combinations of the *f*(*R*) function and of the total neutrino mass

### 2.2 Massive neutrinos and the large-scale structure

Motivated by the apparent violation of energy, momentum and spin in

where *p* and *z* and *z* is given by

After neutrinos with mass

the number density per flavour is fixed by the temperature, so that the universe is currently filled by a relic neutrino background, uniformly distributed, with a density of 113 part/cm^{3} per species and an average temperature of 1.95 K. As neutrinos are non-relativistic particles at late times, they contribute to the total matter density of the universe

Currently, several observations provide limits on the total neutrino mass under the assumption of standard GR [43]. Depending of their mass, neutrinos can affect different quantities such as the matter-radiation equality and at the same time imprint features in cosmological observables like the clustering, the matter power spectrum, the halo mass function and the redshift-space distortions [18, 44, 45, 46, 47].

## 3. Halo mass function and clustering analysis

Considering the impact of MG and massive neutrinos in the clustering, a powerful cosmological test to discriminate among these scenarios is provided by the redshift-space distortions (RSD), that is, the shift in the position of the tracers due to their peculiar motions. For this purpose, cosmological simulations have become a powerful tool for testing theoretical predictions and to lead observational projects. In this context, the formation and evolution of cosmic structures can be understood as a dynamical system of many particles, which trace the underlying mass distribution in a certain cosmological model. The N-body simulations, methods and algorithms have progressed continuously, achieving a high resolution to resolve finer structures with millions of particles, reducing the gap between theory and observations. For a detailed description on fundamentals of cosmological simulations, see, e.g., [48, 49, 50, 51].

Since the formation and evolution of cosmic structures is based on the growth of small fluctuations in the density field, it is expected that the amplitude of these initial perturbations have the correct value at late times to match the observed clustering today. An analytical development based on perturbation theory makes possible to follow the growth of structures to a certain extent using the linear approximation, being valid as long as

In this section we show a complementary analysis to the one performed in [52] in order to investigate the clustering in the context of modified gravity with massive neutrinos. We used a subset of the DUSTGRAIN-*pathfinder* runs [36], which implement the Hu-Sawicki *f*(*R*) model including massive neutrinos and whose cosmological parameters are consistent with Planck 2015 constraints [53].

### 3.1 Halo mass function

As CDM haloes form from collapsing regions that detach from the background density field, their abundance can be related to the volume fraction of a Gaussian density smoothed on a radius *R* above a critical collapse threshold

where *R*(*M*) and *M* is a Gaussian function with variance

Another approach to determine

where *A* is an amplitude of the mass function and *a*, *b*, *c* and *d* are free parameters that depend on halo definition. The variance

where *k* and *W* is the Fourier transform of the real-space top-hat window function of radius *R*. A fundamental feature of the mass function is that it decreases monotonically with increasing masses; furthermore, its dependency on cosmology is encoded in the variance *V* such as

where *A* is the area,

Figure 3 shows the mass function of CDM haloes measured for all models of the DUSTGRAIN-*pathfinder* runs at six different redshifts *f*(*R*) models both with and without neutrinos reproduce in very well agreement this pattern, but only at really high masses, significant differences appear.

Figure 4 compares the halo mass functions of the different DUSTGRAIN-*pathfinder* simulations, computed at *f*(*R*) and massive neutrinos on the dynamical evolution of the matter density field results in different halo formation epochs and different number density of collapsed systems. In particular, the

### 3.2 Clustering analysis

To quantify the halo clustering, we used the two-point correlation function (2PCF) that can be defined as the joint probability of finding a pair of objects at certain spatial separation given a tracer distribution. To measure the full 2PCF denoted as

where *DR* represent the normalised number of data-data, random-random and data-random pairs, respectively. This estimator is almost unbiased with minimum variance; this is the reason why it is preferred over the other estimators regarding the clustering measurements. Since the possible deviations from GR are more evident on small scales, we consider an intermediate non-linear range from 1

where each coefficient corresponds to the *l*th multipole moment:

Figure 5 shows the RSD effects on the iso-correlation curves of the 2D two-point correlation function (2PCF) in the plane (*pathfinder* simulations, in real space (*left panel*), and for the corresponding sample in redshift space (*right panel*). The contours are drawn at the iso-correlation levels

The symmetry of the 2PCF in real space means that the full clustering signal is encoded in the monopole moment *pathfinder* project, at three different redshifts *f*(*R*) with and without massive neutrinos] and the ΛCDM model. The monopole moments of the 2PCF of the

Another important feature to take into account in the clustering analysis is related to the bias that is introduced when the ΛCDM model is wrongly assumed to predict the DM clustering of a *f*(*R*) universe with massive neutrinos [58]. This effective halo bias, *f*(*R*) and

with the CDM power spectrum obtained by rescaling the total matter power spectrum with the corresponding transfer functions,

The impact on the bias when the CDM prescription is not considered can be appreciated in Figure 7. In most cases this correction is small with the exception of the

### 3.3 Modelling the redshift-space distortions

In a realistic case, spectroscopic surveys observe a combination of density and velocity fields in redshift space. The observed redshift is a combination of cosmological effects plus an additional term caused by the peculiar motions along the line of sight of the observer. This combination makes the redshift-space catalogues appear distorted with respect to the real-space ones, and they can be reproduced from N-body simulations since the positions and velocities are known. Currently, the modelling of the redshift-space distortions provide a powerful tool to test the gravity theory by exploring the spatial statistics encoded in the 2PCF, which is anisotropic due to the dynamic distortions.

We consider the first two even multipoles of the 2PCF, *f*(*R*) gravity and massive neutrinos on RSD, we performed a Bayesian analysis to set constraints on the linear growth rate ^{1} [62].

The Kaiser formula is a good description of the RSD only at very large scales, where non-linear effects can be neglected, but it does not describe accurately the non-linear regime. Thus, with the aim of extracting information from the RSD signal at non-linear regime and considering the increasing precision of recent and upcoming surveys, many more approaches have been proposed. There is a vast literature that shows the efforts to model the RSD beyond the linear Kaiser model [63, 64, 65, 66], some of them making use of a phenomenological description of the velocity field and others, instead, taking into account higher orders in perturbation theory since, in principle, there is no reason to stop at linear order. Other approaches do a combination of both frameworks. A simple alternative to model the redshift-space 2PCF at small scales consists of extending the Kaiser formula, by adding a phenomenological damping factor that plays the role of a pairwise velocity distribution. It can account for both linear and non-linear dynamics. Therefore, to construct the likelihood, we consider this model sometimes called *dispersion model* [67], which introduces a damping function to describe the distortions in the clustering at small scales (Fingers-of-God). This model is enough accurate to quantify the relative differences between *f*(*R*) models with massive neutrinos and ΛCDM. For the Bayesian analysis, the dispersion model is fully described by three parameters,

Figure 8 shows the normalised covariance matrices *pathfinder.* The figure shows the constraints for all models considered in this work at

From the *f*(*R*) models studied are statistically indistinguishable from ΛCDM, and further studies are required to break this degeneracy.

## 4. Conclusions

In this chapter we have introduced the theoretical framework of modern cosmology in present massive neutrinos. We emphasize on the structure formation and on the statistical description of the density field as well as the measurements of galaxy clustering and discuss the redshift-space distortions and the differences between clustering in real and redshift space, considering that in the recent years, the spatial distribution of matter on cosmological scales has become one of the most efficient probes to investigate the properties of the universe, such as test gravity theories on large scales, to explore the dark sector and the origin of the accelerated expansion of the universe as well as a probe to constrain alternative cosmological models.

In the context of models based on modified gravity and massive neutrino cosmologies, we investigated the spatial properties of the large-scale structure by exploiting the DUSTGRAIN-*pathfinder* simulations that follow, simultaneously, the effects of *f*(*R*) gravity and massive neutrinos. These are two of the most interesting scenarios that have been recently explored to account for possible observational deviations from the standard ΛCDM model. In particular, we studied whether redshift-space distortions in the 2PCF multipole moments can be effective, breaking the cosmic degeneracy between these two effects. We analysed the redshift-space distortions in the clustering of dark matter haloes at different redshifts, focusing on the monopole and quadrupole moments of the two-point correlation function, both in real and redshift space. The deviations with respect to ΛCDM model have been quantified in terms of the linear growth rate parameter. We found that multipole moments of the 2PCF from redshift-space distortions provide a useful probe to discriminate between ΛCDM and modified gravity models, especially at high redshifts (

## Acknowledgments

We thank Lauro Moscardini, Federico Marulli and Alfonso Veropalumbo for the continuous development of the CosmoBolognaLib and their suggestions during this project. We also thank Carlo Giocoli and Marco Baldi for the crucial work performing the DUSTGRAIN-*pathfinder* runs.

## Conflict of interest

The authors declare no conflict of interest.

## Notes

- Freely available at the public GitHub repository https://github.com/federicomarulli/CosmoBolognaLib.