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4Mon. Not. R. Astron. Soc.000, 1–14 (2011) Printed 7 February 2014 (MN LATEX style file v2.2)

Cosmic Degeneracies I: Joint N-body Simulations of ModifiedGravity and Massive Neutrinos

Marco Baldi1,2,3, Francisco Villaescusa-Navarro4, Matteo Viel4,5, Ewald Puchwein6,7,Volker Springel7,8, Lauro Moscardini1,2,3

1Dipartimento di Fisica e Astronomia, Alma Mater Studiorum Universita di Bologna, viale Berti Pichat, 6/2, I-40127 Bologna, Italy;2INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy;3INFN - Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy;4INAF - Osservatorio Astronomico di Trieste, Via Tiepolo 11,I-34143, Trieste, Italy;5INFN/National Institute for Nuclear Physics, Via Valerio 2, I-34127 Trieste, Italy;6Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA,UK;7Heidelberger Institut fur Theoretische Studien (HITS), Schloss-Wolfsbrunnenweg 35, 69118 Heidelberg, Germany;8Zentrum fur Astronomie der Universitat Heidelberg, ARI,Monchhofstrasse 12-14, 69120 Heidelberg, Germany.

7 February 2014

ABSTRACT

We present the first suite of cosmological N-body simulations that simultaneously in-clude the effects of two different and theoretically independent extensions of the standardΛCDM cosmological scenario – namely anf(R) theory of Modified Gravity (MG) and acosmological background of massive neutrinos – with the aimto investigate their possibleobservational degeneracies. We focus on three basic statistics of the large-scale matter distri-bution, more specifically the nonlinear matter power spectrum, the halo mass function, andthe halo bias. Our results show that while these two extendedmodels separately determinevery prominent and potentially detectable features in all the three statistics, when we allowthem to be simultaneously at work these features are strongly suppressed. In particular, whenan f(R) gravity model withfR0 = −1 × 10−4 is combined with a total neutrino mass ofΣimνi = 0.4 eV, the resulting matter power spectrum, halo mass function, and bias atz = 0are found to be consistent with the standard model’s predictions at the. 10%, . 20%,and. 5% accuracy levels, respectively. Therefore, our results imply an intrinsic theoreticallimit to the effective discriminating power of present and future observational data sets withrespect to these widely considered extensions of the standard cosmological scenario.

Key words: dark energy – dark matter – cosmology: theory – galaxies: formation

1 INTRODUCTION

The primary scientific goal of a wide range of present and futureobservational initiatives in the field of cosmology – such ase.g.BOSS (Ahn et al. 2013), PanStarrs (Kaiser et al. 2002), HETDEX(Hill et al. 2008), DES (Abbott et al. 2005), LSST (Ivezic et al.2008) and Euclid1 (Laureijs et al. 2011) – consists in combiningdifferent probes of the geometric and dynamical evolution of theUniverse to unveil possible deviations from the expected behaviourof the standardΛCDM cosmological scenario. The latter relieson the assumption of a cosmological constantΛ as the source ofthe observed accelerated expansion of the Universe (Riess et al.1998; Perlmutter et al. 1999; Schmidt et al. 1998) and on the ex-istence of some yet undetected Cold Dark Matter (CDM) particle

1 www.euclid-ec.org

with negligible non-gravitational interactions that could enhancethe growth of cosmic structures from the tiny density perturba-tions observed in the primordial Universe through detailedCosmicMicrowave Background (CMB) observations (see e.g.Smoot et al.1992; Bennett et al. 2013; Ade et al. 2013) to the highly structuredUniverse that we observe today (see e.g.Abazajian et al. 2009).

Although the standardΛCDM cosmological model has so farbeen found to be consistent with observational data at ever increas-ing levels of accuracy (see e.g.Blake et al. 2011; Ade et al. 2013),its theoretical foundations remain poorly motivated in theabsenceof a clear understanding of the physical origin of the cosmologicalconstantΛ and of a direct (or indirect) detection of the elementaryparticle associated with CDM. This lack of a firm theoreticalandobservational basis for the standardΛCDM picture has motivatedthe exploration of a large number of alternative scenarios both con-

c© 2011 RAS

2 M. Baldi et al.

cerning the origin of the accelerated expansion and the nature ofthe CDM cosmic field.

The former range of extended models includes on one handgeneralised Dark Energy (DE) scenarios where the cosmic acceler-ation is driven by a classical scalar field (see e.g.Wetterich 1988;Ratra & Peebles 1988; Armendariz-Picon et al. 2001; Caldwell2002; Feng et al. 2005) possibly characterised by non-trivial clus-tering (see e.g.Creminelli et al. 2009; Sefusatti & Vernizzi 2011;Batista & Pace 2013) or interaction (see e.g.Wetterich 1995;Amendola 2000; Farrar & Peebles 2004; Amendola et al. 2008;Baldi 2011) properties, and on the other hand several possibleModified Gravity (MG) models includingScalar-Tensor theoriesofgravity (as e.g.f(R) models, Buchdahl 1970; Starobinsky 1980;Hu & Sawicki 2007; Sotiriou & Faraoni 2010), theDGP scenario(Dvali et al. 2000), or theGalileonmodel (Nicolis et al. 2009). Inthe present work we will focus on the second class of extensionsof the standard cosmology, and in particular on the widely inves-tigated parameterisation off(R) theories of gravity proposed byHu & Sawicki (2007).

The latter range of models, instead, includes various exten-sions of the CDM paradigm characterised by different assumptionson the nature of the fundamental dark matter particle, both con-cerning its phase-space distribution – as for the case of Warm DarkMatter (WDM) models (see e.g.Bode et al. 2001; Viel et al. 2013)or mixed dark matter scenarios (Maccio’ et al. 2012) – and its in-teraction properties (see e.g.Loeb & Weiner 2011; Baldi 2013).While the fundamental composition of the dark matter fraction ofthe total cosmic energy budget has very little impact on the back-ground expansion history of the universe, it can significantly affectthe growth of density perturbations both in the linear and inthenonlinear regimes. In particular, one of the most significant exten-sions of the standard cosmological scenario for what concerns thestructure formation processes amounts to dropping the assumptionof massless neutrinos that is commonly adopted in both analyticaland numerical investigations of the late-time universe. The discov-ery of the neutrino oscillation phenomena (Cleveland et al. 1998)has revealed in an unambiguous way that at least two of the threeneutrino families are massive, such that a fraction of the total mat-ter density of the Universemustbe associated with the cosmic neu-trino background. In this respect, the inclusion of massiveneutrinosinto any cosmological scenario should no longer be regardedas one(amongst many) possible extension of the basic standard model intothe realm of exotic physics, but rather as a necessary ingredient inorder to faithfully reproduce reality. As such, any realistic cosmo-logical modelling – both for the standard cosmological constantΛand for alternative DE or MG scenarios – should no longer avoid toproperly take into account the possibility that part of the dark mat-ter energy density be made of non-relativistic massive neutrinos.

In recent years, significant progress has been made inincluding the effects of massive neutrinos into cosmologicalN-body codes employed to study structure formation processesfrom the linear to the highly nonlinear regime in the contextofstandardΛCDM cosmologies (see e.g.Brandbyge et al. 2008;Brandbyge & Hannestad 2009; Brandbyge & Hannestad 2010;Viel et al. 2010; Agarwal & Feldman 2011; Bird et al. 2012;Wagner et al. 2012). Here, by “standardΛCDM cosmologies”we refer to models where the accelerated cosmic expansion isdriven by a cosmological constant and where a fixed total matterdensity is made up by a fraction of standard CDM particles andby a fraction of massive neutrinos. Such works have highlighteda number of effects that appreciably modify observable quantities(as e.g. the nonlinear matter power spectrum, the abundance

of collapsed halos as a function of their mass, the pattern ofredshift-space distortions, or the clustering propertiesof darkmatter halos, see e.g.Viel et al. 2010; Castorina et al. 2013;Marulli et al. 2011; Villaescusa-Navarro et al. 2013a, respectively)in massive neutrinos cosmologies as compared to their masslessneutrinos counterparts, thereby possibly biasing the inferenceof cosmological parameters from these observables. Similarly,significant progress has been made in implementing alternative DEand MG models in N-body algorithms (see e.g.Baldi 2012, for arecent review), which have allowed self-consistent cosmologicalsimulations in the context of a wide range of such non-standardscenarios for the accelerated cosmic expansion. However, noattempt has been made so far to combine such two efforts andinvestigate the effects of massive neutrinos on the formation andevolution of linear and nonlinear cosmic structures in the contextof alternative DE and MG models.

In the present paper – which is the first in a series of worksaimed at studying the intrinsic observational degeneracies betweendifferent and independent extensions of the standardΛCDM sce-nario – we attempt for the first time to investigate the joint ef-fects of f(R) MG theories and massive neutrinos in the nonlin-ear regime of structure formation by means of a specifically de-signed N-body code that is capable of simultaneously includingboth these additional ingredients in its integration scheme. The lat-ter is a modified version of the widely used TreePM N-body codeGADGET (Springel 2005) which has been obtained by combiningtheMG-GADGET implementation off(R) theories recently devel-oped byPuchwein et al.(2013) with the massive neutrinos moduleby Viel et al. (2010).

With such code in hand, we have performed for the first timea series of combined simulations for one specific realisation off(R) MG with different amounts of massive neutrinos, and com-pared the outcomes of our runs in terms of some basic statisticsof the Large-Scale Structures (LSS) distribution to the same setof neutrino masses acting within a standardΛCDM cosmology.This work therefore extends the earlier studies ofHe (2013) andMotohashi et al.(2013) to the fully nonlinear regime of structureformation. Our results highlight a very strong degeneracy betweenf(R) MG models and massive neutrinos in all the observables thatwe have considered, such that a suitable combination off(R) pa-rameters and neutrino masses might be hardly distinguishable fromthe standard cosmological scenario even when the two separate ef-fects would result in a very significant deviation from the referencecosmology. This degeneracy should be properly taken into accountwhen assessing the effective discriminating power of present andfuture observational surveys with respect to individual extensionsof the standard cosmological model.

The present paper is organised as follows. In Section2 weintroduce thef(R) MG models considered in our numerical simu-lations. In Section3 we review the basics of massive neutrinos andtheir role in cosmological structure formation. In Section4 we de-scribe the numerical setup of our cosmological simulationsand ofthe initial conditions generation. In Section5 we present our resultson the nonlinear matter power spectrum, the halo mass function,and the halo bias. Finally, in Section6 we draw our conclusions.

2 F (R) MODIFIED GRAVITY

In the present work, we will consider Modified Gravity cosmolo-gies in the form off(R) extensions of Einstein’s General Relativity

c© 2011 RAS, MNRAS000, 1–14

Cosmic Degeneracies I:f(R) and massive neutrinos 3

Run Theory of Gravity Σimνi [eV] ΩCDM Ων MpCDM

[M⊙/h] Mpν [M⊙/h]

GR-nu0 GR 0 0.3175 0 6.6× 1011 0GR-nu0.2 GR 0.2 0.3127 0.0048 6.47× 1011 9.86× 109

GR-nu0.4 GR 0.4 0.308 0.0095 6.4× 1011 1.97 × 1010

GR-nu0.6 GR 0.6 0.3032 0.0143 6.27× 1011 2.96 × 1010

fR-nu0 fR0 = −1× 10−4 0 0.3175 0 6.6× 1011 0fR-nu0.2 fR0 = −1× 10−4 0.2 0.3127 0.0048 6.47× 1011 9.86× 109

fR-nu0.4 fR0 = −1× 10−4 0.4 0.308 0.0095 6.4× 1011 1.97 × 1010

fR-nu0.6 fR0 = −1× 10−4 0.6 0.3032 0.0143 6.27× 1011 2.96 × 1010

Table 1.The suite of cosmological N-body simulations considered inthe present work, with their main physical and numerical parameters. HereΩCDM andΩν indicate the energy density ratio to the critical density ofthe universe for CDM and neutrinos, respectively, whileMp

CDMandMp

ν stand for the mass ofCDM and neutrino particles in the different N-body runs.

(hereafter GR). Such models are characterised by the action

S =

∫

d4x√−g

(

R+ f(R)

16πG+ Lm

)

, (1)

where the Ricci scalarR that appears in the standard Einstein-Hilbert action of GR has been replaced byR + f(R). In Eq. (1),G is Newton’s gravitational constant,Lm is the Lagrangian den-sity of matter, andg is the determinant of the metric tensorgµν .These models have been widely investigated in the literature,both in terms of their theoretical predictions at linear (see e.g.Pogosian & Silvestri 2008) and nonlinear (see e.g.Oyaizu et al.2008; Schmidt et al. 2009; Li et al. 2012a; Puchwein et al. 2013;Llinares et al. 2013) scales, and in terms of possible present (seee.g.Lombriser et al. 2012) and forecasted future (Zhao et al. 2009)observational constraints. Within such framework, the quantityfR ≡ df(R)/dR represents an additional scalar degree of free-dom obeying an independent dynamic equation that in the quasi-static approximation and in the limitfR ≪ 1 takes the form2 (seeagainHu & Sawicki 2007):

∇2fR =1

3(δR − 8πGδρ) , (2)

whereδR andδρ are the perturbations in the scalar curvature andmatter density, respectively.

Within the range of possible functional forms off(R), we willfocus on the specific realisation proposed byHu & Sawicki (2007)which has the advantage of providing a close match to the standardΛCDM background expansion history. In this model, the functionf(R) takes the form

f(R) = −m2 c1(

Rm2

)n

c2(

Rm2

)n+ 1

, (3)

wherem2 ≡ H20ΩM, with H0 being the present-day Hubble pa-

rameter andΩM the dimensionless matter density parameter de-fined as the ratio between the mean matter density and the crit-ical density of the universe. In Eq. (3) c1, c2, andn are non-negative constant free parameters that fully specify the model.By imposing the conditionc2(R/m2)n ≫ 1, Eq. (3) becomesf(R) = −m2c1/c2 + O((m2/R)n). It is then possible to ob-tain a background evolution close to that of aΛCDM model bysetting the term−m2c1/c2 equal to−2Λ, whereΛ is the desiredcosmological constant, which is equivalent to imposing a relationbetween the two free parametersc1 andc2:

c1c2

= 6ΩΛ

ΩM

, (4)

2 Whenever not explicitly stated otherwise, we always work inunits wherethe speed of light is assumed to be unity,c = 1.

whereΩΛ ≡ Λ/3H20 is the present dimensionless density param-

eter of the cosmological constant. Under these assumptions, thescalar degree of freedomfR takes the approximate form:

fR ≈ −nc1c22

(

m2

R

)n+1

. (5)

By evaluating the present average scalar curvature of the uni-verseR0 within a standardΛCDM cosmology and combining withEq. (5) one gets an equation for the background value of the presentscalar degree of freedomfR0. Then, by fixing the value offR0 itis possible to obtain an independent relation betweenc1 andc2 asa function ofn, that combined with Eq. (4) completely determinesboth parameters. Therefore, the model can be fully specifiedbyfixing n and fR0 rather thanc1 andc2. Following the conventionadopted in most of the literature we will stick to the former choicethroughout the rest of the paper.

The above equations further simplify after fixing the value ofn, which is set to unity in all the cosmological realisations pre-sented in this work. Forn = 1, then, the model is fully specified bythe single parameterfR0, and the time evolution of the backgroundscalar degree of freedomfR is given by:

fR(a) = fR0

(

R0

R(a)

)2

= fR0

(

1 + 4 ΩΛ

ΩM

a−3 + 4 ΩΛ

ΩM

)2

, (6)

while the curvature perturbation takes the form

δR = R(a)

√

fR(a)

fR− 1

. (7)

In f(R) models, the gravitational potentialΦ satisfies(Hu & Sawicki 2007)

∇2Φ =16πG

3δρ− 1

6δR , (8)

where the second term on the right-hand side is responsible for thespatial dependence of the fifth-force arising as a consequence of themodification of the laws of gravity in addition to the standard New-tonian force. To carry out our analysis, we will resort to theMG-GADGET implementation (Puchwein et al. 2013) of f(R) modelsinto the widely-used parallel Tree-PM N-body codeGADGET3thatwill be briefly described in Section4. This numerical tool allowsto solve Eq. (2) for a generic density distribution given by a set ofN-body particles, and then compute the total force arising on eachparticle through Eq. (8), by including in the source term the cur-vature perturbationδR obtained according to Eq. (7). We refer theinterested reader to theMG-GADGETcode paper for a more detailedpresentation of the numerical implementation.

c© 2011 RAS, MNRAS000, 1–14

4 M. Baldi et al.

Parameter ValueH0 67.1 km s−1 Mpc−1

ΩM 0.3175ΩDE 0.6825Ωb 0.049As 2.215 × 10−9

ns 0.966

Table 2. The set of cosmological parameters adopted in the present work,consistent with the latest results of the Planck collaboration (Ade et al.2013). Herens is the spectral index of primordial density perturbationswhileAs is the amplitude of scalar perturbations at the redshift of the CMB.

3 MASSIVE NEUTRINOS AND STRUCTUREFORMATION

Neutrinos were initially postulated in 1930 by Pauli as a solution tothe apparent violation of energy, momentum and spin conservationin the β-decay process. In 1956 their existence was corroboratedfor the first time in the famous experiment carried out by Cowan& Reines (Cowan et al. 1956). The measurements of the Z bosonlifetime have then pointed out that the number of light neutrinofamilies is three. Moreover, since the discovery of the neutrino os-cillation phenomena (Cleveland et al. 1998) it is known that at leasttwo of the three neutrino families are massive, in contrast to the par-ticle standard model assumption. Whereas measuring the absolutemasses of the neutrinos is very difficult, the neutrino oscillationscan be used to measure the mass square differences among the neu-trino mass eigenstates. The most recent measurements from solar,atmospheric and reactor neutrinos result in:m2

12 = 7.5 × 10−5

eV2 and|m223| = 2.3×10−3 eV2 (Fogli et al. 2012; Forero et al.

2012, respectively), withm1, m2 andm3 being the masses of thethree neutrino mass eigenstates. Unfortunately, the experimentalmeasurements do not allow us to determine the sign of the quantitym2

23. This gives rise to two different possible mass orderings (orhierarchies): thenormalhierarchy wherem1 < m2 < m3, and theinvertedhierarchy wherem3 < m1 < m2.

Knowing the absolute masses of the neutrinos is of critical im-portance, since they represent a door to look for physics beyond thestandard model. Whereas the neutrino oscillations have been usedto set lower limits on the sum of the neutrino masses,Σimνi >

0.056 and 0.095 eV for the normal and inverted hierarchies respec-tively, the tighter upper bounds on the masses of the neutrinos comefrom cosmological observations.

In the very early universe neutrinos were in thermal equi-librium with the photons, the electrons and then positrons.Whilein thermal equilibrium, their momentum distribution followed thestandard Fermi-Dirac distribution. Then, it is easy to showthat onceneutrinos decouple from the primordial plasma their momentumdistribution follows

n(p)dp =4πgν

(2πhc)3p2dp

ep

kBTν + 1, (9)

wheren(p) is the number of cosmic neutrinos with momentumbetweenp andp + dp, gν is the number of neutrino spin statesand kB is the Boltzmann constant. The temperature of the cos-mic neutrino background and that of the CMB are related by

Tν(z = 0) =(

411

)1/3Tγ(z = 0) (see for instanceWeinberg

2008) such that the temperature of the neutrino cosmic backgroundat redshiftz is given byTν(z) ∼= 1.95(1+z) K. The current abun-dance of relic neutrinos, obtained by integrating Eq. (9), results in113 neutrinos per cubic centimeter per neutrino family. Thefrac-tion of the total energy density of the universe made up by massive

neutrinos can also be easily derived from Eq. (9):

Ων =Σimνi

93.14h2eV. (10)

During the radiation-dominated era, neutrinos constituted animportant fraction of the total energy density of the universe, play-ing a crucial role in setting the abundance of the primordialele-ments. The impact of massive neutrinos on cosmology is very wellunderstood at the linear order (Lesgourgues & Pastor 2006): on onehand, massive neutrinos modify the matter-radiation equality time,and on the other hand, they slow down the growth of matter per-turbations. The combination of the above two effects produces asuppression on the matter power spectrum at small scales (see forinstanceLesgourgues & Pastor 2006). The signatures left by neu-trino masses on both the matter power spectrum and the CMB havebeen widely used to set upper limits on their masses. Differentstudies point out thatΣimνi < 0.3 eV with a confidence level of2σ (Zhao et al. 2013; Xia et al. 2012; Riemer-Sorensen et al. 2012;Joudaki 2012; Ade et al. 2013).

Even though the impact of neutrino masses is well under-stood at the linear order, the impact of massive neutrinos onthefully non-linear regime has so far received little attention. Re-cent studies using N-body simulations with massive neutrinos haveinvestigated the neutrino imprints on the nonlinear matterpowerspectrum (Brandbyge et al. 2008; Brandbyge & Hannestad 2009;Brandbyge & Hannestad 2010; Viel et al. 2010; Bird et al. 2012;Agarwal & Feldman 2011; Wagner et al. 2012). Such studies havepointed out that in the fully non-linear regime massive neutrinosinduce a scale- and redshift-dependent suppression on the matterpower spectrum which can be up to∼ 25% larger than the predic-tions of the linear theory.

The mean thermal velocity of the cosmic neutrinos, obtainedfrom Eq. (9), is given by∼ 160(1 + z) eV

mνkm/s. For neutrinos

with Σimνi = 0.3 eV their mean velocity today is then∼ 500km/s, i.e. low enough to cluster within the gravitational potentialwells of galaxy clusters. The neutrino clustering has been stud-ied in several works (Singh & Ma 2003; Ringwald & Wong 2004;Brandbyge et al. 2010; Villaescusa-Navarro et al. 2011a, 2013a).In Villaescusa-Navarro et al.(2011a), for instance, the authorsstudied the possibility of detecting the excess of mass in the out-skirts of galaxy clusters due to the extra mass contributionaris-ing from the clustering of massive neutrinos. Furthermore,inVillaescusa-Navarro et al.(2013a) it was shown that the neutrinoclustering also affects the shape and amplitude of the neutrino mo-mentum distribution of Eq. (9) at low redshift.

Massive neutrinos also leave their imprint on the halo massfunction (HMF). In the pioneering work ofBrandbyge et al.(2010)the authors computed the HMF in cosmologies with massiveneutrinos using N-body simulations. They compared their resultswith the HMF obtained by using the Sheth-Tormen HMF andfound that reasonable agreement between the results of the N-body simulations and the Sheth-Tormen HMF (Sheth & Tormen1999) was achieved if the latter was computed using the totalmatter power spectrum but the mean CDM density of the uni-verse,ρCDM. These findings were subsequently corroborated byVillaescusa-Navarro et al.(2013a); Marulli et al. (2011). However,in the recent work byIchiki & Takada(2012) it was suggested thatan even better agreement would be obtained by using the linearCDM power spectrum instead of the total matter linear power spec-trum. This claim has been tested against N-body simulationsre-sulting in an excellent agreement (see e.g.Castorina et al. 2013;Costanzi et al. 2013). Moreover, inCastorina et al.(2013) it was

c© 2011 RAS, MNRAS000, 1–14

Cosmic Degeneracies I:f(R) and massive neutrinos 5

shown that this is the only way of parameterising the HMF in analmost universal form, i.e. independently of the underlying cosmo-logical model.

The impact of neutrino masses on the LSS of the universe cov-ers a wide range of aspects. InVillaescusa-Navarro et al.(2011b)it was suggested that massive neutrinos could play an importantrole in the dynamics of cosmological voids. The reason is that thelarge thermal velocities of the neutrinos make them less sensitiveto the gravitational evolution of voids. In other words, onewouldnaively expect cosmic voids to be empty of CDM, but not of cos-mological neutrinos. The contribution of neutrinos to the void’s to-tal mass has then important consequences in terms of its evolution.In Villaescusa-Navarro et al.(2011b) the authors used such argu-ment to study the possible signature left by massive neutrinos onLyman-α voids.

In a series of recent papers (Villaescusa-Navarro et al.2013b; Castorina et al. 2013) it has also been shown that on largescales the bias between the spatial distribution of dark matterhalos and that of the underlying matter exhibits a dependenceon scale for cosmological models with massive neutrinos. Suchscale-dependence is highly suppressed if the bias is computed withrespect to the spatial distribution of the CDM instead of thetotalmatter.

As briefly summarised in the present section, the effects in-duced by massive neutrinos on the LSS of the universe are numer-ous. In this work we further explore such effects in non-standardcosmologies. More specifically, we study the combined effects onstructure formation of a cosmic background of massive neutrinoswith different values of the total neutrino mass and of a modifi-cation of standard gravity in the form of thef(R) gravitationalaction introduced in Eq. (1). The main aim of the present work istherefore to investigate whether the simultaneous existence of anextended theory of gravity and of a massive neutrino backgroundmight impact on the observational capability of detecting and/orconstraining any of such two extensions of the standard cosmolog-ical scenario. Then, as will become clear in the rest of the paper,the upper bounds on the total neutrino mass quoted in the presentsection that have been obtained through cosmological observationsmight not be valid anymore if the underlying theory of gravity isdifferent from standard GR. Analogously, present and future con-straints on the nature of gravity might be strongly biased inthepresence of a significant massive neutrino background.

4 COSMOLOGICAL SIMULATIONS

The aim of the present work is to investigate the joint effects ofModified Gravity and massive neutrinos on the basic statisticalproperties of structure formation, like the nonlinear matter powerspectrum, the halo mass function, and the large-scale halo bias,with the goal of quantifying the level of degeneracy betweenthesetwo independent extensions of the standard cosmological scenariothat is usually adopted for large N-body simulations, i.e theΛCDMcosmology with massless neutrinos. Furthermore, we aim to assessthe discriminating power of present and future observational datawith respect to such extended cosmologies. In order to include inour comparison both the linear and nonlinear regimes of structureformation to investigate whether a suitable combination ofsuch dif-ferent regimes might help in breaking – or at least alleviating –possible degeneracies, we need to rely on dedicated N-body simu-lations that include at the same time the effects off(R) modifica-

10-6

10-4

10-2

100

P(k

)

10-4 10-2 100 102

k [h/Mpc]

0.70.8

0.9

1.0

1.11.2

P(k

)/P

(k|m

ν=0)

z=99

mν = 0.6 eVmν = 0.4 eVmν = 0.2 eVmν = 0 eV

Figure 1. The linear matter power spectra adopted for the initial conditionsof the simulations with different total neutrino mass.

tions of gravity and of a cosmic neutrino background with differ-ent possible values of the total neutrino massΣimνi . To this end,we have combined theMG-GADGET implementation off(R) mod-els by Puchwein et al.(2013) with the massive neutrinos moduledeveloped byViel et al. (2010), both included in the widely-usedTree-PM N-body codeGADGET3. More specifically, the simula-tions presented in this work are based on explicitly modelling theneutrinos with particles, as opposed to adopting the approximateFourier-based technique (where they are added as an extra-force ink-space). Using particles has the disadvantage of having significantPoisson noise in the neutrino density field, but it is ultimately themore general technique capable of accounting correctly forall non-linear effects. Furthermore, our numerical setting ensures that noiseissues are unimportant in the regime that we are analysing. Thecombination of these two different modules has required somechanges to the original independent implementations in order toallow the handling of very high memory allocation requirementsarising when applying the multi-grid acceleration scheme of MG-GADGET(see againPuchwein et al. 2013, for a detailed descriptionof the multi-grid scheme) to two particle species with very differentspatial distributions.

With such combined N-body code at hand, we have performeda suite of intermediate-resolution N-body simulations on aperiodiccosmological box of1 Gpc/h aside filled with an equal numberN = 5123 of CDM and neutrino particles, for a range of dif-ferent cosmological models characterised by two possible theo-ries of gravity – namely standard GR andf(R) with n = 1 andfR0 = −1 × 10−4 – and different values of the total neutrinomassesΣimνi = 0 , 0.2 , 0.4 , 0.6 eV. The full range of cos-mologies covered by our simulations suite is summarised in Ta-ble 1. All runs are carried out using the latest Planck constraints(Ade et al. 2013) on the background cosmological parameters,which are summarised in the upper part of Table2, and on the am-plitudeAs and spectral indexns of the linear matter power spec-trum atzCMB ≈ 1100 (listed in the bottom part of Table2), therebyresulting in slightly different values ofσ8(z = 0) for the differentcosmologies.

Initial conditions have been generated with a modified ver-sion of the widely-usedN-GENIC code that allows to include anadditional matter component – besides the standard CDM and gas

c© 2011 RAS, MNRAS000, 1–14

6 M. Baldi et al.

0.01 0.10 1.00k [h/Mpc]

0.4

0.6

0.8

1.0

1.2

1.4

1.6

P(k

)/P

(k) fi

duci

al

z = 0

GR, mν = 0.6 eVGR, mν = 0.4 eVGR, mν = 0.2 eVGR, mν = 0 eV

CAMB+HALOFITPlanck best fit± 2σ (σ8)

0.01 0.10 1.00k [h/Mpc]

0.4

0.6

0.8

1.0

1.2

1.4

1.6

P(k

)/P

(k) fi

duci

al

z = 0.6

0.01 0.10k [h/Mpc]

0.4

0.6

0.8

1.0

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Figure 2. The nonlinear matter power spectrum ratio with respect tothe fiducial model for different values of the total neutrinomass as la-belled. The different panels refer toz = 0 (top), z = 0.6 (middle), andz = 1.3 (bottom). The grey shaded area represents the region obtainedwith CAMB and HALOFIT by setting all cosmological parameters to theirfiducial Planck values exceptσ8 which is allowed to vary within its 2-σconfidence interval.

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CAMB+HALOFITPlanck best fit± 2σ (σ8)

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Figure 3. As Fig.2 but for the combined simulations off(R) gravity andmassive neutrinos.

c© 2011 RAS, MNRAS000, 1–14

Cosmic Degeneracies I:f(R) and massive neutrinos 7

0.01 0.10 1.00k [h/Mpc]

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0)

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z = 1.3

Figure 4. The matter power spectrum suppression due to the presence ofmassive neutrinos in both GR (solid lines) andf(R) (dashed lines) models ascompared to the massless neutrino case at different redshifts: z = 0 (left), z = 0.6 (middle), andz = 1.3 (right).

2 4 6 8 101+z

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M(G

R,m

ν=0)

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Figure 5. The evolution of the matter density perturbations amplitude as a function of redshift normalised to the fiducial model atthree different scales in thelinear (k = 0.1h/Mpc, left), mildly-nonlinear (k = 0.5h/Mpc, middle), and nonlinear (k = 1h/Mpc, right) regimes of structure formation.

particle types – with the density perturbations and the thermal ve-locity distribution of a cosmic neutrino background as computedby the public Boltzmann codeCAMB3 (Lewis et al. 2000) for anygiven total neutrino massΣimνi . In particular, the initial condi-tions were generated atz = 99 by perturbing the positions of theCDM and neutrino particles, that were set initially into a cubic reg-ular grid, using the Zel’dovich approximation. The linear matterpower spectra employed to set up initial conditions for the varioustotal neutrino masses are displayed in Fig.1, along with the ratioto the fiducial massless neutrino case. Since at such high redshiftthe MG fifth-force is fully screened, for each total neutrinomasswe used the same initial conditions for both the GR and thef(R)simulations.

We incorporate the effects of baryons by using a transferfunction that is a weighted average of the CDM and baryon transferfunctions as given by CAMB, even if in the present simulationswe do not include baryonic particles as a separate species intheN-body runs. The neutrino particles also receive an extra velocitycomponent arising from random sampling the neutrino momentumdistribution at the starting redshift. In all the differentruns, thetotal matter density is kept fix at the Planck value ofΩM = 0.3175which is split into the contribution of CDM and massive neutrinos,with the latter being given by Eq. (10). As a result, the CDMdensity and the corresponding mass of simulated CDM particlesdecreases whenΣimνi is increased, as summarised in Table1.

3 www.cosmologist.info

5 RESULTS

We now describe the main outcomes of our numerical investigationof combinedf(R) and massive neutrinos cosmologies, concerningthe nonlinear matter power spectrum, the CDM halo mass function,and the halo-matter bias.

5.1 The nonlinear matter power spectrum

For each of our cosmological simulations, we have computed thetotal (i.e. CDM plus neutrinos) nonlinear matter power spectrumby determining the density field on a cubic cartesian grid withtwice the resolution of the PM grid used for the N-body integra-tion (i.e.10243 grid nodes) through a Cloud-in-Cell mass assign-ment. This procedure provides a determination of the nonlinearmatter power spectrum up to the Nyquist frequency of the PMgrid, corresponding tokNy = πN/L ≈ 3.2 h/Mpc. The obtainedpower spectrum is then truncated at thek-mode where the shotnoise reaches20% of the measured power. With the simulatedpower spectra, we can estimate the separate effects off(R) mod-ified gravity and of massive neutrinos on the linear and nonlinearregimes of structure formation at different redshifts. Furthermore,we are able to compute for the first time the joint effects of both in-dependent extensions of the standard fiducial cosmologicalmodelon the amplitude of linear and nonlinear density perturbations.

In Fig. 2 we display the ratio of the nonlinear matter powerspectrum of standard GR cosmologies with different neutrinomasses to the fiducial case of massless neutrinos, for the three dif-ferent redshiftsz = 0 , 0.6 , 1.3. As expected, the massive neu-

c© 2011 RAS, MNRAS000, 1–14

8 M. Baldi et al.

trino component suppresses the power spectrum amplitude both atlinear and nonlinear scales, with a progressively more significantimpact for larger values of the total neutrino massΣimνi . This isconsistent with numerous previous investigations of structure for-mation processes in the presence of a massive neutrino component(see e.g.Brandbyge et al. 2008; Viel et al. 2010; Bird et al. 2012;Agarwal & Feldman 2011; Wagner et al. 2012).

In the uppermost panel of Fig.3, instead, we show the samepower ratio to the fiducial model for the combined simulations off(R) MG with different neutrino masses atz = 0. As one cansee in the figure, for the massless neutrino run (red line) thepowerspectrum ratio shows the typical shape already found by several au-thors (see e.g.Oyaizu et al. 2008; Li et al. 2012b; Puchwein et al.2013) for the samefR0 = −1 × 10−4 MG scenario investigatedhere, which is characterised by a growing enhancement at linearand mildly nonlinear scales that reaches a peak of about50% atk ∼ 0.6h/Mpc (atz = 0), followed by a decrease at progressivelymore nonlinear scales. The latter decrease is produced by the shiftof the transition scale between the 1-halo and 2-halo term domina-tion in the density power spectrum associated with the effectivelylarger value ofσ8 in thef(R) cosmology arising at low redshiftsas a consequence of the MG fifth-force (see e.g.Zhao et al. 2011).Therefore, our puref(R) run is fully consistent with previousfindings for the same cosmological model and with the resultsofPuchwein et al.(2013) obtained with theMG-GADGET code. Thesame quantity is displayed in the remaining panels of Fig.3 for thesame redshifts shown in Fig.2.

One of the main results of the present paper is then representedby the remaining curves of Fig.3 that refer to the impact on thematter power spectrum of different neutrino masses (green,blue,and magenta forΣimνi = 0.2 , 0.4 , 0.6 eV, respectively) in thecontext of an underlyingf(R) cosmology. As one can see from thefigures, the effect off(R) on the matter power spectrum is stronglysuppressed by the presence of massive neutrinos such that the MGcosmology with neutrino masses ofΣimνi = 0.4 eV does not de-viate more than≈ 10% (at z = 0) from the fiducial scenario. Inboth Figs.2 and 3 the grey shaded areas correspond to the ratioof theΛCDM nonlinear matter power spectrum computed with theHALOFIT package (Smith et al. 2003) within the CAMB code bysetting all cosmological parameters to their fiducial Planck valuesexcept forσ8 which is allowed to vary within its 2-σ confidenceinterval (according toAde et al. 2013). In other words, the greyshaded areas provide a visual estimate of the discriminating powerof presently available cosmological data. This is clearly just arough indicative estimate since we are varying only one parameter,but still provides a quantitative idea of how the differences amongthe various models compare with deviations allowed by present ob-servational uncertainties on standard cosmological parameters.

These results clearly show thatf(R) MG models are stronglydegenerate with massive neutrinos, such that a combinationof bothextensions of the standard cosmological model might prevent orsignificantly dim the prospects to constrain any of them througha direct observational determination of the matter power spectrumat low redshifts, unless an independent measurement of the totalneutrino mass from particle physics experiments becomes avail-able. This conclusion extends the previous results ofHe (2013)(based on CMB observations at high-z) and the recent analysisof Motohashi et al.(2013) (which combines both high-z CMB dataand low-z linear power spectrum measurements) to the low-z non-linear regime of structure formation by employing for the first timespecifically designed N-body simulations that allow to consistentlyevolvef(R) MG cosmologies in the presence of massive neutrinos.

This degeneracy therefore poses a further theoretical limitation onthe constraints that can be inferred on cosmological parameters andextended cosmological models from accurate observationaldeter-minations of the matter power spectrum, besides the widely dis-cussed degeneracy with the uncertain effects of baryonic physicalprocesses at small scales, such as e.g. the feedback from ActiveGalactic Nuclei (see e.g.Semboloni et al. 2011; Puchwein et al.2013, for an estimate of the biasing effects of AGN feedback mech-anisms on the determination of standard cosmological parametersand on the peculiar signatures off(R) MG models, respectively).Nonetheless, as one can see by comparing the different panels ofFig. 3, the redshift evolution of the power spectrum might providea way to disentangle the two effects if sufficiently precise obser-vational measurements of the matter power spectrum up to a co-moving wavelength ofk ≈ 10 h/Mpc can be obtained for differentvalues ofz . 1.3.

We defer a more complete survey of otherf(R) cosmolo-gies to a follow-up paper, as in this work we are mainly inter-ested in assessing the degeneracy betweenf(R) gravity and mas-sive neutrinos at a semi-quantitative level. However, it isnatu-ral to expect that milder realisations off(R) theories – as e.g.fR0 =

−1× 10−5 ,−1× 10−6

– will result in a similar kindof degeneracy with lower values of the neutrino mass, and mighttherefore be even more difficult to disentangle observationally.

In order to compare directly the suppression effect of massiveneutrinos in the two different theories of gravity, in Fig.4 we dis-play the ratio of the matter power spectra in the massive and mass-less neutrino cases for GR (solid lines) andf(R) (dashed lines)cosmologies, for the same redshifts displayed in Figs.2 and3. Asone can see in the plots, the relative suppression induced bymas-sive neutrinos is roughly the same for GR andf(R) gravity, withsome small deviations at the most nonlinear scales, which can beascribed to the different stage of evolution of the large-scale struc-tures in which massive neutrinos evolve for the different cosmolo-gies. This result suggests that a sufficiently accurate estimate of thefull nonlinear matter power spectrum in models that combinef(R)gravity with a massive neutrino background might be obtained byapplying fitting functions of the neutrino suppression obtained withstandard GR simulations on top of the nonlinear matter powerspec-trum computed with puref(R) runs. However, such an approxima-tion should be properly tested for a wider range of neutrino massesandf(R) gravity realisations before being considered sufficientlyrobust.

To conclude our investigation of the matter power spectrum,we have computed the growth factor of density perturbationsatthree different scalesk = 0.1 , 0.5 , 1.0 h/Mpc by computing thesquare root of the ratio of the power spectrum amplitudes at the var-ious snapshots of our simulations suite to the amplitude atz = 0.The results of this procedure are displayed in the three panels ofFig. 5, where we plot the ratio of the growth factor of the differentmodels under investigation to the fiducial case of GR and masslessneutrinos. As one can see in the figures, GR andf(R) models have(as expected) an opposite trend in the growth of density perturba-tions as compared to the fiducial case at all scales. Interestingly,while at linear and mildly nonlinear scales (k = 0.1 ; 0.5 h/Mpc)the hierarchy of the models follows the expected linear trend at allredshifts, at the most nonlinear scales (k = 1h/Mpc) the differ-ent models are found to be more entangled at low redshifts, whilethe growth factor slope at high redshifts shows again the expectedlinear behaviour. This highlights how the complex nonlinear inter-play of the different gravity and neutrinos models producesnon-trivial effects on the evolution of density perturbations at very small

c© 2011 RAS, MNRAS000, 1–14

Cosmic Degeneracies I:f(R) and massive neutrinos 9

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Figure 6. The ratio of the Spherical Overdensity HMF in GR simulationswith different neutrino masses as compared to the fiducial case of GeneralRelativity and massless neutrinos. Different panels referto different red-shifts as for Figs.2 and3. The grey shaded areas correspond to the variationof the halo mass function obtained with the standard Jenkinsfitting formula(Jenkins et al. 2001) by varyingσ8 within its 2-σ confidence interval ac-cording to the latest Planck cosmological constraints (Ade et al. 2013).

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Figure 7. As Fig.6 but for the combined simulations off(R) gravity andmassive neutrinos.

c© 2011 RAS, MNRAS000, 1–14

10 M. Baldi et al.

scales. A detailed study of such highly nonlinear effects would re-quire much higher resolution simulations than those available toour present suite, and we defer it to future works.

5.2 The Halo Mass Function

For all the simulations of our sample we have identified dark mat-ter halos by means of a Friends-of-Friends (FoF) algorithm withlinking lengthℓ = 0.2 × d, whered is the mean interparticle sep-aration. In this case, we have considered only CDM particlesinthe linking procedure, such that the obtained halos do not includeneutrino particles. This is to avoid spurious mass contaminationarising from unbound neutrino particles (see e.g.Castorina et al.2013; Costanzi et al. 2013). We emphasise that the contributionof massive neutrinos to the total mass of the dark matter haloisnegligible for the masses considered in this paper, as it wasre-cently verified with N-body simulations in the context of thestan-dardΛCDM cosmology (Villaescusa-Navarro et al. 2011a, 2013a;LoVerde & Zaldarriaga 2013). We have directly checked that thisresult still holds also forf(R) gravity models by comparing spher-ical overdensity halo masses with and without the contribution ofmassive neutrino particles in ourf(R) simulation with the largestneutrino mass (fR-nu0.6), finding deviations below1% over thewhole mass range of the halo sample. Furthermore, for each FoFhalo we have identified its gravitationally bound substructures bymeans of theSUBFIND algorithm (Springel et al. 2001) and we as-sociate to each FoF halo the virial massM200 of its primary sub-structure computed as the mass of a spherical region centredon theparticle with the minimum potential of the halo enclosing a meanoverdensity200 times larger than the critical density of the uni-verse. With this catalog of halos at hand we have computed theHalo Mass Function (HMF) by binning the halos of each cosmo-logical simulation in10 logarithmically equispaced mass bins overthe mass range3.0× 1013 M⊙/h− 1.0× 1015 M⊙/h, where thelower mass bound is given by the minimum halo mass resolved bythe FoF algorithm for the fiducial cosmology.

In Fig. 6 we plot for different redshifts the ratio of the Spheri-cal Overdensity HMF obtained withSUBFIND for GR gravity withdifferent values of the total neutrino mass to the fiducial case ofGR and massless neutrinos. As expected, and consistently withprevious findings (see e.g.Brandbyge et al. 2010; Marulli et al.2011; Ichiki & Takada 2012; Villaescusa-Navarro et al. 2013a;Castorina et al. 2013; Costanzi et al. 2013), the presence of mas-sive neutrinos significantly reduces the abundance of halosover thewhole mass range allowed by our sample, with the suppressionbe-ing more severe for more massive halos and for larger values of theneutrino mass. This effect shows a rather weak redshift dependencefor z . 1.3.

We now consider the combined effect of massive neutrinosand of anf(R) MG theory on the HMF. In Fig.7 we display theratio of the halo abundance for thefR0 = −1 × 10−4 MG modeldiscussed in this paper with different values of the neutrino massas compared to the fiducial cosmology with GR and massless neu-trinos. As the plots show, thef(R) model alone significantly en-hances the abundance of halos, in particular at large masses. Onthe other hand, when MG is associated to a non-vanishing neutrinomass this enhancement is significantly reduced over the wholemass range available within our sample, with the largest neutrinomass included in our investigation (Σimνi = 0.6 eV) even result-ing in an overall suppression of the halo abundance at all massesand at all redshifts.

Analogously to what we displayed in the power spectrum ra-

tio plots above, in Figs.6 and7 the grey shaded areas correspond tothe mass function ratios obtained by computing the HMF throughthe standard Jenkins fitting formula (Jenkins et al. 2001) for thePlanck cosmological parameters of Table2 but allowingσ8 to varywithin its 2-σ confidence interval. This region therefore providesa visual indication of the present discriminating power of obser-vational determinations of the halo abundance at differentredshiftswith respect to the range of cosmological models discussed in thiswork.

The results show again that also for the abundance of darkmatter halos there is a strong degeneracy between the effects of amodification of gravity and of a non-vanishing neutrino mass, suchthat a suitable combination of these two independent extensionsof the standard cosmological model can result in an overall abun-dance of dark matter halos hardly distinguishable from thatof thestandard fiducialΛCDM cosmology with massless neutrinos.

In analogy to what we have done above for the nonlinear mat-ter power spectrum, in Fig.8 we show the suppression effect on theHMF of massive neutrinos of different masses as compared to themassless case in the GR (solid lines) andf(R) (dashed lines) sim-ulations. Also in this case, one can see that the neutrinos-inducedsuppression is roughly the same in the two theories of gravity, eventhough at low redshifts thef(R) model systematically shows aslightly lower suppression than the GR case. This again appearsto be related to the more evolved large-scale distribution that char-acterisesf(R) cosmologies as compared to GR at low redshifts,which makes the impact of massive neutrinos slightly less effec-tive.

Finally, in Fig.9 we display the relative abundance – as com-pared to the fiducial model – of halos above a given mass thresh-old M as a function of redshift and halo mass, for the same threeredshifts considered in previous figures and for three values of thethreshold massM =

1013 , 1014 , 1015

M⊙/h. Therefore, thisratio shows the evolution of halo counts as a function of redshiftin the different models. From the left panel, we notice that massiveneutrinos always suppress halo number counts at all masses andredshifts, while the massless neutrinosf(R) model (red curve inthe right panel) always shows a higher number counts than itsGRcounterpart. Also here, it is interesting to notice that thefiducialmodel lies in between theΣimνi = 0.2 eV and theΣimνi = 0.4eV cases for all mass thresholds and for all redshifts, thereby con-firming the degeneracy observed in all the other statistics discussedso far.

5.3 Halo-matter bias

Both galaxies and dark matter halos are biased tracers of theun-derlying matter distribution. In this section we focus on the biasbetween the spatial distribution of dark matter halos and that ofthe underlying matter for the different cosmological models con-sidered in this paper. We compute the bias as the ratio of the halo-matter cross-power spectrum,Phm, to the matter auto-power spec-trumPmm

bhm(k) =Phm(k)

Pmm(k). (11)

We have used this estimator for the bias since it does not suf-fer from shot-noise (Dekel & Lahav 1999; Hamaus et al. 2010;Baldauf et al. 2013; Smith et al. 2007; Baldauf et al. 2010). Ourhalo catalogue consists of FoF halos with masses above2.3 ×1013 h−1M⊙. A detailed description of the method used to com-

c© 2011 RAS, MNRAS000, 1–14

Cosmic Degeneracies I:f(R) and massive neutrinos 11

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Figure 8. The ratio of the Spherical Overdensity HMF for the differentneutrino masses with respect to the massless case for a GR (solid lines) andf(R)

(dashed lines) theory of gravity at different redshifts:z = 0 (left), z = 0.6 (middle) andz = 1.3 (right).

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M=1014 MO • /h

M=1015 MO • /h

Figure 9. The ratio of halo number counts above three different mass thresholds with respect to the fiducial model as a function of redshift for the GR (left)and thef(R) (right) simulations with different neutrino masses.

pute the power spectra can be found inVillaescusa-Navarro et al.(2013b).

In Fig. 10 we show the bias of each cosmological model nor-malised to the bias of the GR massless neutrino model. Similarly toprevious figures, we show the results atz = 0 (top), z = 0.6 (mid-dle) and z = 1.3 (bottom). In both GR andf(R) cosmologicalmodels the effect of massive neutrinos goes in the same direction,enhancing the bias of dark matter halos at all scales. This isdue tothe fact that massive neutrinos induce a suppression on the mat-ter power spectrum which has the effect that dark matter halos of agiven mass are rarer in a massive neutrino cosmology in compari-son with a universe with massless neutrinos. Therefore, we expectthe bias of objects of the same mass to be higher in a cosmologicalmodel with massive neutrinos.

In the GR cosmological model we find that massive neutrinosinduce a significant scale-dependent bias on large scales asfoundin Villaescusa-Navarro et al.(2013b); Castorina et al.(2013). Thisis clearly seen in the GRΣimνi = 0.6 eV model (solid magentaline). In contrast, we find that for thef(R) cosmological modelsincorporating massive neutrinos the large-scale bias doesnot ex-hibit any significant scale-dependence, thereby recovering the flatbehaviour that characterises the standard scenario. We will explore

this point in further detail in a follow-up paper by using a largersuite of N-body simulations.

At z = 0 the bias of thef(R)MG model withΣimνi = 0.4 eVonly differs from that of the fiducial GR massless neutrino modelby less than3% at all scales. Atz = 0.6 the bias of the 0.4 eVf(R) MG model only departs by less than a5% from the bias ofthe fiducial model. Larger differences among the above two modelsarise atz = 1.3 (∼ 15%), although these differences still do notdepend on scale at any redshift.

Whereas atz = 0.6 the bias of thef(R) MG model withΣimνi = 0.2 eV is almost perfectly (below2%) degenerate withthe massless neutrinos GR cosmology, larger departures occur atz = 0 (∼ 5%) andz = 1.3 (∼ 10%). Also in this case, we findthat the differences between the above two models does not dependon scale at any redshift.

Therefore we conclude that the degeneracies we find for thematter power spectrum and the halo mass function betweenf(R)MG models with massive neutrinos and the massless GR cosmol-ogy also show up in the halo bias. The degeneracy is almost per-fect at redshiftsz = 0 andz = 0.6, while it is slightly broken atz = 1.3. Interestingly, the differences we find for the bias betweenthef(R) MG model withΣimνi = 0.4 eV neutrinos and the mass-

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12 M. Baldi et al.

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GRfR0 = -1e-04

z = 1.3

Figure 10. The ratio of the halo bias for different neutrino masses inthe GR andf(R) cosmologies considered in the present paper (solid anddashed curves, respectively) over the bias obtained in the fiducial run withGR and massless neutrinos. The three panels refer to different redshifts(z = 0 , 0.6 , 1.3) as for previous figures.

less neutrinos GR cosmology does not depend on the scale at anyredshift.

6 CONCLUSIONS

Identifying and computing the characteristic signatures of extendedcosmological models beyond the standardΛCDM scenario repre-sents a necessary step in order to fully exploit the unprecedentedaccuracy of several present and future observational initiatives. Onone hand, such an identification will provide a robust interpretativeframework for possible (and somewhat hoped for) observationalfeatures in tension with the predictions of the standard cosmologi-cal scenario; on the other hand, the quantitative estimation of thedeviations expected in different observables as a consequence ofspecific extensions of the standard model allows an assessment ofthe effective discriminating power of present and future data setswith respect to such extended scenarios. Given the high level ofaccuracy and the wide range of scales involved, a comparisonbe-tween theoretical predictions and observational data cannot be re-stricted to the background expansion history and the linearregimeof structure formation, but needs to be self-consistently extended tothe fully nonlinear regime, thereby requiring the use of large dedi-cated numerical simulations.

In recent years, a wide number of specifically-designed nume-rical tools have been developed with this aim, allowing to predictthe impact of several extended cosmological models on various ob-servables. However, most numerical implementations have beendesigned to investigate one specific class of extended cosmologiesat a time, without allowing for the possibility that the realuniversemight deviate from our standard model in more than one aspectatthe same time. In other words, while detailed and robust predictionsat the nonlinear level are starting to be produced for individual sce-narios that extend the minimalΛCDM cosmology by challengingonly one among its fundamental assumptions (be it the natureofthe Dark Energy, the theory of gravity, the statistical properties ofthe primordial density field, or the composition of the Dark Mat-ter fraction), these predictions usually require that all the otherstandard assumptions still hold, thereby discarding possible degen-eracies among different and independent extensions of the standardmodel.

In the present paper – which is the first in a series of worksdevoted to the investigation of the possible degeneracies amongdifferent and independent extensions of the standard cosmologicalmodel – we have gone beyond this oversimplified setup by includ-ing for the first time in the same numerical treatment the combinedeffects of a modified theory of gravity (in the form of anf(R)model) and of a cosmological background of massive neutrinos. Tothis end, we have combined the recentMG-GADGET implementa-tion of f(R) gravity by Puchwein et al.(2013) with the massiveneutrinos algorithm developed byViel et al. (2010), allowing us toperform the first cosmological simulations of massive neutrinos inthe context of an underlyingf(R) cosmology.

With such code at hand, we have run a suite of intermediate-resolution simulations of structure formation and we haveinvestigated the impact of this twofold extension of the standardmodel on three basic statistics of the large-scale dark matter distri-bution, namely the nonlinear matter power spectrum, the halo massfunction, and the halo bias. Interestingly, our results have revealeda very strong degeneracy between the observational signaturesof f(R) gravity and those of a massive neutrino background inall these three statistics, thereby confirming that the conservative

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Cosmic Degeneracies I:f(R) and massive neutrinos 13

assumption of testing different extensions of the standardmodelon an individual basis is clearly too restrictive and might lead tosignificant misinterpretations of present and future observationaldata. In particular, our investigation most remarkably shows thata suitable combination of the characteristic parameters off(R)gravity and of the total neutrino mass, which would individuallydetermine very significant deviations from the standard predictionsin all three observables, results hardly distinguishable from theexpectations of the standard cosmology. As a consequence, thediscriminating power of present and future observational con-straints based on these statistics (or on any combination ofthem)with respect to a possible modification of the laws of gravityorto a non-negligible value of the neutrino mass might be stronglyweakened if the two effects are both simultaneously at work.Similarly, null tests of the standard cosmological picturebasedon the same three observables might not be able to conclusivelyexclude any of the two individual extended scenarios since theircombination provides nearly indistinguishable predictions ascompared to the reference model.

More specifically, we have computed the deviation with re-spect to the fiducial scenario (i.e.ΛCDM with massless neutri-nos) of the total nonlinear matter power spectrum for a rangeofsimulations including both separate realisations off(R) gravityand non-negligible neutrino masses, and their combination. As ex-pected, the former simulations show significant deviationswith re-spect to the fiducial case, in full agreement with previous resultsin the literature. Massive neutrinos determine a scale- andredshift-dependent suppression of the matter power spectrum at both linearand nonlinear scales, with a maximum deviation from the mass-less case atk ∼ 1 h/Mpc and atz = 0, reaching about45% forthe case of a total neutrino massΣimνi = 0.6 eV, which is thelargest value considered in the present work. On the other hand,f(R) modified gravity determines an enhancement of the nonlin-ear power at similar scales, with a peak of about55% at k ∼ 0.6h/Mpc and atz = 0 for the specific model assumed in this work(fR0 = −1 × 10−4). However, when the two effects are simulta-neously included in the simulations their combined impact on thematter power spectrum is strongly suppressed with respect to theseparate individual realisations. More quantitatively, the combina-tion of ourf(R) modified gravity cosmology with a neutrino back-ground with total massΣimνi = 0.4 eV does not deviate morethan∼ 10% in the nonlinear matter power spectrum atz = 0 fromthe fiducial model over the whole range of scales available withinour simulations suite. The combined deviation is slightly larger forhigher redshifts, but never exceeds∼ 15% up to z = 1.3. Theseresults show very clearly the significant level of degeneracy be-tween these two independent extensions of the standard cosmolog-ical model.

Similarly, we have computed the halo mass function for all thesimulations of our suite and compared the abundance of collapsedstructures in all the runs with the fiducial case of standard grav-ity and massless neutrinos. Again, the two separate extensions ofthe standard cosmological model show very significant deviationsin the abundance of structures at all redshifts, with the massiveneutrinos determining a significant suppression of the halomassfunction, especially at large masses, while our realisation of f(R)modified gravity tends to produce a much larger number of halos,with an enhancement of the halo mass function up to a factor∼ 2for the largest masses available in our simulations atz = 0. Alsoin this case, however, when the two extensions are simultaneouslyincluded in the numerical integration their overall combined effect

is much weaker than for the separate cases just described. Inter-estingly, the same combination of parameters that results the mostdegenerate with the fiducial model for what concerns the nonlin-ear matter power spectrum is found to provide also the most de-generate mass function, thereby preventing the possibility to usea combination of such two observables in order to disentangle thedifferent scenarios.

Finally, we have performed the same type of comparison fora third basic statistics of the large-scale matter distribution: thehalo-matter bias. Also in this case, bothf(R) modified gravityand massive neutrinos determine significant deviations in thehalo bias as a function of scale and redshift. Consistently withprevious studies, we find that massive neutrinos introduce aclearscale-dependence of the halo bias as compared to the almostscale-independent behaviour of the fiducial cosmology, with largerscales showing a lower value of the bias, and with a slope of thescale-dependence that increases with the total neutrino mass. Wefind that a similar phenomenon occurs also forf(R) modifiedgravity, although with a much weaker amplitude and an oppositetrend of the scale-dependence. Again, when the two extensionsof the standard cosmological model are combined in the samesimulations, the impact on the bias is significantly reduced, bothin its overall amplitude and in its scale-dependence. Also in thiscase, atz = 0 the most effective suppression of the predicteddeviation occurs for the same combination of parameters thatmaximally suppress the deviation in the other statistics. However,the combined bias seems to show a more significant redshift evo-lution as compared to the previous observables, thereby providinga possible way to break the (otherwise ubiquitous) degeneracyamong the different cosmological models. This possibilitywill befurther investigated in more detail in a follow-up work.

To conclude, we have presented in this work the first cosmo-logical simulations that combine the effects off(R) modified grav-ity and of a cosmological background of massive neutrinos. Ourinvestigation is part of a long-term programme aimed at studyingand quantifying possible degeneracies between different and mu-tually non-exclusive extensions of the standardΛCDM cosmolog-ical scenario. In this first study, we have demonstrated thatas-sessing the expected features arising in different observables as aconsequence of a specific extension of the standard cosmologicalframework without properly taking into account possible degenera-cies with other independent extensions might lead to significantlyoverestimating the constraining power of present and future obser-vational data sets. In particular, we have shown that with a suita-ble (and not unreasonable) choice of parameters a combination off(R) modified gravity and of massive neutrinos might result in alarge-scale matter distribution which is hardly distinguishable froma standardΛCDM cosmology in several of its basic statistics.

Therefore, our results suggest an intrinsic theoretical limit tothe effective discriminating power of present and future cosmolog-ical observations in the absence of an independent determination ofthe masses of neutrinos coming from particle physics experiments.

ACKNOWLEDGMENTS

We acknowledge financial contributions from contracts ASI/INAFI/023/12/0, by the PRIN MIUR 2010-2011 “The dark Universe andthe cosmic evolution of baryons: from current surveys to Euclid”and by the PRIN INAF 2012 “The Universe in the box: multi-scale simulations of cosmic structure”. MB is supported by the

c© 2011 RAS, MNRAS000, 1–14

14 M. Baldi et al.

Marie Curie Intra European Fellowship “SIDUN” within the 7thFramework Programme of the European Commission. FVN andMV are supported by the ERC Starting Grant “cosmoIGM”. EPand VS acknowledge support by the Deutsche Forschungsgemein-schaft through Transregio 33, “The Dark Universe”. EP also ac-knowledges support by the ERC grant “The Emergence of Struc-ture during the epoch of Reionization”. The numerical simulationspresented in this work have been performed and analysed on a num-ber of supercomputing infrastructures: the MareNostrum3 cluster atthe BSC supercomputing centre in Barcelona (through the PRACETier-0 grant “SIBEL1”); the Hydra cluster at the RZG supercom-puting centre in Garching; the COSMOS Consortium supercom-puter within the DiRAC Facility jointly funded by STFC, the LargeFacilities Capital Fund of BIS and the University of Cambridge; theDarwin Supercomputer of the University of Cambridge High Per-formance Computing Service (http://www.hpc.cam.ac.uk/), pro-vided by Dell Inc. using Strategic Research InfrastructureFundingfrom the Higher Education Funding Council for England.

REFERENCES

Abazajian K. N., Adelman-McCarthy J. K., Agueros M. A., et al.,2009, ApJS, 182, 543

Abbott T., et al., 2005, arXiv:astro-ph/0510346Ade P., et al., 2013, arXiv:1303.5076Agarwal S., Feldman H. A., 2011, MNRAS, 410, 1647Ahn C. P., Alexandroff R., Allende Prieto C., et al., 2013,arXiv:1307.7735

Amendola L., 2000, Phys. Rev., D62, 043511Amendola L., Baldi M., Wetterich C., 2008, Phys. Rev. D, 78, 2,023015

Armendariz-Picon C., Mukhanov V. F., Steinhardt P. J., 2001,Phys. Rev., D63, 103510

Baldauf T., Seljak U., Smith R. E., Hamaus N., Desjacques V.,2013, arXiv:1305.2917

Baldauf T., Smith R. E., Seljak U., Mandelbaum R., 2010, Phys.Rev. D, 81, 6, 063531

Baldi M., 2011, MNRAS, 411, 1077Baldi M., 2012, Phys.Dark Univ., 1, 162Baldi M., 2013, MNRAS, 428, 2074Batista R., Pace F., 2013, arXiv:1303.0414Bennett C., et al., 2013, Astrophys.J.Suppl., 208, 20Bird S., Viel M., Haehnelt M. G., 2012, MNRAS, 420, 2551Blake C., Kazin E., Beutler F., Davis T., Parkinson D., et al., 2011,Mon.Not.Roy.Astron.Soc., 418, 1707

Bode P., Ostriker J. P., Turok N., 2001, Astrophys. J., 556, 93Brandbyge J., Hannestad S., 2009, JCAP, 5, 2Brandbyge J., Hannestad S., 2010, JCAP, 1001, 021Brandbyge J., Hannestad S., Haugbølle T., Thomsen B., 2008,JCAP, 8, 20

Brandbyge J., Hannestad S., Haugbølle T., Wong Y. Y. Y., 2010,JCAP, 9, 14

Buchdahl H. A., 1970, MNRAS, 150, 1Caldwell R., 2002, Phys.Lett., B545, 23Castorina E., Sefusatti E., Sheth R. K., Villaescusa-Navarro F.,Viel M., 2013, arXiv:1311.1212

Cleveland B. T., Daily T., Davis Jr. R., et al., 1998, ApJ, 496, 505Costanzi M., Villaescusa-Navarro F., Viel M., et al., 2013,arXiv:1311.1514

Cowan Jr. C. L., Reines F., Harrison F. B., Kruse H. W., McGuireA. D., 1956, Science, 124, 103

Creminelli P., D’Amico G., Norena J., Vernizzi F., 2009, JCAP,0902, 018

Dekel A., Lahav O., 1999, ApJ, 520, 24Dvali G., Gabadadze G., Porrati M., 2000, Phys.Lett., B485,208Farrar G. R., Peebles P. J. E., 2004, ApJ, 604, 1Feng B., Wang X.-L., Zhang X.-M., 2005, Phys.Lett., B607, 35Fogli G. L., Lisi E., Marrone A., Montanino D., Palazzo A., Ro-tunno A. M., 2012, Phys. Rev. D, 86, 1, 013012

Forero D. V., Tortola M., Valle J. W. F., 2012, Phys. Rev. D, 86,7, 073012

Hamaus N., Seljak U., Desjacques V., Smith R. E., Baldauf T.,2010, Phys. Rev. D, 82, 4, 043515

He J.-h., 2013, arXiv:1307.4876Hill G. J., et al., 2008, ASP Conf. Ser., 399, 115Hu W., Sawicki I., 2007, Phys. Rev., D76, 064004Ichiki K., Takada M., 2012, Phys. Rev. D, 85, 6, 063521Ivezic Z., et al., 2008, arXiv:0805.2366Jenkins A., et al., 2001, Mon. Not. Roy. Astron. Soc., 321, 372Joudaki S., 2012, arXiv:1202.0005Kaiser N., et al., 2002, Proc. SPIE Int. Soc. Opt. Eng., 4836,154Laureijs R., Amiaux J., Arduini S., et al., 2011, arXiv:1110.3193Lesgourgues J., Pastor S., 2006, Phys. Rept., 429, 307Lewis A., Challinor A., Lasenby A., 2000, Astrophys. J., 538, 473Li B., Hellwing W. A., Koyama K., Zhao G.-B., Jennings E., et al.,2012a, arXiv:1206.4317

Li B., Zhao G.-B., Teyssier R., Koyama K., 2012b, JCAP, 1201,051

Llinares C., Mota D. F., Winther H. A., 2013, arXiv:1307.6748Loeb A., Weiner N., 2011, Phys. Rev. Lett., 106, 171302Lombriser L., Koyama K., Zhao G.-B., Li B., 2012,arXiv:1203.5125

LoVerde M., Zaldarriaga M., 2013, arXiv:1310.6459Maccio’ A. V., Ruchayskiy O., Boyarsky A., Munoz-Cuartas J.C.,2012, arXiv:1202.2858

Marulli F., Carbone C., Viel M., Moscardini L., Cimatti A., 2011,MNRAS, 418, 346

Motohashi H., Starobinsky A. A., Yokoyama J., 2013,Phys.Rev.Lett., 110, 12, 121302

Nicolis A., Rattazzi R., Trincherini E., 2009, Phys.Rev., D79,064036

Oyaizu H., Lima M., Hu W., 2008, Phys. Rev., D78, 123524Perlmutter S., et al., 1999, Astrophys. J., 517, 565Pogosian L., Silvestri A., 2008, Phys. Rev. D, 77, 2, 023503Puchwein E., Baldi M., Springel V., 2013, MNRAS, 436, 348Ratra B., Peebles P. J. E., 1988, Phys. Rev., D37, 3406Riemer-Sorensen S., Parkinson D., Davis T., Blake C., 2012,arXiv:1210.2131

Riess A. G., et al., 1998, Astron. J., 116, 1009Ringwald A., Wong Y. Y. Y., 2004, JCAP, 12, 5Schmidt B. P., et al., 1998, Astrophys.J., 507, 46Schmidt F., Lima M. V., Oyaizu H., Hu W., 2009, Phys. Rev., D79,083518

Sefusatti E., Vernizzi F., 2011, JCAP, 1103, 047Semboloni E., Hoekstra H., Schaye J., van Daalen M. P., Mc-Carthy I. G., 2011, MNRAS, 417, 2020

Sheth R. K., Tormen G., 1999, Mon. Not. Roy. Astron. Soc., 308,119

Singh S., Ma C.-P., 2003, Phys. Rev. D, 67, 2, 023506Smith R. E., et al., 2003, Mon. Not. Roy. Astron. Soc., 341, 1311Smith R. E., Scoccimarro R., Sheth R. K., 2007, Phys. Rev. D, 75,6, 063512

c© 2011 RAS, MNRAS000, 1–14

Cosmic Degeneracies I:f(R) and massive neutrinos 15

Smoot G. F., Bennett C., Kogut A., Wright E., Aymon J., et al.,1992, Astrophys.J., 396, L1

Sotiriou T. P., Faraoni V., 2010, Rev.Mod.Phys., 82, 451Springel V., 2005, Mon. Not. Roy. Astron. Soc., 364, 1105Springel V., White S. D. M., Tormen G., Kauffmann G., 2001,MNRAS, 328, 726

Starobinsky A. A., 1980, Phys. Lett., B91, 99Viel M., Becker G. D., Bolton J. S., Haehnelt M. G., 2013, Phys.Rev. D, 88, 4, 043502

Viel M., Haehnelt M. G., Springel V., 2010, JCAP, 1006, 015Villaescusa-Navarro F., Bird S., Pena-Garay C., Viel M., 2013a,JCAP, 3, 19

Villaescusa-Navarro F., Marulli F., Viel M., et al., 2013b,arXiv:1311.0866

Villaescusa-Navarro F., Miralda-Escude J., Pena-GarayC., QuilisV., 2011a, JCAP, 6, 27

Villaescusa-Navarro F., Vogelsberger M., Viel M., Loeb A.,2011b, arXiv:1106.2543

Wagner C., Verde L., Jimenez R., 2012, ApJ, 752, L31Weinberg S., 2008, Cosmology, Oxford University PressWetterich C., 1988, Nucl. Phys., B302, 668Wetterich C., 1995, Astron. Astrophys., 301, 321Xia J.-Q., Granett B. R., Viel M., et al., 2012, JCAP, 6, 10Zhao G.-B., Li B., Koyama K., 2011, Phys.Rev., D83, 044007Zhao G.-B., Pogosian L., Silvestri A., Zylberberg J., 2009,Phys.Rev. D, 79, 8, 083513

Zhao G.-B., Saito S., Percival W. J., et al., 2013, MNRAS

c© 2011 RAS, MNRAS000, 1–14