Dark Matter, Modified Gravity and the Mass of the Neutrino
aa r X i v : . [ a s t r o - ph ] J un Dark Matter, Modified Gravity and the Mass of the Neutrino.
P. G Ferreira , C. Skordis , C. Zunckel Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK Perimeter Institute, Waterloo, Ontario N2L 2Y5, Canada (Dated: October 23, 2018)It has been suggested that Einstein’s theory of General Relativity can be modified to accomodate mismatchesbetween the gravitational field and luminous matter on a wide range of scales. Covariant theories of modifiedgravity generically predict the existence of extra degrees of freedom which may be interpreted as dark matter.We study a subclass of these theories where the overall energy density in these extra degrees of freedom issubdominant relative to the baryon density and show that they favour the presence of massive neutrinos. Forsome specific cases (such as a flat Universes with a cosmological constant) one finds a conservative lower boundon the neutrinos mass of m ν > . eV. I. INTRODUCTION
There is compelling evidence that the baryons in the Universeare unable to generate the gravitational potentials that we ob-serve on a wide range of scales. A simple paradigm can beused to explain this mismatch between light and gravity: theUniverse is filled with an appreciable amount of matter whichis cold (i.e. has non-relativistic velocities today) and does notinteract with light. It has been shown that Cold Dark Matter(CDM) can explain a host of observation, from dynamics ofclusters to the formation of the cosmic web [1].The CDM paradigm has been proposed within the contextof Newtonian gravity and Einstein’s theory of General Rela-tivity. It has been argued that these theories may not be validon all scales. Indeed, proposals for modifying gravity havebeen shown to fit much of the currently available data [2].A plethora of covariant theories have been studied in detail;TeVeS gravity, modified Einstein-Aether theories, conformalgravity, higher derivative actions, etc, have been advocated aspossible rival theories to the CDM scenario [3, 4, 5, 6, 7, 8, 9].There has been considerable effort in studying the cosmologi-cal consequences of these theories [10, 11, 12, 13]. Given thelevel of precision of current cosmological data, it is possible tofind severe constraints on these alternative theories and com-pare their ability to describe nature with the CDM scenario.There is an important, generic feature of covariant theo-ries of modified gravity which is often overlooked: althoughthey tamper with the gravitational sector of the equations ofmotion, they also inevitably lead to the introduction of ex-tra degrees of freedom which may be interpreted as an exoticform of dark matter. Let us exemplify. Theories which mod-ify the Einstein-Hilbert action by, for example, replacing theRicci scalar, R , by a function of different curvature invari-ants, f ( R, R αβ R αβ , · · · ) , introduce higher derivative terms,and hence new modes. These new modes will contribute to theoverall energy density. This is patently obvious in the case oftheories where f is simply a function of R ; such theories canbe mapped onto normal Einstein gravity with an additionalscalar field. This also true of conformal gravity, where the ac-tion is now constructed from the Weyl tensor. A field mustbe added to fix the scale of gravity and the resulting low en-ergy equations are fourth order [14]. More modern attempts atconstructing theories of modified gravity have the same char-acteristics in a much more explicit way. In TeVeS [5], a scalar field and a vector field is introduced which not only mod-ify the gravitational field equations but also source the verysame field through their stress energy tensor. In generalizedEinstein-Aether theories, a time-like vector field is introduced[7].Given what we have just said, there is an obvious ques-tion: aren’t these extra degrees of freedom simply a contrivedform of dark matter? It is conceivable that the extra degreesof freedom in modified theories of gravity may play such arole. If so, dark matter has been introduced through the backdoor. It turns out that the role of extra degrees of freedomin theories of modified gravity is more complicated than onemight expect. In Skordis et al [10], it was shown that the extradegrees of freedom in TeVeS can make a negligible contribu-tion to the background (or overall) energy density. Indeed, ifTeVeS is to be consistent with big bang nucleosynthesis, thefractional energy density in these extra degrees of freedom, Ω X , must be under a percent. Yet even though Ω X ≪ , fluc-tuations in the extra fields could have a significant impact onthe growth of structure. In particular, due to the modified na-ture of gravity, they could source the growth of gravitationalpotentials and sustain them through Silk damping at recom-bination. These results were corroborated in Dodelson andLiguori [15], where the fluctuations in the vector field werefound to play an important role.Hence some theories of modified gravity can fit current ob-servations of large scale structure, either from galaxy surveysor the cosmic microwave background, even though Ω X ≪ .We would like to point out that the latter property is notgeneric. In some incarnations Ω B ≪ Ω X ≃ where Ω B is the fractional energy density in baryons. These theories endup being a hybrid of the two paradigms, modified gravity anddark matter, and in principle should be harder to distinguishfrom dark matter theories (although there are some sugges-tions of specific tests) [7, 8].In this paper we will try to expand on an important fea-ture of TeVeS pointed out in Skordis et al : if one assumesthat the Universe is flat and the only form of non relativis-tic matter consists of baryons (consistent with Big BangNucleosynthesis), the angular power spectrum of the CosmicMicrowave Background (CMB) will differ significantly fromobservations. The only way to resolve this discrepancy is tointroduce some form of non-relativistic matter, and the onlyone allowed within the known menagerie of fundamental con-stituents of the Universe is a massive neutrino. To matchobservations of the CMB, neutrinos with a mass of approxi-mately eV are needed. This result is clearly a hint of a moregeneral statement that may be made about theories of modifiedgravity in which the extra degrees of freedom play a subdomi-nant role: if these theoriess are to agree with measurements ofthe CMB then they require the presence of massive neutrinos.We wish to see if this implies a lower bound on the mass ofthe neutrino. II. AN APPROXIMATE THEORY AND COSMOLOGICALOBSERVABLES
Let us consider a generic modified theory of gravity in thelimit of homogeneity and isotropy. The physical metric (i.e.the metric which is minimally coupled to the matter fields)can be parametrized in terms of a scale factor, a ( t ) which hasa logarithmic derivative, H = d ln( a ) /dt . The energy densityof the Universe can be split into the normal degrees of free-dom, ρ (such as baryons, photons, neutrinos and dark energy)and the extra degrees of freedom that arise from the modifica-tions, ρ X . The modified Friedman equations look somewhatlike F ( a, H ) H = 8 πG ρ + ρ X ) (1)where G is Newton’s constant and F H is a function thatarises from varying the action for a particular theory. In fact itis convenient to rewrite the equation in a more familiar formby defining an effective Newton’s constant G eff = G/F . Forthe purpose of what follows we use a parametrization suchthat G eff ≃ G eff ( a ) ; with a sufficiently flexible choice ofparameters we can encompass cases where G eff depends on a , H , etc.We consider a sub class of the theories, in which ρ X < ρ B ,(where ρ B is the Baryon density). We consider a parametriza-tion such that ρ X ≃ f B ρ B + f R ρ R (2)where ρ R is the energy density in radiation. We have that f B < and the correct abundance of light elements requiresthat f R < − . We also include in ρ , a component that be-haves like dark energy, ρ DE , with an equation of state w < .We find it convenient to parametrize the equation of state ofthe dark energy component as w = w + w z/ (1 + z ) . Modi-fications to the gravitational sector may lead to accelerated ex-pansion at late times (such as those proposed in [4, 7, 8, 12]),meaning that the dark energy could also arise from the extrafields in the modified gravity sector. Our dark energy termincludes all of these possibilities.With the evolution of the scale factor in hand, there area few observables that we may now calculate. Let us startoff with the position of the first peak of the angular powerspectrum of the Cosmic Microwave Background (CMB). It isa direct measure of the angular diameter distance and henceof the expansion history of the Universe from recombination until today and is the centrepiece of the analysis of this pa-per. Schematically we have the following picture [16]. Be-fore recombination (which occured at time t ∗ ), photons andbaryons were tightly coupled and underwent acoustic oscilla-tions. During tight coupling the photon density contrast in theconformal Newtonian gauge, obeys the differential equation ¨ δ γ + 3 ρ b ρ b + 4 ρ γ ˙ aa ˙ δ γ + k c s δ γ = S [Φ , ¨Φ , Ψ] (3)where c s = ρ γ ρ b +4 ρ γ ) is the sound speed, S [Φ , ¨Φ , Ψ] is asource (a function of the gravitational potentials, Φ and Ψ )and derivatives are with conformal time τ . In the WKB ap-proximation [16] the two linearly independent solutions to thehomogeneous part are cos kr s and sin kr s , and depend on thesound horizon r s ( τ ) = R τ c s dτ . The important thing is thatthese solutions are valid for any theory of gravity for whichphotons and baryons see the same physical metric. All modi-fications are to gravity or additional fields and implicitely alterthe inhomogenous part through the source term S which onlydepends on the potentials Φ and Ψ of that same physical met-ric.The physical scale, d ∗ of the acoustic waves is set bythe sound horizon at time t ∗ , i.e. d ∗ ≃ R t ∗ c s ( t ) dt . Af-ter recombination, photons decoupled from the baryons andfreestreamed towards us, travelling a distance given by d = ca R t t ∗ dt/a = R a a ∗ da ( a H ) where the subscript labels to-day. The angular size on the sky of the sound horizon, θ ∗ , isgiven by θ ∗ ≃ a d ∗ a ∗ d The sound horizon at last scattering leavesa very distinct signature on the angular power spectrum of theCMB: a series of peaks and troughs. The spectrum generatedat recombination is related to the spectrum today via a projec-tion through a spherical Bessel function j ℓ ( k ( τ − τ ∗ ) , where adτ = dt , (an ultraspherical bessel function in the curvedcase). Once again this is independent of the theory of gravity.The position of the peaks and troughs in the angular power-spectrum is primarily dependent on the cosmological shift pa-rameter, R which is related to the angular diameter distanceand is given by R = 12 s Ω M ( a )Ω K ( a ) sin yy = p | Ω K ( a ) | Z a a ∗ daa ( P i Ω i ( a )) / (4)where Ω i ( a ) is the fractional energy density of component i as a function of scale factor ( K corresponds to the curvatureand M to non relativistc matter) [17].Very few assumptions have gone into this calculation: theUniverse underwent recombination and the horizon structureis a result of the expansion rate of the Universe. Current mea-surements of the CMB have reached a level of precision suchthat it is practically impossible to deviate from this simple pic-ture [19]. Attempts at changing these fundamental assump-tions inevitably lead to radical departures from this simple pic-ture and a gross mismatch to the data. So any theory of mod-ified gravity must lead to the basic picture of the CMB that Model 68 % CL 95 % CL Λ CDM . ≤ m ν ≤ .
17 0 . ≤ m ν ≤ . w CDM . ≤ m ν ≤ .
59 0 . ≤ m ν ≤ . w CDM + Ω κ . ≤ m ν ≤ . m ν ≤ . CDM+ G ( z ) 0 . ≤ m ν ≤ . m ν ≤ . w CDM + G ( z ) 1 . ≤ m ν ≤ .
33 0 . ≤ m ν ≤ . w CDM + Ω κ + G ( z ) m ν ≤ . m ν ≤ . CDM + G ( α, γ, z ) 0 . ≤ m ν ≤ . m ν ≤ . TABLE I: Results for different cosmological models for a compila-tion data set. we infer from the data. Hence we can use our Eq. 1 to workout R for theories of modified gravity with the caveat that thefractional energy densities must be rescaled by the effectiveNewton’s constant, i.e. we must replace Ω i by ( G eff /G )Ω i in Equation 4. Throughout this analysis we consider a con-servative bound on the shift parameter: . < R < . [18].Another useful observable, as measured from the Hubblediagram of distant supernovae, is the luminosity distance, d L .It is related to the angular diameter distance, d A , describedabove, through d L = (1 + z ) d A , where z = a /a definesthe redshift at a given value of the scale factor. While theCMB gives us one measure of d A at z ≃ , the Hubble di-agram of distant supernovae gives us a series of measurementsof d L out to z ≃ . . Again, as above we can use our modifiedFriedman equations to calculate d L and compare to the data.Given that we do not have to use any information about pertur-bations about the background, we make even fewer assump-tions. We use the group of supernovae, termed the ‘gold’ set,from the HST/GOODS programme [20], complemented bythe recently discovered higher redshift supernovae, reportedin [21].Finally, we consider two more measurements. We take intoaccount the constraints on Ω B from the abundance of lightelements. We use Ω B h = 0 . ± . , where the Hubbleconstant is defined to be H = 100 h km s − Mpc − . Lastlywe consider current constraints from the Key Project of theHubble Space Telescope (HST) on the expansion rate today.We use H = 72 ± km s − Mpc − . III. EXPLORING PARAMETER SPACE
We are interested in seeing if the presence of massive neu-trinos is a generic feature of the class of models that we areconsidering. We assume three families of neutrinos with iden-tical masses, and we take the mass of each family, m ν , as thefree parameter. An obvious first case to study is a generaliza-tion of the TeVeS result from Skordis et al , i.e. a EuclideanUniverse with a cosmological constant and a constant effec-tive Newton’s constant. Indeed we find that the posterior for m ν is positive, centered at m ν ≃ . eV and we can set alower bound on the neutrino mass at the 95 % confidence level(CL) of m ν > . eV, as shown in table (I). The value of m ν proposed in Skordis et al lies comfortably in that range . < m ν < . eV.Relaxing the assumption that the acceleration is driven bya cosmological constant (i.e. freeing up w and w ), leadsto a lower bound of m ν > . eV, slightly stronger than inthe previous case. The supernovae data strongly constrain theparameters describing the nature of dark energy in the range ≤ z ≤ . , the era where its contribution is dominant, andfavour an effective w ( z ) < − . This means that the contri-bution of the dark energy component diminishes more rapidlythan Λ as a function of z . To compensate, the neutrinos arerequired to be relativistic at the surface of last scattering andhence considerably more massive, giving rise to the observedshift in the distribution to higher masses. The increased free-dom gives a broader distribution. When the supernovae dataset is removed, a larger contribution to the total energy fromthe dark energy component is allowed, weakening the lowerbound on m ν , as shown in Table (II). However, for this simpleclass of theories, strong statements can now be made: there isa definite lower bound on the mass of the neutrino, as can beseen from Table (I).Up until now, studies of theories of modified gravity havebeen undertaken in the context of Euclidean Universes. Re-laxing the assumption of spatial flatness greatly broadens theposterior distribution of m ν , in particular, extending it to asmuch as . eV at the 95 % CL. These models correspond toclosed Universes where Ω K < . In addition the dark en-ergy parameters are less strongly peaked (due to the degen-eracy with curvature). Models in which the Universe is openare however favoured, leading to a weakened lower bound of m ν > . eV but only at the CL. In the absence of thesupernovae data, Ω K is weakly constrained accordance withthe increased freedom. This leads to a generally broader m ν distribution with a similar peak.We have been exploring the effect of the extra degrees offreedom but we should expect modifications to the left handside of equation 1. We have parameterized this in terms of G eff that depends on the scale factor. In principle, the timedependence of G eff can be more complex, depending on thenormal matter fields as well as the extra degrees of freedom.Furthermore, for any given theory of modified gravity, theBianchi identities as well as the various couplings between G eff and the remaining sector, impose specific constraints onits time evolution [22]. I.e. we do no have complete freedomto vary G eff .In what follows, we will be conservative and jettison anyconstraints that come from consistency but we will considertwo types of relatively general behaviour which encompassewhat we have found in a wide range of models. One simpleparametrization is G eff = G (1 + z ) n For example for TeVeS one finds that, for a sufficiently small n one can adequately mimic the behaviour of G eff . Note thatthis parametrization does lead to a monotonically changing G eff , all the way back to recombination and so must reallyonly be considered an approximation- if not, it might lead tosubstantial changes to the peak structure in the CMB at re-combination and we have argued that this is clearly not thecase.A variable G eff can have a substantial effect on allowedneutrino masses. In particular, in a Euclidean Universe withcosmological constant, it lowers the required mass contribu-tion from neutrinos significantly to m ν < . eV at the σ level such that the massless case is no longer ruled out.This implies an anti-correlation between n and m ν . Extend-ing the model further to include dark energy again requireslarger neutrino masses (lower bound of . eV at 95 % CL).However in non-flat Universe case, the freedom in parameterspace means that a wide range of masses are tolerable, includ-ing the massless scenario and m ν = 1 . eV at the 2 σ level.We note that allowing for the possibility of a time-dependent G eff ( z ) parameterized as above and admitting spatial curva-ture will have similar effects on the Hubble equation. The pri-mary effect of G eff is to shift the distribution of m ν to lowervalues, while Ω K increases the range of neutrino masses thatcan be tolerated. This is explicitly illustrated in Figure (1).The plot (a) compares the 68 % and 95 % confidence intervalsin ( m ν , Ω K ) space when G eff is time-independent (dashedlines) and dynamical (solid lines) and shows the shift in theallowed regions to more negative values of Ω K . Figure (1b)illustrates the impact of spatial curvature on constraints on m ν in the presence of a time-dependent G eff . The σ and σ re-gions are significantly reduced when curvature is admitted.Another possible parameterization is if G eff switches be-tween two values at some point in the past. For example,if G eff is approximately six times larger during the baryondominated era than it is now, the background evolution willbe essentially equivalent to that of dark matter dominated Uni-verse at that time. To mimic this effect we consider G eff = G (cid:18) αz γz (cid:19) (5)At low redshift, G eff starts at G and increases linearly with z . At large redshift ( z > /γ ), G(z) tends towards a constant G α/γ . We limit the change in G(z) from z = 0 to recombi-nation by imposing the condition that α/γ < such that itsdoes not change by factor of more than . We find that withthis parametrization, which is reminiscent of a number of dif-ferent models, that the results are almost identical to that ofthe previous proposal for G eff . Indeed, it is the very late timebehaviour of G eff that plays a significant role in changing theobservables and in that respects the two parametrizations arevery similar. IV. DISCUSSION
It has been claimed that modified theories of gravity in-evitably require the presence of massive neutrinos and thatthese may be sufficiently massive to be measurable with upand coming neutrino experiments such as KATRIN [23]. Thisclaim has been triggered by two pieces of anecdotal evidence.Firstly that the simplest TeVeS model needs neutrinos to fit theangular power spectrum of the CMB as shown in [10]. Andsecondly, that attempts at reconciling observed and inferred
Model 68 % CL 95 % CL Λ CDM m ν ≤ . m ν ≤ . w CDM . ≤ m ν ≤ .
97 0 . ≤ m ν ≤ . w CDM + Ω κ . ≤ m ν ≤ . m ν ≤ . CDM + G ( z ) m ν ≤ . m ν ≤ . w CDM + G ( z ) 0 . ≤ m ν ≤ . m ν ≤ . w CDM + Ω κ + G ( z ) m ν ≤ . m ν ≤ . CDM + G ( α, γ, z ) m ν ≤ . m ν ≤ . TABLE II: Results for different cosmological models for a compila-tion data set where the supernovae data is excluded.(a)(b)FIG. 1: (a) The 68 % and 95 % confidence regions in ( m ν , Ω K ) spacefrom the compilation data set excluding the supernovae data. The re-gion with dashed lines corresponds to a model where n = 0 (ie. G ( z ) = G ), while the region with solid lines corresponds to a Uni-verse where n = 0 (ie. G ( z ) = G ( z, n ) ). (b) The 68 % and 95 % confidence regions in ( m ν , n ) space. The region with dashed linescorresponds to a model where Ω K = 0 , while the region with solidlines corresponds to a Universe where Ω K = 0 . The plots illustratethe general trend of G eff to shift the distribution of m ν to lowervalues, while Ω K increases the range of neutrino masses that can betolerated. masses of clusters requires the presence of a massive neutrinohalo [24, 25]. In this letter we have attempted to extend theremit of the first piece of evidence.We have found that, although for a restricted set of models,we can place a lower bound on the mass of the neutrino, formore general ranges of parameters, it is possible to satisfy thesubset of cosmological constraints without having to invokemassive neutrinos. This is not to say that specific models with,for example, a variable effective Newton’s constant might notlead to a tight constraint on the neutrino mass. But it is clearlynot possible to make a definitive statement on the mass of theneutrino for general theories of modified gravity. Theoriesmust be studied case by case and we have shown how this canbe done in an economical way.It may be possible to come up with constraints on the neu-trino masses from a different set of observables, related to the second piece of evidence. For example, in the simplest pic-ture of a cluster in these theories, neutrinos seem to be in-evitable to be able to make up dynamical mass measurementsand weak lensing observations. This simple picture is incom-plete and much of the work that has been done on clusters inthe context of modified gravity has opted to ignore the extradegrees of freedom [26]. They can play a significant role and,in the same way as for large scale observations, may substan-tially weaken cluster constraints on the neutrino mass. A moredetailed analysis of these systems must be undertaken beforedefinitive conclusions can be inferred. Acknowledgments : We thank A. Cooray, A. Melchiorri, G.Starkman and T. Zlosnik for discussions. C. Zunckel is sup-ported by a Domus A scholarship awarded by Merton College.Research at Perimeter Institute for Theoretical Physics is sup-ported in part by the Goverment of Canada through NSERCand by the Province of Ontario through MRI. [1] J. Peacock, Cosmological Physics, CUP (1999).[2] M. Milgrom, Astrophys. J, 270, 365-370 (1983), R.H.Sanders,S.S. McGaugh, Ann. Rev. Astron. Astrophys., 40, 263-317(2002).[3] R.H.Sanders, Astrophys. J. 480, 492-502 (1997).[4] P.D. Mannheim, Astrophys. J. 342,635 (1989).[5] J.D.Bekenstein, Phys. Rev. D70, 083509 (2004).[6] T. Jacobson, D. Mattingley, Phys. Rev. D64, 024028 (2001),C.Eling, T. Jacobson, D. Mattingley, gr-qc/0410001 (2004).[7] T.G. Zlosnik, P.G. Ferreira, G.D. Starkman Phys. Rev. D75,044017 (2007).[8] H.S. Zhao arXiv:0710.3616 (2007).[9] C. Skordis, arXiv:0801.1985.[10] C. Skordis at al , Phys.Rev.Lett. 96, 011301 (2006).[11] C. Skordis, Phys.Rev.D74,103513 (2006).[12] F.Bourliot et al
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