Equilibrium configurations of 11eV sterile neutrinos in MONDian galaxy clusters
aa r X i v : . [ a s t r o - ph . C O ] S e p Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 8 November 2018 (MN L A TEX style file v2.2)
Equilibrium configurations of eV sterile neutrinos inMONDian galaxy clusters G. W. Angus , ⋆ , B. Famaey , , A. Diaferio , Dipartimento di Fisica Generale “Amedeo Avogadro”, Universit`a degli studi di Torino, Via P. Giuria 1, I-10125, Torino, Italy Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Torino, Italy AIfA, Universit¨at Bonn, 53121 Bonn, Germany Observatoire Astronomique, Universit´e de Strasbourg, CNRS UMR 7550, F-67000 Strasbourg, France
ABSTRACT
Modified Newtonian Dynamics (MOND) can fit a broad range of galaxy kinematicdata, but struggles with clusters of galaxies. MONDian clusters need dark matter, andhere we test the 11 eV /c sterile neutrino - used to fit the first three acoustic peaksof the cosmic microwave background - by investigating their equilibrium distributionsin 30 groups and clusters over a wide range of temperatures. We do this by firsttaking the known sterile neutrino density, necessary for hydrostatic equilibrium of theintracluster medium (or to produce the observed lensing map). Then, we solve forthe sterile neutrino velocity dispersion, needed for their own hydrostatic equilibrium,through the equation of state for a partially degenerate neutrino gas. The velocitydispersion is a unique, continuous function of radius determined by the density andmass of the sterile neutrinos particles. Knowing both the sterile neutrino density andvelocity dispersion tells us the Tremaine-Gunn phase-space limit at all radii. We findthat all 30 systems serendipitously reach the Tremaine-Gunn limit by the centre,which means a portion of the dynamical mass must always be covered by the brightestcluster galaxy. Interestingly, the typical fitted K-band mass-to-light ratio is unity andat most 1.2, which is very consistent - although leaving no margin for error - withstellar population synthesis models. Amidst the sample there are several special casesincluding the Coma cluster (for which dark matter was first proposed), NGC 720(where geometrical evidence for dark matter was found) and the bullet cluster (wheredark matter - of some kind - in clusters was directly proven to exist). We demonstratethat 11 eV /c sterile neutrinos are unlikely to influence spiral galaxy rotation curves, asthey don’t influence even some very massive early-types (NGC 4125 and NGC 6482).Finally, we conclude that it is intriguing that the minimum mass of sterile neutrinoparticle that can match the cosmic microwave background is identical to the minimummass found here to be consistent with equilibrium configurations of MONDian clustersof galaxies. Sterile neutrinos are hypothetical additions to the standardmodel of particle physics. They are right handed, neutralleptons which interact only via gravity, which earns themthe “sterile” prefix, contrary to the active neutrinos whichalso participate in the weak interaction. Gravity aside, ster-ile neutrinos also can interact with the active neutrinos viathe quantum mechanical phenomenon of neutrino oscilla-tions. This behaviour has been investigated by the LiquidScintillator Neutrino Detector (Aguilar et al. 2001) and theMiniBoone experiment (Maltoni & Schwetz 2007). However,no concrete evidence was convincingly found to suggest theexistence, or non-existence, of sterile neutrinos from thedisappearance of active neutrinos. The basis of some ap-peals to sterile neutrinos are for aesthetic reasons, since the active neutrinos are entirely left-handed; and others makeuse of them in the so called “see-saw mechanism” whichcan give rise to the small masses of the active neutrinos(Lindner et al. 2002). Here we make no claims of a deepertheory for sterile neutrinos, but rather continue to investi-gate a startling coincidence.A recent analysis of the Cosmic Microwave Background(CMB) by Angus (2009) has demonstrated that the acous-tic peaks in the angular power spectrum as measured byWMAP (Dunkley et al. 2009) and ACBAR (Reichardt et al.2009) can convincingly be generated by a single, thermal (byvirtue of neutrino oscillations in the early Universe) sterileneutrino (SN) with mass m ν s = 11 eV /c . This SN is astraight substitution for the cold dark matter (CDM) of theconcordance cosmological model (Spergel et al. 2007), suchthat Ω ν s = 0 . m ν s = 0 . b = 0 .
047 and the spec- c (cid:13) G. W. Angus, B. Famaey, A. Diaferio tral index n s = 0 . a o = 1 . × − ms − ,around and below which dynamics do not follow from stan-dard Newtonian theory. In particular, the true modulus ofgravity, g ( r ) = V c ( r ) r − , is not linearly related to the New-tonian gravity, g n ( r ) = GM ( r ) r − , but instead g = 12 g n (cid:20) r a o g n (cid:21) (1)This ensures adherence to two critical axioms: that gravityis Newtonian in regions of strong gravity, and that when g << a o , g ∝ /r meaning rotation curves are flat at theperiphery of spiral galaxies.The additional gravity afforded by MOND, replacesthe need for DM in dwarf spheroidals (Angus 2008;Milgrom 1995; S´anchez-Salcedo & Hernandez 2007;Serra et al. 2009), spiral (e.g. McGaugh & de Blok1998; Sanders & Noordermeer 2007; Famaey & Binney2005; McGaugh 2008) and X-ray dim elliptical galaxies(Milgrom & Sanders 2003; Angus et al. 2008b) often withremarkable accuracy. Furthermore, rotation curves of tidaldwarf galaxies have been observed by Bournaud et al.(2007) and the independent analysis by Gentile et al.(2007) and Milgrom (2007) reveal not only their consistencywith MOND (with zero adjusted parameters), but alsoconstitutes a direct falsification of dark matter being madeonly of CDM in galaxies: assuming the data are reliable.What is more, Kroupa et al. (2005) have advocated thatthe Milky Way’s dwarf spheroidal galaxies are also tidaldwarf galaxies. This convincing demonstration of MOND’spredictive ability does not, however, extend to the realmsof clusters of galaxies.An obvious difference between galaxies and clusters ofgalaxies is simply the scale involved. If a typical galaxy isone or two tens of kpc across, clusters have accurate mea-surements of the gravitational potential (through the intr-acluster medium or weak gravitational lensing) on scalesten times that and therefore volumes many thousand timesgreater. Studies of the dynamics of groups and clustersin MOND (e.g. The & White 1988; Sanders 1994, 1999;Aguirre et al. 2001; Sanders 2003; Pointecouteau & Silk2005; Angus et al. 2007; Sanders 2007; Feix et al. 2008;Milgrom 2008b) have all shown that there is a huge cen-tral mass deficit and Angus et al. (2008a) demonstratedthat active neutrinos, even at the experimental maximum ( ∼ eV /c ), cannot clump densely enough. A more plausi-ble solution is the 11 eV /c SN proposed by Angus (2009)to fit the first three peaks of the CMB.The feasible particle mass of SN is fully determined byfitting the CMB power spectrum, and there is virtually nofreedom available above 5% of the 11 eV /c mass. It is theparticle mass, and particle mass alone, that sets the prop-erties of the SNs in clusters and galaxies. Therefore, it is ahighly intriguing corollary that the mass required to matchthe CMB is in the tiny range of neutrino masses (perhaps11 − eV /c ) that can both free stream out of galaxies(MOND does not require any DM in galaxies) and clumpdensely enough in galaxy clusters to account for the seriousmass deficit exposed there.We now have the basis of a predictive cosmologicalmodel, where we have made two positive trades. Firstly weexchange Newtonian dynamics for Modified Newtonian Dy-namics, which helps explain in detail the origin of the massdiscrepancy in all galaxies, and with fewer freedoms than theoften contrived CDM halos (as demonstrated by McGaugh2005; Kuzio de Naray et al. 2009). Secondly, we swap CDMfor an 11 eV /c SN. The added bonus of SNs over CDM, isthat knowing only the particle mass gives fixed predictionsfor the CMB and for structure formation (with a small de-pendence on the µ -function in only the latter case).In the very early Universe, neutrino decoupling (bothactive and sterile) occurs at a temperature of kT ∼ MeV .This means that the 11 eV /c SNs (as well as the sub eV /c active neutrinos) are ultra-relativistic during decou-pling, which freezes in their Fermi-Dirac distribution andtheir cosmological abundance is fixed (e.g. Peacock 1999).Typical CDM candidates, like 100 GeV neutralinos (e.g.Hofmann et al. 2001) or > keV /c SNs (see the reviewby Boyarsky et al. 2009) are non-relativistic during decou-pling, and self-annihilations must be used to tune the cos-mological abundance. This applies to all CDM candidateswhich means knowing the mass of a CDM particle tells usthe mass and not the cosmological abundance. Free param-eters, like the interaction cross-section, must explain whythey have the correct cosmological abundance.In addition, while searching for an explanation of theanomalous low energy excess of electron neutrinos observedby the Miniboone experiment, Giunti & Laveder (2008)found agreement with the data by postulating a perfectlyplausible renormalisation of the original flux of muon-neutrinos and oscillations (that are energy dependent) fromelectron neutrinos to 11 ± eV /c SNs. This hypothesisclearly warrants further investigation, and will be testablewith the T2K neutrino experiment (Hastings 2009) whichwill have a near detector at L = 280 m giving it a similarL/E (E being neutrino energy) as MiniBoone (M. Laveder,private communication).In this paper, we seek to investigate the 11 eV /c SN’sinfluence in clusters of galaxies in MOND. Primarily, weneed to ascertain that every cluster has an equilibrium dis-tribution of neutrinos that can account for all the gravi-tating mass. This is by no means guaranteed since thereis a maximum density set by the Tremaine-Gunn limit(Tremaine & Gunn 1979; see § M/L c (cid:13) , 000–000 quilibrium configurations of eV sterile neutrinos in MONDian galaxy clusters of the BCG to keep the SN density below the Tremaine-Gunn limit is too high), they would be excluded. To thisend we have taken a relatively large sample of 30 relaxedgalaxy groups and clusters to gauge their consistency withthe missing mass problem of clusters of galaxies in MOND. We take the temperature and density profiles of the intra-cluster medium (ICM) in 7 clusters of galaxies with tem-peratures in the range of 1 . − keV from the sample ofVikhlinin et al. (2006), a further 13 groups of galaxies fromthe sample of Gastaldello et al. (2007), the cluster A 2589from Zappacosta et al. (2006) and 3 X-ray bright, isolatedearly type galaxies from the sample of Humphrey et al.(2006). This covers the full sample used in Angus et al.(2008a), but in addition, we make an analysis of the twoclusters comprising the bullet cluster (Clowe et al. 2004,2006; Bradaˇc et al. 2006; Angus et al. 2007) working fromthe NFW halos fitted to the weak-lensing convergence map.We also take a fitted NFW profile for the cluster A 1689from the strong lensing analysis of Halkola et al. (2006),take 3 estimates for the density profile of the Coma clus-ter (Gavazzi et al. 2009; Kubo et al. 2007), a pair of clus-ters in the Lynx field (Jee et al. 2006) and the fossil groupRXJ 1416 (Khosroshahi et al. 2006 ).For the Gastaldello et al. (2007) sample, we only had X-ray data for 10 of the 13, so for the other 3 (A 2717, IC 1860,MS 1160) we started from their fitted NFW (Navarro et al.1997) profiles and included the brightest cluster galaxy.In Vikhlinin et al. (2006), the authors give fully analyt-ical descriptions of the ICM temperature and density allow-ing us to solve the equation of hydrostatic equilibrium togive the dynamical mass of each cluster (see § β -models to the ICM density and have fitted the sameanalytical models defined in Vikhlinin et al. (2006) to thetemperature. These models are rather sophistocated, manyparameter models that maximise the accuracy of the fits.The masses of the brightest cluster galaxies(BCGs) are taken either from the Gastaldello et al.(2007); Vikhlinin et al. (2006); Humphrey et al. (2006);Zappacosta et al. (2006) K-band luminosity estimates,the observations of Lin & Mohr (2004) or from K-bandmagnitudes on the NED database as per Angus et al.(2008a).Naturally, there are errors associated with the best fitdensity and temperature profiles, which we do not explicitlyinvestigate here. It is enough to say that the observationaluncertainties surrounding the Newtonian dynamical mass(from which everything is deduced) are no larger than theuncertainties in the triaxiality of the cluster or the interpo-lating function used to find the MONDian dynamical mass,so we waste little time debating them. Moreover, we use asample containing 30 of some of the most relaxed systemsso that a general consensus can be reached.
The density of 11 eV /c SNs (or any DM candidate) inMONDian clusters of galaxies required to provide hydro-static equilibrium of the ICM is given by the following steps,as per Angus et al. (2008a). Firstly, we take the observeddensity and temperature of the ICM, ρ x ( r ) and kT x ( r ) re-spectively, and numerically take their logarithmic derivativewith respect to radius to find the true gravity as a functionof radius g ( r ) = − kT x ( r ) wm p r (cid:20) d ln ρ x ( r ) d ln r + d ln kT x ( r ) d ln r (cid:21) , (2)where w = 0 .
62 is the mean molecular weight of the ICM.The gravity simply is related to the total MOND enclosedmass by M m ( r ) = r G − g ( r ) µ ( g/a o ) , (3)and the interpolating function is the simple one µ ( g/a o ) = g/a o g/a o , (4)(see Famaey et al. 2007 for a discussion of how it fares inspirals). This single line is the only stage at which MONDis involved.After subtracting the mass of the ICM ( M x ( r ) = R r π ˆ r ρ x (ˆ r ) d ˆ r ), we are left with the SN mass distribu-tion and the unsubtracted BCG, M bcg + M ν s ( r ) = M m ( r ) − . M x ( r ), where the mass of the brightest cluster galaxy( M bcg = M/L K × L K,bcg ). The 1.15, which is of virtually noconsequence, multiplies the ICM mass to include the contri-bution of galaxies in the cluster. We invert this to give theSN density, there we ignore the BCG until later, althoughit is vitally important, for reasons that will become obviouslater. ρ ν s ( r ) = (4 πr ) − ddr M ν s ( r ) . (5) At this point we have a deduced density of SNs in a clusterof given temperature profile: at least it is the density exactlyrequired for hydrostatic equilibrium of the ICM (or in thecase of lensing, to create the observed convergence map). Inaddition to this constraint, there is an equation for hydro-static equilibrium of the SNs themselves, which is crucial tothis analysis. This equation invokes the equation of state ofneutrinos (active or sterile; see e.g. Sanders 2007) and whatthis gives us are equations that define the density and pres-sure of a partially degenerate neutrino gas, coupled via thehydrostatic equilibrium relation ddr P ν s = − ρ ν s ( r ) g ( r ) . (6)To express the density, we start with the equilibriumoccupation number f = 1 exp ( x − χ ) + 1 , (7)where x and χ are respectively the ratios of neutrino energyand chemical potential to temperature. A large positive or c (cid:13) , 000–000 G. W. Angus, B. Famaey, A. Diaferio negative χ denotes strong degeneracy or non-degeneracy re-spectively. From this we find the phase space density n = g ν h − m ν s f. (8)Here, h = 4 . × − eV.s is Planck’s constant and g ν s =2takes into account the anti-particles, whilst rememberingthat neutrinos have only a solitary helicity state. One im-mediately sees that there is an absolute upper limit to theallowed density of SNs in phase-space, corresponding to fulldegeneracy ( χ = + ∞ ) and known as the Pauli limit. Thestarting momentum distribution of neutrinos in the earlyuniverse corresponds to half this limit.It should be clear here that in groups and clusters ofgalaxies, SNs of 11 eV /c mass are non-relativistic since theyare travelling at velocities of the order 100-1000 km s − (e.g.Fig 1 central panels). Therefore, neutrino energy is relatedto momentum by E = p / m ν s . The number density of neu-trinos is given by R fd p and multiplying this by neutrinomass ( m ν s = 11 eV /c ), converting momentum to energyand temperature to velocity dispersion ( kT ν s = m ν s σ ν s ) wefind our equation for neutrino mass density ρ ν s ( r ) = 4 √ πg ν h − m ν s σ ν s ( r ) F / ( χ ) , (9)where the SN velocity dispersion and temperature are σ ν s and kT ν s respectively and F / ( χ ) = R ∞ x / f ( χ ) dx . In anon-relativistic neutrino gas, the pressure is equal to 2/3 theinternal energy per unit volume ( U ν s = R E ( p ) fd p ) giving P ν s ( r ) = 8 √ π g ν h − m ν s σ ν s ( r ) F / ( χ ) , (10)where F / ( χ ) = R ∞ x / f ( χ ) dx .It emerges that there are two variables here that mustbe set in order for the neutrinos to exist in hydrostatic equi-librium. Primarily, χ ( r ), must be fixed so that the fixed den-sity of Eq 5 is matched by Eq 9. In case this is not obvious,we solve Eq 9 for F / ( χ ), for which there is a unique, con-tinuous χ ( r ) once the neutrino velocity dispersion is given.There is no independent method, apart from cosmologicalsimulations of the collapse of the baryonic plus SN two-fluidmixture to estimate the chemical potential, χ , so we choosehere to fit it to the data. Whether detailed numerical sim-ulations of cluster formation in a MOND cosmology willreproduce this chemical potential will be a crucial test ofthe equilibrium models presented hereafter.This χ ( r ) is then transfered to Eq 10, and although χ ( r ) is used to balance ρ ν ( r ) and ensure it remains un-changed when σ ν s is varied, it influences the pressure, P ν s ,non-linearly. Secondly, Eq 6 must be satisfied, however, itis not satisfied if we make the assumption that the sterileneutrino velocity dispersion (VD) is identical to the ICMVD. In general, there would be too much pressure becausethe neutrinos would be too hot for their given distribution. A priori , there is no reason that the 11 eV /c SNsshould have precisely the same temperature as the ICM.Both fluids are orbiting in the same gravitational potential,but there is no rapid exchange mechanism to bring them intomutual equilibrium, as there was in the very early Universe.In fact, this is likely to be related to the formation epochof the cluster, where the SN halos would have formed atrelatively high redshift while the Universe was more dense.Later, the ICM would have fallen into the deep potentialwell, created by the SN halo, and therefore it is logical that the ICM should be hotter than the SNs. In addition to prob-ing the χ ( r )’s fitted, numerical simulations of galaxy clustersin MOND will, in time, be able to tell us if the difference inneutrino and ICM VD is realistic.From Eqs 9 & 10 one can see that the SN VD influencesthe product of the density and gravity (the right hand sideof Eq 6) non-linearly with respect to the gradient of theneutrino pressure. Essentially, to force the SN distributioninto hydrostatic equilibrium, with its fixed density, we canmodify the SN VD. However, as we shall discover in the nextsection, this cannot be achieved by a simple scaling from theICM VD, but instead by solving for the neutrino VD, σ ν s ,as a unique, continuous function at every radius. In principle, for hydrostatic equilibrium of the SNs, we need ddr P νs ( r ) ρ νs ( r ) g ( r ) = − σ ν s ( r ) ddr σ ν s ( r )+ σ ν s ( r )5 F / ( r ) ddr F / ( r ) = − F / ( r ) F / ( r ) g ( r ) . (11)Using the substitution ǫ ( r ) = σ ν s ( r ), leading to ǫ ′ ( r ) =2 σ ν s ( r ) σ ′ ν s ( r ) and the integrating factor F / ( r ) . , we canreduce this first order linear differential equation to σ ν s ( r ) = 35 F / ( r ) − . Z r ∞ F / (ˆ r ) F / (ˆ r ) . g (ˆ r ) d ˆ r. (12)To find pressure and density in the first instance, wemust submit a trial σ ν s to Eq 12: typically we try σ x ,where the ICM VD is defined by σ x = c × (cid:18) kT x wm p (cid:19) / . (13)Here, c = 3 × km s − is the speed of light, kT x is the ICMtemperature in keV , w = 0 .
62 is again the mean molecularweight and m p = 9 . × keV is the mass energy of aproton. From the trial solution, we iterate rapidly towardsconvergence. The final σ ν s ( r ) gives hydrostatic equilibriumto better than 1 per cent at all radii and most importantly isunique - set only by the SN mass and its indirectly observeddensity from Eq 5. We now have a unique correlation between the density ofSNs and their velocity dispersions. This is important becausethese two variables define the phase space distribution of theSNs, to which there is a fundamental limit.Liouville’s theorem states that (in the absence of en-counters) flow in phase space is incompressible and that eachelement of phase-space density is conserved along the flowlines. However, this only applies to the fine-grained phasespace density of an infinitesimal region. Rather, for the ob-servable, which is the coarse-grained (macroscopic) phase-space density, we simply must not exceed the maximum ofthe fine-grained one.Thus, the SN phase-space density must not increaseduring collapse, from its starting value of g ν h − m ν s (whichis half the Pauli degeneracy limit), to its current maximum c (cid:13) , 000–000 quilibrium configurations of eV sterile neutrinos in MONDian galaxy clusters value of ρ ν s ( r )[2 πσ ν s ( r )] − . (where we assume the veloc-ity distribution is locally Gaussian everywhere with dis-persion σ ν s ). This limit is called the Tremaine-Gunn limit(Tremaine & Gunn 1979; hereafter TG) and rearranging itin terms of the critical density for an 11 eV /c SN, where1 eV /c = 8 . × − M ⊙ and s c = 9 . × − pc (where s is obviously 1 second) gives us ρ ν s ,TG ( r ) = 3 . × (cid:18) σ ν s ( r ) c (cid:19) M ⊙ pc − . (14)The important thing to bear in mind here is that this isthe exact value of the TG limit because it is calculated fromthe unique, derived SN VD exactly necessary for hydrostaticequilibrium and not by assuming some relation between theICM and the neutrinos, as is often the case (Angus et al.2008a; Natarajan & Zhao 2008; Gentile et al. 2008). There-fore, we have the TG limit as a function of radius, since weknow the exact value of the VD from solving Eq 12.Let us note however that departures from Gaussianityin the velocity distribution could perhaps somewhat affectthis physical upper limit on the SN density.The consequence of the TG limit is that, therefore if thedensity of SNs required for cohesion of the cluster gas ex-ceeds TG limit (in particular at the centre) then the MONDplus 11 eV /c SN hypothesis would be ruled out.
In Figs 1-6 we present the densities, VDs and enclosed massprofiles for the 30 groups and clusters. In the left hand pan-els we plot the SN density (solid line type), ICM density(dotted) and the TG limit of the cluster (dashed) againstradius. In the middle panels we plot both the observed ICM(dotted) and the derived (from Eq 12) SN (solid) VD as func-tions of radius. In the right hand panels, we plot each clus-ter’s enclosed Newtonian dynamical mass (dashed), MONDdynamical mass (dot-dashed), BCG total mass with unity
M/L (solid) and ICM mass (dotted).There are a few salient features to observe: primarily,the SN VD (or temperature) is in most cases 20-50 per centlower than the ICM VD. The simple explanation, alluded toearlier, is that the SN halos presumably formed at relativelyhigh redshift, through only their mutual gravitation. On theother hand, the ICM fell from large distances, through thealready present, deep potential well of the SNs and thushad greater potential energy to transfer to kinetic energy,although this still has to be demonstrated with numericalsimulations.Taking a closer look at the SN densities and VDs, theyhave conspicuous kinks around 20-30 kpc . This radius, r tg ,is where the density reaches the TG limit, . If no phase spacelimit existed, the SN density would continue to increase to-wards the Pauli limit and the neutrino equation of statewould begin to substantially change from non-degeneracy(large negative χ ) P ν s ∝ ρ ν s σ ν s to the degenerate one(large positive χ ) P ν s ∝ ρ / ν s (where the TG limit occursat χ ∼ . eV /c SN hypothesis is. If no BCG existed in any of our clusters,they would immediately fail.We know the luminosity of most BCGs in the K-band which gives an excellent indication of mass (see e.g.Bell & de Jong 2001; Conroy et al. 2009; Gastaldello et al.2007; Humphrey et al. 2006) and we know a few in otherbands (B,V,R) or not at all. Therefore, we must modify the
M/L K (given in table 1) to exactly match M m at r tg , whichbasically means the BCG is picking up all the “slack” leftwhen the SN reaches the TG limit and can no longer accountfor the full dynamical mass.There are some points to bear in mind, firstly the high-est M/L K used was 1.2 and the lowest 0.1. In the latter case,of a very low M/L K , this can be due to the total luminositynot being enclosed by r tg which is often ∼ − kpc ).In the case of M/L K ∼
1, this can be considered a fittingparameter, since there is no freedom in the cluster to havea
M/L K lower than the value used because the SN densityreaches the TG limit at the centre of every cluster. Further-more, the M/L K cannot be significantly larger or it wouldbe in disagreement with the typical M/L K demonstratedby Bell & de Jong (2001). In the cases where M/L K < . M/L K to be larger and simply for the SNto be lower. Nevertheless, the TG limit will still be reachedin every cluster, the only difference will be that if we are un-derestimating the BCG mass then r tg will simply be lowerfor that particular cluster. The only reason we fix the den-sity to be equal to the TG limit for all radii smaller than r tg is to highlight the maximum amount of luminosity theBCG could lose (e.g. if observations are incorrect) and stillprovide the central density required.Generally, it is interesting to note that the SN halosneeded to fit galaxy clusters here have a density slope sim-ilar to that of the ICM in the central parts, but becomingsharper at intermediate distance, which is accompanied bya relatively flat velocity dispersion. At the edges of the clus-ters, the ICM density becomes larger than the SN density(which falls to zero) and there is an apparent sharp decreaseof the sterile neutrino velocity dispersion to zero also. Thisis merely a numerical artefact of the sterile neutrino densitybeing set to zero at the edges of clusters, under the assump-tion of hydrostatic equilibrium. Since the sterile neutrinodensity will in reality fall to a very small number, but stillgreater than zero, the velocity dispersion will actually beisothermal, as we would expect. c (cid:13) , 000–000 G. W. Angus, B. Famaey, A. Diaferio
Below we discuss some pertinent observations about indi-vidual groups or clusters.
Analysis of the CMB strongly favours the hypothesis thatnon-baryonic DM exists in the Universe, and the bullet clus-ter (Clowe et al. 2006) compounds it (although Milgrom2008b has suggested that the DM of clusters could be ultracold and collisionless clumps of molecular gas which wouldsatisfy the constraint of this particular cluster). As two giantgalaxy clusters crashed into each other at incredible speed(Markevitch et al. 2002, but see also Angus & McGaugh2008 for why this might pose a problem for Λ
CDM ), mu-tual ram pressure from the 2 ICMs dragged one anotherout of the clusters leaving only the galaxies and DM topass through and emerge on the opposite sides on the sky.A weak-lensing reconstruction required two DM halos tooverlay the positions of the galaxies with NFW parame-ters for the main and sub cluster respectively - M =15 . , . × M ⊙ , r = 2 . , . c = 1 . , . . × M ⊙ stellar mass to the sub cluster (labeled “Bullet 2”) to coverthe dynamical mass in the central 30 kpc , which is verysignificant. The stellar mass quoted in Clowe et al. (2006)at the weak-lensing peak of the sub cluster is (5 . ± . × M ⊙ within 100 kpc , from I-band observations assuminga M/L I of 2. We need only add a trivial stellar mass of 1 . × M ⊙ to the main cluster, which is interesting since onecan see from the left hand panel of Fig 1 from Clowe et al.(2006) that there is no obvious BCG candidate associatedwith it. In fact, the two giant ellipticals of the main clusterhighlighted by Clowe et al. (2006) are roughly 50 kpc (tothe northern one) and 75 kpc (to the eastern one) from thelensing peak, but can easily offer the necessary stellar mass.On the other hand, the centre of the BCG of the sub clusteris only roughly 25 kpc from the lensing peak and significantlight is spilling over within 10 kpc . Furthermore, since BCGsusually dominate the stellar mass in the central 100 kpc ofclusters, the majority of the 5 . ± . × M ⊙ is likelyassociated with it, making the 3 . × M ⊙ plausible. Noticealso that the 3 . × M ⊙ is not required within 10 kpc ,but rather by 30 kpc . Within 10 kpc less than 1 . × M ⊙ is required. Another intriguing point about BCG masses is the oneused for the group RGH 80, which has two equally mas-sive central galaxies: NGC 5098a and NGC 5098b with L K = 2 . × L ⊙ and L K = 2 . × L ⊙ respectively. Inour previous paper (Angus et al. 2008a) we were only inter-ested in the central 100 kpc , therefore, were able to combinethe two galaxy luminosities together. However, only one ofthe galaxies is at the very centre (see Fig 9 of Mahdavi et al. 2005 or Fig 1 of Randall et al. 2009), and the other has aprojected separation of around 50 kpc and possibly a con-siderable line of sight distance. Before removing the secondgalaxy, the TG limit was considerably larger than the max-imum central density, but after discounting it, the TG limitis reached at 20 kpc . We found no galaxy luminosities for the two clusters A 1689and A 2390, but they require BCGs with total masses of atleast 2 . . × M ⊙ respectively, which is a predic-tion of this model. Interestingly, A 1689 has been used ex-tensively to argue against CDM by Broadhurst & Barkana(2008) because the observed NFW concentration parameteris considerably larger than that expected from cosmologicalsimulations. We find the equilibrium model with 11 eV /c SNs nicely reaches the TG limit at 20 kpc , even though ithas one of the highest central SN densities of all our sample.
The two clusters in the Lynx field are in the process ofmerging (like the bullet cluster) and thus the mass profilesare potentially overlapping. Nevertheless, their offset seemsto be sufficiently large to get a reasonable estimate fromweak lensing and this has also been sanity checked with X-ray hydrodynamics (Jee et al. 2006). Both the larger clusterand smaller cluster are fitted with the same NFW profile: M = 2 . × M ⊙ , r = 0 .
75 Mpc and concentration c = 4. The galaxy luminosities within 500 kpc of each of thetwo lensing peaks are 1.5 and 0.8 × L ⊙ in the B-band.We require 6 . × M ⊙ for each BCG, which is easy af-fordable by the luminosity of both clusters because typicalB-band M/L s can range between 5 and 10.
The historical significance of the Coma cluster with respectto the dark matter problem is probably far more signifi-cant than its scientific significance in this present case, butwe include it out of curiosity. It was originally analysed inMOND by The & White (1988) and was actually concludedto be more or less consistent with MOND, although thegalaxies required radially biased orbits and the accelera-tion constant, a o had to be increased by at least a factorof 2 from the one used here. The problem with Coma isthat it is not particularly relaxed and also its sphericity isquestionable. For example, Neumann et al. (2003) showedthat there is ongoing merging, which makes measurementsof the central mass profile uncertain. Therefore, the onlyway to get a decent estimate of the dark halo of Comais to use weak lensing (like we have done with the bulletcluster), but even then the assumption of spherical symme-try is dubious. The best study of Coma was performed byGavazzi et al. (2009), but unfortunately found only a ratherspeculative NFW profile of M = 5 . +4 . − . × M ⊙ and r = 1 . +0 . − . Mpc with no prior on the concentration pa-rameter (found to be c = 5 . +3 . − . ). With a prior on theconcentration parameter (set to be c = 3 . +1 . − . ) they c (cid:13) , 000–000 quilibrium configurations of eV sterile neutrinos in MONDian galaxy clusters found M = 9 . +6 . − . × and r = 2 . +0 . − . Mpc . Fi-nally, we added an estimate from Kubo et al. (2007): M =27 . ± . × , r = 2 . ± . Mpc and c = 3 . +13 . − . .The mass of the central galaxy would need to be 1 . , . . × M ⊙ respectively for the three cases. The K-bandluminosity of the central galaxy in Coma is 10 L ⊙ , so thereis plenty galactic mass to supply what is needed. We included one fossil group RXJ 1416 in the sample since ithas a very minor galaxy component (aside from the BCG).Khosroshahi et al. (2006) fitted an NFW profile to X-raydata from Chandra and XMMN with parameters M =3 . × M ⊙ , r = 1 . Mpc and c = 11 . ± . M/L of the BCG which hasluminosity L R = 7 × L ⊙ . The M/L R need only be 1.2,whereas 5 is a typical value for an old stellar population inthat band. So here again, as with several of the other massivesystems, the full luminosity of the BCG is not required al-though the SN halo reaches the TG limit at the centre. Thismight be because the BCG is extended and only a fractionof the light is enclosed within 20 or 30 kpc (where the SNhalo reaches the TG limit). Alternatively, if we used the fullmass of the BCG, the SN density would simply reach theTG limit at a smaller radius. These two X-ray bright early type galaxies were taken fromthe sample of Humphrey et al. (2006). Interestingly from thepoint of view of SNs in galaxies (which could disrupt thegood fits to rotation curves i.e. the basis of MOND) theirdynamical masses are comfortably covered by the BCG masseverywhere. From their density figures you can see the SNsare at the TG limit everywhere, but the SNs make virtuallyno impact on their mass profiles. In fact, if the SN densityfor NGC 4125 was 8 × − M ⊙ pc − all the way to the centrefrom 10 kpc , the enclosed mass of SNs would be 3 . × M ⊙ in comparison to roughly 1 . × M ⊙ for the BCG (at thatradius), which is a factor of 30. Therefore, there is no reasonto believe that SNs will influence the internal dynamics ofindividual galaxies in a meaningful way. These are the two most problematic groups.Buote & Canizares (1994, 1996); Buote et al. (2002)observed that the twisting of X-ray isophotes around theelliptical galaxy NGC 720 compared to the intrinsic ellip-ticity of the galaxy (which outweighs the gas by more thantwo orders of magnitude) could only be compatible with thepresence of DM at least four times the galaxy mass (by theedge of the galaxy). This SN halo would have an ellipticitydistinct to that of the galaxy and the gas would tracethe superimposed potential, hence generating the twistingisophotes. The luminosity of NGC 720 is L K = 1 . × L ⊙ and requires M/L K = 1 . kpc , so cannotcontribute any more mass. The problem is that NGC 720 has a relatively young stellar population, although there isa age gradient from ∼ Gyr in the centre to ∼ Gyr by ∼ kpc (Humphrey et al. 2006). This leads to significantsystematic uncertainty in constraining the M/L K , which isgiven as 0 . ± .
11 and 0 . ± .
07 for the Salpeter andKroupa IMF respectively. A recent re-evaluation with dataof superior resolution (D. Buote private communication)puts the Kroupa value at 0 . ± .
18 (meaning the Salpetervalue will be somewhat larger), but this still falls well shortof the necessary galactic mass. This could be a seriousproblem if the low
M/L K could be confirmed, but for nowit is a prediction of this model that the true M/L K (whenthe correct IMF, age distribution and merger history aretaken into account) will be close to 1.2.NGC 720 is not the only system that is very sen-sitive to the observations. NGC 1550 was studied byGastaldello et al. (2007) with both Chandra and XMM-Newton and both sets of temperature data are plot-ted in their Fig 3. Later it was observed again byKawaharada et al. (2009) using XMM-Newton, in a studywhich suggested there was evidence for a recent merger. Ifwe use either set of XMM-Newton data points, the M/L K would need to be at least 1.6, whereas if we only subscribeto the Chandra data, the M/L K need only be 1.2. Follow upobservations could provide a very strong test of the model,but Chandra’s greater spatial resolution makes it the morereliable data set. µ -function Angus (2009) demonstrated that 11 eV /c SNs were requiredto fit the first three acoustic peaks of the CMB and that 2species of 5 . eV /c SNs were totally inadequate (as werethree 2 . eV /c ordinary neutrinos). There was also verylittle freedom beyond 5 per cent of 11 eV /c , without intro-ducing unjustified free parameters.For further instruction to the importance of the SNmass to be 11 eV /c , we calculated the M/L K necessary toproduce an equilibrium SN distribution in the group ESO-306 (which is by no means the most constraining) in twodifferent scenarios. Firstly, trying a 9 . eV /c SN and sec-ondly keeping an 11 eV /c SN mass but using the standard µ -function of MOND: where instead of using Eq 4 we try µ ( g/a o ) = g/a o √ g/a o ) . Whereas using the simple µ -functionand an 11 eV /c SN requires
M/L K = 1, the 9 . eV /c SNrequires
M/L K = 1 . µ -function wouldneed 1.4. Given that these sorts of high M/L K would bethe rule, rather than the exception, they seem incompat-ible with stellar population synthesis models like those ofBell & de Jong (2001).This evidence in favour of the simple µ -function is notan isolated case. In Famaey & Binney (2005) and McGaugh(2008), the simple function was prefered from a fit tothe Milky Way’s rotation curve (a high surface brightnessgalaxy) as was the case in the large sample of high surfacebrightness galaxies carried out by Sanders & Noordermeer(2007). This preference is also being found by an ongoingstudy of ultra high resolution rotation curves observed bythe THINGS (Walter et al. 2005) collaboration (G. Gentile,private communication). The main problem with the sim-ple function is that it cannot be used all the way to the c (cid:13) , 000–000 G. W. Angus, B. Famaey, A. Diaferio strong gravity regime. In particular, it would produce toohigh a modification in the inner Solar System, which is ex-cluded, for example, by measures of the perihelion precessionof Mercury. The solution is to use a µ -function that wouldrapidly interpolate between the simple and standard µ forvery large values of the gravitational acceleration, but thisis not applicable here. It is an encouraging result that the 11 eV /c SNs haveunique and continuous equilibrium models for such a largedynamic range of cluster properties, but what is remarkableis that at the centre of every cluster, the TG limit is reached.For each given cluster, there is a specific, maximum densityprofile that can exist in equilibrium: a good example beingNGC 5129 (in Fig 3), which must have the maximum al-lowed density from 40 kpc for the SN halo to both be inhydrostatic equilibrium and provide the correct dynamicalmass as measured by the ICM properties.Accordingly, it is the TG limit and the degenerate prop-erties of the SNs (since they are fermions) that sets the dy-namical properties of all clusters. No relation, even remotelylike this exists if the cluster DM is cold or non-fermionic.
It is apparent from the examples of NGC 4125 andNGC 6482 (see § − M ⊙ pc − was allowed in the MONDfits to the rotation curves of Ursa Major galaxies. This mag-nitude of SN density is typically found within 1 Mpc in thevery massive clusters and within 100 − kpc from the cen-tre of groups. At the centre of clusters, the situation is ob-viously different, but stable spiral galaxies are never presentthere since tides would rip them apart before they fall to thecentre. Therefore, this will not disturb the MOND Tully-Fisher relation since field spirals should be far from any SNhalos.However, SNs are required to exist in the centres of somevery massive ellipticals. This is clear from the right handpanels of the enclosed mass profiles of the clusters A 478(Fig 1), A 907 and others. In these figures one can see themass of the BCG (the solid line) is smaller than the MONDenclosed mass. Thus, a considerable mass of SNs is requiredinside the limits of the stellar orbits, but always significantlyless than the mass in stars. In its original form, MOND should be able to explain allgalaxy dynamics without dark matter. Although rotationcurves have always yielded excellent results, the data forlensing studies (beginning with Zhao et al. 2006) of indi-vidual galaxies has not been as promising. For instanceTian et al. (2009) show that the weak-lensing of single, iso-lated galaxies is perfectly compatible with MOND up toa particular galaxy luminosity (from L r = 0 . − . × L ⊙ ). Thereafter, the lensing data implies the need for DM, which is exactly as we might expect if these moreluminous galaxies are embedded in a low density but ex-tended SN halo (akin to those in the Humphrey et al. 2006sample, see § In this paper we have shown explicitly how to calculate theequilibrium configurations of neutrinos in MONDian galaxyclusters. The density of sterile neutrinos is fixed by the prop-erties of the ICM (or the observed lensing map), but deriva-tion of the sterile neutrino velocity dispersion allows a spe-cific density profile to exist in hydrostatic equilibrium. Wehave presented the detailed properties of 30 typical galaxygroups and clusters over a wide range of masses and tem-peratures (and even redshift vis-a-vis the bullet cluster, thelynx cluster and A 1689) and have shown that not only canwe elucidate velocity dispersion profiles that allow the sterileneutrinos to exist in hydrostatic equilibrium, but also thatthe Tremaine-Gunn limit sets the central density.It would appear to be a very strong coincidence thatby doing little more than fixing the mass of a sterile neu-trino to be 11 eV /c , we can serendipitously, explain theformation of the acoustic peaks in the CMB and specifythe exact properties of systems that require DM in MOND.In particular, regardless of cluster mass, the velocity dis-persion of sterile neutrinos necessary to impose hydrostaticequilibrium allows the density to reach its maximum (whichis a function of both sterile neutrino particle mass, velocitydispersion and cluster mass indirectly) at the centre of thecluster. The only stipulation is whether these equilibriumconfigurations are stable, which should be the next check.Out of the 30 systems, there are two which need to bemonitored. NGC 720 requires a K-band mass-to-light ratioof 1.2, which is possible for an old stellar population, but thecurrent best population synthesis model suggests 0 . ± . § eV /c sterile neu-trino model comes not from the rich clusters, but ratherfrom smaller groups or individual galaxies with bright X-ray halos.In a similar sense to how the NFW density profile de-duced from N-body simulations of CDM structure forma-tion have been shown to be inadequate descriptions of somegalaxy and galaxy cluster DM halos (Broadhurst & Barkana2008; de Blok & McGaugh 1998; Gentile et al. 2004) wesuggest that our highly regular density and velocity disper-sion profiles shown in Figs 1-6 must be used to judge the c (cid:13) , 000–000 quilibrium configurations of eV sterile neutrinos in MONDian galaxy clusters consistency of simulated structure growth in MOND cosmo-logical simulations. Presumably, it is guaranteed that theobjects that condense out of the background will reach theTremaine-Gunn limit at the centre (which makes it so al-luring that all the clusters studied here do), but it is notcertain that the phase space densities will fall with radiusin the manner shown here, nor that the chemical potentialwill have the correct values.Given that we fit the CMB and have these interest-ing results for clusters of galaxies, another crucial test ofthis model (after checking stability) will be to see if MONDN-body cosmological simulations, with 11 eV /c sterile neu-trinos can form structures resembling those shown here andmatch the linear matter power spectrum. It is worth pointingout that Skordis et al. (2006) computed the matter powerspectrum with MOND-like gravity and three 2 . eV /c active neutrinos (the a o used was 3 . × larger than theone used typically to fit rotations curves and used here)and showed the extra matter density from the neutrinos(Ω ν = 0 .
17 compared to Ω b = 0 .
05) coupled with theMOND gravity could come relatively close to providing agood match to the matter power spectrum measured by theSDSS (Tegmark et al. 2004). The higher matter density pro-vided by the SNs (Ω ν s = 0 .
225 and Ω b = 0 . a o , could very well providea superior fit to the matter power spectrum. To resolve thiswe need cosmological numerical simulations, which are stillin their infancy (see Llinares et al. 2008).Relativistic MOND theories beginning with TeVeS(Bekenstein 2004, also see the review by Skordis 2009) andBSTV (Sanders 2005), which led to new ideas like Gener-alised Einstein-Aether (see e.g., Zlosnik et al. 2007, 2008)are still highly complex and only address the galactic darkmatter problem (not the cluster, or cosmological dark mat-ter problem; nor the dark energy problem). Therefore, onecaveat we would add to our conclusions is that this cosmo-logical model still requires dark energy in the same coinci-dental amount as Λ CDM . However, as elaborated upon inMilgrom (1999) and Milgrom (2008a), MOND and dark en-ergy must be two sides of the same coin that leads seamlesslyto 2 πa o ≈ c (Λ / / . There is some progress in this direc-tion (F¨uzfa & Alimi 2007; Bruneton et al. 2009; Li & Zhao2009; Blanchet & Le Tiec 2009) but it has not yet been con-vincingly or efficiently shown. GWA’s research is supported by the University of Torinoand Regione Piemonte. Partial support from the INFN grantPD51 is also gratefully acknowledged. Support from thePRIN2006 grant “Costituenti fondamentali dell’Universo”of the Italian Ministry of University and Scientific Researchis gratefully acknowledged by AD. BF acknowledges the fi-nancial support of the Alexander von Humboldt foundation.We thank both David A. Buote and Fabio Gastaldello forproviding the ICM temperature and density data in elec-tronic format.
Cluster L K (10 L ⊙ ) Min M/L K A 262 4.1 1.0AWM 4 7.5 0.3ESO 306 7.0 1.0ESO 552 8.2 0.3MKW 4 7.2 1.1NGC 1550 2.1 1.2NGC 5044 2.9 1.1NGC 5129 5.0 0.6NGC 533 1.2 1.0RGH 80 2.9 0.8A 478 8.4 1.2A 907 16.5 1.1A 1413 18.3 0.4A 1991 6.9 0.65A 2029 20.1 0.2A 2390 18 . ∗ ...RXJ 1159 10.3 0.4NGC 720 1.7 1.2NGC 4125 1.8 0.9NGC 6482 3.2 1.0A 2589 2.3 (V) 0.65A 2717 5.4 0.2IC 1860 4.4 1.0MS 0116 5.8 0.25A 1689 2 . ∗ ...Coma ∼ . ∗ ...Bullet 2 3 . ∗ ... Table 1.
Here we list the K-band luminosities of our BCGs alongwith the minimum K-band mass-to-light ratio required to fit thedynamical mass. The clusters with a ( ∗ ) lack information aboutthe BCG, so in the luminosity column, we have entered the re-quired BCG luminosity with unity M/L . The clusters with a lumi-nosity followed by (V),(B) or (R) have their luminosity measuredin that band, and not the K-band. The Coma cluster has 3 sepa-rate mass profiles, so the 3 M/Ls refer to the profiles in the orderthey are taken in Fig 5. The clusters are separated into samples:the top set are from the Gastaldello et al. (2007) sample; then theVikhlinin et al. (2006) sample; Humphrey et al. 2006; Zappacostaet al. (2006); miscellaneous NFW fits; the bullet cluster.
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Figure 3.
As per Fig 1. c (cid:13) , 000–000 quilibrium configurations of eV sterile neutrinos in MONDian galaxy clusters Figure 4.
As per Fig 1 except certain lines corresponding to the intra-cluster medium are absent because the sterile neutrino densitieswere deduced from weak or strong lensing analyses.c (cid:13) , 000–000 G. W. Angus, B. Famaey, A. Diaferio
Figure 5.
As per Figs 1-4 except that they are for A 1689 (Halkola et al. 2006), the Coma cluster, the two clusters that comprise theLynx cluster (Jee et al. 2006) and the fossil group RXJ 1416 (Khosroshahi et al. 2006), from which the masses are taken in NFW form. Forthe Coma cluster we use 3 different measurements for the NFW mass profile as discussed in detail in § Figure 6.
As per Figs 1-5 except that they are for the two clusters that comprise the bullet cluster (Clowe et al. 2006), from which themasses of the two clusters are taken in NFW form. This case is discussed in detail in § (cid:13) , 000–000 quilibrium configurations of eV sterile neutrinos in MONDian galaxy clusters c (cid:13)000