# Subjecting dark matter candidates to the cluster test

###### Abstract

Galaxy clusters, employed by Zwicky to demonstrate the existence of dark matter, pose new stringent tests. If merging clusters demonstrate that dark matter is self-interacting with cross section , MACHOs, primordial black holes and light axions that build MACHOs are ruled out as cluster dark matter. Recent strong lensing and X-ray gas data of the quite relaxed and quite spherical cluster A1835 allow to test the cases of dark matter with Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distribution, next to Navarro-Frenck-White profiles. Fits to all these profiles are formally rejected at over , except in the fermionic situation. The interpretation in terms of (nearly) Dirac neutrinos with mass of eV/ is consistent with results on the cluster A1689, with the WMAP, Planck and DES dark matter fractions and with the nondetection of neutrinoless double -decay. The case will be tested in the 2018 KATRIN experiment.

## I Introduction

The existence of dark matter (DM), or some equivalent effect, is beyond doubt and proves the existence of new degrees of freedom. The usual suspects are WIMPs, axions and sterile neutrinos. The standard model of cosmology CDM explains Big Bang Nucleosynthesis (BBN), the Cosmic Microwave Background (CMB) and the Baryon Acoustic Oscillations (BAO), as recently supported by the Dark Energy Survey troxel2017dark (). But there are several issues, such as: The DM particle has been sought intensly but not found arcadi2017waning (), neither is there a hint for supersymmetry at the LHC. BBN faces the Li problem cyburt2016big (), the CMB has a small Hubble constant freedman2017cosmology () and faces foreground issues verschuur2016nature (); vavryvcuk2017missing (). Red-and-dead galaxies require early structure formation croton2008red (), as does a dusty galaxy at with some in gas strandet2017ism (). Lyman- clouds are supposed to be stabilized by a high temperature plasma, which should be easy to detect but never was.

These and other sobering results motivate to reconsider other DM options, like primordial black holes (PBHs) or MACHO dark matter. PBHs were thought to be ruled out, but became fashionable again after the discovery of gravitational waves from BH mergers, to meet fresh criticism koushiappas2017dynamics (). A MACHO can be e.g. a planet or a solar mass object, that may consist of normal matter, but also stand for a self-gravitating Bose-Einstein condensate (BEC) of axions or axion-like particles (ALPs). From another angle, our studies of lensing by the cluster A1689 consistently yield good fits for neutrino DM nieuwenhuizen2009non (); nieuwenhuizen2013observations (); nieuwenhuizen2016dirac ().

Supposing that DM does not exist but that Newton’s law gets modified below a critical value of the acceleration has been fruitful for the description for galactic rotation curves milgrom1983modification (). However, it has been demonstrated that these theories, in particular MOND, Emergent Gravity, and MOG, run into serious troubles for galaxy clusters. The fairly relaxed cluster Abell 1689 posed problems for these theories nieuwenhuizen2017zwicky (), as did a second relaxed cluster, A1835 nieuwenhuizen2017modified (). To function in clusters, MOND and EG would need additional DM, e.g., in the form of eV thermal neutrinos. This hot DM is known to induce free streaming in the early Universe, thus suppressing structure formation. They are considered as ruled; in fact the sum of neutrino masses is estimated to lie in the 0.1 – 0.3 eV range. Nevertheless, a rarely considered question is: has structure formation indeed been linear?

With the road for non-Newtonian gravity essentially closed in our contribution to FQMT’15 nieuwenhuizen2017zwicky (), the way forward is to study implications of particle dark matter theories in galaxy clusters. In contrast to CMB and BAO theories, relaxed clusters have simple physics: one may assume that some kind of equilibrium has been reached, so that the history needs not be considered. As such, they put important bench marks.

The paper is composed as follows. In section 2 we consider the effect of DM self-interaction. In section 3 we discuss data for the cluster A1835 and their binning. This is applied to NFW fits in section 4 and to thermal fits in section 5. The paper ends with a summary and an outlook. Throughout the paper we use the reduced Hubble constant .

## Ii On dark matter self-interaction

### ii.1 MACHOs and PBHs

In clusters there are too few baryons to account for all the DM but MACHOs may consist of axions or ALPs, or be PBHs. Let us look at a specific cluster, the “train wreck” cluster Abell 520. It reflects the past collision of at least three sub-clusters, which are on their exit. Surprisingly, it has a central starless core of a few times and mass-to-light ratio 860 jee2012study (); clowe2012dark (); jee2014hubble (). This has been modelled by self-interacting DM (SiDM) with an elastic scattering cross section of . A similar estimate comes from the Bullet Cluster markevitch2004direct (). MACHOs and PBHs can not have this; for 1 Earth mass, e. g., they would need the gigantic value . If SiDM exists, MACHOs are ruled out as the cluster DM. For both clusters the existence of SiDM has been questioned, however robertson2016does (); peel2017sparse (). But also the cluster A3827 yields a mild indication for self-interaction, ( , where is an inclination angle massey2017dark ().

### ii.2 WIMPs

The same argument applies to WIMPs, though in a much weaker form. Intuitively, scattering occurs by contact interaction if particles come within their Compton radius. The condition then leads to

(1) |

which would explain why no WIMP has been observed in the GeV regime. To go beyond this puts a constraint on theories.

### ii.3 Sterile neutrinos

In recent years attention has been payed to sterile neutrinos, so-called warm DM. In particular the report of a 3.5 keV -ray line, possibly related to a 7 keV sterile neutrino, has been inspiring bulbul2014detection (); boyarsky2014unidentified (). For elastic scattering the value may not look problematic, but actually they should hardly interact at all, since sterile-sterile neutrino scattering happens indirectly via their mixing with standard ‘active’ neutrinos. For an active-sterile mixing angle , the cross section can be estimated as lesgourgues2013neutrino (). With keV and cappelluti2017searching () it follows that . If sterile neutrinos are to make up SiDM, they need an another, strong scattering mechanism.

### ii.4 Axions and axion-like particles

ALPs may be as light as eV; with eV masses they will be thermal; if heavier, they act as WIMPs. Light ones may form Bose-Einstein condensates (BECs). It has been proposed that very light ones, eV, build BECs which act as MACHOs hui2016hypothesis (). However, MACHO scenarios can not act as SiDM.

Let us see whether perhaps the whole cluster DM can be one Mpc-sized BEC constituted by ALPs. Its ground state wavefunction satisfies the Schrödinger equation

(2) |

and the Poisson equation, which relates the gravitational potential to the mass density of the Galaxies, the of the X-ray gas and the of the DM,

(3) |

Here with normalisation . In the cluster centre the mass density is known to stem mainly from the brightest cluster galaxy, so . Hence the potential is harmonic, . With it has a frequency . This problem is solved in every quantum mechanics textbook. Its characteristic length

(4) |

is tiny on the cluster scale, so the condensate must basically act as a point mass, maximally equal to

(5) |

which even for eV is less than and thus negligible. Extended DM distributions must thus have many BECs acting as MACHOs, a scenario discussed already. Hence light axions and ALPs are problematic as SiDM.

## Iii A1835 data and their binning

For the cluster A1835 theories of DM can be tested on recent data for , the mass in a cylinder around the cluster centre nieuwenhuizen2017modified (). From the observed strong lensing arclets mass maps are generated; this being an underdetermined problem, an ensemble of compatible mass maps is produced and from them their values at radii with , such that kpc. In the centre only a few arclets occur, hence only of the contain data for and their covariances nieuwenhuizen2017modified (); the index is relabelled accordingly. The matrix has a big spread of eigenvalues, roughly between and . The standard definition of involves but small eigenvalues should not matter and have to be regularised. Hereto we shall merely employ the data themselves.

As first step to eliminate the small eigenvalues, the data points are grouped in bins with in principle points, but not all bins can be full. Choosing or we minimize bias around the bin 9, which has the smallest errors. We can now relabel the index , according to the bin number and the location inside the bin; this defines , and . As bin centre we take the geometrical average .

As a new step, we divide out the theoretical value in the binning. Given a theoretical or empirical , the data is binned as

(6) |

The standard binning with would do less justice to the data than the best fit, and hence lead to a loss of information. Moreover, the binning (6) makes the choice of as good as any other. The binned covariances read

(7) |

has eigenvalues typically from 0.07 to , hardly better than . The way to proceed is by noting that eq. (6) puts forward a measure for the intra-bin fluctuations,

(8) |

This is actually a square; without absolute values, it would vanish. As final step, we add the as diagonal regulator and define the total binned covariance matrix ,

(9) |

The eigenvalues of go down to , so further regularization with an ad hoc constant limousin2007combining (); nieuwenhuizen2013observations (); nieuwenhuizen2016dirac () is not needed. As measure for the goodness of the fit we take

(10) |

It differs from the standard in that the data and the covariances are binned employing the fit function .

To estimate the errors in fit parameters we assume that the data involve Gaussian errors. Denoting and the errors by , the leading Gaussian errors of are collected symbolically as

(11) |

where , and likewise for . The covariances are defined from as and the errors in the as .

## Iv NFW fits

We first apply this to the Navarro-Frenk-White (NFW) profile navarro1997universal (),

From any mass density , the tested quantity is

(13) |

As best fit to we find for NFW with

(14) |

Using this corresponds to concentration . With , and , the case is formally ruled out at 6.7 .

The generalization “gNFW” involves a power jing2002triaxial (),

(15) |

The best gNFW fit again occurs for ,

(16) |

and , so that . This fit has , and is formally ruled out at 6.8 . The unexpected value is caused by the small errors of the data around 100 kpc, see fig. 1. They arise since the lensing arclets produce mass maps with nearly the same there. For small , on the other hand, there are fewer arclets and larger errors, while for large the relative errors increase as usual.

## V Thermal particles

### v.1 Generalities

We turn to thermal bosons for spieces of mass and chemical potential at temperature . Setting , the Bose-Einstein mass density reads

(17) |

with . For this describes isothermal classical particles and for thermal fermions.

The data for the X-ray gas in A1835 fit well to nieuwenhuizen2017modified ()

(18) |

with km/s; kpc. We model the galaxy mass density as limousin2007combining ()

(19) |

Solving the Poisson equation (3) we may now determine from (13), which can also be expressed as nieuwenhuizen2009non ()

(20) |

With and varying less than , this relation is numerically better behaved.

### v.2 Isothermal classical particles or objects

### v.3 Thermal bosons

Let us return to the BE case (17) for axions, ALPs and dark photons. It is instructive to minimize for at fixed , so that . The worst case occurs at ,

(22) | |||

Its and mean formal ruling out at . For taking increasingly negative values, diminishes untill for the BE distribution approaches a MB one, with its large from (21). Hence minimizing as function of will drive the best boson fit towards the classical isothermal limit, where it is still formally ruled out at .

### v.4 Thermal fermions

After all these negative findings, we test eq. (17) for fermions. Successful fermion fits to data sets of the cluster A1689 have been reported nieuwenhuizen2009non (); nieuwenhuizen2013observations (); nieuwenhuizen2016dirac (). For A1835 this case again yields a good fit. For the value with and is perfectly acceptable and proves the adequacy of our approach. The parameters are

(23) | |||||

For the DM density the parameters are reasonably constrained, but for the galaxies not. The mass takes the value

(24) |

The fit is presented in fig. 1. The residues have a systematic trend, again induced by the small errors around 100 kpc and minimized by choosing bin 10 as the one with 5 points.

In our approach the dark matter is fitted together with the galaxies. One may wonder whether this induces a bias towards fermions. However, dropping the galaxies mass density and only fitting the last 6 bins again brings fermions as best fit, be it with mass of eV.

### v.5 Interpretation in terms of neutrinos

The fermionic case likely refers to neutrinos and anti-neutrinos. Indeed, they act as relativistic degrees of freedom during the BBN, which poses new issues, so it is economic that some of them are known particles.

Active neutrinos are in principle Majorana particles, but with eV mass, neutrinoless double -decay should have been discovered. Indeed, GERDA gives as most recent result – eV gerda2017background (), where, in the usual notation lesgourgues2013neutrino (),

(25) |

For equal and , the known mixing angles lesgourgues2013neutrino () yield the value , so that in general . Violating this bound for any , our neutrinos must be of (nearly) Dirac type nieuwenhuizen2016dirac (). Up to the small effects of neutrino oscillations, the active neutrinos have (nearly) equal mass, also 3 sterile partners with (nearly) this mass and a (nearly) zero sterile Majorana mass matrix lesgourgues2013neutrino (). With the antineutrinos there are fermion species or 3 + 3 fermion families.

The number density is for each species lesgourgues2013neutrino (), so if the cold dark matter fraction actually stems from neutrinos, the WMAP value hinshaw2013nine () corresponds to eV and the Planck value ade2016planck () to eV. DES Y1 troxel2017dark () implies eV. Within 1.5 these cases are covered by (24) and support our findings for A1689 nieuwenhuizen2009non (); nieuwenhuizen2013observations (); nieuwenhuizen2016dirac ().

## Vi Summary

After recalling that modifications of Newton’s law do not solve the dark matter problem in galaxy clusters nieuwenhuizen2017zwicky (); nieuwenhuizen2017modified (), we consider the performance of the most studied DM candidates in clusters. An important question is whether DM is self-interacting (SiDM). If this is indeed the case, its elastic cross section puts strong constraints: MACHOs and primordial black holes are ruled out, together with light axion-like particles that have to build MACHOs. It would also put constraints on other particle models, for instance, axions and sterile neutrinos should, at best, scatter very weakly. Hence the establishment or ruling out of SiDM in cluster collision is of major interest.

In contrast to CMB and BAO analyses, relaxed clusters provide a simple cosmological test, because their history has just led to a certain relaxed shape for the DM and can be disregarded. To compare to our previous works on A1689, we consider here the cluster A1835, for which strong lensing and X-ray data were presented nieuwenhuizen2017modified (). We introduce a new, parameter-free method to regularize the small eigenvalues of the covariance matrix: binning and accounting for the intra-bin variations. We present results for one particular way of binning and fitting; other ones produced the same trend. Within this approach we analyze several options for dark matter. NFW models and classical isothermal models do not fare well for the small errors in the data and seem eliminated at more than ; hence even if DM turns out not to be self-interacting, MACHOs and PBHs seem to be ruled out. Thermal bosonic models perform even less well unless they are in their classical isothermal limit; this severely questions whether thermal axions or ALPs can constitute the DM.

Thermal fermionic DM, however, does offer a good match. They have to represent (nearly) Dirac neutrinos with a mass of 1.5 – 1.9 eV; also the 3 right handed sterile partners have (nearly) this mass and a (nearly) vanishing Majorana mass matrix. The exclusion of more than one sterile neutrino in oscillation experiments giunti2016light () would not concern them.

If neutrinos indeed have a such a large mass, nonlinearities will be needed in the plasma phase to circumvent the free-streaming road block of linear structure formation. But the notorious the Li problem in the BBN may as well require nonlinearities. The latter could be restricted to the cluster scale and down to the galaxy scale or lower, and have not much impact on the CMB. Neutrinos with eV mss have no impact inside galaxies, but the solution could lie in MOND milgrom1983modification () or gravitational hydrodynamics nieuwenhuizen2009gravitational ().

An effect similar to dark matter self-interaction in cluster-cluster collision may be caused by the Pauli principle acting in the collision of such quantum degenerate “neutrino stars”.

## Vii Outlook

The question raised by previous studies of A1689 and now confirmed for A1835 becomes pressing: What is the reason for singling out degenerate fermions as best fit for cluster lensing? While it is desirable to study more clusters, preferably relaxed spherical ones, one may already wonder: Is there a conspiracy, or is simply the neutrino, after all, just the dark matter particle, and CDM only an effective theory? And is the neutrino a Dirac fermion just having its right handed partner? The answer will come from the test of the electron antineutrino mass in the KATRIN experiment ottenweinheimer2008 (); for the prediction of 1.5 – 1.9 eV two months of data taking in 2018 cho2017weighing () should suffice. If such a detection is indeed made, the neutrino sector of the standard model is basically determined and the cluster dark matter riddle solved.

Acknowledgments We thank A. Morandi, M. Limousin and E.F.G. van Heusden for discussion.

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