Standard Model neutrinos are not usually considered plausible dark matter
candidates because the usual treatment of their decoupling in the early
universe implies that their mass must be sufficiently small to make them
``hot'' dark matter. In this paper we show that decoupling of Standard Model
neutrinos in low reheat models may result in neutrino densities very much less
than usually assumed, and thus their mass may be in the keV range. Standard
Model neutrinos may therefore be warm dark matter candidates.Comment: 5 pages, 5 figures, LaTeX file uses revtex packag

A.D. Dolgov

Antonio Riotto

D.V. Semikoz

Dmitry V. Semikoz

Edward W. Kolb

G. Smoot

G.F. Giudice

Gian F. Giudice

Igor I. Tkachev

J.R. Bond

M. Kawasaki

S.S. Gerstein

English

arXiv

Standard Model neutrinos are not usually considered plausible dark matter candidates because the usual treatment of their decoupling in the early universe implies that their mass must be sufficiently small to make them ``hot'' dark matter. In this paper we show that decoupling of Standard Model neutrinos in low reheat models may result in neutrino densities very much less than usually assumed, and thus their mass may be in the keV range. Standard Model neutrinos may therefore be warm dark matter candidates.Standard Model neutrinos are not usually considered plausible dark matter candidates because the usual treatment of their decoupling in the early universe implies that their mass must be sufficiently small to make them ``hot'' dark matter. In this paper we show that decoupling of Standard Model neutrinos in low reheat models may result in neutrino densities very much less than usually assumed, and thus their mass may be in the keV range. Standard Model neutrinos may therefore be warm dark matter candidates

Giudice, Gian F.

Kolb, Edward W.

Riotto, Antonio

Semikoz, Dmitry V.

Tkachev, Igor I.

CERN Document Server

Standard Model Neutrinos as Warm Dark MatterGian F. Giudice(1), Edward W. Kolb(2), Antonio Riotto(3), Dmitry V. Semikoz(4;5) and Igor I. Tkachev(1;5)(1)CERN Theory Division, CH-1211 Geneva 23, Switzerland(2)NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500,and Department of Astronomy and Astrophysics, Enrico Fermi Institute,The University of Chicago, Chicago, Illinois 60637-1433(3)Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy(4)Max Planck Insitut fu¨r Physik, Fo¨hringer Ring 6, 80805 Mu¨nchen, Germany(5)Institute for Nuclear Research of the Academy of Sciences of Russia, Moscow 117312, Russia(December, 2000)Standard Model neutrinos are not usually considered plausible dark matter candidates because theusual treatment of their decoupling in the early universe implies that their mass must be sufficientlysmall to make them “hot” dark matter. In this paper we show that decoupling of Standard Modelneutrinos in low reheat models may result in neutrino densities very much less than usually assumed,and thus their mass may be in the keV range. Standard Model neutrinos may therefore be warmdark matter candidates.PACS: 98.80.Cq; FNAL-Pub-00-317/A; SNS-PH/00-20; MPI-PhT-2000-49I. INTRODUCTIONCosmological dark matter in the form of neutrinos withmasses in the eV range is the quintessential example ofhot dark matter (HDM) [1]. Indeed, for a standard relicthat decouples from the surrounding thermal bath whenstill relativistic, the current abundance can be easily es-timated to be [2]ΩXh2 ’ 78gXg(TD) mXkeV, (1)where gX is the number of degrees of freedom of theparticle and g(TD) is the total eective number of rel-ativistic degrees of freedom at the decoupling tempera-ture TD. Relativistic standard model neutrinos decou-ple from chemical equilibrium at a temperature of a fewMeV, when g(TD) ’ 10.75. Since current observationsindicate that the dark matter density amounts to approx-imately ΩX ’ 0.3, one usually concludes that the massof standard model neutrinos cannot be larger than about30h2 eV [2,3].Since neutrinos must be light in order to avoid overclos-ing the universe, they were moving at nearly the speedof light at redshift z  106 when the cosmic horizon rstencompassed 1012M, the amount of dark matter con-tained in the halo of a large galaxy like the Milky Way.This implies that the free streaming of light neutrinosdestroyed any fluctuations smaller than that of a super-cluster (about 1015M). Cosmological structure forms ina light-neutrino dominated universe in a top-down sce-nario, in which superclusters of galaxies form rst, withgalaxies and clusters forming through a process of frag-mentation. However, in this scenario galaxies form toolate and their distribution is much more inhomogeneousthan observations indicate. An even more serious prob-lem is that when normalized to the low amplitude of thecosmic microwave background fluctuations detected bythe COBE satellite [4], the HDM spectrum is only be-ginning to reach nonlinearity at the present epoch andthe free-streaming wavelength cuto in the amplitude ofthe spectrum should be considerably smaller to form anystructure by the present. Because of these diculties, theidea of standard model neutrinos as dark matter has beengradually abandoned in favor of scenarios where most ofthe mass in the universe is in the form of cold dark matter(CDM).The purpose of this paper is to show that in low-reheatcosmologies, standard model neutrinos may play the roleof warm dark matter (WDM), avoiding the pitfalls ofHDM, and may be responsible for the cosmological struc-tures we observe.The key point of our observation is that the limit to thecontribution of neutrinos to the present density need notconstrain neutrino masses to be in the eV region. On thecontrary, neutrino number densities in low-reheat modelsmay be more than an order of magnitude lower than usu-ally assumed, and their mass, m , more than an orderof magnitude higher, perhaps in the keV range. Fluc-tuations corresponding to suciently large galaxy haloswith masses around 1011M may survive free streamingand standard model neutrinos may act as WDM. Be-fore launching into more details, let us see why standardmodel neutrinos may be heavier than usually thought.II. ON THE NEUTRINO DENSITYThe traditional computation of the abundance of stan-dard model neutrinos in the early universe is based onthe simple, but untested, assumption that light neutri-nos were in chemical equilibrium at temperatures largerthan TD. In other words, in the standard cosmological1bound, m < 30h2 eV, it is tacitly assumed that the uni-verse had gone through a radiation-dominated phase withtemperatures larger than about an MeV with active neu-trinos in equilibrium. The assumption of an initial con-dition of neutrinos in thermal and chemical equilibriumin a radiation-dominated universe is then equivalent tothe hypothesis that the maximum temperature obtainedduring the (last) radiation-dominated era, which will re-fer to as the reheating temperature TRH , is much largerthan the decoupling temperature. The fact that we haveno physical evidence of the radiation-dominated era wellbefore the epoch of nucleosynthesis is a simple, but cru-cial, point. It was shown in Ref. [5] that the reheatingtemperature of order 1 MeV is still compatible with light-element nucleosynthesis. A priori, one should considerTRH as an unknown quantity that can take any value aslow as 1 MeV.It is usually assumed that the radiation-dominated eracommences after a period of inflation, and that the colduniverse at the end of inflation becomes the hot universeof the radiation-dominated era in a process known asreheating. The reheating process need not be instanta-neous. On the contrary, before the radiation-dominatedphase there may have been a prolonged phase duringwhich the energy density of the universe was dominatedby some component other than radiation. This compo-nent is often represented by a coherent oscillating eldsuch as the inflaton eld, but one could just as easilyimagine that the universe is dominated by some unsta-ble massive particle species. During reheating there isa slow formation of a thermal bath of relativistic par-ticles. The temperature of this thermal bath has a pe-culiar behavior [2]. It reaches a maximum temperatureTmax  TRH(H2I M2Pl/T4RH1=4 (HI is the value of theHubble rate at the beginning of the reheating process)and then has a less steep dependence on the scale fac-tor a than in the radiation-dominated era, T  a−3=8.During this phase entropy is continuously created (theuniverse is reheating!) and the Hubble rate scales likeH  [g(T )/g1=2 (TRH)](T 4/T 2RHMPl).At a given temperature, the expansion is faster forsmaller reheat temperatures. When the temperature de-creases to TRH , the universe enters the radiation phase,and one recovers the more familiar Hubble law, H T 2/MPl.Let us now assume that the largest temperature ofthe universe during the radiation-dominated phase isvery small, of the order of a few MeV. Since neutrinoshave only weak interactions, it is very dicult for thethermal scatterings during the reheating stage to gen-erate standard model neutrinos through processes likee+e− ! νν and to bring neutrinos into chemical equi-librium. Furthermore, decreasing the reheat tempera-ture increases the rate of the expansion of the universe,making it more and more dicult for the weak inter-actions to bring the neutrinos to chemical equilibrium.Therefore, if the reheating temperature is small enough,standard model neutrinos produced during the reheatingstage never go into chemical equilibrium. In this case,neutrinos are present in the thermal bath at the begin-ning of the radiation-dominated phase, but they have anumber density n that is smaller than the equilibriumnumber density [6]. This simple argument shows thatthe present abundance of neutrinos may be much smallerthan predicted assuming that the largest temperature ofthe radiation-dominated universe was much larger than afew MeV. Thus, standard model neutrinos heavier thanabout 30h2 eV are perfectly compatible with cosmology.The above expectation was conrmed in Ref. [6], wherethe eective number density of neutrinos was computedby solving the corresponding Boltzmann equation ob-tained under the assumptions of Maxwell{Boltzmannstatistics and local thermodynamic equilibrium in thecalculation of the thermal averaged cross section. Thisamounts to assuming that electrons have an equilibriumBoltzmann distribution function and to neglect all thePauli blocking factors. In this paper we have performeda more rened and correct computation by solving di-rectly the kinetic equations for the neutrino phase-spacedistribution f(p, t).We will use the numerical code developed in Ref. [7],where the collision integrals are analytically reduced totwo dimensions [8]. The kinetic equations for neutrinoshave the form∂fi(p1, t)∂t−H(t)p1 ∂fi(p1, t)∂p1= Ii;coll, (2)where the collision integral Icoll is dominated by two-body reactions 1+2! 3+4, and is given by the expres-sionIcoll=S2E1X Z d3p22E2(2pi)3d3p32E3(2pi)3d3p42E4(2pi)3(2pi)4δ(4)(p1 + p2 − p3 − p4)F (f1, f2, f3, f4)jMj212!34. (3)Here F = f3f4(1−f1)(1−f2)−f1f2(1−f3)(1−f4), jMj2is the square of the weak-interaction amplitude summedover spins of all particles except the rst one, and S is thesymmetrization factor which includes (1/2!) for each pairof identical particles in the initial and nal states and afactor of 2 if there are two identical particles in the initialstate. Finally, there is a summation over all possible setsof leptons 2, 3, and 4. Notice that a similar approachwas taken in Ref. [5], even though there the interactionsamong neutrinos as well as the electron mass were ne-glected and electrons were assumed to have a Boltzmanndistribution. (These approximations allow the reductionof the collision integrals in the kinetic equations to one-dimensional integrals, thus simplifying the numerical cal-culations.) Furthermore, in Ref. [5] the emphasis was onthe impact of very low reheating temperatures on stan-dard big-bang nucleosynthesis, and therefore neutrinoswere taken to be massless.We have assumed that reheating is due to the decayinto light states of a particle φ, which might be the infla-ton eld, a modulus eld, or any unstable particle which2dominated the energy density of the universe before theradiation-dominated phase. The time evolution of the ρenergy density is given bydρdt= −Γρ − 3Hρ, (4)where Γ is the decay rate of the φ eld, which maybe expressed in terms of the reheat temperature TRHas Γ = 3H = 3(T 2RH/MPl)[8pi3g(TRH)/90]1=2 whereg(TRH) is the number of relativistic degrees of freedomat TRH . Notice that in Refs. [5,6], the value of g in the(Γ{TRH) relation was xed to 10.75. This is not neces-sarily the actual value of g(TRH); we will return to thispoint below. Also, in Ref. [6], the denition of TRH isΓ = H , rather than Γ = 3H .We imposed the covariant energy conservation _ρ(t) =−3H(ρ + P ), where ρ is the total energy density isρ = ρ(t) +pi2T 4γ15+2pi2Zdq q2pq2 + m2eexp (E/Tγ) + 1+1pi2Zdq q3fe(q) +2pi2Zdq q3fµ(q), (5)and a similar expression holds for the pressure P . Theinitial density of the scalar eld ρ is not relevant sinceit just denes the initial time of the evolution.FIG. 1. The relative neutrino densities as functions of timefor TRH = 1 MeV.At the beginning of the evolution, the electromagneticinteractions are much faster than the neutrino interac-tion rates and therefore, before neutrinos are produced,the plasma reaches temperatures higher than the nal re-heating temperature TRH [2,6]. Later, neutrinos νi of agiven flavor start being produced by electron-positron an-nihilations (e+e− ! νiνi) and by neutrino-(anti)neutrinoannihilations (νj νj ! νiνi with i 6= j). While tau- andmuon-neutrinos are produced only by neutral current in-teractions, electron-neutrino production has a contribu-tion from charged current interactions as well. However,since the present bound on me is in the eV range, we willnot be interested in νe’s as dark matter. At late times,the neutrino distribution reaches some dynamical shapewhich diers from the equilibrium one. We have checkedthat the neutrino abundances follow the same evolutioncurve regardless of their initial abundances and the nalneutrino distribution is insensitive to the initial condi-tions. The time dependence of neutrino density (at latetimes) is shown in Fig. 1.FIG. 2. The relative number density of different neutrinospecies to the equilibrium neutrino number density as a func-tion of the reheating temperature.FIG. 3. The final distribution functions for νe, ν and νfor TRH = 1 MeV as function of the dimensionless momentump/Tγ . The dotted line corresponds to the standard modelneutrino distribution function.The nal tau- and muon-neutrino number density(they are the same) normalized to the neutrino num-ber density in equilibrium n/nEQ (where nEQ =2ξ(3)T 3 /pi2) is plotted in Fig. 2 as a function of TRH .This ratio becomes smaller than unity for TRH smallerthan about 8 MeV, signalling a departure from equi-librium. The nal momentum distribution function forTRH = 1 MeV is plotted in Fig. 3. For comparison, wehave plotted the equilibrium neutrino distribution func-tion taking into account the fact that the neutrino tem-perature is a factor 1.4 times smaller then the photontemperature Tγ due to e annihilation after neutrino de-coupling.3What is relevant is that the abundance of tau- andmuon-neutrinos is about a factor of 2.7  10−2 smallerthan the standard abundance for reheating temperaturesaround 1 MeV. This in turn implies that the standardupper bound of 30h2 eV on m no longer applies andthe abundance of tau- and muon-neutrinos for 1 MeV <TRH < 3 MeV can be expressed asΩτ h2 = Ωµh2 = m4 keV  TRH1 MeV3, (6)where we used a power-law approximation n/nEQ 0.024(TRH/1 MeV)3. For larger reheat temperatures,3 MeV < TRH < 8 MeV, one needs a slightlymore complicated expression to t n : n/nEQ ’0.44 tan−1 [(TRH(MeV)− 2.6)/1.17] + 0.43.FIG. 4. The actual value of g∗ at TRH as a function of TRH .As mentioned above, the value of g in the (Γ{TRH)relation is actually a function of TRH . The value ofg(TRH) provides an indication of the relative abundanceof neutrinos in the thermal bath. The value of g(TRH)as a function of TRH is given in Fig. 4 and allows torelate the decay rate to the reheat temperature, Γ−1 =(MeV/TRH)2g(TRH)−1=21.6 sec ’ 0.56(MeV/TRH)2:16sec.Our result in Eq. (6) is slightly smaller than the relicneutrino abundance computed in Ref. [5]. For instance,Fig. 2 of Ref. [5] shows a value of Ωh2 a factor of 1.6larger than what is given by Eq. (6), for TRH = 2 MeV.The calculation in Ref. [5] makes the approximation of(i) neglecting the electron mass in the collision integrals,(ii) assuming a Boltzmann distribution for the electrons,(iii) neglecting ν{ν interactions. We nd that the rstapproximation is well justied, but the other two give er-rors of up to 30% (depending on the value of TRH), whichhowever are opposite in sign and roughly cancel eachother. The numerical discrepancy is largely explained bythe fact that the authors of Ref. [5] x g(TRH) = 10.75,and therefore their result should be multiplied by a cor-rection factor [10.75/g(TRH)]3=4. For the same reason,the limit on TRH derived in Ref. [5] from nucleosynthesisshould also be rescaled by a factor [10.75/g(TRH)]1=4.With this procedure we nd that nucleosynthesis givesslightly more stringent lower bounds on TRH : TRH > 1.2MeV (at 68% CL) and TRH > 0.8 MeV (at 95% CL).The result of Eq. (6) is similar to that found in Ref. [6](allowing for the diering denition of TRH mentionedabove). This small dierence traces to the assumptionsmade in Ref. [6] of Maxwell{Boltzmann statistics andlocal thermodynamic equilibrium in the calculation of thethermal averaged cross section.III. STANDARD MODEL NEUTRINOS ASWARM DARK MATTERCDM reproduces the observable universe at largescales, but appears to be in conflict with observationson sub-galactic scales. CDM produces too many dwarfgalaxies and overdense galactic cores compared to ob-servations. The resolution of this diculty has beensearched for along dierent routes. Many suggestions canbe united under recipe of reducing the power on smallscales. In the WDM scenario, this reduction occur natu-rally via the mechanism of free streaming. The comovingsmoothing scale can be estimated as the comoving hori-zon at matter-radiation equality times the rms velocityof dark matter particles at that time [9]R  0.2 (Ωh21=3 (keV/mX)4=3 Mpc. (7)Interestingly, ΩX also takes correct value within thesame range of mX if the reheating temperature is of order1 MeV [see Eq. (6)]. This numerical coincidence deservesfurther study and detailed numerical investigation.FIG. 5. The effective chemical potential as a function ofreheat temperature.Encouraging preliminary results were obtained in Ref.[10]. To aid such studies, we parameterize our nal dis-tribution function of neutrinos as a thermal distributionwith some eective \temperature" and a \chemical po-tential." The dierence between the eective tempera-ture and the standard-model neutrino temperature is in-4signicant (only a few percent) and the modication tothe distribution function can be attributed to the chem-ical potential, which is shown in Fig. 5.At large TRH the chemical potential approaches thestandard-model value µ = 0.01T. At small TRH it canbe tted as µ/T = 8.8 exp(−0.84TRH/MeV).To conclude, the combination of a standard model neu-trino of mass of a few keV and a reheat temperatureof about an MeV will result in neutrinos as candidateWDM. Assuming Ω = 0.3 in one species of neutrinoand h = 0.65, then R = 0.4(TRH/MeV)4 Mpc. As dis-cussed above, any reheat temperature above about anMeV is consistent with BBN.We want to stress that the hypothesis of neutrinosas WDM is not inconsistent at present with the recentdata on atmospheric and solar neutrino anomalies. At-mospheric neutrino data could be accomodated by oscil-lations between quasi-degenerate ν and ν states and insuch a case solar neutrino observations would require νeto oscillate into a sterile state. Alternatively, the solarneutrino data might be explained by conversion of νe intoan active state and the atmospheric neutrino decit bythe conversion of ν into a sterile state.ACKNOWLEDGMENTSThe work of E.W.K. was supported in part by NASA(NAG5-7092).[1] For a recent review, see J. R. Primack and M. A. Gross,“Hot dark matter in cosmology,” astro-ph/0007165.[2] E. W. Kolb and M. S. Turner, The Early Universe,(Addison-Wesley, Menlo Park, Ca., 1990).[3] S.S. Gerstein and Ya.B. Zeldovich, Pis’ma ZhETF 4, 174(1966).[4] G. Smoot et al., Astrophys. J. 396, L1 (1992).[5] M. Kawasaki, K. Kohri, N. Sugiyama, Phys.Rev.D62,(2000) 023506.[6] G. F. Giudice, E. W. Kolb and A. Riotto, hep-ph/0005123, to be published in Phys. Rev. D 62, 023506(2000).[7] A. D. Dolgov, S. H. Hansen and D. V. Semikoz, Nucl.Phys. B503, 426 (1997); Nucl. Phys. B543, 269 (1999).[8] D. V. Semikoz and I. I. Tkachev, Phys. Rev. D 55, 489(1997).[9] J. R. Bond and A. Szalay, Ap. J. 274, 443 (1984).[10] P. Bode, J. Ostriker and N. Turok, astro-ph/0010389.5

Standard Model Neutrinos as Warm Dark Matter

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Standard model neutrinos as warm dark matter

Standard Model Neutrinos as Warm Dark Matter

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