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Axion Miniclusters and Bose Stars

arXiv:hep-ph/9303313v1 29 Mar 1993FERMILAB–PUB–93/066-AMarch 1993Axion Miniclusters and Bose StarsEdward W. Kolb(1),(2) and Igor I. Tkachev(1),(3)(1)NASA/Fermilab Astrophysics CenterFermi National Accelerator Laboratory, Batavia, IL 60510(2)Department of Astronomy and Astrophysics, Enrico Fermi InstituteThe University of Chicago, Chicago, IL60637(3)Institute for Nuclear Research of the Academy of Sciences of Russia, Moscow117312, RussiaEvolution of inhomogeneities in the axion field around the QCDepoch is studied numerically, including for the first time important non-linear effects. It is found that perturbations on scales corresponding tocausally disconnected regions at T ∼1 GeV can lead to very dense axionclumps, with present density ρa >∼10−8 g cm−3.

This is high enough forthe collisional 2a →2a process to lead to Bose–Einstein relaxation in thegravitationally bound clumps of axions, forming Bose stars.PACS number(s): 98.80.Cq, 14.80.Gt, 05.30.Jp, 98.70.–femail: rocky@fnas01.fnal.gov; tkachev@fnas13.fnal.gov

The invisible axion is one of the best motivated candidates for cosmic dark mat-ter, despite being subject to strong cosmological and astrophysical constraints on itsproperties (1010 GeV <∼fa <∼1012 GeV for the axion decay constant; 10−5 eV <∼ma <∼10−3 eV for the axion mass) [1]. As dark matter, axions would play a role in theevolution of primordial density fluctuations and formation of large scale structure.

Inaddition to its generic properties, axions also have unique features as dark matter.For instance, large amplitude density fluctuations produced on scales of the horizonat the QCD epoch [2] lead to tiny gravitationally bound “miniclusters” [3]. It wasfound that the density in miniclusters exceeds by ten orders of magnitude the localdark matter density in the Solar neighborhood [3].

This might have a number ofastrophysical consequences, as well as implications for laboratory axion searches [4].In previous studies, spatial gradients of the axion field in the equations of motionwere neglected. This is a reasonable assumption for temperatures below the QCDscale where the evolution of coherent axion oscillations can be treated as pressureless,cold dust.

However, we find that just at the crucial time when the inverse mass of theaxion is approximately the size of the horizon, gradient terms become important, anda full field-theoretical approach is needed. Here we present the results of a numericalstudy of the evolution of the inhomogeneous axion field around the QCD epoch.Though we only consider spherically symmetric configurations, the importance ofthe combined effect of the field gradients and the non-liner attractive self interactionshould also occur if we relax spherical symmetry.

The resulting axion clumps aremuch denser than previously thought, reaching the critical conditions for Bose starformation [5].The axion field θ(x) is created during the Peccei-Quinn symmetry breaking phasetransition at T ∼fa, uncorrelated on scales larger than the horizon at this time [6].For T <∼fa, the field becomes smooth on scales up to the horizon, H−1(T), where1

H is the expansion rate. This continues until T = T1 ≈1 GeV when the axion massswitches on, i.e., when ma(T1) ≈3H(T1).

Coherent axion oscillations then transformfluctuations in the initial amplitude into fluctuations in the axion density.Since the initial amplitude of coherent axion oscillations on the horizon scaleH−1(T1) is uncorrelated, one expects typical positive density fluctuations on thisscale will satisfy ρa ≈2¯ρa, where ¯ρa is mean cosmological density of axions [3]. Atthe temperature of equal matter and radiation energy density, Te = 5.5 Ωah2 eV [7],these fluctuations are already non-linear and will separate out as miniclusters withρa ≈3 (10 eV)4 ≈10−14 g cm−3 [3].

The minicluster mass will be of the order ofthe dark-matter mass within the Hubble length at temperature T1, Mmc ∼10−9 M⊙.The radius of the cluster is Rmc ∼1013cm, and the gravitational binding energy willresult in an escape velocity ve/c ∼10−8. Note that the mean phase-space densityof axions in such a gravitational well is enormous: n ∼ρam−4a v−3e∼1048f 412, wheref12 ≡fa/1012 GeV.We will show below that due to non-linear effects, a substantial number of regionscan have axion density at T > Te many times larger than 2 ¯ρa.Let us parametrize the energy density of a single fluctuation as ρa(Te < T

The energy density insidea given fluctuation is equal to the radiation energy density at T = Φ(θi)Te. At thattime the fluctuation becomes gravitationally non-linear and collapses.

Consequently,at Teρa(θi) ∼Φ4(θi)¯ρa(Te),(1)will be the minicluster density after it separates out as a bound object.Even arelatively small increase in Φ(θi) is important because the density depends upon the2

fourth power of Φ(θi).Ref. [2] demonstrated that due to anharmonic effects for fluctuations with θi closeto π, some correlated regions can have values of Φ(θi) larger than just a factor of two.The reason is simple: the closer θi is to the top of the axion potential,V (θ) = m2a(T)f 2a(1 −cos θ) ≡Λ4a(T)(1 −cos θ),(2)the later axion oscillations commence.

However this effect alone is not very significant.In the range 0.1 <∼ξ <∼10−3 we can parametrize it as Φ(θi) ≈1.5(θi/π)2ξ−0.35, whereξ ≡(π −θi)/π, and Φ(θi) is significantly larger than 2 only for field values very finelytuned to the top of the potential. Moreover, the axion field is not exactly coherenton the horizon scale, and small fluctuations might spoil this picture.At temperatures T ≫T1, the potential is negligible in the equations of motioncompared to the gradient terms which force the field to be homogeneous on scalesless than the horizon.

At T ≪T1, on the contrary, gradients can be neglected andone can treat the evolution of fluctuations as that of a pressureless gas. Clearly atT ∼T1, both the gradient terms and the potential are important, and in order tofind the energy density profile at freeze out one has to trace the inhomogeneous fieldevolution through the epoch T ∼T1.It is convenient to work in conformal coordinates with metric ds2 = a2(η)(dη2 −d⃗x 2).

During radiation dominance a ∝η and η ∝T −1. The dependence of the axionmass upon the temperature at T > ΛQCD can be found in the dilute-instanton-gasapproximation [8], and can be parametrized as a power law, m2a(η) = m2a(η∗)(η/η∗)n,where n = 7.4 ± 0.2 [2].

Introducing the field ψ = ηθ, the equations of motion fora spherically symmetric axion fluctuation in an expanding Universe is of the form¨ψ −ψ′′ −2ψ′/r+ ¯ηn+3 sin (ψ/η) = 0, where ¯η is the reduced conformal time parameter¯η = η/η∗, and ma(η∗) = H(η∗). The radial coordinate r is defined in the comoving3

FIG. 1.

Energy density contrast in a fluctuation with initial radius r0 = 1.8 andθi , and the widthof transient region, ∆r.The important common feature is that the final densitydistribution develops a sharp peak in the center.

The larger the gradients of initialconfiguration, the higher the final peak, e.g., the peak grows with increase in |θi |.4

FIG. 2.

Energy density profiles at ¯η = 4 for identical initial fluctuations evolved withdifferent Lagrangians. Solid line: axion case; dashed line: V (θ) ∝θ2/2; dotted line:V (θ) ∝θ2/2 + θ4/4; dash-dotted line: axion potential with field gradients switchedoff.The peak also grows with decreasing width of the transient region.

We present herethe results of runs with initial amplitude of the field outside the fluctuation equal tothe r.m.s. value of the misalignment angle, i.e., θ>i = π/√3, and width of transientlayer ∆r ∼0.6.Energy density profiles as a function of time are presented in Fig.

1 for a typicalcase. At ¯η = 1 there are two waves, incoming and outgoing, both propagating withthe velocity of light.

At approximately ¯η = 2 the incoming wave reaches the centerand the outgoing wave reaches r ≈3.5. At later times the wave front does not move5

significantly because the axion mass effectively switches on at ¯η ≈2, and the edge ofthe fluctuation “freezes.”One reason for energy density growth at later times is the continuing increase ofthe axion mass. However the relative density contrast in the center with respect tothe unperturbed homogeneous environment continues to increase up to the final timeof integration, ¯η = 4.

This is entirely a non-linear effect. One can see this in thefollowing way: The average pressure over a period of homogeneous axion oscillationsin potential Eq.

(2) is negative, and is equal to ⟨P⟩≃−Λ4a(T)θ40/64, where θ0 isthe amplitude of the oscillations [9].In other words, the axion self-interaction isattractive. The larger the amplitude of oscillations inside the fluctuation, the morenegative will be the pressure inside, and consequently, fluctuations with excess axionswill contract in the comoving volume.

In addition, matter with a smaller pressuresuffers less redshift in the energy density. To see this effect we present in Fig.

2 thefinal density profiles correspondin g to identical initial field distributions evolved withdifferent potentials: the axion potential of Eq. (2); the axion potential with gradientsartificially switched off; a pure harmonic potential, V (θ) ∝θ2/2, where ⟨P⟩= 0;and the potential V (θ) ∝θ2/2 + θ4/4, where ⟨P⟩> 0.

Note that for the harmonicpotential, at ¯η = 4 the maximal density excess is only about 3, i.e., ten times smallerthan for the axion potential.The dependence of the energy density contrast in the center upon the initial radiusis shown on Fig. 3.

In the whole range of values of r0 plotted, the energy density takesits maximum value just in the center of the final configuration. Only if r0 < 1.55or r0 > 2.05 does the final energy density profile have a maximum at some non-zero radius.

In a sense, the initial radius of the fluctuation in the plotted range ismore or less tuned in such a way that the arrival of the incoming wave at the centeris synchronized with the switching on of the axion mass. However, there is nothing6

FIG. 3.

Dependence of density contrast in the center of a fluctuation at ¯η = 4 uponthe initial radius of a fluctuation for several values of the initial misalignment angleinside the fluctuation.unnatural in this “synchronization,” since as larger and larger scales enter the horizonin an expanding Universe there will always be a scale for which the incident wave ofa disappearing fluctuation reaches the center just at the moment of freeze out.Quantitatively, the assumption of spherical symmetry is very important. How-ever, in general any isolated contrast in the initial misalignment angle will decay viaincoming and outgoing waves which will not possess spherical symmetry.

The overallpicture will be the same as in the spherical case, but the values of the maximal energycontrast in the final configuration at a given θ

3) due to the7

attractive self-interaction resulting in negative pressure. This has nothing to do withthe symmetry of the fluctuation, and we may expect to find large density contrastsin regions where the field values happen to be close to π initially [10].The effect of the field gradients is important not only in the discussion of theformation of high density peaks, but also in the careful estimate of the mean densityof axion matter.

We found that the total excess mass of axions within a fluctuation,compared to the homogeneous background, does not vary much, and is equal approx-imately to half of the excess mass if gradients in the equations of motion would beneglected. This deficit might be attributed to the redshift at early times, ¯η <∼2, whenaxions are still relativistic.The energy density contrast plotted in Figs.

1 and 2 will coincide with the factorΦ(θi) in Eq. (1) if we assume that the mean cosmological density of axions correspondsto homogeneous oscillations with initial amplitude equal to the r.m.s.

value of themisalignment angle. As we have noted already, the energy density in an axion clumpafter it separates out from the general expansion will be Φ4(θi) times larger than theenergy density at Te.

So a density contrast of 30 will correspond to roughly a factorof 106 in the energy density of the cluster at T < Te.All axion miniclusters could be, in principle, relevant to laboratory axion searchexperiments, since for a minicluster with Φ as small as 2, the density is 1010 larger thanthe local galactic halo density. However, the probability of a direct encounter witha clump is small.

The interesting question arises, could there be any astrophysicalconsequences of very dense axion clumps? Below we shall discuss the possibility of“Bose star” formation inside axion miniclusters.The physical radius of an axion clump at Te is larger by many orders of magnitudethan the de Broglie wavelength of an axion in the corresponding gravitational well.Consequently, gravitational collapse of the axion clump and subsequent virialization8

can be described in the usual terms of cold dark matter particles. In a few crossingtimes some equilibrium (presumably close to an isothermal) distribution of axionsin phase space will be established.It is remarkable that in spite of the apparentsmallness of axion quartic self-couplings, |λa| = (fπ/fa)4 ∼10−53f −412 , the subsequentrelaxation in an axion minicluster due to 2a →2a scattering can be significant as aconsequence of the huge mean phase-space density of axions [5].

In the case of Bose-Einstein statistics the inverse relaxation time is (1+ ¯n) times the classical expression,or τ −1R∼¯n veσρa/ma, where σ is the corresponding cross section.For part iclesbounded in a gravitational well, it is convenient to rewrite this expression in the form[5]τR ∼m7aλ−2a ρ−2a v2e. (3)The shallower the gravitational well at a given density of axions, the larger the meanphase space density, and consequently the smaller the relaxation time due to the v2edependence in Eq.

(3). Note also the dependence of the inverse relaxation time uponthe square of the particle density.The relaxation time (3) is smaller then the present age of the Universe if theenergy density in the minicluster satisfiesρ10 > 106v−8qf12,(4)where ρ10 ≡ρ/(10 eV)4 and v−8 ≡ve/10−8.

If this occurs, then an even denser corein the center of the axion cloud should start to form. An analogous process is theso-called gravithermal instability caused by gravitational scattering.

This was studiedin detail for star clusters, where the “particles” obey classical Maxwell–Boltzmannstatistics. Axions will obey Bose–Einstein statistics, with equilibrium phase-spacedensity n(p) = ncond +[eβE −1]−1, containing a sum of two contributions, a Bose con-densate and a thermal distribution.

The maximal energy density that non-condensed9

axions can saturate is ρther ∼m4av3e, which corresponds to ¯nther ∼1. Consequently,given the initial condition ¯n ≫1, one expects that eventually the number of particlesin the condensate will be comparable to the total number of particles in the regionwhere relaxation is efficient.

Under the influence of self-g ravity, a Bose star [9,11]then forms [5]. One can consider a Bose star as coherent axion field in a gravitationalwell, generally with non-zero angular momentum in an excited energy state [9].Comparing Eqs.

(1) and (4), we conclude that the relaxation time is smaller thanthe present age of the Universe and conditions for Bose star formation can be reachedin miniclusters with density contrast Φ(θi) >∼30 at the QCD epoch. For examples ofsuch density contrasts, see Figs.

1 and 3.Under appropriate conditions stimulated decays of axions to two photons in adense axion Bose star are possible [9,12] (see also [13]), which can lead to the formationof unique radio sources—axionic masers.In view of results of present paper weconclude that the questions of axion Bose star formation, structure and possibleastrophysical signatures deserves detailed study.In conclusion, we have presented a numerical study of the evolution of inhomo-geneties in the axion field around the QCD epoch, including for the first time impor-tant non-linear effects. We found that the non-linear effects can lead to a much largercore density of axions in miniclusters than previously estimated.

The increase in thedensity may be sufficiently large that axion miniclusters might exceed the criticaldensity necessary for them to relax to form Bose stars.It is a pleasure to thank H. Feldman, J. Frieman, A. Kashlinsky, A. Klypin, D.Pogosyan, A. Stebbins, M. Turner, and R. Watkins for useful discussions. This workwas supported in part by the DOE and NASA grant NAGW-2381 at Fermilab.10

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