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Draft version March 18, 2009Preprint typeset using LATEX style emulateapj v. 10/09/06

DARK STARS: A NEW LOOK AT THE FIRST STARS IN THE UNIVERSE

Douglas Spolyar1, Peter Bodenheimer2, Katherine Freese3, and Paolo Gondolo4

Draft version March 18, 2009

ABSTRACT

We have proposed that the first phase of stellar evolution in the history of the Universe may beDark Stars (DS), powered by dark matter heating rather than by nuclear fusion, and in this paperwe examine the history of these DS. The power source is annihilation of Weakly Interacting MassiveParticles (WIMPs) which are their own antiparticles. These WIMPs are the best motivated darkmatter (DM) candidates and may be discovered by ongoing direct or indirect detection searches (e.g.FERMI/GLAST) or at the Large Hadron Collider at CERN. A new stellar phase results, powered byDM annihilation as long as there is DM fuel, from millions to billions of years. We build up the darkstars from the time DM heating becomes the dominant power source, accreting more and more matteronto them. We have included many new e!ects in the current study, including a variety of particlemasses and accretion rates, nuclear burning, feedback mechanisms, and possible repopulation of DMdensity due to capture. Remarkably, we find that in all these cases, we obtain the same result: the firststars are very large, 500-1000 times as massive as the Sun; as well as pu!y (radii 1-10 A.U.), bright(106 ! 107L!), and cool (Tsurf <10,000 K) during the accretion. These results di!er markedly fromthe standard picture in the absence of DM heating, in which the maximum mass is about 140 M!

(McKee & Tan 2008) and the temperatures are much hotter (Tsurf >50,000K). Hence DS should beobservationally distinct from standard Pop III stars. In addition, DS avoid the (unobserved) elementenrichment produced by the standard first stars. Once the dark matter fuel is exhausted, the DSbecomes a heavy main sequence star; these stars eventually collapse to form massive black holes thatmay provide seeds for the supermassive black holes observed at early times as well as explanations forrecent ARCADE data (Sei!ert et al. 2009) and for intermediate mass black holes.Subject headings: Dark Matter, Star Formation, Accretion

1. INTRODUCTION

Spolyar et al. (2008; hereafter Paper I) proposed a newphase of stellar evolution: the first stars to form in theUniverse may be Dark Stars, powered by dark matterheating rather than by nuclear fusion. Here dark mat-ter (DM), while constituting a negligible fraction of thestar’s mass, provides the energy source that powers thestar. The first stars in the Universe mark the end of thecosmic dark ages, provide the enriched gas required forlater stellar generations, contribute to reionization, andmay be precursors to black holes that coalesce and powerbright early quasars. One of the outstanding problems inastrophysics is to investigate the mass and properties ofthese first stars. Depending on their initial masses, thestars lead very di!erent lives, produce very di!erent ele-mental enrichment, and end up as very di!erent objects.In this paper, we find the mass, luminosity, temperature,and radius of dark stars as a function of time: our re-sults di!er in important ways from the standard pictureof first stars without DM heating.

The DM particles considered here are Weakly Inter-acting Massive Particles (WIMPS), such as the lightestsupersymmetric particles or Kaluza-Klein dark matter

Electronic address: ktfreese@umich.eduElectronic address: peter@ucolick.orgElectronic address: dspolyar@physics.ucsc.eduElectronic address: paolo@physics.utah.edu

1 Physics Dept., University of California, Santa Cruz, CA 950642 UCO/Lick Observatory and Dept. of Astronomy and Astro-

physics, University of California, Santa Cruz, CA 950643 Michigan Center for Theoretical Physics, Physics Dept., Univ.

of Michigan, Ann Arbor, MI 481094 Physics Dept., University of Utah, Salt Lake City, UT 84112

(for reviews, see Jungman et al. 1996; Bertone et al.2005; Hooper & Profumo 2007). The search for theseparticles is one of the major motivations for the LargeHadron Collider at CERN. The particles in considerationare their own antiparticles; thus, they annihilate amongthemselves in the early universe and naturally providethe correct relic density today to explain the dark mat-ter of the universe. The first stars are particularly goodsites for DM annihilation because they form in a veryhigh density DM environment: they form at early timesat high redshifts (density scales as (1 + z)3) and in thehigh density centers of DM haloes. The first stars format redshifts z " 10 ! 50 in DM halos of 106M! (for re-views see e.g. Ripamonti & Abel 2005; Barkana & Loeb2001; Yoshida et al. 2003; Bromm & Larson 2004; seealso Yoshida et al. 2006) One star is thought to forminside one such DM halo. These halos consist of 85%DM and 15% baryons in the form of metal-free gas madeof H and He. Theoretical calculations indicate that thebaryonic matter cools and collapses via H2 cooling (Pee-bles & Dicke 1968; Matsuda et al. 1971; Hollenbach &McKee 1979) into a single small protostar (Omukai &Nishi 1998) at the center of the halo.

As our canonical values, we use the standard #!v$ =3 % 10"26 cm3/s for the annihilation cross section andm! = 100 GeV for the WIMP particle mass. In thispaper we examine a variety of possible WIMP massesand cross sections, and find that our main result holdsindependent of these properties. We consider m! = 1GeV, 10 GeV, 100 GeV, 1 TeV, and 10 TeV. This varietyof WIMP masses may be traded for a variety of valuesof the annihilation cross section, since DM heating scales

2

as #!v$/m!.Paper I found that DM annihilation provides a power-

ful heat source in the first stars, a source so intense thatits heating overwhelms all cooling mechanisms; subse-quent work has found that the heating can dominate overfusion as well once it becomes important at later stages(see below). Paper I outlined the three key ingredientsfor Dark Stars: 1) high dark matter densities, 2) the an-nihilation products get trapped inside the star, and 3)DM heating wins over other cooling or heating mecha-nisms. These same ingredients are required throughoutthe evolution of the dark stars, whether during the pro-tostellar phase or during the main sequence phase.

WIMP annihilation produces energy at a rate per unitvolume

QDM = #!v$"2!/m!, (1)

where "! is the mass density of the WIMPs. Once thegas density of the collapsing protostar exceeds a criticalvalue, most of the annihilation energy is trapped in thestar. For a 100 GeV particle, when the hydrogen densityreaches " 1013 cm"3, typically 1/3 of the energy is lost toneutrinos that escape the star, while the other 2/3 of theenergy is trapped inside the star. Hence the luminosityfrom the DM heating is

LDM "2

3

!

QDMdV (2)

where dV is the volume element. From this point for-wards, the DM heating beats any cooling mechanisms.This point is the beginning of the life of the dark star, aDM powered star which lasts until the DM fuel runs out.The DM constitutes a negligible fraction of the mass ofthe dark star (less than one percent of the total stellarmass) and yet powers the star. This ”power of darkness”is due to the 67% e"ciency of converting the DM massinto an energy source for the star, in contrast to the "1%e"ciency of converting nuclear mass into energy in thecase of fusion.

In this paper we ascertain the properties of the darkstar as it grows from an initially small mass at its incep-tion to its final fate as a very large star. At the beginning,when the DM heating first dominates inside the star, thedark star mass for our canonical case of 100 GeV massparticles is 0.6M! with a radius of 17 AU (Abel et al.2002; Gao et al. 2007). Then further matter can raindown upon the star, causing the dark star to grow. Bythe time the star reaches about 3 M!, with a radius of 1–10 AU, it can be approximated as an object in thermaland hydrostatic equilibrium, with dark matter heatingbalancing the radiated luminosity. There is " 1000M!

of unstable gas (" Jeans mass) which can in principle ac-crete onto the star, unless there is some feedback to stopit. We build up the dark star from a few solar masses byaccretion, in steps of one solar mass. At each stage wefind the stellar structure as a polytrope in hydrostaticand thermal equilibrium. We continue this process ofbuilding up the DS until the DM runs out.

As seen in equation (1), a key element in the anni-hilation is knowledge of the DM density inside the DS.The initial DM density profile in the 106M! halo is notenough to drive DM heating. We consider three mecha-nisms for enhancing the density. First, as originally pro-posed in Paper 1 and confirmed in Freese et al (2008b),

adiabatic contraction (AC) drives up the density. As thegas collapses into the star, DM is gravitationally pulledalong with it. Second, as the DS accretes more baryonicmatter, again DM falls in along with the gas. Eventuallythe supply of DM due to this continued AC runs out, andthe star may heat up until fusion sets in. However, thereis the possibility of repopulating the DM in the DS due toa third source of DM, capture from the ambient medium(Iocco 2008; Freese et al. 2008c). Capture only becomesimportant once the DS is already large (hundreds of solarmasses), and only with the additional particle physics in-gredient of a significant WIMP/nucleon elastic scatteringcross section at or near the current experimental bounds.Capture can be important as long as the DS resides ina su"ciently high density environment. In this case theDS lives until it leaves the womb at the untouched cen-ter of a 106M! DM halo. Unfortunately for the DS,such a comfortable home is not likely to persist for morethan 10-100 Myr (if the simulations are right), though inthe extreme case one may hope that they survive to thepresent day. In this paper we consider all three mecha-nisms for DM enhancement, including DM capture. Weremark at the outset that AC is absolutely essential tothe existence of the DS. None of these three mechanismswould operate were it not for some AC driving up thedensity in the vicinity of the DS.

Previously (Freese et al. 2008a), we considered onespecific case, that of 100 GeV particle mass and a con-stant accretion rate of 2 % 10"3M!/yr with no capture,assuming that the DS stellar structure can be describedby a series of n = 3/2 polytropes. We found that ittakes " 0.5 million years to build up the star to its fi-nal mass, which we found to be very large, " 800M!.While the accretion was in process, the dark stars werequite bright and cool, with luminosities L " 106L! andsurface temperatures Tsurf < 10, 000K. These resultsare in contrast to the standard properties of the Popula-tion III stars (the first stars). Standard Pop III stars arepredicted to form by accretion onto a much smaller pro-tostar (10"3M!; Omukai & Nishi 1998), to reach muchhotter surface temperatures Tsurf &50,000 K during ac-cretion, and at the end of accretion to be far less massive(" 100 ! 200M!) due to a variety of feedback mech-anisms that stop accretion of the gas (McKee and Tan2008). These di!erences in properties between dark starsand standard Pop III stars lead to a universe that canlook quite di!erent, as discussed below.

In this paper we present the results of a complete studyof building up the dark star mass and finding the stel-lar structure at each step in mass accretion. We includemany new e!ects in this paper compared to Freese et al.(2008a). (1) We generalize the particle physics parame-ters by considering a variety of particle masses or equiv-alently annihilation cross sections. (2) We study twopossible accretion rates: a variable accretion rate fromTan & McKee (2004) and an alternate variable accretionrate found by O’Shea & Norman (2007). (3) Whereaspreviously we considered only n = 3/2 polytropes, whichshould be used only when convection dominates, we nowallow variable polytropic index to include the transitionto radiative transport (n = 3). (4) We include gravityas a heat source. In the later stages of accretion, as theDM begins to run out and the star begins to contract,the gravitational potential energy is converted to an im-

3

portant power source. (5) We include nuclear burning,which becomes important once the DM starts to run out.(6) We include feedback mechanisms which can preventfurther accretion. Once the stellar surface becomes hotenough, again when the DM is running out, the radia-tion can prevent accretion. (7) We include the e!ectsof repopulating the DM inside the star via capture fromthe ambient medium. The result of this paper is that,for all the variety of parameter choices we consider, andwith all the physical e!ects included, with and withoutcapture, our basic result is the same. The final stellarmass is driven to be very high and, while DM reigns, thestar remains bright but cool.

Other work on dark-matter contraction, capture, andannihilation in the first stars (Freese et al. 2008c; Taosoet al. 2008; Yoon et al. 2008; Iocco et al. 2008; Ri-pamonti et al. 2009) has concentrated on the case ofthe pre-main-sequence, main-sequence, and post-main-sequence evolution of stars at fixed mass, in contrast tothe present work, which allows for accretion from thedark-matter halo.

We also cite previous work on DM annihilation in to-day’s stars (less powerful than in the first stars): Krausset al. (1985); Bouquet & Salati (1989); Salati & Silk(1989); Moskalenko & Wai (2007); Scott et al. (2007);Bertone & Fairbairn (2007); Scott et al. (2009).

2. EQUILIBRIUM STRUCTURE

2.1. Basic Equations

At a given stage of accretion, the dark star has a pre-scribed mass. We make the assumption that it can bedescribed as a polytrope in hydrostatic and thermal equi-librium. The hydrostatic equilibrium is defined by

dP

dr= !"

GMr

r2;

dMr

dr= 4#r2"(r) (3)

and by the polytropic assumption

P = K"1+1/n (4)

where P is the pressure, " is the density, Mr is themass enclosed within radius r, and the constant K isdetermined once the total mass and radius are specified(Chandrasekhar 1939). Pre-main-sequence stellar mod-els are adequately described by polytropes in the rangen = 1.5 (fully convective) to n = 3 (fully radiative).Given P and " at a point, the temperature T is definedby the equation of state of a mixture of gas and radiation

P (r) =Rg"(r)T (r)

µ+

1

3aT (r)4 = Pg + Prad (5)

where Rg = kB/mu is the gas constant, mu is the atomicmass unit, kB is the Boltzmann constant, and the meanatomic weight µ = (2X + 3/4Y )"1 = 0.588. We takethe H mass fraction X = 0.76 and the He mass fractionY = 0.24. In the resulting models T &10,000 K exceptnear the very surface, so the approximation for µ assumesthat H and He are fully ionized. In the final modelswith masses near 800 M! the radiation pressure is ofconsiderable importance.

Given a mass M and an estimate for the outer ra-dius R#, the hydrostatic structure is integrated outwardfrom the center. At each point "(r) and T (r) are usedto determine the Rosseland mean opacity $ from a zero-metallicity table from OPAL (Iglesias & Rogers 1996),

supplemented by a table from Lenzuni et al. (1991) forT < 6000 K. The photosphere is defined by the hydro-static condition

$P =2

3g (6)

where g is the acceleration of gravity. This point cor-responds to inward integrated optical depth % ' 2/3.When it is reached, the local temperature is set to Te!

and the stellar radiated luminosity is therefore

L# = 4#R2#!BT 4

e! (7)

where R# is the photospheric radius.The thermal equilibrium condition is L# = Ltot, where

the total energy supply for the star (see below) is domi-nated during the DS phase by the DM luminosity LDM .Thus the next step is to determine LDM from equation(2) and from the density distribution of dark matter (dis-cussed in the next subsection). Depending on the valueof LDM compared to L#, the radius is adjusted, and thepolytrope is recalculated. The dark matter heating isalso recalculated based on the revised baryon density dis-tribution. The radius is iterated until the condition ofthermal equilibrium is met.

2.2. DM Densities

The value of the DM density inside the star is a keyingredient in DM heating. We start with a 106 M! halocomposed of 85% DM and 15% baryons. We take an ini-tial Navarro, Frenk, & White profile (1996; NFW) witha concentration parameter c = 2 at z = 20 in a standard#CDM universe. We note at the outset that our resultsare not dependent on high central DM densities of NFWhaloes. Even if we take the initial central density to be acore (not rising towards the center of the halo), we obtainqualitatively the same results of this paper; we showedthis in Freese et al (2008b).

Enhanced DM density due to adiabatic contraction:Further density enhancement is required inside the starin order for annihilation to play any role. Paper I rec-ognized a key e!ect that increases the DM density: adi-abatic contraction (AC). As the gas falls into the star,the DM is gravitationally pulled along with it. Given theinitial NFW profile, we follow its response to the chang-ing baryonic gravitational potential as the gas condenses.Paper I used a simple Blumenthal method, which as-sumes circular particle orbits (Blumenthal et al. 1986;Barnes & White 1984; Ryden & Gunn 1987) to ob-tain estimates of the density. Subsequently Freese etal. (2008b) did an exact calculation using the Youngmethod (Young 1980) which includes radial orbits, andconfirmed our original results (within a factor of two).Thus we feel confident that we may use the simple Blu-menthal method in our work. We found

"! " 5(GeV/cm3)(nh/cm3)0.81, (8)

where nh is the gas density. For example, due to thiscontraction, at a hydrogen density of 1013/cm3, the DMdensity is 1011 GeV/cm3. Without adiabatic contrac-tion, DM heating in the first stars would be so small asto be irrelevant.

Enhanced DM due to accretion: As further gas accretesonto the DS, driving its mass up from " 1M! to manyhundreds of solar masses, more DM is pulled along with

4

it into the star. At each step in the accretion process,we compute the resultant DM profile in the dark star byusing the Blumenthal et al. (1986) prescription for adi-abatic contraction. The DM density profile is calculatedat each iteration of the stellar structure, so that the DMluminosity can be determined.

The continued accretion of DM sustains the dark starfor " 106 years. Then, unless the DM is supplementedfurther, it runs out, that is, it annihilates faster than itcan be resupplied. At this point the 106M! DM halo hasa tiny “hole” in the middle that is devoid of DM. Onemight hope to repopulate the DM by refilling the “hole,”but this requires further study. On a conservative note,this paper disregards this possibility. Without any fur-ther DM coming into the star (which is now very large),it would contract and heat up to the point where fusionsets in; then the star joins the Main Sequence (MS). Thisis one possible scenario which we examine in the paper.Alternatively, there is another possible mechanism to re-plenish the DM in the dark star that may keep it livinglonger: capture.

Enhanced DM due to capture: At the later stages, theDM inside the DS may be repopulated due to capture(Iocco 2008; Freese et al. 2008c). This mechanism re-lies on an additional piece of particle physics, scatteringof DM particles o! the nuclei in the star. Some of theDM particles bound to the 106M! DM halo occasionallypass through the DS, and if they lose enough energy byscattering, can be trapped in the star. Then more DMwould reside inside the DS and could continue to providea heat source. This capture process is irrelevant duringthe initial stages of the DS, and only becomes impor-tant once the gas density is high enough to provide asignificant scattering rate, which happens when the DSis already quite large (> 100M!). For capture to be sig-nificant, there are two assumptions: (1) the scatteringcross section must be high enough and (2) the ambientdensity of DM must be high enough. Regarding the firstassumption: Whereas the existence of DS relies only ona standard annihilation cross section (fixed by the DMrelic density), the scattering cross section is more specu-lative. It can vary over many orders of magnitude, andis constrained only by experimental bounds from directdetection experiments (numbers discussed below). Re-garding the second assumption: The ambient densityis unknown, but we can again obtain estimates usingthe adiabatic contraction method of Paper 1. Indeed itseems reasonable for the ambient DM density to be highfor timescales up to at most tens-hundreds of millions ofyears, but after that the 106M! host for the dark starsurely has merged with other objects and the central re-gions are likely to be disrupted; simulations have not yetresolved this question. Then it becomes more di"cult forthe DS to remain in a high DM density feeding groundand it may run out of fuel.

We therefore consider two di!erent situations: adia-batic contraction with and without capture; i.e., withand without additional capture of DM at the later stages.In the latter case we focus on “minimal capture,” wherethe stellar luminosity has equal contributions from DMheating and from fusion, as described further below. Inboth cases, we find the same result: the dark stars con-tinue to accrete until they are very heavy.

2.3. Evolution

The initial condition for our simulations is a DS of 3M!, roughly the mass where the assumption of nearlycomplete ionization first holds. We find an equilibriumsolution for this mass, then build up the mass of the starin increments of one solar mass.

We allow surrounding matter from the original bary-onic core to accrete onto the DS, with two di!erent as-sumptions for the mass accretion: first, the variable ratefrom Tan & McKee (2004) and second, the variable ratefrom O’Shea & Norman (2007). The Tan/McKee ratedecreases from 1.5% 10"2 M!/yr at a DS mass of 3 M!

to 1.5 % 10"3 M!/yr at 1000 M!. The O’Shea/Normanrate decreases from 3 % 10"2 M!/yr at a DS mass of 3M! to 3.3 % 10"4 M!/yr at 1000 M!. The accretionluminosity arising from infall of material from the pri-mordial core onto the DS is not included in the radiatedenergy. It is implicitly assumed to be radiated away inmaterial outside the star, for example an accretion disk(McKee & Tan 2008), and in any case, at the large radiiinvolved here, this energy is small compared with theDM annihilation energy.

We remove the amount of DM that has annihilated ateach stage at each radius. We continue stepping up inmass, in increments of 1 M!, until we reach 1000 M!, theJeans mass of the core (Bromm & Larson 2004) or untilall of the following conditions are met: 1) the total massis less than 1000 M!, 2) the dark matter available fromadiabatic contraction is used up; 3) nuclear burning hasset in, and 4) the energy generation from gravitationalcontraction is less than 1% of the total luminosity of thestar. The last two of these conditions define the zero-agemain sequence (ZAMS). The time step is adjusted foreach model to give the correct assumed accretion rate.The final mass can be less than 1000 M! because offeedback e!ects of the ionizing stellar radiation, whichcan reverse the infall of accreting gas (McKee & Tan2008). The feedback e!ect is included in an approximatemanner, starting when the surface temperature reaches50,000 K.

At the beginning of the evolution, the Schwarzschildstability criterion for convection shows that the model isfully convective; thus the polytrope of n = 1.5 is a goodapproximation to the structure. However as more mass isadded and internal temperatures increase, opacities dropand, once the model has reached about 100 M!, radia-tive zones start to appear at the inner radii and moveoutward. Beyond about 300 M! the model is almostfully radiative. The evolutionary calculation takes intoaccount this e!ect approximately, by gradually shiftingthe value of n up from 1.5 to 3 in the appropriate massrange. In the later stages of evolution, the constant valuen = 3 is used.

2.4. Energy Supply

The energy supply for the star changes with time andcomes from four major sources:

Ltot = LDM + Lgrav + Lnuc + Lcap. (9)

The general thermal equilibrium condition is then

L# = Ltot. (10)

The contribution LDM from adiabatically contractedDM dominates the heat supply first. Its contribution

5

been discussed previously and is given in equation (2).Later, once the adiabatically contracted DM becomessparse, the other e!ects kick in: gravity, nuclear burning,and captured DM. We discuss each of these contributionsin turn.

2.4.1. Gravity

At the later stages of evolution, when the supply ofDM starts to run out, the star adjusts to the reductionin heat supply by contracting. For a polytrope of n = 3,the total energy of the star is

Etot = !3

2

GM2

R+ Eth + Erad (11)

where Eth is the total thermal energy and the radiationenergy Erad =

"

aT 4dV . Using the virial theorem

!3

2

GM2

R+ 2Eth + Erad = 0, (12)

we eliminate Eth from equation (11) and obtain the grav-itational contribution to the radiated luminosity

Lgrav = !d

dt(Etot) =

3

4

d

dt

#

GM2

R

$

!1

2

d

dtErad . (13)

2.4.2. Nuclear Fusion

Once the star has contracted to su"ciently high in-ternal temperatures " 108K, nuclear burning sets in.For a metal-free star, we include the following three en-ergy sources: 1) the burning of the primordial deuterium(mass fraction 4%10"5) at temperatures around 106 K, 2)the equilibrium proton-proton cycle for hydrogen burn-ing, and 3) the triple-alpha reaction for helium burning.At the high internal temperatures (2.75% 108 K) neededfor the star to reach the ZAMS, helium burning is an im-portant contributor to the energy generation, along withthe proton-proton cycle. We stop the evolution of thestar when it reaches the ZAMS, so changes in abundanceof H and He are not considered. Then Lnuc =

"

&nucdMwhere &nuc is the energy generation rate in erg g"1 s"1; itis obtained by the standard methods described in Clay-ton (1968). The astrophysical cross section factors for theproton-proton reactions are taken from Bahcall (1989,Table 3.2), and the helium-burning parameters are takenfrom Kippenhahn & Weigert (1990).

2.4.3. Feedback

During the initial annihilation phase, the surface tem-perature Te! remains at or below 104 K. However oncegravitational contraction sets in, it increases substan-tially, reaching ' 105 K when nuclear burning starts.In this temperature range, feedback e!ects from the stel-lar radiation acting on the infalling material can shut o!accretion (McKee & Tan 2008). These e!ects includephotodissociation of H2, Lyman ' radiation pressure,formation and expansion of an HII region, and photo-evaporation of a disk. These e!ects become importantat a mass of ' 50 M! in standard Pop III calculations ofaccretion and evolution, but the e!ect is suppressed un-til much higher masses in the DM case because of muchlower Te! . Noting that 50 M! on the standard metal-freeZAMS corresponds to Te! ' 50,000 K (Schaerer 2002),we apply a linear reduction factor to the accretion rate

above that temperature, such that accretion is shut o!completely when Te! =100,000 K. This cuto! generallydetermines the final mass of the star when it reaches theZAMS.

2.4.4. Capture

The contribution Lcap is determined by calculating theannihilation rate of captured DM

Lcap = 2m!$cap, (14)

where

$cap =

!

d3x"2cap#!v(r)$/m! (15)

and "cap(r) is the density profile of captured DM. DMheating from captured DM has previously been discussedin detail by Freese et al. (2008c) and Iocco (2008).

The density profile of the captured DM has quite a dif-ferent shape than the adiabatically contracted (AC) DMdensity profile discussed previously. Whereas the ACprofile is more or less constant throughout most of thestar (see Fig. 3), the captured DM profile is highly con-centrated toward the center with an exponential fallo!in radius, as we will now show. Once DM is captured,it scatters multiple times, thermalizes with the star, andsinks to the center of the star. The DM then assumes theform of a thermal distribution in the gravitational wellof the star

"cap = m!ncap = fEqfTh"o,ce"["(r)m!/Tc] . (16)

Here Tc, the star’s central temperature, is used since theDM has the same temperature as the stellar core; ((r) isthe star’s gravitational potential; "o,c is the central den-sity of the captured DM; and fTh and fEq are discussedbelow. One can immediately see another di!erence withthe AC profile: for captured DM the profile is more cen-trally concentrated for larger DM particle mass, whereasthe density profile for adiabatically contracted DM is in-dependent of particle mass.

The quantity fTh is the probability that a capturedDM particle has thermalized with the star.

fTh = 1 ! e"t1/#Th. (17)

t1 is the time since a DM particle has been captured and%Th is the thermalization time scale of captured DM.

Captured DM does not immediately thermalize withthe star; it requires multiple scatters. The thermalizationtime scale for the DM is

%Th =m!/mH

2!scvescnH(18)

as previously discussed in Freese et al. (2008c). Here vescis the escape velocity from the surface of the star, nH isthe average stellar density of hydrogen, and mH is theproton mass. We have only considered spin-dependentscattering, which includes scattering o! hydrogen withspin 1/2, but not o! Helium which has spin 0.

Heating from captured DM is unimportant until thestar approaches the MS; the thermalization time scaleis very long (millions of years) and fTh ' 0. Once theDM supplied from adiabatic contraction runs out, thestar begins to contract, and the thermalization time scalebecomes very short, on the order of a year. Thus, the starrapidly transitions from fTh ' 0 to fTh ' 1. The DM is

6

either in thermal equilibrium or it is not. With this inmind, t1 has been set to 100 yrs for all of the simulations.Only %Th is recalculated at every time step, which is areasonable prescription; t1 set between a few thousandyears and a few years makes no quantitative di!erenceupon the evolution of the DS.

The ratio fEq gives the ratio of DM particles currentlyinside of the star due to capture, N , to the maximumnumber of DM particles that could possibly be present,No,

fEq =N

No. (19)

At the start of the simulation, the DS is too di!use tocapture any DM and fEq ' 0. Once the star goes ontothe main sequence, the maximum number is reached andfEq ' 1.

We can now give an explicit expression of fEq by solv-ing for the total number of particles N bound to thestar. This number is determined by a competition be-tween capture and annihilation,

N = C ! 2$ , (20)

where ‘dot’ refers to di!erentiation in time. The totalnumber of DM particles captured per unit time C wascalculated following Gould (1987a). The total numberof DM particles annihilated per unit time $ contains afactor of two since two particles are annihilated per an-nihilation.). Equation (20) can be solved for N ,

N = No tanh(t2/%eq) (21)

where No =%

C/CA, and %eq = 1/(

CCA, and CA =2$/N2 (Griest & Seckel 1987).

When the annihilation rate equals the capture rate, Ngoes to No. This transition occurs on the equilibriumtime scale %eq. The equilibrium time scale is recalculatedat each time step. The time t2 in equation (21) countsthe time since capture is turned on. In equation (16), wenow set

fEq =N

No= tanh(t2/%eq). (22)

Conceptually t2 is not the same as t1 but at a practicallevel it is reasonable to set t2 = t1. In the first place, t1refers to the time since a single DM particle has beencaptured and in fact t1 is di!erent for each captured DMparticle. On the other hand, t2 is a single time for allcaptured and thermalized DM and starts when captureturns on. Regardless, once the star descends towards themain sequence, fEq and fTh both go to unity. The con-nection is not a coincidence. First %eq is a factor of afew to a hundred less than %Th throughout the simula-tion. In addition, t2 > t1 except for the very first DMparticle captured. Fixing t2 = t1 does not change thebehavior of fEq in relation fTh as just described. Hence,for simplicity the simulation assumes t1 = t2.

In the limit that fEq and fTh go to unity, then $cap =C/2. Hence, Lcap simplifies to

Lcap =2

3m!C (23)

where the 2/3 accounts for the loss of neutrinos as pre-viously discussed in connection with equation (2). Alsoroughly Lcap ) "!!sc and is independent of m!. Hence

Lcap scales with both the ambient background DM den-sity "!, and the scattering cross section of the DM o! ofbaryons !sc.

Physically reasonable values for the scattering crosssection and background DM densities imply that Lcap

can a!ect the first stars once they approach the MS. Di-rect and indirect detection experiments require !sc *4 % 10"39 (Savage et al. 2004; Chang et al. 2008; Sav-age et al 2008; Hooper et al. 2009); a WIMP’s !sc canbe many orders of magnitude smaller than the exper-imental bound. The ambient background DM density,"!, can be very high, on the order of 1014(GeV/cm3)(as obtained from Eq. (8)) due to adiabatic contraction.To demonstrate the “minimal” e!ect of capture, we set"! = 1.42%1010(GeV/cm3) and !sc = 10"39cm2, chosenso that about half of the luminosity is from DM captureand the other half from fusion for the standard 100 GeVcase at the ZAMS. These parameters have been used inall cases where capture has been turned on. Even min-imal capture shows that DM capture lowers the star’ssurface temperature and increases the mass of the firststars.

We note that the product of ambient density and crosssection could be very di!erent than the values used inthis paper. If this product is much smaller than the val-ues considered in the ’minimal capture’ case, then fusioncompletely dominates over capture and one may ignoreit. On the other hand, if this product is much largerthan the ’minimal capture’ case, then fusion is negligibleand the DS may continue to grow by accretion and be-come much larger than discussed in this paper; this casewill be discussed elsewhere. The ’minimal’ case is thusthe borderline case between these two possibilities, andlasts only as long as the DS remains embedded in a highambient DM density assumed here.

3. RESULTS

Here we present our results for the evolution of aDS, starting from a 3M! star powered by DM annihi-lation (defined as t = 0), building up the stellar massby accretion, and watching as the other possible heatsources become important. We have considered a rangeof DM masses: 1 GeV, 10 GeV, 100 GeV, 1 TeV and 10TeV, which characterizes the plausible range for WIMPmasses. Since the DM heating scales as #!v$/m!, thisrange of particle masses also corresponds to a range ofannihilation cross sections. To account for the uncer-tainties associated with the accretion rate, the study hasused two di!erent accretion rates: one from Tan & Mc-Kee (2004) and the other from O’Shea & Norman (2007)(as discussed in section 2.3 above). We consider thecase of (i) no capture and (ii) “minimal capture” (de-fined as above), with "! = 1.42 % 1010(GeV/cm3) and!sc = 10"39cm2, chosen so that about half of the lumi-nosity is from DM capture and the other half from fusionfor the standard 100 GeV case at the ZAMS. In this pa-per we will not treat the case of maximal capture, wherethe DM heating would dominate (over all other sourcesincluding fusion) so that the DS would continue to growas long as it is fed more DM .

We here show the results of our simulations. First webriefly outline our results and present our tables and fig-ures. These are then discussed in detail, case by case, inthe remainder of the results section.

7

TABLE 1Properties and Evolution of Dark Stars for m! = 1 GeV, M from Tan & McKee, !!v" = 3 # 10!26 cm3/s. For the lower

stellar masses, identical results are obtained for the cases of no capture and minimal capture; hence the first four rowsapply equally to both cases. The fifth and sixth rows apply only to the no capture case while the last two rows apply

only when capture is included.

M" L" R" Te! !c Tc MDM !o,DM t(M#) (106L#) (1012cm) (103K) (gm/cm3) (106K) (1031g) (gm/cm3) (103yr)106 9.0 240 5.4 1.8 ! 10!8 0.1 36 4.5 ! 10!11 19371 17 270 5.9 5.2 ! 10!8 0.20 64 2.0 ! 10!11 119690 6.0 110 7.5 5.9 ! 10!6 1.0 10 6.8 ! 10!11 280756 3.3 37 10 2.2 ! 10!4 3.4 1.2 4.8 ! 10!10 310793 4.5 5.1 31 8.9 ! 10!2 25 " 0.0 " 0.0 330820 8.4 0.55 111 115 276 " 0.0 " 0.0 430

Alternative propertiesfor final two masseswith capture included:

793 4.6 5.7 30 7.1 ! 10!2 24 8.4 ! 10!4 2.94 ! 10!8 328824 8.8 .58 109 102 265 8.1 ! 10!5 2.14 ! 10!6 459

TABLE 2Properties and Evolution of Dark Stars for m! = 100 GeV, M from Tan & McKee, !!v" = 3 # 10!26 cm3/s. Entries as

described in Table 1.

M" L" R" Te! !c Tc MDM !o,DM t(M#) (106L#) (1012cm) (103K) (gm/cm3) (106K) (1031g) (gm/cm3) (103yr)106 1.0 70 5.8 1.1 ! 10!6 0.4 16 1.8 ! 10!9 19479 4.8 84 7.8 2.0 ! 10!5 1.4 41 2.0 ! 10!9 171600 5.0 71 8.5 4.1 ! 10!5 1.8 36 1.9 ! 10!9 235716 6.7 11 23 2.0 ! 10!2 15 2.7 3.8 ! 10!8 303756 6.9 2.0 56 2.8 78 " 0.0 " 0.0 330779 8.4 0.55 110 120 280 " 0.0 " 0.0 387

Alternative propertiesfor final two masseswith capture included:

756 (c) 6.8 2.0 55 2.55 76 7.5 ! 10!5 5.3 ! 10!5 328787(c) 8.5 0.58 108 105 270 2.2 ! 10!5 5.3 ! 10!4 414

TABLE 3Properties and Evolution of Dark Stars for m! = 10 TeV, M from Tan & McKee, !!v" = 3 # 10!26 cm3/s. Entries as

described in Table 1.

M" L" R" Te! !c Tc MDM !o,DM t(M#) (106L#) (1012cm) (103K) (gm/cm3) (106K) (1031g) (gm/cm3) (103yr)106 0.1 22 6.0 3.8 ! 10!5 1.3 7.8 1.8 ! 10!8 19310 0.59 21 9.3 4.4 ! 10!4 3.6 15 1.5 ! 10!7 88399 3.3 6.6 25 3.6 ! 10!2 16 6.6 9.9 ! 10!7 134479 3.0 2.9 32 0.5 40 2.1 1.1 ! 10!6 172552 5.2 0.56 97 79 220 " 0.0 " 0.0 230552 5.3 0.47 107 136 270 " 0.0 " 0.0 256

Alternative propertiesfor final two masseswith capture included:

550 (c) 5.2 0.6 95 66 210 8.5 ! 10!12 5.3 ! 104 230553 (c) 5.4 0.48 105 127 260 5.7 ! 10!12 1.2 ! 105 266

8

Tables 1–3, each for a fixed m!, present the evolu-tion of some basic parameters of the star throughout theDS phase: stellar mass M#, luminosity L#, photosphericradius R#, surface temperature Te! , the star’s baryoniccentral density "c, central temperature Tc, the amountof DM inside of the star MDM, the DM’s central density"o,DM, as well as the elapsed time t since the start of thecalculation. MDM and "o,DM give the mass and densityof DM either from adiabatic contraction (the first 6 rows)or from capture (the last two rows).

In the tables, the stellar masses were chosen to roughlycharacterize important transitions for the DS. The firstmass was chosen at roughly the time when a dark starbegins to take on the size, temperature, and luminos-ity which are characteristic of the DS phase, which isaround 100 M!. The second mass is given when the DSacquires its maximum amount of DM, which varies fromcase to case. The third mass is towards the end of the DSadiabatic phase. The fourth row characterizes the star’sproperties just prior to running out of adiabatically con-tracted DM. The fifth row is given just after the star runsout of adiabatically contracted DM, while the sixth givesthe star’s properties on the ZAMS. Capture only becomesimportant once the adiabatically contracted DM has runout. Prior to running out, the stellar evolution is thesame with or without DM capture. For the case withcapture included, the seventh and eighth row should beused instead of the fifth and sixth row.

Further features of the calculations are shown in Fig-ures 1–4, which give, respectively, (1) the evolution ofall cases without capture in the Hertzsprung-Russell dia-gram, (2) the luminosity as a function of time for the 100GeV case with and without capture, (3a) the evolution ofthe density distribution of the baryonic and dark matterfor the 100 GeV case without capture, (3b) the distribu-tion of dark matter heating rate for the 100 GeV casewith capture, and (4) the evolution of various quantitiesfor all cases without capture.

3.1. Canonical Case without Capture

Our canonical case takes m! = 100 GeV, the accretionrate as given by Tan & McKee (2004), and #!v$ = 3 %10"26 cm3s"1 for the calculation of the annihilation rate.

The initial model (defined to be t = 0) has a mass of3 M! with a surface temperature of 4100 K, a radius of3 % 1013 cm, a luminosity of 4.5 % 104 L! and a fullyconvective structure. The DS stays fully convective forstellar masses below 50 M! at which point a small ra-diative zone appears. The star makes a transition fromconvective to radiative in the M# = (100!200) M! massrange, with a radiative zone growing outward from thecenter, and then becomes fully radiative (but for a smallconvective region at the surface) for M# > 300 M!. Thestar stays almost fully radiative for the remainder of theDS phase. Above 750 M!, hydrogen and helium burningset in, and a new convective zone appears at the centerof the star.

Evolution of Dark Matter Density: The DM densitiesinside of the star change due to annihilation and the evo-lution of the dark star’s baryonic structure. Figure 3plots the baryonic and DM density profiles for the stan-dard case at several di!erent stellar masses. As the star’smass grows, the DM and baryonic densities also increase.The DM profile for the last model, which has a stellar

mass of 779 M!, is not plotted because at this point allof the adiabatically contracted DM has been used up.

The adiabatically contracted DM density profile ismore or less constant with radius and then falls o! to-ward the surface of the star. The DM densities for theearlier models show a density increase toward the cen-ter of the star due to the original NFW profile, e.g. inthe 300 M! model. The slight peak flattens out at higherstellar masses as can be seen in the 500 M! models, sinceregions with a higher DM density annihilate more rapidlythan regions with a lower DM density. Once the peak isworn away, the central DM profile stays flat and ulti-mately decreases.

The DM and baryonic densities evolve slowly for muchof the DS phase. In fact between 100 M! and 600 M!,the DM central density barely changes (Table 2). Ini-tially, the DM mass grows (Fig. 4e) and reaches a max-imum when the total mass is about 450 M!, at whichpoint the star has about 0.2 M! of DM. At this point,the DM heating becomes insu"cient to support the star.The star begins to slowly contract and boosts the DMdensity, but the boost is still modest up to a time of2 % 105 years or until the star reaches 600 M!.

Around 600 M!, the star begins to rapidly shrink.Note the rapid dominance of gravitational energy at thistime (Fig. 2). The DM densities shoot up, but the totalamount of DM inside of the star precipitously drops. By750 M!, the DM has run out and the star begins to tran-sition into a standard Pop. III star. The star reaches theMS at around 780 M! at which point feedback e!ectscut o! any further accretion.

Evolution of temperature and energy generation: Thesurface properties of the DS evolve rapidly between theinitial model at 3 M! and 100 M!. The luminosityshoots up by a factor of 25, the radius by a factor of3, and the star’s surface temperature goes from 4100 Kto 5800 K.

The star looks radically di!erent from a metal-free staron the MS. Even at 100 M!, the DS is still very cool witha surface temperature of around 5800 K vs 100, 000 K (forthe standard ZAMS) and has a central temperature ofonly about a half a million degrees Kelvin vs 200 milliondegrees (ZAMS). The star is also much more extended,' 7 % 1013 cm vs ' 5 % 1011 cm (ZAMS).

Between 100 ! 600 M!, the star’s luminosity evolvesless rapidly, increasing by a factor of 5 to a value of5 % 106 L!. The star’s surface temperature reaches7800 K, while the radius remains almost constant, pass-ing through a broad maximum at about 8.5 % 1013 cm.The central temperature is still quite cool (1.8 % 106 K)compared to a metal-free ZAMS star. Most of the star isstill too cool to even burn deuterium and lithium. DMcontinues to power the star, as is apparent in Figure 2.

Above 600 M!, the DS begins to transform into a stan-dard metal-free star. As the DM begins to run out thesurface temperature begins to shoot up and the stellar ra-dius shrinks. Just prior to running out of DM, at 716 M!,the star’s surface reaches 23, 000 K, the radius has shrunkto 1013 cm and the central temperature has shot up to 15million K. Near the beginning of this rapid contractionthe star is su"ciently hot to burn deuterium rapidly, ascan be seen from the spike of nuclear heating in Figure 2just before 0.3 Myr; this spike lasts about 50,000 years.The burning of deuterium, however, remains an insignifi-

9

Fig. 1.— Evolution in the H-R diagram, for 5 di!erent particle masses as indicated in the legend. The luminosity is given in solar unitsand the temperature is in Kelvin. The dots indicate a series of time points, which are the same for all cases. The open dots distinguish the100 GeV (B) case from the 100 GeV case. All cases are calculated with the accretion rate given by Tan & McKee (2004) except the curvelabelled 100 GeV (B), which is calculated with the rate given by O’Shea & Norman (2007). The metal-free zero-age main sequence (Z = 0ZAMS) is taken from Schaerer (2002). The peak below 104K in the 1 GeV case is due to the overwhelming DM luminosity in this case.

cant power source for the star. As the DS runs out of DMgravity and nuclear burning begin to turn on as energysources for the star. Just at the point where all three en-ergy sources (DM heating, gravity, and nuclear burning)all contribute simultaneously, the luminosity increases byabout 50 % and then decreases again; one could call thisa “flash” which lasts about 20,000 years and is the “lastgasp” of adiabatically contracted DM annihilation.

Near 750 M! the energy generation in the star becomesdominated by, first, gravity, and then nuclear burning.The star’s surface temperature is just above 50,000 K(Table 2) and feedback e!ects have started to cut o! theaccretion of baryons. Eventually, the star begins to settleonto the MS and nuclear burning becomes the dominantpower source. The surface temperature has shot up to109,000 K, and finally feedback e!ects have stopped anyfurther accretion. On the MS, the radius is 5.5%1011 cmand the central temperature has gone up to 280 millionK. At the final cuto!, which is our zero-age main se-quence model, the star has a mass of 779 M!.

The final model on the metal-free MS compares wellwith that produced by a full stellar structure code(Schaerer 2002). Schaerer’s radius is slightly larger at8.4 % 1011 cm vs. 5.5 % 1011 cm for the polytrope. The

polytrope’s surface temperature is slightly too high at109,000 K vs 106,000 K and the polytrope’s luminosityis too low by a factor of about 2. The di!erence is notsurprising; we have not done the radiative energy trans-port. Remarkably, the numbers are quite close despitethe polytropic assumption.

3.2. General Case without Capture

In addition to the canonical case, we have studied manyother parameter choices as described previously: a vari-ety of WIMP masses (or, equivalently, annihilation crosssections) and a variety of accretion rates. In this sec-tion we still restrict ourselves to the case of no capture.In all cases, we obtain the same essential results. Whileone can see in Figure 4 that there is some scatter of thebasic properties of the star, yet we find the followinggeneral results for all cases: DS evolution occurs over anextended period of time ranging from (2!5)%105 years.All cases have an extended radius (2%1013!3%1014)cm,are cool (Te! <10,000 K) for much of the evolution, andhave a large final mass (500 ! 1000) M!. All of the DScases studied are physically very distinct from standardPop. III stars. The standard Pop. III star is physicallymore compact with a radius of only ' 5 % 1011cm, and

10

Fig. 2.— Luminosity evolution for the 100 GeV case as a functionof time (lower scale) and stellar mass (upper scale). The solid(red) top curve is the total luminosity. The lower curves give thepartial contributions of di!erent sources of energy powering thestar a) (upper frame) without capture, and b) (lower frame) with‘minimal’ capture. In both frames, the total luminosity is initiallydominated by DM annihilation (the total and annihilation curvesare indistinguishable until about 0.3 Myr after the beginning of thesimulation); then gravity dominates, followed by nuclear fusion. Inthe lower frame, capture becomes important at late times.

hotter with a surface temperature of '100,000 K. Theestimated final mass for the standard Pop III.1 star isalso smaller, ' 100 M!, because the high surface tem-perature leads to radiative feedback e!ects that preventaccretion beyond that point (McKee & Tan 2008).

Figure 1 shows the evolution curves in the H-R diagramfor a variety of cases. The figure shows the evolution forvarious particle masses with the Tan/McKee accretionrate, and compares the 100 GeV case with a calculationusing the O’Shea/Norman accretion rate.

The DS evolution curves on the HR diagram are qual-itatively similar for all of the di!erent cases. The DSspends most of its lifetime at low temperatures below10,000 K. Eventually, DM begins to run out, the starcontracts and heats up, and gravity and nuclear burn-ing begin to turn on. Just at the point where all three

Fig. 3.— a) (upper frame): Evolution of a dark star for m! = 100GeV as mass is accreted onto the initial protostellar core of 3 M#

(for the case of no capture). The set of upper (lower) solid curvescorrespond to the baryonic (DM) density profile (values given onleft axis) at di!erent stellar masses and times. b)(Lower frame):di!erential luminosity rdLDM/dr as a function of r for the massesindicated.

power sources are important, the luminosity ”flashes”(as seen in Fig. 2), i.e., increases by a 50 % for about20,000 years and then goes back down. Then, DM runsout. For the lower particle masses, as seen in Fig. 1, theluminosity drops slightly as gravity becomes the mainand dominant power source. As the surface temperaturecontinues to increase, nuclear burning becomes impor-tant when Te! ' 30, 000 K. Above this temperature (seeFig. 1), the stellar luminosity again increases as the starapproaches the ZAMS. Above 50,000 K, feedback e!ectsturn on and begin to stop the accretion of baryons. Oncethe star reaches 100,000 K, accretion completely stops,and the star goes onto the ZAMS.

While the basic DS evolution is similar for all cases,we now discuss the variation due to di!erent parame-ters. The evolution curves di!er depending upon theDM particle mass. For larger m!, the stellar mass andluminosity decrease such that the evolution curve on the

11

Fig. 4.— Dark Star evolution (for the case of no capture) illustrating dependence on DM particle mass and accretion rate(O’Shea/Normanvs. Tan/McKee). Di!erent curves are the same as explained in the legend of Fig. 1. The quantities plotted are a) stellar mass (upperleft), b) total radius (upper right), c) surface temperature (middle left), d) central stellar (baryon) density (middle right), e) amount of DMinside of the star (lower left), and f) radiative gradient at the center of the star (lower right). In the latter plot, central regions above thehorizontal line are unstable to convection; hence this plot illustrates the onset of a central radiative zone.

H-R diagram for the 1 TeV case is beneath that for the100 GeV case. The reduced luminosity is a consequenceof the reduced energy production at the higher particlemasses, since DM heating is inversely proportional to m!(see eq. (1)). Then at higher particle masses a smallerradius is needed to balance the dark-matter energy pro-duction and the radiated luminosity.

In the Tables and in Figure 3, one can see the den-sity profiles of the various components in the DS. TheDM densities also depend upon m!. As shown in theTables, the average DM density in the star is an increas-ing function of m!. For example, at the time the DSreaches 100 M!, the central DM densities vary between2 % 10"8 g/cm3 for m! = 10 TeV and 5 % 10"11 g/cm3

for m! = 1 GeV. These densities are many orders ofmagnitude lower than the baryon densities in the DS, ascan be seen in Figure 3. Here one can see “the powerof darkness:” Despite the low DM densities in the star,DM annihilation is an incredibly e"cient energy sourcesince nearly all of the particle mass is converted to heatfor the star. We also wish to point out that both baryonand DM densities in the DS are far lower than in metalfree main sequence stars of similar stellar mass; ZAMSstars have densities > 1 gm/cm3.

One can also see the dependence of the DS evolution

upon accretion rate. Our canonical case, obtained us-ing the accretion rate from Tan & McKee (2004), can becontrasted with that using the O’Shea/Norman accretionrate. For 100 GeV particles, the DS evolution is identicalfor the two accretion rates until Te! " 7, 000 K (105 yr).After that the luminosity of the O’Shea/Norman caseshoots higher as a result of the higher accretion rate.Consequently the DS with the O’Shea/Norman accre-tion rate has a shorter lifetime than the DS with theTan/McKee accretion rate (350 vs. 400 thousand years);a larger final mass (916M! vs. 780M!) and higher lu-minosity.

The 1 GeV case displays some interesting behavior:one can see a spike in the evolutionary track on the H-Rdiagram. The DS reaches a maximum luminosity of 2 %107 L! and then subsequently drops by a factor of nearlyten, crossing beneath the 100 GeV case. Subsequentlythe 1 GeV curve then again crosses the 100 GeV casebefore settling onto the MS. The large early luminosityis due to extreme e!ectiveness of DM heating for the 1GeV case; DM heating is inversely proportional to m!(eq. 1).

The stellar structure of all of the cases studied is similarto our canonical case, but the transition from a convec-tive to a radiative structure happens at di!erent times,

12

depending upon the DM particle mass. The lower rightpanel in Fig. 4 plots the temperature gradient in theinnermost few zones at the center of the DS, to showthe onset of radiative energy transfer at the center of thestar. Models above the line have convective cores whilemodels below have radiative cores. As we increase theDM mass, the onset of radiative transport happens ear-lier, at lower stellar masses. For our canonical 100 GeVcase, a small radiative zone appears when the mass ofthe DS reaches 50M!. The m! = 1 GeV case, however,does not begin to form a radiative zone until the mass isnearly 400 M!. The larger radii and cooler internal tem-peratures, compared with the canonical 100 GeV case,result in higher interior opacities, favoring convection.

The timing of the transition from convective to radia-tive can be understood from the stellar structure. For m!greater than the canonical case, the DM heating is lesse!ective since it is inversely proportional to m!. Thestar compensates by becoming more compact, therebyincreasing the DM density and the internal temperature.In the relevant temperature range, the opacity scales as"T"3.5, and " scales as T 3. The smaller radius actuallyresults in a decrease in luminosity as well, so the radiativegradient

d logT

d logP=

3$LP

16#GacT 4Mr(24)

is reduced, tending to suppress convection relativelyearly. Conversely if m! is less than 100GeV, the starmust be more pu!y, which increases the radiative gradi-ent. Thus the transition will occur at a later stage.

The cases studied have some di!erences in terms oflifetime, luminosity, and final mass. At the time theDS has grown to 100 M!, the 1 GeV case is an orderof magnitude brighter than the 100 GeV case. Similarly,the 100 GeV case is about an order of magnitude brighterthan the 10 TeV case. The DS lifetime grows shorter forincreasing m!. The lifetime for the 1 GeV case is 4.3%105

years while the lifetime for the 10 TeV case is only 2.6%105 years. The reduced amount of fuel available in the 10TeV case more than compensates for the lower luminosityin determining the lifetime. Also, the 1 GeV case hasa higher final mass than the 10 TeV case (820 M! vs552 M!); clearly a longer lifetime allows more mass toaccrete.

3.3. DM Evolution-Adiabatic Contraction Only

Here we discuss the evolution of the DS once its DMfuel begins to run out. In this section, we do not considercapture; in the next section we will add the e!ects ofcapture as a mechanism to refuel the DM inside the star.

Initially, as baryons accrete onto the DS, adiabatic con-traction pulls DM into the star at a su"ciently high rateto support the star, but eventually, adiabatic contractionfails to pull in enough DM at high enough densities tosupport it.

Adiabatic contraction eventually fails for two main rea-sons. DM must be brought inside the star, first, to re-place the DM which has annihilated away and, second,to support an ever-increasing stellar luminosity as thestellar mass increases due to accretion. In fact, an ac-cretion rate that decreases as a function of time (Tan &McKee 2004) only compounds the problem, leading to arelatively shorter life span since even less DM is brought

into the star. With an insu"ent amount of DM at highenough densities, the star must contract to increase theDM heating rate to match the stellar luminosity. Thecontraction leads to higher surface temperatures suchthat feedback radiative e!ects eventually shut o! accre-tion altogether.

As the star contracts, a large fraction of DM previouslyinside of the star ends up outside of the star. The star’scontraction amounts to at least two orders of magnitudein radius, leaving a large fraction of the DM outside thestar. Thus DM annihilation accounts for only a fractionof the DM “lost” from the star. For instance in the 100GeV case, only a fifth of the DM is lost due to annihila-tion; the rest is “left outside”.

The “left outside” e!ect can be understood in a simpleway. DM heating is mainly a volume e!ect as can be seenin Figure 3. Due to adiabatic contraction, "DM does notscale perfectly with baryon density (eq. 8). For instance,the total amount of DM inside of star of fixed mass (evenwithout DM annihilation) decreases as the radius of thestar shrinks.

Without a way to replenish the DM such as with cap-ture, the DS will eventually run out of DM due to thecombination of the “left outside” e!ect and DM annihi-lation.

3.4. DS evolution with Capture included

The DM inside the DS can be replenished by cap-ture. DM particles bound to the larger halo pass throughthe star, and some of them can be captured by los-ing energy in WIMP/nucleon scattering interactions.We remind the reader that we have defined our caseof ’minimal capture’ to have background DM density"! = 1.42 % 1010(GeV/cm3) and scattering cross sec-tion !sc = 10"39cm2 chosen so that about half of theluminosity is from DM capture and the other half fromfusion for the standard 100 GeV case at the ZAMS.

With capture, the stellar evolution is the same as inthe cases previously discussed without capture through-out most of the history of the DS. The contribution to theheating from capture Lcap only becomes important onceDM from adiabatic contraction runs out. Once this hap-pens, gravity still initially dominates, but for the ’mini-mal capture’ case we have considered, Lcap is more im-portant than Lnuc until the star eventually reaches a new“dark” main sequence, where the two contributions be-come equal. During this phase the star is more extendedand cooler than without capture, which allows for morebaryons to accrete onto the star once the first dark starphase driven by adiabatic contraction ends. However forthe standard minimal case the final mass with capture isat most 1% larger than that without.

In a future paper we will consider the case of muchhigher capture rates than the minimal ones we havestudied here. The capture rate depends on the ambi-ent DM density in which the DS is immersed and on theWIMP/nucleus scattering cross section. High ambientdensity could easily arise; in fact the adiabatic contrac-tion arguments applied to the DS from the beginningindicate that this is quite likely. On the other hand, thescattering cross section may be significantly lower thanthe current experimental bound adopted in our minimalcapture case. In any case, with a su"ciently high capturerate, say two or more orders of magnitude higher than

13

the minimal value considered here, the star can stay DMpowered and su"ciently cool such that baryons can inprinciple continue to accrete onto the star indefinitely, orat least until the star is disrupted. This latter case willbe explored in a separate paper where it will be shownthat the dark star could easily end up with a mass on theorder of several tens of thousands of solar masses and alifetime of least tens of millions of years.

4. DISCUSSION

We have followed the growth of equilibrium protostel-lar Dark Stars (DS), powered by Dark Matter (DM) an-nihilation, up to the point when the DS descends ontothe main sequence. Nuclear burning, gravitational con-traction, and DM capture are also considered as energysources. During the phase when dark-matter heatingdominates, the objects have sizes of a few AU and cen-tral T ' 105 ! 106 K. Su"cient DM is brought into thestar by contraction from the DM halo to result in a DSphase of at least several hundred thousand years (withoutDM capture). Because of the relatively low Te! (4000–10,000 K), feedback mechanisms for shutting o! accre-tion of baryons, such as the formation of HII regions orthe dissociation of infalling H2 by Lyman-Werner pho-tons, are not e!ective until DM begins to run out. Theimplication is that main-sequence stars of Pop. III arevery massive. Regardless of uncertain parameters suchas the DM particle mass, the accretion rate, and scatter-ing, DS are cool, massive, pu!y and extended. The finalmasses lie in the range 500–1000 M!, very weakly de-pendent on particle masses, which were assumed to varyover a factor of 104.

One may ask how long the dark stars live. If thereis no capture, they live until the DM they are able topull in via adiabatic contraction runs out; the numericalresults show lifetimes in the range 3 % 105 to 5 % 105 yr.If there is capture, they can continue to exist as long asthey reside in a medium with a high enough density ofdark matter to provide their entire energy by scattering,capture, and annihilation.

In addition, the properties of the DM halo a!ect thenature of the DS. For instance, a larger concentration pa-rameter allows for more DM to be pulled in by adiabaticcontraction. The additional DM prolongs the lifetimeof the DS phase and increases the DS final mass. Thecanonical case with a concentration parameter c = 3.5rather than x = 2 has a final mass of 870 M! vs. 779 M!

and has a lifetime of 442,000 yrs vs. 387,000 yrs. Fur-thermore, the concentration parameter of each DS halowill be di!erent, depending upon the history of each halo.Thus the Initial Mass Function (IMF) of the ZAMS aris-ing from the growth of dark stars will be sensitive to thehalo parameters. This e!ect will be studied thoroughlyin a separate publication.

DS shine with a few 106L! and one might hope todetect them, particularly those that live to the lowestredshift (z = 0 in the most optimistic case). One mayhope that the ones that form most recently are detectableby JWST or TMT and di!erentiable from the standardmetal-free Pop. III objects. DS are also predicted to haveatomic hydrogen lines originating in the warmer photo-

spheres, and H2 lines arising from the infalling material,which is still relatively cool.

The final fate of DS once the DM runs out is also dif-ferent from that of standard Pop III stars, again leadingto di!erent observable signatures. Standard Pop III starsare thought to be " 100 ! 200M!, whereas DS lead tofar more massive MS stars. Heger & Woosley (2002)showed that for 140M! < M < 260M!, pair instabilitysupernovae lead to odd-even e!ects in the nuclei pro-duced, an e!ect that is not observed. Thus if Pop III.1stars are really in this mass range one would have to con-strain their abundance. Instead, we find far more mas-sive Dark Stars which would eventually become heavymain sequence (ZAMS) stars of 500-1000 M!. Heger &Woosley find that these stars after relatively short life-times collapse to black holes. Or, if they rotate veryrapidly, Ohkubo et al (2006) argue that they could be-come supernovae, leaving behind perhaps half their massas black holes. In this case the presumed very bright su-pernova could possibly be observable. In either case, theelemental abundances arising from pair instability SN ofstandard Pop III (less massive) stars are avoided so thatone might avoid the constraint on their numbers. Otherconstraints on DS will arise from cosmological consider-ations. A first study of their e!ects (and those of theresultant MS stars) on reionization have been done bySchleicher et al. (2008a,2008b), and further work in thisdirection is warranted.

The resultant black holes from DS could be very impor-tant. DS would make plausible precursors of the 109M!

black holes observed at z = 6 (Li et al. 2007; Pelopessyet al. 2007); of Intermediate Mass Black Holes; of blackholes at the centers of galaxies; and of the black holesrecently inferred as an explanation of the extragalacticradio excess seen by the ARCADE experiment (Sei!ertet al. 2009). However, see Alvarez et al. (2008) whopresent caveats regarding the growth of early black holes.In addition, the black hole remnants from DS could playa role in high-redshift gamma ray bursts thought to takeplace due to accretion onto early black holes (we thankG. Kanbach for making us aware of this possibility).

In the presence of prolonged capture, the DS could endup even larger, possibly leading to supermassive starsthat consume all the baryons in the original halo, i.e.,possibly up to 105M! stars. The role of angular momen-tum during the accretion must be accounted for prop-erly and will a!ect the final result; in fact, even for the1000M! case studied in this paper, we intend to examinethe possible e!ects of angular momentum. This possibil-ity will be further investigated in a future publication (inpreparation), both from the point of view of the stellarstructure, as well as the consequences for the universe.

We acknowledge support from: the DOE and MCTPvia the Univ. of Michigan (K.F.); NSF grant AST-0507117 and GAANN (D.S.); NSF grant PHY-0456825(P.G.). K.F. and D.S. are extremely grateful to ChrisMcKee and Pierre Salati for their encouragement of thisline of research, and to A. Aguirre, L. Bildsten, R.Bouwens, N. Gnedin, G.Kanbach, A. Kravtsov, N. Mur-ray, J. Primack, C. Savage, J. Sellwood, J. Tan, and N.Yoshida for helpful discussions.

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