Dark stars: Implications and constraints from cosmic reionization and extragalactic background radiation

0.63 MB·English

by  Dominik R. G. Schleicher

Dark stars: Implications and constraints from cosmic reionization and extragalactic
background radiation
Dominik R. G. Schleicher, Robi Banerjee, Ralf S. Klessen∗
Institute of Theoretical Astrophysics / ZAH,
Albert-Ueberle-Str. 2,
D-69120 Heidelberg,
Germany
Darkstarspoweredbydarkmatterannihilationhavebeenproposedasthefirstluminoussources
in theuniverse. These stars arebelieved to form in thecentral darkmatter cusp of low-mass mini-
halos. Recentcalculations indicatestellarmassesupto∼1000M⊙ and/orhaveverylonglifetimes.
The UV photons from these objects could therefore contribute significantly to cosmic reionization.
9 Here we show that such dark star models would require a somewhat artificial reionization history,
0 based on a double-reionization phase and a late star-burst near redshift z ∼ 6, in order to fulfill
0 theWMAPconstraint on theoptical depthaswell as theGunn-Petersonconstraint at z∼6. This
2 suggests that, if dark stars were common in the early universe, then models are preferred which
n predict a number of UV photons similar to conventional Pop. III stars. This excludes 800 M⊙
darkstars thatentera main-sequencephaseand other models that lead to astrong increase in the
a
J numberof UVphotons.
We also derive constraints for massive as well as light dark matter candidates from the observed
7
X-ray,gamma-rayandneutrinobackground,consideringdarkmatterprofileswhichhavebeensteep-
2
ened during the formation of dark stars. This increases the clumping factor at high redshift and

] gives rise to a higher dark matter annihilation rate in theearly universe. Wefurthermore estimate
h thepotentialcontributionfrom theannihilation productsintheremnantsofdarkstars,which may
p providea promising path to constrain such models further, butwhich is currently still uncertain.
-
o
r
t I. INTRODUCTION FERMI satellite [122] will shed more light on such ques-
s
tions and may even distinguish between such scenarios
a
[ due to specific signatures in the anisotropic distribution
Growing astrophysical evidence suggests that dark
of this radiation [12].
2 matter in the universe is self-annihilating. X-ray obser-
v vationsfromthecenterofourGalaxyfindbright511keV Thefirststarshavebeensuggestedtohavehighmasses
9 emission which cannot be attributed to single sources of the order 100 M , thus providing powerful ionizing
1 [1, 2], but can be well-described assuming dark matter sources in th∼e early u⊙niverse [13, 14]. The effect of dark
5
1 annihilation [3]. Further observations indicate also an matter annihilation on the first stars has been explored
. excessofGeVphotons[4], ofmicrowavephotons[5], and recently in different studies. Spolyar et al. [15] showed
9
of positrons [6]. A commonfeature ofthese observations that an equilibrium between cooling and energy deposi-
0
8 is that the emission seems isotropic and not correlated tion from dark matter annihilation can always be found
0 to the Galactic disk. However,there is usually some dis- during the collapse of the proto-stellar cloud. This has
: crepancybetweenthe modelpredictionsandthe amount beenexploredfurtherbyIocco[16]andFreeseetal.[17],
v
ofobservedradiation,whichmaybedue touncertainties who considered the effect of scattering between baryons
i
X in the dark matter distribution, astrophysical processes and dark matter particles, increasing the dark matter
r and uncertainties in the model for dark matter annihila- abundance in the star. Iocco et al. [18] considered dark
a tion [7]. star masses in the range 5 M 600 M and cal-
∗ ⊙
≤ ≤
It is well-known that weakly-interacting massive dark culated the evolution of the pre-main-sequence phase,
matterparticlesmayprovideanaturalexplanationofthe finding that the dark star phase where the energy in-
observed dark matter abundance [8, 9]. Calculations by put from dark matter annihilation dominates may last
Ahn et al. [10] indicated that the extragalactic gamma- up to 104 yr. Freese et al. [19] examined the formation
ray background cannot be explained from astrophysical process ofthe star in more detail, considering polytropic
sources alone, but that also a contribution from dark equilibria and additional mass accretion until the total
matter annihilation is needed at energies between 1-20 Jeans mass of 800 M⊙ is reached. They find that this
GeV. It is currently unclear whether this is in fact the process lasts fo∼r 5 105 yr. They suggest that dark
∼ ×
case or if a sufficient amountof non-thermalelectrons in stars are even more massive than what is typically as-
active galactic nuclei (AGN) is available to explain this sumed for the first stars, and may be the progenitors for
background radiation [11]. Future observations with the the first supermassiveblack holes at high redshift. Iocco
et al. [18], Taoso et al. [20] and Yoon et al. [21] have
calculated the stellar evolution for the case in which the
dark matter density inside the star is enhanced by the
∗Electronicaddress: [email protected] capture of addition WIMPs via off-scattering from stel-
2
larbaryons. Ioccoetal.[18]followedthestellarevolution A. Main-sequence dominated models
untiltheendofHeburning,Yoonetal.[21]untiltheend
ofoxygenburningandTaosoetal.[20]untiltheendofH After an initial phase of equilibrium between cooling
burning. Yoonet al. [21]also took the effects of rotation and heating from dark matter annihilation [15, 16, 17],
into account. The calculations found a potentially very the dark star will contract further while the dark mat-
long lifetime of dark stars and correspondingly a strong terannihilatesawayandthe heatingratethus decreases.
increase in the number of UV photons that may con- Thisdurationofthisadiabaticcontraction(AC)phaseis
tribute to reionization. Dark starsinthe Galactic center currently controversial: While Iocco et al. [18] find it to
have been discussed by Scott et al. [22, 23]. be in the range of (2 20) 103 yr, Freese et al. [19] re-
quireabout106 yr. H−oweve×r,withasurfacetemperature
Such models for the stellar population in the early
of 6000 K, the stars are rather cold in this phase, and
universe imply that the first luminous sources produce
∼
thuswillnotcontributesignificantlytoreionization. The
muchmoreionizing photons,andreionizationstarts ear-
uncertainty in the duration of the AC phase is therefore
lier than for a population of conventional Pop. III stars.
not crucial in this context.
Infact,werecentlydemonstratedthatreionizationbased
Iftheelasticscatteringcrosssectionaswellasthedark
on massive Pop. III can well reproduce the observed
reionizationopticaldepth [24]. Increasingthe number of matter density around the star are sufficiently large, the
starwillenteraphasewhichisdominatedbythecapture
ionizingphotonsperstellarbaryonmaythusreionizethe
universe too early and produce a too large reionization of further dark matter particles. Such a scenario will be
discussedinmoredetailinthe nextsubsection. Here,we
optical depth. This can only be avoided by introducing
a transition to a stellar population which produces less assume that the elastic scattering cross section is either
ionizing photons, such that the universe can recombine toosmall,orthatthedarkmatterreservoirnearthestar
afterthefirstreionizationphase. Wethereforeconsidera is not sufficient to maintain the capture phase for long.
double-reionizationscenario in order to re-obtainthe re- Then, the star will enter the main-sequence phase (MS),
quiredopticaldepth. Wediscusssuchmodelsin III and in which the luminosity is generatedby nuclear burning.
demonstratethat some models ofdark starsrequ§irecon- Stars with 1000 M⊙ are very bright in this phase,
siderable fine-tuning in reionization models in order to andemit ∼4 104 hydrogen-ionizingphotonsperstellar
∼ ×
be compatible with the reionization optical depth from baryon during their lifetime [28, 29]. We will refer to
the WMAP [123] 5-year data [25, 26] and to complete stars of such type, which have only a short or even no
reionization at redshift z 6 [27]. In IV, we show how phase driven by dark matter capture, as MS-dominated
∼ § models.
such scenarios can be tested via 21 cm measurements.
For the case of MS-dominated models, we will focus
Afurtherconsequenceoftheformationofdarkstarsis essentially onthe very massive stars suggestedby Freese
thesteepeningofthedensityprofilesinminihalos[16,19], etal.[19]. ForstarsinthetypicalPop.III massrange,it
thus increasingthe darkmatter clumping factor withre- has been shownelsewhere [e. g. 24] that they are consis-
spect to standard NFW models. In V, we estimate the tentwithreionizationconstraints. Astarwith 800M
increase in the clumping factor duri§ng the formation of forming in a dark matter halo of 106 M co∼rrespond⊙s
⊙
darkstarsandcomparethecalculationwithourexpecta- to a star formation efficiency of 1∼%, which we adopt for
tion for conventionalNFW profiles and heavy dark mat- this case.
ter candidates. In VI, we perform similar calculations
§
for the light dark matter scenario. Further discussion
and outlook is provided in VII.
B. Capture-dominated models
§
For a non-zerospin-dependent scattering cross section
between baryons and dark matter particles, stars can
capture additional WIMPs which may increase the dark
II. THE MODELS matter density inside the star. For a cross section of the
order 5 10−39 cm2 and an environmental dark mat-
ter dens×ity of 1010 GeV cm−3, this contribution be-
∼
As discussed in the introduction, various models have comes significant and alters the stellar evolution during
been suggested for dark stars. The main difference be- themainsequencephase. Wewillrefertosuchascenario
tweenthesemodelscomesfromconsideringorneglecting as a capture-dominated (CD) model. These phases have
scattering between dark matter particles and baryons. been studied in detail by Iocco et al. [18], Taoso et al.
In addition, it is not fully clear how important a phase [20]andYoonetal.[21]. Theyfoundthatthe numberof
of dark matter capture via off-scattering from baryons ionizing photons produced by such stars may be consid-
actually is, depending on further assumptions on the erablyincreasedwith respectto high-massstars without
dark matter reservoir. In the following, we will thus dis- darkmatterannihilationeffects,whichismostlyduetoa
tinguish between main-sequence dominated models and longerlifetime. Inparticularfor darkmatterdensities of
capture-dominated models. (1 5) 1010 GeVcm−3 ,thenumberofproducedioniz-
− ×
3
ing photons may be increased by up to two orders mag- where C(z) = 27.466 exp( 0.114z+0.001328z2) is the
−
nitude, while it decreases rapidly for larger dark matter clumpingfactor[42],n thenumberdensityofionized
e,H+
densities,andthenumberofionizingphotonsperbaryon hydrogen, α the case A recombination coefficient [43],
A
even drops below the value for Pop. II stars at.threshold H(z) the Hubble function, n the mean neutral hydro-
H
densitiesof1 1012GeVcm−3. AsYoonetal.[21]found gen density in regions unaffected by UV feedback and
×
only a weak dependence on stellar rotation, we will not dn /dz the UV photon production rate. Our model
ph
explicitly distinguish between models with and without consistsofordinarydifferentialequations(ODEs)forthe
rotation in the follow. evolutionoftemperatureT andionizedfractionx inthe
i
For the calculation of reionization, we will focus on overall neutral medium. For the application considered
some representativemodels of Yoon et al. [21] in the fol- here, the dominant contribution to the effective ionized
lowing. However,wepointoutthattherearestillsignifi- fractionxeff =QH++(1 QH+)xi andtheeffectivetem-
cantuncertaintiesinthesemodels,inparticularthedark perature Teff =104 K Q−H++T(1 QH+) comes indeed
−
matterparametersandthelifetimesofthestars. Thelat- fromtheUVfeedbackofthestellarpopulation,i. e. from
ter should be seen as upper limits, as they assume that thehotionizedphase. AccordingtoGnedinandHui[44]
a sufficient reservoir of dark matter is available in the and Gnedin [45], we introduce the filtering mass scale as
stellar neighborhood to allow for ongoing dark matter
capture. This may however be disrupted by dynamical M2/3 = 3 ada′M2/3(a′) 1 a′ 1/2 , (2)
processes. Anapparentdisagreementofdarkstarsinthe F aZ0 J " −(cid:18)a(cid:19) #
early universe with our reionization model may thus in-
dicate thatthe stellar lifetimes areindeedsmallerdue to where a=(1+z)−1 is the scale factor and M the ther-
J
such processes. mal Jeans mass, given as
3
c n −1/2
s
M =2M . (3)
III. REIONIZATION CONSTRAINTS J ⊙ 0.2 km/s 103 cm−3
(cid:18) (cid:19) (cid:16) (cid:17)
Here, c is the sound speed evaluated at temperature
s
Inthissection,webrieflyreviewourreionizationmodel
T , in order to take into account the backreaction of
eff
and discuss reionization histories for main-sequence and
heating on structure formation. In this framework, the
capture-dominatedmodels. Thesecalculationsimplicitly
production of UV photons can be described as
assumeannihilationcrosssectionsoftheorder10−26cm2
and dark matter particle masses of the order 100 GeV, dn /dz df
ph coll
ξ , (4)
the values whicharetypicallyadoptedindarkstar mod- n ∼ dz
H
els. In such models, dark matter annihilation does not
contribute to cosmic reionization [24]. The chemistry whereξ =AHef∗fescNion,withAHe =4/(4 3Yp)=1.22,
−
in the pre-ionization era is thus unchanged and well- Nion the number of ionizing photons per stellar baryon,
described by previous works [30, 31, 32, 33], such that f∗isthestarformationefficiencyandfesctheescapefrac-
the initial conditions for star formation are unchanged. tion of UV photons from their host galaxies. The quan-
Consideringhigher annihilationcrosssectionsessentially tity fcoll denotes the fraction of dark matter collapsed
yields an additional contribution to the reionization op- into halos, and is given as
tical depth, which would sharpen the constraints given
δ (z)
below. f =erfc c , (5)
coll
√2σ(M )
(cid:20) min (cid:21)
where M =min(M ,105 M ), δ =1.69/D(z) is the
min F ⊙ c
A. General approach
linearized density threshold for collapse in the spherical
top-hat model and σ(M ) describes the power associ-
min
Our calculation of reionization is based on the frame- ated with the mass scale M .
min
work developed by Schleicher et al. [24], which we have A relevant question in this context is also the role of
implemented in the RECFAST code [124] [34, 35]. We Lyman-Werner (LW) feedback, which may suppress the
will review here only those ingredients which are most star formation rate in low-mass halos. The role of such
relevantforthiswork. Duringreionization,theIGMcon- feedback has been addressed using different approaches.
sistsofatwo-phasemedium,i.e. ahotionizedphaseand For instance, Machacek et al. [46], O’Shea and Norman
arathercoldandoverallyneutralphase. Therelativesize [47] and Wise and Abel [48] have addressed this ques-
ofthesephasesisdeterminedfromthevolume-fillingfac- tion employing numerical simulations in a cosmological
tor QH+ of the H+ regions [36, 37, 38, 39, 40, 41] as a context, assuming a constant LW-background radiation
function of redshift, given by field. Thesesimulationsindicatedthatsuchfeedbackcan
delayed star formation considerably.
dQH+ QH+C(z)ne,H+αA dnph/dz More self-consistent simulations show, however, that
= + , (1)
dz H(z)(1+z) n the above calculations overestimated the role of LW-
H
4
with observations. Alternatively, as explained in the in-
troduction, a transition in the stellar population might
help to alleviate the problem for high-mass dark stars.
We will explore this possibility in more detail to work
out whether such a scenario is conceiveable.
NumericalsimulationsbyDoveetal.[53],Ciardietal.
[54] and Fujita et al. [55] indicated rather high es-
cape fractions of order 100% for massive Pop. III stars.
Wood and Loeb [56] found rather low escape fractions
below 10%, while radiation hydrodynamics simulations
by Whalen et al. [57] show that such stars can eas-
ily photo-evaporate the minihalo. Here we adopt the
point of view that indeed massive stars can photoe-
vaporate small minihalos, but that the escape fraction
will be reduced to 10% in atomic cooling halos that
havevirialtemperat∼ures largerthan 104 K.Thus,we set
FIG.1: Theevolutionoftheeffectiveionizedfractionxeff,for
reionizationmodelswithmain-sequencedominateddarkstars fesc = 1 if the filtering mass is below the mass scale
(see Table I). Models MS 1 and MS 2 can be ruled out by M =5 107M 10 3/2 thatcorrespondstothevirial
reionizationconstraints,whilemodelsMS3andMS4require c × ⊙ 1+z
a sudden increase in the star formation rate by a factor of temperatureof10(cid:16)4 K[(cid:17)58,59],andfesc =0.1intheother
30 at redshift 6.5. It appears more realistic to assume lower case. To reflect the expected stellar mass of 800 M ,
⊙
∼
masses and star formation efficiencies to reconcile dark star we choose a star formation efficiency of f 1%, an
∗
models with observations. order of magnitude higher than what we expec∼t for con-
ventional Pop. III stars [24].
Assuming that reionization is completely due to these
feedback. Considering single stellar sources and neglect-
MS-dominateddarkstars(modelMS1),wefindthatthe
ing self-shielding, Wise and Abel [49] showed that LW-
universe is fully ionized at redshift z =15.5 and the
reion
feedback only marginally delays star formation in halos
reionization optical depth is τ 0.22, i. e. signifi-
reion
that already started collapsing before the nearby star ∼
cantly larger than the WMAP 5 optical depth (see Fig.
ignites. More detailed simulations taking into account
1). Such a model is clearly ruled out.
self-shielding show that the star formation rate may be
To reconcile the presence of such massive dark stars
changed by only 20% in the presence of such feedback
withobservations,onecouldinvokeadouble-reionization
[50]. This is due to the rapid re-formation of molecular
scenario,assumingatransitiontoadifferentmodeofstar
hydrogeninrelic HII regions,which leadsto abundances
of the order 10−4. Such abundances effectively shield formation induced by the strong UV feedback of MS-
dominateddarkstars. Infact, evenfor conventionalstar
against LW-feedback and make it ineffective [51]. This
formation models, it is discussed that such UV feedback
is the point of view adopted here, which may translate
mayleadtoalessmassivemodeofstarformation[60,61,
into an uncertainty of 20% in the star formation rate.
∼ 62]. In addition, chemical enrichment should facilitate
In fact, in scenarios involving dark matter annihilation,
suchatransitionaswell[41,63, 64,65,66, 67],although
H formationandself-shieldingcouldbeevenfurtheren-
2
it is unclear how well metals will mix with the pristine
hanced compared to the standard case [52].
gas. We assume that the transition to a low-mass star
The models have to reproduce the reionizationoptical
formation mode with a Scalo-type IMF [68] happens at
depthgivenbyτ =0.087 0.017[26]andfullyionization
± redshift 15.5, when the universe is fully ionized and UV
at z 6 [27]. In the following, we will try to construct
∼ feedbackfullyeffective. ForthesubsequentPop.IIstars,
appropriate reionization histories for the different dark
we assume a star formation efficiency of f = 5 10−3
star models. and N =4 103 UV photons per stellar∗baryo×n.
ion
×
Corresponding photon escape fractions are highly
uncertain. Observations of Steidel et al. [69] indicate
B. Reionization with MS-dominated dark stars
an escape fraction of 10% at z 3, while others
∼
find detections or upper limits in the range 5 10%
−
As shown previously [24], MS-dominated dark stars [70, 71, 72, 73]. We adopt the generic value of 10% for
with 1000 M would significantly overproduce the simplicity, though our results do not strongly depend on
⊙
∼
reionization optical depth if this type of stars had been this assumption. For this scenario, to which we refer as
common throughout the early universe. If, on the other model MS 2, we find an optical depth τ =0.082well
reion
hand, MS-dominated stars only had mass scales of within the WMAP constraint,but the universe does not
∼
100M ,comparabletoconventionalPop.IIIstars,reion- getfully ionizeduntil redshiftzero. This scenariois thus
⊙
izationcouldnotdiscriminatebetweenthemandconven- rejected based on the constraint from quasar absorption
tionalPop.IIIstars,anddarkstarswouldbecompatible spectra [27].
5
To fulfill both the WMAP constraint as well as
full-ionization at z 6, we need to introduce an addi-
∼
tional transition in our model. At redshift z = 6.5,
burst
we increase the star formation efficiency to 15%. This
might be considered as a sudden star burst and results
in full-ionization at z = 6.2. In this case, we find
τ = 0.116, which is within the 2σ range of the
reion
WMAP data. However, we are not aware of astrophysi-
cal models that provide a motivation for such a sudden
star burst that increases the star formation rate by a
factor of 30. Based on gamma-ray burst studies, Yu¨ksel
et al. [74] showed that the cosmic star formation rate
does not change abruptly in the redshift range between
redshift zero and z = 6.5. Such a sudden burst is
burst
thus at the edge of violating observation constraints.
FIG.2: Theevolutionoftheeffectiveionizedfractionxeff,for
To improve the agreement with WMAP, one can con-
reionization models with capture-dominated dark stars (see
sider to shift the first transition to z = 18 where
PopII TableII). ModelsCD1a,CD1b,CD2a,CD2bandCD3are
full ionization is not yet reached (model MC 4), which ruledoutduetoreionizationconstraints,whiletheremaining
yieldsanopticaldepthτreion =0.086,ingoodagreement models require an artificial star burst.
with WMAP. At this redshift, 68% of the universe are
already ionized, so UV feedback might already be active
and induce a transition in the stellar population. The baryon, N , is model-dependent and changes with the
ion
results are given in Fig. 1 and summarized in Table I. environmental dark matter density, ρ . We select three
X
representativemodelsofYoonetal.[21],whichassumea
Model zPopII zburst τreion zf spin-dependent scattering crosssection of 5 10−39 cm2
×
MS 1 - - 0.22 15.5 (see Table II). In general, stellar models depend on the
MS 2 15.8 - 0.078 never productofthisscatteringcrosssectionwiththethreshold
MS 3 15.5 6.5 0.116 6.2 dark matter density at the stellar radius [20, 21]. Lower
elastic scattering cross sections therefore correspond to
MS 4 18. 6.5 0.086 6.2
going to smaller threshold densities at the same elastic
scattering cross section.
TABLEI:Reionization modelsforMS-dominateddarkstars. In the models CD 1 and 2,N is largerthan for con-
ion
TheparameterszPopIIandzburstgivethetransitionredshifts ventional Pop. III stars, while in the model CD 3, it is
to a mode of Pop. II star formation and to the sudden star even less than in the case of Scalo-type Pop. II stars.
burst,whileτreion isthecalculatedreionizationopticaldepth Suchalowluminosityisunlikelytophoto-evaporatestar-
and z the redshift of full ionization.
f forming halos, and we thus adopt f = 10% for this
esc
case. However,suchScalo-typePop.IIstarsareruledout
However, we find that only models MS 3 and MS 4 as sole sources for reionization [24]. As we show in Fig.
cannot be ruled out observationally. These models re- 2, even with a high star formation efficiency of f =1%,
∗
quiretwoseveretransitionsinthe stellarpopulationand they never ionize the universe completely.
cannot be considered as ”natural”. Improved measure- In principle, one could consider the presence of other
mentsofthereionizationopticaldepthfromPlanck[125] sources to ionize the universe. While dark stars of type
willremovefurtheruncertaintiesandmayruleoutmodel CD3maybethefirststarstoform,onemightenvisiona
MS 3 as well. From a theoretical point of view, it must transition to a stellar population with the power to ion-
be checked whether strong UV feedback can lead to the ize the universe. This transition is unlikely due to UV
requiredtransitionto alow-massstarpopulation,andin feedback, as UV feedback from dark stars is rather weak
addition,theplausibilityofasuddenstarburstnearred- inthis scenario. One thus hasto rely oneffective mixing
shift 6 must be examined as well. In summary, it seems ofthe producedmetallicity, orassume that the firststel-
moreplausibletoconcludethatMS-dominateddarkstars lar clusters in atomic cooling halos contain a sufficient
were less massive than suggested by Freese et al. [17], as number of massive stars to reionize the universe [63].
already hinted by Schleicher et al. [24].
For the other two models, N is significantly larger
ion
andweadopttheprocedurefromtheprevioussubsection,
suchthatf depends onthe filteringmass. We adopta
esc
C. Reionization with CD dark stars star formation efficiency of f = 0.1%. We examine the
∗
reionization models given in Table II, which essentially
For CD dark star models, the situation is complicated follow the philosophy of the models from the previous
by the fact that the number of UV photons per stellar section. Wecalculatethereionizationhistoryforthecase
6
where these dark stars are sole sources (CD 1a, CD2a)
and find that the optical depth is considerably too high.
Wethendeterminetheredshiftwheretheuniverseisfully
ionized and assume a transition to Pop. II stars at this
redshift. In addition, to obtainfull ionizationat redshift
6, we assume a late star burst as in the models MS 3
andMS4. This approachcorrespondsto the models CD
1b and CD 2b, and yields optical depth that are at least
within the 2σ error of WMAP 5. In the models CD 1c
and CD 2c, we improve the agreement with WMAP by
introducing the Pop. II transition at an earlier redshift.
The results are given in Fig. 2. Again, it turns out
that somewhat artificialmodels are requiredto allow for
an initial population of CD dark stars. The best way to
reconcile these models with the constraints from reion-
ization might be to focus on those models that predict
a parameter N which is closer to the Pop. III value
ion
of 4 104. This may be possible, as the transition from
×
the models CD 1 and 2 to CD 3 is likely continuous,
and an appropriate range of parameters may exist to re-
concile models with observations. This would require a
ρ between 1011 GeV cm−3 and 1012 GeV cm−3. As
X
mentioned earlier, the apparent violation of reionization
constraints by some models depends also on the uncer-
taintiesinthestellarlifetime. Ifthedarkmatterreservoir
near the star is destroyed earlier due to dynamical pro-
cesses,thelifetimemaybesignificantlyreduced. Also,we
stressthattheconclusionsdependontheadoptedelastic
scattering cross section and the dark matter density in
the environment. The discussion here is limited to those
models that have previously been worked out in detail.
IV. PREDICTIONS FOR 21 CM
OBSERVATIONS
While some of the models suggested above essentially
co-incide with standard reionization by mimicing the ef-
fects of conventional Pop. III stars, others may have a
FIG. 3: 21 cm signatures of double-reionization scenarios
very distinctive signature, as they consist of a double-
(here MS 4 from Table I). Given is the evolution after the
reionization phase, and upcoming 21 cm telescopes like first reionization phase, when the H gas is heated from the
LOFAR [126] or SKA [127] can thus verify or rule out previous ionization. Top: HI gas temperature, here identi-
such suggestions. The calculation shown in Fig. 3 is cal to the spin temperature. Middle: Expected mean 21 cm
basedonthedoublereionizationmodelMS4,butclearly brightness fluctuation. Bottom: Frequency gradient of the
themodelsMS3,CD1b,CD1c,CD2bandCD2cyield mean 21 cm brightness fluctuation.
similar results. In such a double-reionization scenario,
the gasis heatedto 104 Kduring the firstreionization
∼ the Lyman series and may couple the spin temperature
epoch. Assuming that the first reionization epoch ends
T of atomic hydrogen to the gas temperature T via
atredshiftz ,thegastemperatureinthenon-ionized spin
PopII
the Wouthuysen-Field effect [75, 76]. In fact, a small
medium will then evolve adiabatically as
amount of Lyman α radiation suffices to set T = T
spin
1+z 2 [77, 78, 79], which we assume here. Also, as the uni-
T 104 K . (6) verseis optically thin to this radiationbackground,even
∼ 1+z
(cid:18) PopII(cid:19) Pop. II sources will suffice to couple the spin temperure
In addition, the previous reionizationphase will have es- tothegastemperature. Themean21cmbrightnesstem-
tablished a radiation continuum between the Lyman α perature fluctuation is then given as
line and the Lyman limit, where the universe is opti-
callythin, apartfromsingleresonancescorrespondingto Ωbh2 0.15 1+z 1/2
δT = 27x (1+δ)
the Lyman series. This radiation is now redshifted into b H 0.023 Ω h2 10
(cid:18) (cid:19)(cid:18) m (cid:19)
7
Reion. model ρX/1012 Nion f∗ zPopII τreion
CD 1a 0.01 GeV cm−3 1.75×105 0.1% - 0.162
CD 1b 0.01 GeV cm−3 1.75×105 0.1% 12.7 0.109
CD 1c 0.01 GeV cm−3 1.75×105 0.1% 14.5 0.089
CD 2a 0.05 GeV cm−3 2.4×106 0.1% - 0.283
CD 2b 0.05 GeV cm−3 2.4×106 0.1% 21.6 0.106
CD 2c 0.05 GeV cm−3 2.4×106 0.1% 23 0.084
CD 3 1 GeV cm−3 1.1×103 1% - 0.004
TABLE II: Reionization models for CD dark stars stars. The number of ionizing photons was determined from the work of
Yoonetal. [21]. TheparameterszPopII andzburst givethetransition redshiftstoamodeofPop.IIstarformation andtothe
suddenstarburst,whileτreion isthecalculatedreionizationopticaldepthandzf theredshiftoffullionization. Thecalculation
assumes a spin-dependent scattering cross section of 5×10−39 cm2. As stellar models depend on the product of this cross
sectionwiththethresholddarkmatterdensity,theeffectofalowerscatteringcrosssectionisequivalenttoasmallerthreshold
density.
T T H(z)/(1+z)
S r
− mK, (7)
× T dv /dr
(cid:18) S (cid:19)(cid:18) || || (cid:19)
where x denotes the neutral hydrogen fraction, δ the
H
fractional overdensity, Ω , Ω the cosmological density
b m
parameters for baryons and total matter, h is related to
the Hubble constantH via h=H /(100km/s/Mpc), T
0 0 r
the radiation temperature and dv /dr the gradient of
|| ||
the proper velocity along the line of sight, including the
Hubble expansion. We further calculate the frequency
gradientof the mean 21 cm brightness temperature fluc-
tuation to show its characteristic frequency dependence.
In Fig. 3, we show the evolutionof the gas temperature,
the mean 21 cm brightness fluctuation and its frequency
gradient for model MS 4.
As pointed out above, we expect similar results for FIG. 4: The predicted gamma-ray background due to di-
other double-reionization models because of the char- rect annihilation into gamma-rays in the presence of adia-
acteristic adiabatic evolution of the gas and spin baticcontraction duringtheformation ofdark stars, andthe
background measured by EGRET (squares) [92]. One finds
temperature. The decrease of the spin temperature
two peaks in the annihilation background for a given parti-
with increasing redshift is a unique feature that is not
cle mass: Onecorresponding to annihilation at redshift zero,
present in other models that like dark matter decay
andonecorrespondingtotheredshiftwheretheenhancement
[80] or ambipolar diffusion heating from primordial
from adiabatic contraction was strongest.
magnetic fields [81, 82, 83], which may also increase the
temperature during and before reionization.
on such scenarios are available from the Galactic cen-
ter and the extragalactic gamma-rayand neutrino back-
V. COSMIC CONSTRAINTS ON MASSIVE
DARK MATTER CANDIDATES grouns [88, 89, 90, 91]. In this section, we consider how
such constraints are affected when the increase in the
annihilation rate due to enhanced dark matter densities
Intypicaldarkstarmodels,itisassumedthatmassive
after the formation of dark stars is taken into account.
dark matter candidates like neutralinos with masses of
the order100GeVannihilateintogamma-rays,electron-
positron pairs and neutrinos [15, 16, 17, 18, 19, 21, 84].
Similar to the constraint on high-redshift quasars from
A. Gamma-ray constraints
the X-ray background [85, 86, 87], the gamma-ray and
neutrino backgrounds allow to constrain the model for
and the amount of dark matter annihilation. As de- We adopt the formalism of Mack et al. [91] who re-
tailedpredictionsforthedecayspectraarehighlymodel- cently addressed the direct annihilation of massive dark
dependent,itistypicallyassumedthatroughly1/3ofthe matter particles into gamma-rays. The background in-
energy goes into each annihilation channel. Constraints tensity I is given from an integration along the line of
ν
8
sight as
c dzP ([1+z]ν,z)
ν
I = , (8)
ν 4π H(z)(1+z)4
Z
where P (ν,z) is the (proper) volume emissivity of
ν
gamma-ray photons, which is given as
m
P =α δ((1+z)ν m ) DM keV σv n2 C ,
ν b − DM keV h i DM γ
(9)
where σv = 3 10−26 cm3 s−1 denotes the thermally-
h i ×
averaged annihilation cross section, α = 1/3 is the
b
adopted branching-ratio to gamma-rays and m the
DM
mass of the dark matter particle in keV. C refers to
γ
the dark matter clumping factor. This clumping factor
depends on the adopted dark matter profile and the as-
FIG.5: Thepredictedneutrinobackgroundduetodirectan-
sumptions regarding substructure in a halo [93, 94, 95, nihilationintoneutrinosinthepresenceofadiabaticcontrac-
96]. Here we use the clumping factor for a NFW dark tion during theformation of dark stars, and theatmospheric
matter profile [97] which has been derived by Ahn and neutrinobackground[99]. Onefindstwopeaksintheannihi-
Komatsu [93, 98]. For z < 20, it is given in the absence lationbackgroundforagivenparticlemass: Onecorrespond-
of adiabatic contraction as a power-law of the form ingtoannihilation atredshiftzero, andonecorrespondingto
the redshift where the enhancement from adiabatic contrac-
C =C (0)(1+z)−β, (10) tion was strongest.
DM DM
where C (0) is the clumping factor at redshift zero
DM
and β determines the slope. For a NFW profile [97], between MF and Mc. Once Mc becomes larger than
CDM(0) 105 and β 1.8. The enhancement due to MF, dark star formation must end naturally. In fact, it
∼ ∼
adiabatic contraction is taken into account by defining may even end before, as discussed in III. To obtain the
§
highest possible effect, we assume that dark stars form
Cγ =CDMfenh, (11) as long as possible. We thus have
whahleorecluthmepfiancgtofarcfteonrhddueesctoribaedsiatbhaetiecnchoanntcreamcteinont o(Af tCh)e. fenh = 1+103fcoll(MfF)(−Mfco)ll(Mc) . (13)
WehaveestimatedthiseffectbasedontheresultsofIocco (cid:18) coll F (cid:19)
et al. [18], comparing a standard NFW profile with the In Fig. 4, we compare the results with EGRET observa-
enhanced profile that was created during dark star for- tions of the gamma-ray background [92]. In the absence
mation. Weonlycomparethemdowntotheradiusofthe ofadiabaticcontraction,thepredictedbackgroundpeaks
dark star and find an enhancement of the order 103. at the contribution from redshift zero [91]. We find that
∼
For the NFW case, the clumping factor would be essen- the enhancement of annihilation due to adiabatic con-
tially unchanged when including smaller radii as well, traction produces a second peak in the predicted back-
while the AC profile is significantly steeper and the con- ground which originates from higher redshifts. In this
tribution from inside would dominate the contribution scenario, particle masses smaller than 30 GeV can thus
to the halo clumping factor. However, as the annihila- be ruled out.
tionproducts are trappedinside the star,it is naturalto
introduce an inner cut-off at the stellar radius. In addi-
tion, we have to consider the range of halo masses and B. Neutrino constraints
redshifts in which dark stars may form. We assume that
the halo mass must be larger than the filtering mass to
The contribution to the cosmic neutrino flux can be
form dark stars. However, there is also an upper mass
obtained in analogy to Eq.(8). As recent works[89, 90],
limit. Halos with masses above
we adopt an annihilation spectrum of the form
3/2
Mc =5×107M⊙ 11+0z (12) Pν =αbδ((1+z)ν−mDM)mkeDVM keVhσvin2DMCneutrino,
(cid:18) (cid:19)
(14)
correspond to virial temperatures of 104 K [58] and are which is analogous to the spectrum for annihilation into
highly turbulent [59]. It seems thus unlikely that stars gamma-rays. The branching ratio to neutrinos is as-
willformontheverycuspofthedarkmatterdistribution sumed to be 1/3 as well, and the annihilation comes
in such halos, and more complex structures may arise. from the same dark matter distribution, thus yielding
We thus assume that dark stars form in the mass range C = C . The atmospheric neutrino background
neutrino γ
9
matter left over in the final remnant can contribute). In
this case, we have a proper volume emissivity
m
DM
P = δ((1+z)ν m ) keVα f f
ν DM 511 r a
− keV
df (M )
coll F
n f , (15)
DM core
× dt
where n is the mean proper number density of dark
DM
matter particles, m the particle mass in keV and
DM
df /dt can be evaluated from Eq. (5). The model-
coll
dependentfactorf determineswhichfractionofthedark
r
matter in the star will be left in the remnant. We adopt
f = 1 to obtain an upper limit. The factor f deter-
r a
mines the fraction of the remaining dark matter which
actually annihilates, which we set to f = 1 as well. As
a
FIG. 6: The maximum gamma-ray background due to direct in VIA, α = 1/4 is the fraction of electron-positron
511
§
annihilation into gamma-rays in the remnants of dark stars, annihilations per one dark matter annihilation process,
and the background measured by EGRET [92]. The actual correspondingtoannihilationviapositroniumformation.
contribution to the gamma-ray background is highly model-
In Fig. 6, we compare the results with EGRET obser-
dependent(see discussion in thetext).
vations [92]. We find that the maximum contribution is
clearly above the observed background.
Whether this maximum contribution can be reached,
hasbeencalculatedfromdifferentexperimentswithgen-
is howeveruncertainandthe previousworkinthe litera-
erallygoodagreement[99,100,101, 102,103]. Iocco[16]
tureonlyallowsonetomakerathercrudeestimates. For
adoptedasimilaratmosphericneutrinoflux forcompari-
instanceIoccoetal.[18]calculatethedensityprofilefora
sonwith the expected neutrino flux from dark stars. We
fiducial100M protostar,findingthatthedensitywithin
⊙
adopt here the data provided by Honda et al. [99] and the star roughly scales with r−2 outside a plateau at a
comparethemtothepredictedbackgroundinFig.5. The radius r 1011 cm. At the stellar radius of 1014 cm,
predicted background is always well below the observed the dark∼matter density is still 1012 GeV c∼m−3. The
background. ∼
timescale to remove this dark matter enhancement by
annihilation is 100 Myr for 100 GeV neutralinos. We
∼
needtoestimatewhichfractionofthedarkmatterinside
C. Emission from dark star remnants the star will be left at the end of its life, where the gas
densityisexpelledbyasupernovaexplosionandthedark
In the previous subsections, we have included the en- matter annihilation from this region may contribute to
hancement of the halo clumping factor down to the stel- the cosmic gamma-ray background.
lar radius, as by definition the annihilation products on Yoon et al. [21] adopted a timescale of 100 Myr, the
smaller scales are trapped inside the star. At the end of typical merger timescale at these redshifts, as the maxi-
their lifetime, these stars may explode and the baryon mumlifetimefordarkstars. Dependingonthescattering
density in the center may be largely depleted. The dark cross section and the environmental density, the actual
matter density has certainly been significantly reduced dark star lifetime may be considerably shorter. Indeed,
duetoannihilationsduringthelifetimeofthestar,butit asweshowedin IIIC,itisdifficulttoreconcilelifetimes
§
may still be enhancedcomparedto the usual NFW case. of 100 Myr with appropriatereionizationscenarios. It
∼
A detailed calculation of this effect is strongly model- is therefore reasonable to assume shorter timescales. In
dependent. As we have seen above, the strongest con- such a case, a reasonable estimate is that 40% of the
∼
straintsareobtainedfor directannihilationinto gamma- dark matter inside the star would be left at the end of
rays, which is the case we pursue here in more detail. its life. This would still be enhanced compared to the
So far, we assumed that dark stars form in halos be- standard NFW profile. In this case, the parameter f
r
tweenthe filteringmassM andthe masscorresponding is 40%, and f may be of order 1, as the annihila-
F a
toavirialtemperatureof104 K,M . Toobtainanupper tio∼n timescale is comparable to the Hubble time. We
c
limit,itissufficienttoassumethatinallhalosaboveM notethatthesenumbersarehighlyuncertain,inparticu-
F
adarkstarremnantwillformatsomepoint. Suchanas- lar regarding the exact evolution of dark matter density
sumption clearly overestimates the total contribution at during the lifetime of the star, the effect of a supernova
low redshift. When the dark star has formed, a fraction explosion on the dark matter cusp as well as the conse-
f 10−6 of the dark matter from the total halo is in quences of minor mergers.
core
∼
the star [19]. For the upper limit, we assume that the There is however also a viable possibility that the
total amount of dark matter in star will contribute to dark matter distribution inside the star is significantly
the X-ray background (in fact, however, only the dark steeper than assumed above. In the case of dark matter
10
capture by off-scattering from baryons, the dark matter and to ensure a sufficient positron production rate. It is
density inside the star follows a Gaussian shape and is known that electron-positronannihilation occurs mainly
highly concentrated in a small region of r 2 109 cm via positronium-formation in our galaxy [108]. In addi-
∼ ×
[16, 20, 21, 104]. The implications are not entirely clear. tion,it wasshown[109]thatdarkmatter annihilationto
If capture of dark matter stops at the end of the life of electron-positronpairsmustbeaccompaniedbyacontin-
the star, the density inside the star will annihilate away uousradiationknownasinternalbremsstrahlung,arising
quickly, and no significant contribution may come from from electromagnetic radiative corrections to the dark
the remnant. If, on the other hand, dark matter cap- matter annihilation process.
ture goes on until the end of the life of the star, a con- Motivatedbythese results,itwasproposedthatinter-
tribution to the background seems viable. In summary, nal bremsstrahlung from dark matter annihilation may
this may provide a potential contribution to the cosmic be responsible for the gamma-ray background at ener-
background,butits strengthisstillhighlyuncertainand giesof1-20MeV[98]. Conventionalastrophysicalsources
should be explored further by future work. cannot explain the observed gamma-ray background at
thesefrequencies[10]. Acomparisonoftheobservedand
predicted background below 511 keV yields constraints
D. Dependence of dark star models on the on the dark matter particle mass [93]. Here we examine
neutralino mass whether and how this scenario is affected if dark stars
form in the early universe. We use a thermally aver-
aged cross section σv 3 10−26 cm3 s−1 to account
We concludethis sectionwitha discussiononthe con-
h i ∼ ×
for the observed dark matter density [8]. This implies
straintsfromcosmicbackgroundsfordifferentneutralino
that σv is velocity-independent (S-wave annihilation).
masses. As dark star models in the literature mostly
h i
While Boehm et al. [3] argue that S-wave annihilation
consider neutralinos of 100 GeV, there are uncertain-
overpredicts the flux from the galactic center, others ar-
ties that need to be addressed when considering differ-
gue that it is still consistent [93, 98]. The cross-section
ent neutralino masses. For models involving the capture
adopted here is well-within the conservative constraints
of dark matter, Iocco [16] states that the mass of the
of Mack et al. [91]. The effect of light dark matter an-
neutralino does not change the annihilation luminosity.
nihilation on structure formation in the early universe
Taosoetal.[20]findthatvariationsduetodifferentneu-
has been studied in various works, e. g. [110, 111, 112].
tralinomassesarelessthan5%. While theseresultsmay
Constraintsfromupcoming21cmobservationshavebeen
holdforhighmasses,SpergelandPress[105]showedthat
exploredby Furlanetto et al. [80] and Vald´es et al.[113],
forneutralinomassesbelow4GeV,theywouldevaporate
while constraints from background radiation have been
from the star, as scattering with baryons can upscatter
considered by Mapelli and Ferrara [114]. The effects of
them as well.
early dark matter halos on reionization have been ad-
In addition, the AC phase may be modified as well, as
dressed recently by Natarajan and Schwarz [115].
the dark matter annihilation rate in this phase is degen-
As in the previous section, we point out that signifi-
erate in the parameter σv /m . Iocco et al. [18] find
DM
h i cantuncertaintiesarepresentwhenconsideringdarkstar
thatthedurationoftheACphasemaychangebyalmost
models for differentdarkmatter masses,asthis question
50% if the dark matter mass is changed by a factor of
is largely unexplored. In particular, we emphasize that
2. The effect of different neutralino masses is therefore
no capturing phase will be presentfor lightdark matter,
uncertain and should be explored in more detail. We
as shown in the work of Spergel and Press [105]. An-
will however assume that the general behaviour involv-
other uncertainty is the question whether to adopt self-
ing adiabatic contraction in the minihalo is still similar,
annihilating dark matter (i. e. Majorana particles) or
such that the calculations below are approximately cor-
particles and antiparticles of dark matter. In the calcu-
rect also for different neutralino masses.
lations below, we assume that light dark matter is self-
annihilating. Otherwise, our results would be changed
by a factor of 0.5.
VI. COSMIC CONSTRAINTS ON LIGHT DARK
MATTER
A. 511 keV emission
Observations of 511 keV emission in the center of our
Galaxy[106]providerecentmotivationtomodelsoflight
TheexpectedX-raybackgroundfrom511keVemission
dark matter [107]. Such observational signatures can
is calculated from Eq. (8). The volume emissivity of 511
be explained assuming dark matter annihilation, while
keV photons is given as
other models still have difficulties reproducing the ob-
servations [3]. The model assumes that dark matter an- P =δ((1+z)ν ν )511 keVα σv n2 C ,
ν − 511 511 h i DM 511
nihilates into electron-positron pairs, which in turn an- (16)
nihilate into 511 keV photons. Direct annihilation of where σv denotes the thermally-averaged annihilation
h i
dark matter into gamma-rays or neutrinos is assumed cross section, α is the fraction producing an electron-
511
to be suppressed to avoid the gamma-ray constraints positron pair per dark matter annihilation process and