Table Of ContentAxion Dark Matter and Cosmological Parameters
O. Erken, P. Sikivie, H. Tam, Q. Yang
Department of Physics, University of Florida, Gainesville, FL 32611, USA
We observe that photon cooling after big bang nucleosynthesis (BBN) but before recombination
can remove the conflict between the observed and theoretically predicted value of the primordial
abundanceof7Li. Suchcoolingisordinarilydifficulttoachieve. However,therecentrealizationthat
darkmatteraxionsformaBose-Einsteincondensate(BEC)providesapossiblemechanism,because
themuchcolderaxionsmayreach thermalcontact with thephotons. Thisproposalpredictsahigh
effectivenumberof neutrinos as measured by thecosmic microwave anisotropy spectrum.
2
PACSnumbers: 95.35.+d
1
0
2
INTRODUCTION effective temperature of the metal-poor stars [7, 8], and
n surface 7Li depletion in the interior of stars due to some
a
mixing or diffusive processes [9], have also been investi-
J TheagreementbetweenobservationsandtheBBNpre-
gated as possible solutions, but are still considered con-
3 dictions for the primordial abundances of light elements
troversial[1].
2 is often touted as a triumph of the standardΛCDM cos-
mological model. Under the assumption that there are
] Thefactthatthenuclearreactionsrelevanttothepro-
O three neutrino species, BBN as a theory requires essen-
duction of both primordial and post-BBN 7Li are now
tially a single input: the baryon-to-photonratio, usually
C quite well understood has led to speculations that the
givenbytheparameterη =1010n /n [1]. Ifonetakes
. 10 B γ anomaly might instead be caused by new physics. Many
h η to be 6.190± 0.145, in accordance with the latest
p 10 explanationshavebeenproposed,suchasthevariationin
WilkinsonMicrowaveAnisotropyProbe(WMAP)results
- time of the deuteron binding energy and of fundamental
o [2,3],theinferredprimordialabundancesofthemajority
couplings [10, 11], and the decay of a relatively long-
r of the light elements (D, 4He, 3He) are remarkably con-
t lived particle in the context of supersymmetry [12]. At
s sistent with BBN predictions, save one exception: that
a this point, none of these explanations have won general
of7Liisapproximatelytwotothreetimeslessthanwhat
[ acceptance in the cosmology community.
the theory predicts. The discrepancy is deemed statisti-
2
cally significant, and there is so far no widely accepted
v In this paper, we propose a way to remove the con-
explanation for the anomaly. In the literature, this is
7 flict between data and theory for the abundance of 7Li,
0 referred to as the “Lithium Problem”.
basedonthe coolingofphotonsbetweenthe endofBBN
5 OneofthemostdifficultissuesinvolvedintestingBBN
and decoupling. Processes that do this are difficult to
4
ishowreliablytoinfertheprimordialabundancesoflight
. come by. Indeed, typical processes arising from new
4 elements from measurements that are available to us.
physics tend to heat up the photons, modifying η in
0 10
Subsequent to BBN, the original relic abundances are
the wrong direction [1]. However, the recent realiza-
1
allsubjectto furthermodificationby complicatedstellar
1 tion that dark matter axions form a BEC at approxi-
: processes. 7Li, for example, can be both depleted and mately 500 eV photon temperature [13] provides a pos-
v
synthesized in stars, as well as produced by cosmic-ray
i sible mechanism [14]. Essentially, the high occupation
X nucleosynthesis. As such, the abundance of 7Li is in-
ofaxionmodes withverylow momenta greatlyenhances
r ferredprimarily from absorptionlines in the atmosphere thestrengthoftheirgravitationalinteractions,suchthat
a
of galactic halo stars with low metallicity, since these
anexchangeofenergybetweenthephotonsandthemuch
stars are very old and have experienced very little nu-
colder axions becomes possible. Photon cooling implies
clear processing (See [1, 3–5] for details).
that η <η , which has the effect of reduc-
10,BBN 10,WMAP
Although these post-BBN effects lead to considerable ing the production of 7Li [5]. If thermal equilibrium be-
complication, they also open up many different avenues tween the photons and axions is achieved, the 7Li abun-
to explain the 7Li anomaly. For many years, it has dance is reduced by approximately a factor 2 (see be-
been hoped that better determination of nuclear param- low), alleviating the discrepancy and perhaps removing
eters will gradually narrow the discrepancy, though it it altogether. However, our proposal predicts a higher
waseventuallyrealizedthatdoesnotseemachievable[6]. abundance for D than present observations indicate and
Quitethecontrary,itwasfoundin[1]thatimproveddata predicts that the effective number of thermally excited
on the neutron life-time and the cross sections p(n,γ)d neutrino degrees of freedom is high: N = 6.77.
eff
and 3He(α,γ)7Be increases the predicted abundance of
7Li, worsening the disagreement. Revisions to stellar Photoncoolingbykineticmixingwithhiddenphotons
evolution, as a consequence of systematic errors in the was proposed in ref. [15].
2
DARK MATTER AXIONS The study of the catastrophe structure of the inner
causticsofgalactichalosprovidesevidencethatthedark
The axionwasoriginallypostulatedto explainthe ab- matter is an axion BEC [13, 19]. Briefly, there is a di-
sence of CP violation in the strong interactions [16]. It chotomyintheclassificationoftheinnercausticsinterms
waslaterrealizedthatthepopulationofaxionsproduced of their catastrophe structure, depending on the angu-
by the turn on of the axion mass during the QCD phase lar momentumdistribution of the infalling particles [20].
transition possesses the right properties to be cold dark Axions in a BEC are in a state of net overall rotation
matter(CDM)[17]. First,theyhavethemeasuredCDM and produce caustic rings, whereas ordinary CDM has
density if the axion mass m is of order 10−5eV/c2 [18]. anirrotationalvelocityfield andproduces tent-likecaus-
Second, the axions thus produced are very cold: their tics. Thereareseveralpiecesofevidencefortheexistence
average momentum is only of order the Hubble expan- of caustic rings at the predicted radii in various galaxies
sionrate(3×10−9eV/¯h)whentheaxionmasseffectively [21,22]. Itisshowninref. [19]thatthephasespacestruc-
turns on and has been redshifting ever since to a mere ture impliedby the evidence forcausticringsis precisely
10−17mctoday. Third,axionsinthismassrangeinteract andinallrespectsthatpredictedbytheassumptionthat
very weakly through all forces other than gravity. These dark matter is a rethermalizing axion BEC.
properties make axions one of the leading candidates for
CDM.The othermaincontendersareweaklyinteracting
PHOTON COOLING
massiveparticles(WIMPs)andsterileneutrinos. Hence-
forth, we set ¯h=c=1.
The Lithium Problem refers to the mismatch between
Observationallyitseemsdifficulttodistinguishamong
the observed and predicted abundance of primordial 7Li
the CDM candidates. However, it was recently real-
ized that axions form a BEC through gravitational self- by a factor 2 or 3 [1, 3, 4]. For η10 >∼ 2.7, the predicted
7Liabundanceincreaseswithη . Hence,acoolingofthe
interactions at approximately a photon temperature of 10
photonsbetweentheendofBBNanddecouplingreduces
500 eV [13, 14]. The relaxation rate of cold dark matter
the discrepancy.
axions through gravitationalself-interactions is of order
The gravitational fields of the cold axion fluid cause
Γ ∼4πGnm2ℓ2 (1) transitions between momentum states of other particle
a
species present. For particles which are bosons or non
where n is the number density of cold axions and ℓ ∼ degenerate fermions, the relaxation rate through gravi-
t a(t) their correlation length. t ∼ 2 · 10−7 s is the tational interactions with the cold axions is of order [14]
1a(t1) 1
time at which the axionmass effectively turns on during ω
the QCD phase transition, and a(t) is the scale factor. Γ∼4πGmnℓ (2)
∆p
Eq. (1) is appropriate when the energy dispersion of the
particlesislessthantheirrelaxationrate. Werefertothis where ω is the typical energy of the particles and ∆p
caseas‘the condensedregime’,todistinguishitfromthe their momentum dispersion. Eq. (2) generalizes Eq. (1)
morecommonlyencountered‘particlekineticregime’de- to other species that are in the presence of the cold ax-
fined by the condition that the energy dispersion of the ions. [Eq. (1) follows from Eq. (2) by setting ω = m
particles is large compared to the relaxation rate. Cold and∆p=ℓ−1 asisappropriateforthecoldaxionsthem-
dark matter axions are in the condensed regime after t . selves.] For photons to cool substantially it is necessary
1
The ratio Γ (t)/H(t), where H(t) is the Hubble rate, is that energy is transferred from the photons to the low
a
of order5·10−7 att but increaseswith time as a(t)−1t, momentum highly occupied axion states and from those
1
and reaches one at approximately 500 eV photon tem- to the relativistic axion states. For both relativistic ax-
perature. At that time the axions thermalize and form ion states and for photons, ∆p ∼ ω and hence their re-
a BEC. Almost all axions go to the lowest energy state laxation rate Γ through gravitational interactions with
r
available. The correlation length grows and becomes of cold axions is of order 4πGnmℓ. Using the Friedmann
order the horizon. In the linear regime of evolution of equation,one finds that Γ /H ∝a(t) before equality be-
r
density perturbations and within the horizon,the lowest tween matter and radiation and remains constant after
energystateistimeindependentandnorethermalization that. At equality, Γ /H| ∼ ℓ(t )/t . If ℓ/t is order
r teq eq eq
isnecessaryfortheaxionstoremaininthelowestenergy one at equality, the photons reach thermal equilibrium
state. Inthatcase,axionBECandordinaryCDMarein- with the axions and hence cool.
distinguishableonallscalesofobservationalinterest[13]. Gravitational interactions conserve particle number
However, beyond first order perturbation theory and/or andthereforeproduceonlykinetic(asopposedtochemi-
uponenteringthehorizon,theaxionsrethermalizetotry cal)equilibriumbetweenthespeciesinvolved. Also,after
and remain in the lowest energy available state. Axion 500 eV photon temperature, the coupling between pho-
BEC behaves differently from CDM then and the result- tons and baryons is in the kinetic, rather than chemi-
ing differences are observable. cal, equilibrium regime [23]. Upon cooling, the photons
3
thatcannotbeaccommodatedinthermallyexcitedstates
enter the ground state, a plasma oscillation with zero
)*#+, %&’(
wavevector. Sincethephotonchemicalpotentialremains
!"# $
zero, the final photon spectrum is Planckian, consistent Steigman 2005
with observation. -./ 0./
Eq. (2) does not apply to degenerate fermions be-
!"# $
cause of Pauli blocking. The cosmic neutrinos are semi-
0./
Iocco et. al.
degenerate since they have a thermal distribution with -./
zerochemicalpotential. Their thermalizationrate is less
than that Γ of relativistic bosons. Since Γ /H ∝tnℓ∝ !"# $
r r
t2a−3(t), that ratio does not grow after equality. Since SPrteiviagtme aCno mm. 0./ -./
the relativistic axions may only reach thermal contact
withthecoldaxionsatequalityandtheneutrinosarede-
2 4 6 8
layedrelativetotherelativisticaxions,webelieveitmost η
10, BBN
likely that neutrinos remain decoupled from the axions,
photons and baryons at all times.
FIG. 1: Values of η10,BBN inferred from the abundances of
It is straightforward to determine how much the pho- 7Li,D,3Heand4He,andthepredictedvaluesinthestandard
tons cool if they reach thermal equilibrium with the ax- cosmologicalmodel(WIMP)andinourproposal(axion). The
ions. Energy conservation implies ρ = ρ +ρ be- data inferred values are taken from refs. [4], [25] and [26].
i,γ f,γ f,a
causethecontributionstotheenergydensityoftheinitial Theerrorbarsindicatetheη10,BBN valuesconsistentwiththe
axions and of the baryons are negligible. The ratio be- estimated 1-σ uncertainties in theobservations.
tween the final and initial photon temperature is thus
(2/3)1/4. Since their number density is proportional to
T3, we find:
inthelightofrecentobservations[26]. Theerrorbarsin-
2 3/4 dicatetherangeofη10,BBN consistentwiththeestimated
η10,BBN = η10,WMAP =4.57±0.11 (3) 1-σ uncertainties in the observations. The axion predic-
3
(cid:18) (cid:19) tionagreesverywellwith the 7Li abundance atthe time
using η10,WMAP = 6.190± 0.145 [2]. Because the 7Li of Steigman’s 2005 review (η10,7Li = 4.50±0.30). How-
abundance is proportional to η2 in the range of in- evermorerecentobservationsindicatealowerprimordial
10,BBN
terest, it is reduced by approximately the factor (2)23 ≃ 7Li abundance, worsening the Lithium Problem.
3
0.55.
Perhaps more problematic is that a smaller η
10,BBN
A number of authors proposed earlier that the dark
predicts an overproduction of D. Traditionally, D has
matterisaBEC[24]. Thephotoncoolingdescribedhere
been the prime choice as a baryometer among the light
may occur in those cases as well. If the particles are
elements, due to its sensitivity to η and simple
10,BBN
in the condensed regime, the relaxation rate is given by
post-BBN evolution (abundance monotonically decreas-
Eq.(1),calculatedwithappropriatevaluesforn, m,and
ing). The major drawback with D is that its abundance
ℓ. However,inmanyoftheseproposals,thecosmological
is inferred from a very small set of (seven) spectra of
history of the dark matter particle is not known, render-
QSO absorption line systems [27]. Worse yet, these few
ing the computation of the relaxation rate difficult.
measurements have a large dispersion, and do not seem
tocorrelatewithmetallicity,obscuringtheexpecteddeu-
terium plateau. Due to the various inadequacies in the
COSMOLOGICAL CONSEQUENCES
D measurements mentioned, we have reservations about
thecommonpracticeofattachingmostsignificanceonD
Effect on light element primordial abundances
inthecomparisonbetweendataandBBNpredictions. In
comparison, 7Li is inferred from a large number of mea-
Whether photon cooling by axion BEC solves the
surements,whicharemore-or-lessconsistent. Also,since
LithiumProblemremainstobeseen. Thedatahavebeen D is more easily destructible than 7Li, it is conceivable
timedependentinadditiontotheusualuncertainties. In
that unknown stellar processes further deplete D.
Fig. 1, we plot the value of η in the standard cos-
10,BBN
mological model, labeled ‘WIMP’, and in the scenario Finally the 3He and 4He inferred η values have
10,BBN
described here, labeled ‘axion’, along with the values in- large error bars and hence carry less statistical weight.
ferred from the observed light element abundances ac- The 4He inferred value has increased recently (5.5 <
cordingto the review by G. Steigmanin 2005[4], the re- η <11accordingtoref. [25]and7.5<η <20
10,4He 10,4He
viewbyF.Ioccoetal. in2008[25]andaprivatecommu- according to ref. [26]) compared to its accepted value a
nication from G. Steigman updating his 2005 estimates few years ago, creating additional uncertainty.
4
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eff
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eff
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eff
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eff
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