1.4 Where ould axions and ALPs hide?
2.1.2 Before or after ination?
quantum state,
N occupation ∼ (2π) 3 4π/3
n φ,0
m 3 0 δv 3 ∼ 10 42 m 1
m 0
3/2 eV m 0
5/2
,
(2.31)where we used
R 0 /R 1 ∼ T 1 /T 0 ∼ p
m 1 m Pl /T 0
. If the ALP self-interations are strong enough to ahieve thermalisation and ALP-number-onserving, as arguedin referenes [87,88℄ for the ase of axions, this high oupation number leads
to the formation of a Bose-Einstein ondensate. This ould imprint interesting
signatures inosmologial observations, likegalatiausti rings [8790℄.
The axion realignment prodution is a partiular ase of what is desribed
above. The
Λ
parameter in this ase isΛ QCD
, and the phase transition thatturns on the axion mass is the QCD phase transition. The axion thermal mass
was rst alulated from the dilute instanton gas approximation [91℄ and more
reently usingthe instanton liquidone [92℄. The theoretial unertainties in these
alulationsmaketheparametersforthethermalaxionmassslightlydierent. For
example,asimpleand reentapproximationinthedilutegas approximation,that
alsoagrees very well athigh temperaturewith the liquid instanton one, is [92℄
m a (T ) = 4.1 × 10 −4 Λ 2 f a
Λ T
3.34
,
(2.32)for
Λ = 400
MeV. This approximation is valid aboveT ∼ 0.1
GeV, sine belowthis value the axion mass approahes the zero-temperature value
m a
given byequation (1.6). The exat shape of the thermal axionmass determines
t 1
, thus itaets the
m 0 /m 1
ratio inequation (2.22a).axionenergy density
ρ a
remainsϕ 2 i
dependent. Assumingaat prioronthe valueof the initial misalignment angle,
ϕ i ∼ O (1)
is the most likely outome. If thisis the ase, it is possible to state that the axion mass should be
m a & 6 µ
eV,and thus
f a . 10 12
GeV, to avoid DM over-prodution. This is the traditionally quoted lower bound, rst obtained in 1982 [7779℄. It is represented in gure 1.2with the Cold DM label. Suessive estimates, with a morerened treatment of
the average of the squared initial angle or of the thermal axion mass, give more
severe boundsbut ofthe sameorder. Fortherealignmentontribution, the reent
referene [92℄ gives a limit whih is stronger by one order of magnitude. The
traditional
f a . 10 12
GeV bound is thereforethe most onservative one.Itseems that this old DM bound ouldbe easilyavoided inthe ase ination
happenedafterthespontaneous symmetrybreaking,ifweaeptane-tunedvery
smallvalue for
ϕ i
. In gure 2.5are plotted the isoontours ofϕ i
that provide thewhole old DM density, if in equation (2.22b) we take
m 1 = m 0
andF (T ) = 1
.However,
ϕ i
has a minimum value due to the quantum utuations generated inthe pseudosalar eld during ination. These unavoidable inhomogeneities in
ϕ i
are of order
δϕ i ∼ H I /(2πf φ )
, whereH I
is the value of the expansion rate atthe endof ination. Thispreludes ne-tuningofthe universe averageof
φ i
belowH I /2π
,andsets aminimumDMabundaneforaspeiedvalueofH I
. Sinetheφ
eld is eetivelymassless during inationinthis senario, these inhomogeneities
orrespond to isourvature perturbations of the gravitational potential. This has
been disussed extensively in the literature in the ontext of axions and of string
axions, see forexample [81,93,94℄.
TheWMAP7observationsoftheprimordialdensityutuationsset very
strin-gentonstraintsonisourvatureperturbationsfromwhihoneanobtainanupper
bound on
H I
, assuminga givenf φ
. WMAP measures [86℄α ≡ h| S 2 |i
h| S 2 |i + h| R 2 |i < 0.077 ,
(2.33)where
h| S 2 |i ≈ π H 2 φ 2 I 2
i
is the isourvature power spetrum, and
h| R 2 |i
the adiabatione, whih is generated by the inaton or by other elds. At the pivot sale
k 0 = 0.002 Mpc −1
, WMAP ndsh| R 2 |i = 2.42 × 10 −9
,whih gives the boundH I . 4 × 10 −5 ϕ i f φ .
(2.34)j i = 10
0
j i = 10
-2
j i = 10
- 4
j i = 10
-6
j i = 10
- 8
H I = 10 10 GeV H I = 10 9 GeV H I = 10 8 GeV H I = 10 7 GeV H I = 10 6 GeV
Τ<10 17 s
KSVZ Axion
-15 -10 -5 0 5 10
-20 -18 -16 -14
-12 9
11
13
15
17
Log 10 m Φ @eVD
Log 10 g Φ @ GeV - 1 D Log 10 f Φ @ GeV D
Figure 2.5: The anthropi window. The solid lines represent the isoontours
of the values of
ϕ i
that provide the whole old DM density, aording toequa-tion(2.22b) ,taking
m 1 = m 0
andF (T ) = 1
. Dashedlinesgivetheisoontoursofthemaximumvalueof
H I
allowed bythelimit (2.34) , alulatedusing thesamevalueof
ϕ i
asbefore. Asa referene, theKSVZaxion parameters areplotted asa dottedline. In thegrey region ALPsareosmologially unstable.
The lower bound for the reheating temperature is
T RH > 4
MeV [95℄. Linkingnaively the two quantities gives
H I ≈ T RH 2 /m Pl > 1.3 × 10 −24
GeV. In the lightof equation (2.34), it is rather a weak bound by itself. However, in the most
ommoninationmodels,
H I
ispreferredto bemuh larger,toallowmehanismsthat require very high initial temperatures of the universe like leptogenesis,
whih needs
T > 10 9
GeV [96℄. The isoontours of the maximumH I
allowed bylimit(2.34)are plottedingure2.5, wherethe valueof
ϕ i
that providesthe wholeDMdensityisused. Agivenvalueof
H I
exludestheregionoftheparameterspaethat lies on the right of its isoontour. Then, along the
H I
isoontourf φ
seletsthe
ϕ i
that is provided by primordialutuations,while onthe left half-planetheinitialmisalignmentallowed is largerand thusless ne-tuned.
In the ase of the axion, the values of
f a
that satisfy the onstraint (2.34) forlarge
H I
are above the GUT sale, whih seems very reasonable from the modelbuilding point of view: they formthe so-alled anthrophi window 1
. Considering
thattheaxionwasintroduedtoavoidne-tuningtosolvethestrong-CPproblem,
the legitimay of the hoie of a very small
ϕ i
to justify an axion sale linked toΛ GUT
orthe stringsaleisquestionable[97℄. However, theanthropipointof view an nd a vindiation regarding the DM density and other signiant physialquantities simply as observational data, from whih it is possible to extrat the
value ofthe initialmisalignmentwithanaposteriorianalysis[98℄. This proedure
islegitimatebeauseoftheintrinsirandomnessof
ϕ i
,andonvertsthene-tuningissue into anobservationalproblem.
Ontheotherhand,iftheSSBtookplaeafterinationthe
φ i
pikeduprandomvalues indierentausally disonneted regions of the universe andthe anthropi
windowloses itssigniane. If thisisthe ase, alsothe gradienteets shouldbe
taken intoaount. Atlarger sales, the DM density averages to aonstant value
orresponding to
h φ 2 i i ∼ π 2 f φ 2 /3
, bearing the mentionedO (1)
orretion due tothe non-harmonibehaviouroflarge initialphases. Theinitialsize ofthe domains
of dierent
φ i
annotbe larger thanL i,dom ∼ 1
H SSB ∼ m Pl f φ 2 p
g ∗ (f φ ) .
(2.35)Non-linear eets, due to the attrative self-interation aused by higher order
terms inthe expansion of the potential (1.25), drive the overabundanes to form
peuliar DM lumps that are alled minilusters [99102℄. These at like seeds
enhaningthe suessivegravitationallumpingthatleads tostruture formation.
The miniluster mass is set by the dark matter mass inside the Hubble horizon
d H = H −1
when the self-interation freezes-out, i.e.M mc ∼ ρ φ (T λ )d H (T λ ) 3
for1
Theadjetiveanthropirefersto the fat that theexistene ofthe humanspeies requires
theuniverseto fullsomeonditionsaboutitsageandomposition,whih tooabundantaxion
the freeze-outtemperature
T λ
. Long-range interations willbeexponentially sup-pressed at distanes longerthan1/m φ
so we an expetT λ
to be of the orderT 1
,withatmostalogarithmidependeneonotherparameters. Thisisindeedthease
forQCDaxions,forwhihtheminilusteringisquenhedsoonaftertheQCDphase
transition that turns on the potential (1.25) [87℄ giving
M mc ∼ 10 −12 M ⊙
, whereM ⊙ = 1.989 × 10 33 g
is the solar mass, and a radiusR mc ∼ 10 11
m [103℄. In theaseofALPs,
M mc
anbelargerifthemassislighter. Theauthorsof[102℄pointedoutthatthepresentdataontheCDMpowerspetrumonstrain
M mc . 4 × 10 3 M ⊙
whihtranslates intoalowerbound intemperature
T λ > 2 × 10 −5 GeV
andintheALP mass
m φ > H(T = 2 × 10 −5 GeV) ∼ 10 −20
eV. If some of these minilusters survive the tidal disruption duringstruture formationthey shouldbeobservableinforthominglensing experiments [102,103℄.