Sommerfeld Enhancement of DM Annihilation: Resonance Structure, Freeze-Out and CMB Spectral Bound

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by  Steen Hannestad

Sommerfeld Enhancement of DM Annihilation:
Resonance Structure, Freeze-Out and CMB Spectral Bound
Steen Hannestad and Thomas Tram
∗
Department of Physics and Astronomy, University of Aarhus, 8000 Aarhus C, Denmark
(Dated: January 6, 2011)
In the last few years there has been some interest in WIMP Dark Matter models featuring a
velocitydependentcross section throughtheSommerfeld enhancementmechanism,which isanon-
relativistic effect due to massive bosons in the dark sector. In the first part of this article, we find
analyticexpressionsfortheboostfactorforthreedifferentmodelpotentials,theCoulombpotential,
thesphericalwellandthesphericalconewellandcomparewiththenumericalsolutionoftheYukawa
potential. Wefindthattheresonancepatternofallthepotentialscanbecastintothesameuniversal
form.
InthesecondpartofthearticleweperformadetailedcomputationoftheDarkMatterrelicdensity
for models having Sommerfeld enhancement by solving the Boltzmann equation numerically. We
calculate the expected distortions of the CMB blackbody spectrum from WIMP annihilations and
1 compare these to the bounds set by FIRAS. We conclude that only a small part of the parameter
1 space can beruled out by theFIRASobservations.
0
2
n I. INTRODUCTION ray excess, was first studied in [17, 18] and later in [19]
a and [20].
J
The most probable candidate for Dark Matter is ar-
5
guably the WIMP, a Weakly Interacting Massive Par-
II. SOMMERFELD ENHANCEMENT
ticle. This is due to what is often called the ’WIMP
]
O miracle’; new physics is required at the TeV scale when
C unitarity breaks down in the Standard Model, and some Sommerfeld enhancement is a consequence of having
symmetry should protect the proton from decaying too light force carriers mediate an attractive interaction in
.
h fast through new diagrams involving these new degrees thedarksector,whichcreatesaYukawa-potentialforthe
p of freedom. The symmetry may ensure that at least one WIMPs. In this work we consider only s-wave annihila-
o- of the new particles is stable on cosmologicaltimescales, tion. We write this potential as
r whilethe annihilationcrosssectionistypicallywithinan λ2 α
st order of magnitude of what is required to explain the V(r)= e−mφr = e−mφr, (1)
−4πr −r
a observed relic abundance.
[ where λ is the coupling parameter, m is the mass of
In the last few years there have been some observa- φ
the force carrier and r is the relative distance between
3 tions [1–9] pointing to anomalous emission within the
v galaxy. It is possible, that these observations may the WIMPs. Since the potential is only a function of
1 the relative distance, it only enters in the reduced one-
be explained by WIMPs annihilating in the galactic
1 particle Schr¨odingerequation for the relative motion. In
halo [10–14]. Common to all these suggestions is that
5 spherical coordinates this is
the necessary annihilation rate requires an annihilation
1
. cross section a few orders of magnitude larger than the 1 k2
8 2ψ = V(r) ψ , (2)
one consistent with a thermal relic density, σv k k
0 3 10 26cm3/s. h iTH ≃ −2µ∇ (cid:18)2µ − (cid:19)
0 −
· where µ=m /2 is the reduced mass. We are interested
1 This apparent discrepancy, as well as the necessary χ
: suppression of hadronic final states, has been suggested in the probability density of ψk when r = 0, since the
v boostfactorforthes-wavecaseisgivenbyS = ψ (0)2,
to stem from new GeV-mass force carriers in the dark k k
i | |
X sector [15, 16] which leads to Sommerfeld enhancement when ψk is normalized to the asymptotic form
r of the Dark Matter pair annihilation cross section. Al- eikr
a though the anomalies mentioned above may turn out to ψ eikz +f(θ) as r . (3)
→ r →∞
be explained by normal astrophysical processes, there
This form of the boost factor can be derived [16] from a
may still be force carriers in the dark sector which are
simple argument: Annihilations are assumed to proceed
light compared to the mass of the Dark Matter particle,
byadeltafunctioninteraction,sotherateofthisprocess
so the mechanism of Sommerfeld enhancement is inter-
mustbeproportionaltothenormsquaredofthereduced
esting and generic. Sommerfeld enhancement applied to
wave function at zero separation, σ ψ(r =0)2. De-
WIMPannihilationsingeneral,notrelatedtothecosmic
noting the V =0 wave function by ψ∝0, w| e must h|ave
σ ψ(0)2
S = = | | . (4)
∗Comments: [email protected] σ0 ψ0(0)2
| |
2
Whenthepotentialisrotationallysymmetric,weknow
thatthesolutionofequation(2)becomesinvariantunder
2.5
rotations around the axis of the incoming particle, and
hence we can expand ψ in products of Legendre poly- −2.5
nomials and a radial wave function R . Only R is
kl k0
2
non-zero at the origin, so we define χ rR , which
k ≡ k0 −3
leads to the equation
1 d2χ α (f)10−3.5 1.5
m dr2 = −re−mφr−mχβ2 χ⇒ (5) log
χ
(cid:16) (cid:17)
m d2χ α m −4
φ = e−x χβ2 χ (6) 1
m dx2 −x − m ⇒
χ (cid:18) φ (cid:19)
−4.5
d2χ α β 2
dx2 =− fxe−x+(cid:18)f(cid:19) !χ, (7) −5 0.5
−3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2
log (α)
whereweintroducedthedimensionlessdistancex m r 10
φ
≡
and the ratio of the two masses f mφ/mχ. Equation FIG.1: Sommerfeldboostfactorinlog10forarelativevelocity
(7), together with the boundary con≡ditions of 150km/s.
χ(0)=0 (8a) where W is the Lambert W-function. To calculate the
amplitude we use the asymptotic waveform (8b):
β
χ sin x+δ , as x . (8b)
→ f →∞ β
(cid:18) (cid:19) χ=Asin x+δ (11a)
f
defines the problem. (cid:18) (cid:19)
dχ β β
e= Acos x+δ , (11b)
dx f f
(cid:18) (cid:19)
A. Numerical solution
e
which leads to this expression for the amplitude A:
In principle we should solve the boundary value prob-
lem (8) numerically, but since the Schr¨odinger equation f dχ 2
A= χ2+ . (12)
is linear, we can exchange the condition (8b) by setting s β dx
(cid:18) (cid:19)
the derivative of χ to unity at x = 0 and solve what is e
now an initial value problem for χ. The solution to the We evolve our waveforem χ to 1.5x , and after this
range
original problem is just given by χ = χ/A, where A is pointwecalculateAateachsucceedingpointuntilAhas
the asymptotic amplitude of χ. The enhancement factor converged.
e e
is then given by
e
R2 (0) 2 1deχ 2 f 2 B. Coulomb potential
S = k0 = k(0) = . (9)
k
k k dr Aβ
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) In the limit of a massless force carrier, the Yukawa
In figure 1 we(cid:12)have pl(cid:12)otted(cid:12)the Somm(cid:12) erfe(cid:12)ld b(cid:12)oost factor
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) potentialbecomesaCoulombpotential. Inthis case,the
for a velocity of 150km/s for the whole (α,f)-parameter
enhancement factor can be calculated in closed form,
space. The vertical lines in figure 1 are ragged by thin,
straight lines of resonance which have, as we shall see π/ǫ β
C
S = , ǫ = (13)
latTero, caarlcauthlaetrelatrhgeeaimmppalicttudoenAtheinfre(9ez),e owuetmpruosctesksn.ow ǫC 1−e−π/ǫC C α
when our waveform χ has reached its asymptotic form. by solving the Schr¨odinger equation in terms of hyper
This must happen when the Yukawa potential becomes geometricfunctions. Thederivationofequation(13)can
much less than the kinetic energy of the particle, so we be found in section A. We want to stress, that the ap-
e
find the position of equality: plicability of this limit does not only depend on f 1,
≪
but is also dependent on α and β. The correct question
α e−xrange = β2 to ask is whether or not the scattering will take place in
x f ⇒ aregimein whichthe potentiallookslike aCoulombpo-
range
fα tential. We want the range of the potential xrange from
xrangeexrange = β2 ⇒ (10), to be less than something like ln(2). This would
keep the exponential in the Yukawa potential at order 1
fα
x =W , (10) during the interaction, so we should be safe in assum-
range β2 ing a Coulomb potential instead. (However, this does
(cid:18) (cid:19)
3
potential is
−2
2.6 d2χ V β 2
0
= + χ (16)
−2.5 2.4 dx2 − mφf (cid:18)f(cid:19) ! ⇒
2.2 d2χ
= K2+ǫ2 , (17)
−3 2 dx2 −
(f)10−3.5 1.8 where we are using x(cid:0)=mφr an(cid:1)d
g
o 1.6 3α β
l K = ǫ= . (18)
−4 1.4 rfL3 f
1.2 This equation is solved by sines and cosines, so if we
define p = √K2+ǫ2, we can write the general solution
−4.5 1
as
0.8
−5 χx<L(x)=Asin(px+γ) (19a)
−3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2
log10(α) χx>L(x)=Bsin(ǫ(x L)+δ), (19b)
−
FIG. 2: Boost factor in log for the Coulomb potential in
10 and using boundary conditions (8), we can set γ = 0
log for a relative velocity of 150km/s.
10
and B = 1. We must now match the wave function and
its derivative at the boundary at x = L. We get the
nottake into accountthe non-perturbativeeffects of res-
equations
onances.) The boost factor (13) is plotted in figure 2. If
wecomparewiththeSommerfeld-caseinfigure1,wecan Asin(pL)=sin(δ) (20a)
see that the lowerleft partof the Sommerfeldparameter Apcos(pL)=ǫcos(δ) (20b)
space is well described by the boost from the Coulomb
and we solve for the amplitude A. We find:
potential. This part of the parameter space is also the
part where the range (10) of the potential is less than 1
A2 = , (21)
about 0.7 as expected. However, we do not find reso- 1+ K 2cos2(pL)
nancesinthe Coulombcaseaswe doforthe Sommerfeld ǫ
enhancement, because the potential is not localized. To which results in a boost fa(cid:0)cto(cid:1)r
learnmoreabouttheseresonances,werelatetheYukawa 1+ K 2
potential to two model potentials which can be solved S = ǫ . (22)
analytically. 1+ K 2(cid:0)cos(cid:1)2(pL)
ǫ
We have plotted this b(cid:0)oos(cid:1)t factor for L = 1 in figure
3 for the same (α,f)-parameter space as before. If we
compare with figure 1, we can see the same type of reso-
C. Spherical Well nances,althoughthey become suppressedin the Yukawa
case when the scattering becomes Coulomb-like. The
structure of (22) tells us something about the behavior
The resonance structure of the Sommerfeld-
of resonances, even for the Yukawa-case. First off, since
enhancement does not appear in the massless limit,
the denominator is always larger than unity, the maxi-
but we can understand it qualitatively by examining
mal boost must be of the order K2/ǫ2. Secondly, if
the spherical well and relating the well to the Yukawa ∼
the cosine is of orderunity, the boost will be orderunity
potential[41]. We look at a potential of the form
as well. So the resonances occur when pL/π n+1/2.
≃
IfweareinterestedinresonancesgivingS oforder100
V(r)= −V0, r≤L/mφ (14) ormore,wecanmaketheapproximationǫ/K ≪1,which
0, r>L/mφ. allows us to use the zeroth order approximation p K.
(cid:26) ≈
Using the definition of K, we find
We set V0 to a value such that the potential integrated KL 3α L α 3
over volume agrees with the Yukawa potential (1). We = = , (23)
π fL3 π f Lπ2
find r r r
so the position of the resonances depends only on the
3αm ratio of α and f. This suggests that we introduce the
φ
V0 = L3 , (15) rotated coordinates
α
u= (24a)
f
where L is the range of the potential in units of m−φ1, v =αf. (24b)
andshouldbeorder1. TheSchr¨odingerequationforthis
4
−2 3
−3.5
3.5
−2.5 −4 2.5
3
−4.5
−3 α f) 2
2.5 (0 −5
g(f)10−3.5 2 = log1−5.5 1.5
lo v) −6
1.5 (0
−4 og1−6.5 1
l
1
−7
−4.5 0.5
0.5 −7.5
−8
−5 0
−3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5
log (α) log (u) = log (α/f)
10 10 10
FIG. 3: Boost factor in log for the spherical well with v = FIG. 5: The Sommerfeld boost factor with v = 150km/s in
10
150km/s. The depth of the well is related to the strength of the new coordinates u and v. We have cropped the plot at
thecorresponding Yukawapotential. the(α,f) limits from figure1
we calculate the Sommerfeld boost factor in the (u,v)

coordinates, and the result is shown in figure 5. As is
−3.5
3.5 clear from the plot, u alone determines the position of
−4
the peaks,just as forthe sphericalwell. This is inagree-
3
−4.5 ment with [19], as our coordinate u is identical to their
α( f)0 −5 2.5 parameter ǫφ.
1
g
=lo −5.5 2
(v)10 −6 1.5 D. Spherical Slope Well
g
o −6.5
l We now examine another model potential, the spher-
1
−7 ical slope well, to see how the position of resonances
−7.5 0.5 change compared to the spherical well. We write the
potential
−8

−1 −0.5 0 0.5log 1(u)=1lo.5g (α2/f) 2.5 3 3.5 V 1 mφr , r L/m
10 10 V(r)= − 0 − L ≤ φ (27)
0, r>L/m ,
FIG.4: Boost factor forthespherical well with v=150km/s (cid:26) (cid:0) (cid:1) φ
in the newcoordinates uand v.
andasbefore,wesetV toavaluesuchthatthepotential
0
Using (23), we find that the resonances happens at integratedovervolumeagreeswiththe Yukawapotential
(1). In this case we find
Lπ2 1
12αm
√u = n+ φ
n V = , (28)
r 3 (cid:18) 2(cid:19)⇒ 0 L3
un = Lπ2 n+ 1 2, n=0,1,... (25) where L is the range of the potential in units of m−φ1,
3 2 andshouldbeorder1. TheSchr¨odingerequationforthis
(cid:18) (cid:19)
potential is
Since the distance between resonances increases only by
n2, it is clear that they will happen closer and closer d2χ V 1 β 2
0
when viewed in a logarithmic plot, just as we saw in the = (L x)+ χ
Yukawa-case. The other parameter v controls the order dx2 − mφf L − (cid:18)f(cid:19) ! ⇒
of the boost factor, since we have d2χ
= K2(L x)+ǫ2
dx2 − − ⇒
K2 3 fα 3 v
= = . (26) d2χ (cid:0) (cid:1)
ǫ2 L3 β2 L3β2 = K2x+p2 ,
dx2 − −
We have shown the boost plot in the rotated coordi- (cid:0) (cid:1)
where we are using x = m r, K = 12α and
nates (u,v) in figure 4. We suspect that this choice of φ fL4
coordinates would also work well in the Yukawa case, so p = √K2L+ǫ2. If we now do the substitqution ξ =
5
K 4/3 K2x p2 , we get the Airy equation
−
−

(cid:0) (cid:1) d2χ −3.5 4
=ξχ, (29)
dξ2
−4 3.5
which has the general solution
−4.5
χ=c1Ai(ξ)+c2Bi(ξ). (30) α( f)10 −5 23.5
g
We are interested in χ at two different values, ξ0 =lo −5.5
ξ(x=0) and ξL ≡ξ(x=L). These are given by ≡ (v)0 −6 2
1
g 1.5
ξ0 =K−4/3 p2 = K−4/3p2 (31a) lo −6.5
− −
ξL =K−4/3(cid:0)K2L(cid:1) p2 = K−4/3ǫ2. (31b) −7 1
− −
−7.5 0.5
Since we have ξ <(cid:0) 0 in both(cid:1) cases, it makes sense to
introduce the variable −8
−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5
ζ = 2( ξ)3/2, (32) log10(u)=log10(α/f)
3 − FIG.6: Boost factorinlog10 forthesphericalslopewellwith
v=150km/s. Thedepthofthewellisrelatedtothestrength
and reexpress the solution χ and its derivative in terms
of thecorresponding Yukawapotential.
of Bessel functions. We have:
χ(ξ)= ξ AJ (ζ)+BJ (ζ) (33a) Using equation (9), we can find the boost factor S:
1/3 1/3
− − −
dχ
dξ(ξ)=−pξ(cid:2)A(cid:2)J−2/3(ζ)+BJ2/3(ζ)(cid:3). (cid:3) (33b) S = pǫ24K−4/3A2 J−2/3(ζ0)+tJ2/3(ζ0) 2 ⇒
wherethenewcoefficientsAandB arerelatedtotheold
ones by p4 J 2/3(cid:2)(ζ0)+tJ2/3(ζ0) 2 (cid:3)
S = − . (38)
A= c1/3+c2/√3 B =c1/3+c2/√3. (34) ǫ4 −(cid:2)J1/3(ζL)+tJ−1/3(ζL)(cid:3)22+
− + J (ζ )+tJ (ζ )
(cid:2) 2/3 L 2/3 L(cid:3)
Since we require χ(ξ ) = 0, equation (33a) can be used −
0
(cid:2) (cid:3)
to relate A and B, since we have
This is a rather complicated expression,but the numer-
ator can be simplified by using the series expansion for
AJ (ζ )=BJ (ζ )
1/3 0 1/3 0
− ⇒ J(z) and the following identity for the gamma function:
J (ζ )
1/3 0
B = A tA. (35)
J (ζ ) ≡
1/3 0 Γ(1/3)Γ(2/3)=Γ(1/3)Γ(1 1/3)
− −
At this pointwe haveone freeparameterAleft to deter- =πcsc(π/3)=2π/√3 (39a)
mine from the solution χx<L. The solution outside the
well is just as before, (19b), and contains one free pa- J (z)=(z/2)ν ∞ (−)k(z/2)2k (39b)
ν
rameter, δ. Both parameters are fixed by matching the k!Γ(ν+k+1)
k=0
solutionand its derivative at the boundary;we have two X
equations with two unknowns:
Having these, we can prove the neat identity
sin(δ)=χx<L(x=L) (36a)
dχx<L J (z)J (z)+J (z)J (z)= √3. (40)
ǫcos(δ)= dx (x=L), (36b) −2/3 −1/3 1/3 2/3 πz
and the interesting parameter A is most easily found by Reinsertingt,thenumeratorofequation(38)cannowbe
inserting the right hand sides in the well known trigono- written as
metric identity
2
ǫ2 =ǫ2sin2(δ)+(ǫcos(δ))2 num.= J 2/3(ζ0)+tJ2/3(ζ0)
−
=ǫ2( ξ )A2 J (ζ )+tJ (ζ ) 2+ =(cid:2)J 1/3(z)−2 (cid:3)
− L − 1/3 L −1/3 L − × 2
+ξL2K4/3A2 (cid:2)J−2/3(ζL)+tJ2/3(ζL) 2(cid:3)⇒ × J−1/3(ζ30)J−2/3(ζ0)+J1/3(ζ0)J2/3(ζ0)
A2 = (cid:2)K4/3/ǫ2 . (cid:3) (37) =(cid:2)π2ζ2J (ζ )2. (cid:3) (41)
2 0 1/3 0
J (ζ )+tJ (ζ ) + −
1/3 L 1/3 L
− − 2
+ J (ζ )+tJ (ζ )
(cid:2) 2/3 L 2/3 L(cid:3) Inserting(41)into(38)andrearrangingthedenominator,
−
(cid:2) (cid:3)
6
we find: E. Hulth´en potential
33K4
S = J (ζ )2 J (ζ )2+J (ζ )2 +
22π2ǫ4p2 1/3 0 −1/3 L 2/3 L Comparingequation(25)and(45),wenoticethatthey
(cid:18) look very similar. Only the fraction in front and the
+J (ζ )2 J (ζ(cid:2))2+J (ζ )2 + (cid:3)
1/3 0 1/3 L 2/3 L ’phase’ is different. It seems likely that we may fit the
− −
+J (ζ )J (ζ )J (ζ ) peaks of the resonances in the Sommerfeld case by an
1/3 0 1(cid:2)/3 0 1/3 L (cid:3)
− − 1× expression:
−
J (ζ ) J (ζ ) . (42)
× −2/3 L − 1/3 L (cid:19) un =Lπ2(n+b)2, (46)
(cid:2) (cid:3)
Wehaveplottedthisboostfactorinfigure6inthe(u,v)- andthisisindeedthecase. WefindthevaluesL=0.1592
coordinates we introduced for the spherical well. We and b = 1.006. The value of b is very close to 1, hinting
see the same behavior as before, but the expression for that a similar treatment is doable in the Yukawa case,
the boost factor is still too complicated for us to deduce yielding a boost factor which depends only on a sine to
wheretheresonanceswillbe. Letustakeanotherlookat lowest order. A model potential called the Hulth´en po-
the denominator of equation (42). We want to estimate tentialadmitsananalyticsolutionforthes-wavecase[42],
thenumericalvalueofthetwoarguments,ζ0 andζL. We andtheSommerfeldboostcomingfromthispotentialwas
got: studied in [21] and later in [22].
2 2p3 The potential looks like
ζ = ( ξ )3/2 = (43a)
0 3 − 0 3K2 Aδe δr
−
ζL = 32(−ξL)3/2 = 32Kǫ32. (43b) VH =−1−e−δr, (47)
where A and δ are parameters. We stress that this is
Ifweareinterestedinregionsofpossiblylargeboostfac-
not a general version of the Yukawa potential, but it is
tors, we may assume K ǫ as we did before, and we
≫ a model potential just like the spherical well and the
conclude that ζ 1 and ζ 1. It turns out, that to
0 ≫ L ≪ spherical slope well. However, unlike these two poten-
a reasonable approximation, we can set the Bessel func-
tials, the Hulth´en potential reproduces the 1/r-behavior
tions equal to their asymptotic values. If we remember
ofthe Yukawapotentialinthe limitr 0,anditdecays
that →
exponentially instead of having a fixed range. By a pro-
(z/2)ν cedure similar[43] to ours, A and δ is found to be A=α
J (z) , z 1
ν ≃ Γ(ν+1) ≪ and δ = kmφ = π2/6. In our notation, the l = 0 case
of the Sommerfeld boost just before equation 44 in [21],
2
J (z) cos(z νπ/2 π/4), z 1 can be written as
ν
≃ πz − − ≫
r
wecanidentify thetwolargesttermsinthe denominator παsinh 2π β
β kf
S = . (48)
of equation(42). We are looking for the largestnegative H (cid:16) (cid:17)
cosh 2π β cos 2π β kαf 1
powersofζL,becausethey willgivethe largestcontribu- kf − kf β2 −
tion to the denominator. We have one term behaving as (cid:16) (cid:17) (cid:16) q (cid:17)
∼ζL−4/3 andone termfromthe crossproductis behaving tEioqnukaαtifon (4β82), c(wanhibche idseruisvueadlleyansiolyt tuhnedcearsteh),ehaoswsuemvepr-,
as ∼ ζL−1, and the rest of the 6 terms has more positive amoreca≤refulderivationrevealsthisequationtobe true
powers. ThesetwoworsttermsmultiplyJ (ζ ),sowe
1/3 0 in any case. We have plotted the Hulth´en boost factor
suspecttohavearesonancepatternthatf−ollowthe zeros
in figure 7, and it looks very similar to the Yukawa case
of J (ζ ), that is:
1/3 0 in figure 1. Inspection of equation (48) reveals the reso-
−
2p3 1 nance pattern to be
J (ζ ) cos π =0
−1/3 0 ∼ (cid:18)3K2 − 12 (cid:19) ⇒ un =k(n+1)2
2p3 1 2K3L3/2 1 1 π2
3K2 − 12π ≃ 3K2 − 12π =π n+ 2 ⇒ = (n+1)2. (49)
(cid:18) (cid:19) 6
2
2 7
KL3/2 = n+ π However, for large n, the resonance pattern is shifted
3 12 ⇒ completely because of the slight difference between π2/6
(cid:18) (cid:19) (cid:18) (cid:19)
3π2 7 2 and the fit value. For a velocity of 150km/s we found
un = L n+ . (45) equation(48)to be within10%ofthe numericalsolution
16 12
(cid:18) (cid:19) at 82% of the plotted parameter space and within 30%
Thus,wehavederivedananalyticequationforthepo- at98%oftheparameterspace. Foravelocityof10km/s,
sitionofresonances,just as wedid forthe sphericalwell, theHulth´enboostfactoriswithin10%inonly43%ofthe
andthisexpressionagreesexactlywiththeresonancesin parameter space and within 30% in 74% of the param-
figure 6. eter space. By using k = 0.1592π2 and simultaneously
7
A. Relic density calculation
2.6
2.4 Tocatchthefullnatureofthermalfreezeoutinmodels
−2.5 2.2 with Sommerfeld enhancement, we must do a full calcu-
lation of the integrated Boltzmann equation because of
2
−3 the non-trivial velocity dependence. It should be noted,
1.8
that we do expect something of the same order of mag-
og(f)10−3.5 11..46 nveitruydceloassehtσov1iTaHt,tshiencteimtheeofSofrmeemzeerofeultd. bWooessttafarcttforromis
l
the integrated Boltzmann equation for Dark Matter,
1.2
−4
1 d(n a3)
a 3 χ = σ v n2 n2 , (50)
−4.5 0.8 − dt h ann i χ,eq− χ
0.6 (cid:8) (cid:9)
where a is the scale factor, n is the number density of
χ
−5 DarkMatter, n is the DarkMatter equilibriumnum-
−3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2 χ,eq
log (α) berdensityand σ v isthethermallyaveragedannihi-
10 h ann i
lationcrosssection. Wenormalizethenumberdensityto
FIG. 7: Sommerfeld boost factor in log for the Hulth´en
potential at a relative velocity of 150km/s1.0 thetotalentropydensitys∝a−3byintroducingY ≡ nsχ:
dY
=s σ v Y2 Y2 (51)
correcting the phase by 0.006 by hand in equation (48), dt h ann i eq−
accordingtothefitparameters,wedidsomewhatbetter. (cid:8) (cid:9)
We also want to substitute the time parameter t by the
The modified formula reproduced the resonance pattern dimensionless evolution parameter x mχ. After some
perfectly, but the agreement of the boost magnitude at ≡ Tγ
manipulations, we find the final form of the Boltzmann
the first peak became worse.
equation:
dY = π mpmχh12 σ v Y2 Y2 . (52)
dx 45 x2 ∗ h ann i eq−
r
(cid:8) (cid:9)
The manipulations leading up to equation (52) can be
III. CONSTRAINING SOMMERFELD
found in section B of the appendix. In our case, σ =
ENHANCEMENT WITH THE CMB
σ S(v,...), where σ is the s-wave annihilation cross
0 0
section which is independent of velocity, and S is the
Sommerfeldboost factor,whichdepends on velocityand
There has been some work on constraining models in-
the model parameters. As was noted by the authors
corporating Sommerfeld Enhancement using the CMB.
of [32, 33], σ is not a completely free parameter but is
WIMPsannihilatingafterrecombinationmaycontribute 0
to reionisation, thereby changing the predicted optical related to α and mχ by dimensional analysis, σ0 ∼ mα22χ.
depth which can be inferred from the CMB anisotropy However,theexactfactorishighlymodeldependent,and
spectrum. This was studied by [23]. Earlier, during the for that reason we keep σ as a parameter.
0
recombinationphase,WIMPannihilationswithSommer- InsertingtheSommerfeldcrosssectioninequation(52)
feldEnhancementalsomodifythestandardscenario. An leads to our final equation:
upperboundonthiseffectcanagainbeinferredfromthe
anisotropyspectrum, as was done in [24–30]. We will fo- dY = mpmχh21σ S(v,...)v Y2 Y2 (53)
cus on WIMP annihilations happening even earlier, in dx x2 ∗ 0h i eq−
the redshift range of approximately 1100<z <2.1 106. λ(x) Y2 Y2 . (cid:8) (cid:9) (54)
· ≡ eq−
Annihilationsoccurringinthisredshiftrangewillnotin-
fluencetheanisotropyspectrum,buttheywilldistortthe The parameter σ(cid:8)0 is then f(cid:9)ound by imposing the bound-
Black Body spectrum. ary condition that the WIMP must make up all of Dark
Matter. In this article we are considering the case where
Common to the above mentioned bounds is the fact
the WIMP is not its own antiparticle, so we impose the
that they use the expected cross section for a thermal
boundaryconditionΩ =2Ω assumingthatthereis
DM χ,0
relic, σv 3 10 26cm3/s, and multiply this with
h iTH ≃ · − no asymmetry between χ and its antiparticle.
the boostfactortoobtainthe effectiveannihilationcross
The equilibriumnumber density Y canbe calculated
eq
section. As was pointed out in [31], this is not strictly
exactly under the assumption that Dark Matter follows
correct, since the freeze out process is modified by the
a Maxwell-Boltzmanndistribution. We got
boostmechanism. Sotoconsistentlyprobetheparameter
space,wemustfirstfindtherelativisticannihilationcross 45 x2
Y =g K (x), (55)
section σ0, which gives the correctrelic abundance. eq i4π4g S 2
∗
8
where K is the modified Bessel function of the second Compton scattering and inverse Compton scattering be-
2
kind. Thederivationofequation(55)canbefoundinsec- longs to the second. Double Compton scattering freezes
tionBoftheappendix. Thevaluesofg inequation(55) out at z 2.1 106 and Compton scattering freezes
S DC
1 ∗ out at z ≃5.4 1·04. Energy which is deposited in the
andh2 inequation(53)dependsonthetemperature,and C ≃ ·
it is fo∗und by interpolationin a precomputedtable. The photon gas after zDC but before zC will be redistributed
to give an entropy maximising Planck spectrum with a
number of internal degrees of freedom, g , was set to 2.
i
chemical potential µ. Energy input after z , (but before
We also need the thermally averaged cross section. If C
recombination), can not equilibrate and will result in a
we approximate the distribution function for the Dark
Compton-y distortion [35] of the Planck spectrum.
Matter gas by the non-relativistic Maxwell-Boltzmann
The relevant quantity for deriving the size of both ef-
distribution
fectsisthustherelativeenergyinputtotheCMBduring
2 these two epochs. Following [36], we write
f(β)= x32β2e−12xβ2, (56)
π
r δρ t2 ρ˙
γ ann
= dt
the thermal averageof S becomes ρ ρ
γ Zt1 γ
S(β,...)v 2x23 1S(β,...)β2e−21xβ2. (57) = t2 2Fρmχ(h1σa+nnzv)i4n2χdt, (61)
h i≃rπ Z0 Zt1 γ,0
Thisdistributionisonlynormalizedto1whenthe upper whereργ istheenergydensityoftheCMBphotonsandz
limit goes to infinity, but if we are in the non-relativistic is the redshift. F denotes the fraction of energywhich is
limitthis isasmallcorrection. (Inthe calculationwe set transfered to the CMB photons. This is independent of
the upper limit dynamically to 4 times the position of redshift when z &2500, and according to table 1 of [30],
the peak of the distribution.) Before freeze out the gas this is between 30% and 90% depending on the annihi-
is not strictly non-relativistic and we should instead use lation channel. The factor of 2 in equation (61) stems
the relativistic distribution: from the fact that we, in consistence with our freeze out
calculation, assume that χ is not its own antiparticle.
x
f(γ)dγ = βγ2(β)e−xγ(β) (58) We introduce the following relations:
K (x) ⇒
2
f(β)dβ = x γ5β2e−γx, (59) H2(t)= 4π3 g T4 (1+z)4 (62a)
K2(x) 45m2 ∗ γ,0 ⇒
p
wBehsesreel-γfuinsctthieonusoufatlhgeamsemcoandfacktionrd.anTdhKe 2thiesrtmhealmaovdeirfiaegde ddzt =−r4π5m2πpg∗−12Tγ−,02(1+z)−3 (62b)
then becomes ρ
n =n (1+z)3 = χ,0(1+z)3
χ χ,0
m
x 1 χ
Sv = S(β,...)γ5β2e−x(γ−1), (60) 3H2m2
h i k2(x)Z0 =Ωχ 0 p(1+z)3 (62c)
8πm
χ
where k (x) = K (x)ex. When γ becomes close to 1
2 2 ρ =ρ (1+z)4 (62d)
we must use the series expansion to calculate 1 γ for γ γ,0
numerical stability. The difference in the require−d cross Tγ =Tγ,0(1+z) (62e)
sectionσ from using (60) comparedto (57) was negligi- σ v =σ S(β,...)v (62f)
0 ann 0
h i h i
ble, however.
The numericalsolutionofequation(54) is notentirely Using equations (62) in equation (61) yields:
trivial, especially not since the equation is stiff and we
needto solveitrepeatedly. InsectionBofthe appendix, δργ = 405 5g−12F σ0 Ω2 m5pH04
we have described the numerical scheme we use for this ργ 64π5rπ ∗ mχ χ" Tγ6,0 #×
problem, which is the same used in the DarkSUSY soft-
z(t1) 1
ware package [34]. We recommend this scheme to others S (z,...) dz
avg
interested in Freeze-Out calculations. ×Zz(t2) 1+z
405 5 1 100GeV σ0
= g−2f
B. Distorting the Black Body spectrum 64π5rπ ∗ (cid:20) mχ (cid:21)(cid:20)10−26cm3/s(cid:21)×
z(t1) 1
(Ω h)2C S (z,...) dz. (63)
When energy is injected into the CMB photons, × χ −7Zz(t2) avg 1+z
two types of processes are needed to restore a black
body spectrum: Number changing processes and equi- Here C 2.8696 10 7 is a numerical constant. Since
7 −
librating processes. Double Compton scattering and the Dar−k M≃atter is·very cold at this time, the distribu-
bremsstrahlung belongs to the first category, while tion function is strongly peaked around its mean value,
9

1.9
−2.5 1.8 −2.5 1.25
1.7
−3 −3 1.2
1.6
log(f)10−3.5 11..45 log(f)10−3.5 1.15
−4 1.3 −4 1.1
1.2
−4.5 −4.5 1.05
1.1
−5 −3 −2.8 −2.6 −2.4 −l2o.2g (−α2) −1.8 −1.6 −1.4 −1.2 1 −5 −3 −2.8 −2.6 −2.4 −l2o.2g (−α2) −1.8 −1.6 −1.4 −1.2 1
10 10
FIG. 8: Required annihilation cross section σ0 in units of FIG. 9: The ratio σ0KD=8MeV/σ0KD=500MeV for a 200GeV
10−26cm3/s to explain the total Dark Matter abundance for WIMP.
a 200GeV WIMP and a kinetic decoupling temperature of
8MeV.
so to a good approximation, we may write S (z) 1.035
avg
S(βmean(z)). We may also assume that the Sommerfel≃d −2.5 1.03
enhancementhassaturatedatthis stage,makingthe ap-
1.025
proximation S(β (z)) S(β (z(t ))), allowing us −3
mean mean 1
≃
to pull S outside the integral which can then be done 1.02
analytically: (f)10−3.5 1.015
g
o
l 1.01
δργ 405 5g−21F 100GeV σ0 −4
ργ ≈ 64π5rπ ∗ (cid:20) mχ (cid:21)(cid:20)10−26cm3/s(cid:21)× 1.005
1+z(t ) 1
(Ω h)2C S(β (z ))ln 1 . (64) −4.5
χ 7 mean 1
− (cid:12)1+z(t2)(cid:12) 0.995
(cid:12) (cid:12)
In these calculations we have assu(cid:12)med a rad(cid:12)iation dom- −5 −3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2
(cid:12) (cid:12) log (α)
inated universe as described by the Friedmann equation 10
(62a), which breaks down at z zeq 3300, well before FIG. 10: The ratio σ0/σ0′ of cross sections for a 200GeV
∼ ≃
recombination. But as can be seen from the analytical WIMP and a 1000GeV WIMP respectively, which decouples
approximation, the dependence on z(t ) is logarithmic, kinetically at xKD =2·103.
2
so the error in doing this is insignificant. When plotting
the µ and y -distortions, we have put F = 1 for conve- and a 1000GeV WIMP for a kinetic decoupling value of
nience, but|si|ncethe dependence is linearin F, itcanbe xKD =2 103forboth. However,inarealparticlephysics
·
reinstated by multiplying the distortions by F. model we expect xKD to have a slight mass dependence,
so the actualeffect of differentmassesmay be somewhat
bigger[38].
C. Results For each point in the parameter space, we solve the
integral in equation (63) on both the interval which is
Foreachpointinthe(α,f)-parameterspace,wesolved relevant for Compton-y distortions as well as the inter-
| |
theBoltzmannequationfromx =1tox =m /T to val relevant for the CMB photons to develop a chemical
1 2 χ γ,0
find the value of σ . This depends weakly on the WIMP potentialµ. Theresultsfora200GeVparticleareshown
0
mass as well as the kinetic decoupling temperature but in figure 11 and 12. Considering the current bounds on
the overall dependence on the Sommerfeld parameters µ- and y-distortions from FIRAS [39] of µ < 9 10−5,
can be seen in figure 8. The kinetic decoupling tempera- y < 1.5 10−5, only a small portion of t|h|e para·meter
| | ·
turewastakenasaparameter,andtheeffectonthefreeze space can be ruled out. But there is a rather large part
out processis shownonfigure 9. We find that this effect of lower right part of the parameter space which is close
can be as large as 30% in agreement with [31, 37]. We at saturating the current bound. The analysis for the µ-
expect σ to be nearly independent of the WIMP mass distortionwasalreadycarriedoutin[37],andourresults
0
and this is confirmed by figure 10 which shows maxi- agrees with theirs.
mally one percent difference between a 200GeV WIMP Asisevidentfromfigure11and12,thetwoboundsare
10
4
3.5
−2.5
3
−3
2.5
(f)10−3.5 2
g
o
l
1.5
−4
1
−4.5
0.5
−5 0
−3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2
log (α)
FIG. 11: The magnitude of the Compton-|y| parameter in 10
log for a 200GeV WIMP and a kinetic decoupling temper- FIG. 13: Logarithm of theannihilation cross section in units
10
atureof 8MeV. of 10−26cm3/s in a halo with velocity dispersion of approxi-
mately 150km/s. White points are ruled out by either µ- or
|y|-bound.
4
3.5
−2.5
3
−3
2.5
(f)10−3.5 2
g
o
l
1.5
−4
1
−4.5
0.5
FIG. 12: The magnitude of thechemical potential µ in log
10
fora200GeVWIMPandakineticdecouplingtemperatureof −5 0
−3 −2.8 −2.6 −2.4 −2.2 −2 −1.8 −1.6 −1.4 −1.2
8MeV. log (α)
10
FIG. 14: Logarithm of theannihilation cross section in units
degenerate in the way that they both tends to rule out
of 10−26cm3/s in a halo with velocity dispersion of approxi-
the resonancesandthe lowerrightpartof the parameter
mately150km/s. Whitepointscanberuledout ifthebound
space. Withthecurrentlimits,the y -boundisalwaysas
| | gets improved bya factor of 500.
strongorstrongerthanthe µ-bound. One wouldsuspect
this to be the case, since the y-distortion happens at a
bound with the anisotropy bound. From [30], we have
latertimethantheµ-distortion,givingtheWIMPsmore
the bound
time to cool. This is indeed true, but only for a small
subset of the parameter space, at the resonances in the
360 10 26cm3/s m
lower left part. For the rest of the parameter space, the σ v < · − χ , (65)
Sommerfeld enhancement has already saturated at this h ann isaturated Frc 1TeV
point, and no further enhancement is possible.
We can consider what would be the allowed possibil- where Frc is the average fraction of energy being trans-
ities for the annihilation cross section in a halo having feredtotheCMBatthetimeofrecombination. Thishas
a fiducial velocity dispersionof 150km/s by removing all different values depending on the annihilation channel,
pointsthatexceedseitherbound. Wehaveplottedthisin as can be seen in table 1 of [30]. However, it is roughly
figure13forour200GeVexampleWIMP.Itis suspected of order 30% for most processes.
that a new FIRAS-like satellite, if built, could bring the Aswediscussedearlier,plugginginthestandardvalue
bound on y down to the order 10 7. In figure 14 we σ v for a thermal relic is not strictly accurate. It
| | ∼ − h ann iTH
have shown what figure 13 would look like with a fidu- is easily fixed however, by using our calculated values of
cial bound of y <10 7. It is also worth comparingthis σ , as well as the thermally averaged Sommerfeld factor
− 0
| |