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arXiv:1110.2491v2 [hep-ph] 28 Dec 2011

Preprint typeset in JHEP style - HYPER VERSION

OU-HEP-111006

Coupled Boltzmann calculation of

mixed axion/neutralino cold dark matter

production in the early universe

Howard Baera, Andre Lessaband Warintorn Sreethawonga

aDept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA

bInstituto de F´ ısica, Universidade de S˜ ao Paulo, S˜ ao Paulo - SP, Brazil

E-mail: baer@nhn.ou.edu, lessa@fma.if.usp.br, wstan@nhn.ou.edu

Abstract: We calculate the relic abundance of mixed axion/neutralino cold dark matter

which arises in R-parity conserving supersymmetric (SUSY) models wherein the strong

CP problem is solved by the Peccei-Quinn (PQ) mechanism with a concommitant ax-

ion/saxion/axino supermultiplet. By numerically solving the coupled Boltzmann equa-

tions, we include the combined effects of 1. thermal axino production with cascade decays

to a neutralino LSP, 2. thermal saxion production and production via coherent oscillations

along with cascade decays and entropy injection, 3. thermal neutralino production and

re-annihilation after both axino and saxion decays, 4. gravitino production and decay and

5. axion production both thermally and via oscillations. For SUSY models with too high a

standard neutralino thermal abundance, we find the combined effect of SUSY PQ particles

is not enough to lower the neutralino abundance down to its measured value, while at

the same time respecting bounds on late-decaying neutral particles from BBN. However,

models with a standard neutralino underabundance can now be allowed with either neu-

tralino or axion domination of dark matter, and furthermore, these models can allow the

PQ breaking scale fato be pushed up into the 1014−1015GeV range, which is where it is

typically expected to be in string theory models.

Keywords: Supersymmetry Phenomenology, Supersymmetric Standard Model, Dark

Matter, Axions.

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1. Introduction

The Standard Model (SM) of particle physics is beset by two afflictions: 1. in the scalar

(Higgs) sector of the theory, quadratic divergences require large fine-tunings of electroweak

parameters which depend on the scale Λ below which the SM is regarded as the correct

effective field theory of nature and 2. in the QCD sector of the theory, the Lagrangian

term

¯θ

32π2FAµν˜Fµν

required by ’tHooft’s solution to the U(1)Aproblem is constrained to a value¯θ ? 10−10

to gain accord with measurements of the neutron EDM[1]. The first of these is solved

by the introduction of softly broken weak scale supersymmetry (SUSY) into the theory[2]

(which receives some indirect support from the measured values of gauge couplings at

LEP[3] and from global fits to precision electroweak data[4]), while the second problem

is solved by the introduction of a global U(1)PQPeccei-Quinn (PQ) symmetry broken by

QCD anomalies[5], which requires the existence of an (“invisible”) axion[6, 7], with mass

expected in the micro-eV or below range[8]. Solving both problems simultaneously requires

supersymmetrization of the SM (usually via the Minimal Supersymmetric Standard Model,

or MSSM) along with the introduction of an axion supermultiplet ˆ a into the theory. The ˆ a

supermultiplet contains an R-parity-even spin-0 saxion field s(x) along with an R-parity-

odd spin-1

2axino ˜ a(x), in addition to the usual pseudoscalar axion field a(x):

L ∋

A

(1.1)

ˆ a =s + ia

√2

+ i√2¯θ˜ aL+ i¯θθLFa,(1.2)

in 4-component spinor notation[2].

In such a theory, it is expected that SM superpartner particles with weak scale masses

should emerge, along with a weak scale saxion, whilst the axino mass is more model de-

pendent, with m˜ a∼ keV-TeV being expected[9]. The axion, saxion and axino couplings

to matter depend on the PQ breaking scale fa1, which is required fa? 109GeV by stellar

cooling calculations[10]. The axion is often considered as a very appealing dark matter

(DM) candidate[11, 12].2

In the MSSM, DM candidates include the lightest neutralino?Z1(a WIMP), the spin-3

matter is tightly constrained and disfavored by the standard picture of Big Bang nucle-

osynthesis (BBN)[15], whilst right-hand neutrino states are expected to exist near the GUT

scale according to the elegant see-saw mechanism for neutrino mass[16]. Many authors thus

expect dark matter to be comprised of the SUSY neutralinos, a natural WIMP candidate

which is motivated by the so-called “WIMP miracle”. However, detailed analyses show

that neutralino dark matter requires a rather high degree of fine-tuning[17] to match the

2

gravitino?G or possibly the superpartner of a right-handed neutrino[14]. Gravitino dark

1Throughout this work we omit the number of generations factor N, which appear along with the PQ

scale, fa/N, in the DSFZ model and in the KSVZ model with more than one heavy quark generation. All

our results can then be trivially generalized replacing fa by fa/N.

2For a somewhat different axion/axino scenario, see Ref. [13].

– 1 –

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WMAP-measured cold DM abundance[18]:

ΩDMh2= 0.1123 ± 0.0035 at 68% CL. (1.3)

In fact, the measured abundance lies in the most improbable locus of values of neutralino

relic density as predicted by general scans over SUSY model parameter space[19].

The PQ-extended Minimal Supersymmetric Standard Model (PQMSSM) offers addi-

tional possibilities to describe the dark matter content of the universe. In the PQMSSM,

the axino may play the role of stable lightest SUSY partner (LSP)[20, 21], while the quasi-

stable axion may also constitute a component of DM[22], giving rise to mixed axion/axino

(a˜ a) CDM. In supergravity theories however, the axino mass is expected to lie at the

weak scale[23], so that the neutralino remains as LSP, and the possibility occurs for mixed

axion/neutralino (a?Z1) CDM.

the relic abundance of neutralinos in the mixed a?Z1CDM scenario. This approach applies

velocity ?σv? is approximately constant with temperature, as occurs for a wino-like or

higgsino-like neutralino[25]. Detailed calculations of the relic abundance of mixed a?Z1

were presented.

The standard calculation of the neutralino Yield Ystd

tralino number density and s is the entropy density) gives

?90/π2g∗(Tfr)?1/2

4?σv?MPTfr

where g∗(Tfr) is the number of active degrees of freedom at temperature T = Tfr, where

3√5?σv?MPm3/2

π5/2T1/2

In a recent paper, Choi et al.[24] presented a semi-analytic approach for estimating

to cases where the thermally averaged neutralino annihilation cross section times relative

CDM were performed in Ref. [26], where formulae for the neutralino and axion abundances

? Z1

≡

n?

Z1

s

(where n? Z1is the neu-

Ystd

?Z1

=

,(1.4)

Tstd

fr= m? Z1/ln[

? Z1

frg1/2

∗ (Tfr)

].(1.5)

is the freeze-out temperature and MP is the reduced Planck mass.

If instead axinos are thermally produced (TP) at a large rate at re-heat temperature

TRafter inflation, then they cascade decay to (stable) neutralinos at decay temperature

?

and can boost the neutralino abundance. The late-time injection of neutralinos into the

cosmic soup at temperatures T˜ a

D< Tfrmay cause a neutralino re-annihilation effect such

that the neutralino Yield is instead given by[24, 26]

?90/π2g∗(T˜ a

Since T˜ a

Dis typically in the MeV-GeV range, i.e. well below Tfr∼ m? Z1/20, the neutralino

abundance after re-annihilation can be highly enhanced relative to the standard cosmo-

logical picture. In addition, one must fold into the relic abundance the axion contribution

T˜ a

D=

Γ˜ aMP/?π2g∗(T˜ a

D)/90?1/4, (1.6)

Yre−ann

?Z1

|T=T˜ a

D≃

D)?1/2

D

4?σv?MPT˜ a

. (1.7)

– 2 –

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arising from coherent axion field oscillations beginning at axion oscillation temperature

Ta∼ 1 GeV.

An additional complication comes from entropy production from axino decay after

Tfr(which may dilute the neutralino abundance) or after Ta(which may dilute the axion

abundance). This may occur in the case where axinos temporarily dominate the energy

density of the universe. Depending on the PQ parameters of the PQMSSM model (fa, m˜ a,

initial axion misalignment angle θi and TR), the dark matter abundance may be either

neutralino- or axion-dominated. In fact, cases may occur where the DM relic abundance

is shared comparably between the two. In the latter case, it might be possible to directly

detect relic neutralino WIMP particles as well as relic axions!

While the semi-analytic treatment of Ref’s [24] and [26] provides a broad portrait of

the mixed a?Z1CDM picture, a number of important features have been neglected. These

• For bino-like neutralinos, ?σv? ∼ a + bT2where a ∼ 0 since we mainly have p-wave

annihilation cross sections. In this case, ?σv? is no longer independent of temperature,

and the simple formulae 1.4 and 1.7 are no longer valid.

include the following.

• In Ref’s [24] and [26], the effects of saxion production and decay in the early universe

are neglected. In fact, saxion thermal production or production via coherent oscilla-

tions (CO)[27], followed by late time saxion decay, may inject considerable entropy

into the early universe, thus diluting all relics present at the saxion decay temper-

ature Ts

D. Saxions may also add to the neutralino abundance via decays such as

s → ˜ g˜ g, followed by gluino cascade decays. There exists the possibility of saxion and

axino co-domination of the universe. In this case, there might be a second neutralino

re-annihilation taking place at Ts

D.

• The treatments of [24] and [26] invoke the “sudden decay” approximation for late-

decaying axinos, whereas in fact the decay process is a continuous one proceeding in

time until the decaying species is highly depleted (all have decayed).

• The treatments of [24] and [26] largely ignore the effect of gravitino production and

decay in the early universe.

To include the above effects into a calculation of the mixed a?Z1relic abundance, one

a full solution of the coupled Boltzmann equations which govern various abundances of

neutralinos, axinos, axions, saxions, gravitinos and radiation.

Toward this end, in Sec. 2 we present a simplified set of coupled Boltzmann equations,

which we use to calculate the relic abundance of mixed axion/neutralino dark matter.

More details about the approximations made and each term present in our equations are

discussed in Appendix A.

In Sec.3, we present various numerical results for the mixed a?Z1 CDM scenario

saxion field, adjusting the parameters of the PQMSSM can only increase the neutralino

must go beyond the semi-analytic treatment presented in Ref’s [24, 26], and proceed with

using the full set of Boltzmann equations. We find that, even after the inclusion of the

– 3 –

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abundance, and not decrease it, while at the same time respecting bounds on late-decaying

neutral particles from BBN. This result is the same as found in Refs. [24] and [26], but

now corresponds to a much stronger statement, since the saxion entropy injection had

been neglected in the previous works. Furthermore, our results also apply to models with

bino-like neutralinos, which could not be studied in the semi-analytical framework used in

Refs. [24] and [26].

Since the neutralino abundance can be only enhanced in the PQMSSM, in models such

as mSUGRA, those points which are excluded by a standard overabundance of neutralinos

are still excluded in the PQMSSM! This rather strong conclusion does depend on at least

three assumptions: 1. that thermal axino production rates are not suppressed by low-lying

PQ-charged matter multiplets[28]3, 2. that saxion decay is dominated by gluon and gluino

pairs and 3. that the assumed saxion field strength s(x) ≡ θsfais of order the PQ-breaking

scale fa, i.e. that θs∼ 1.

We also examine several cases with a standard underabundance of neutralino dark

matter. In these cases, again the neutralino abundance is only increased (if BBN constraints

are respected).Thus, adjustment of PQMSSM parameters can bring models with an

underabundance of neutralinos into accord with the measured DM relic density. In these

cases, the DM abundance tends to be neutralino-dominated. Also, in these cases, solutions

exist where the PQ scale fais either near its lower range, or where fais much closer to

MGUT, with fa∼ 1014GeV typically allowed. This is much closer to the scale of fawhich is

thought to arise from string theory[31]. In Sec. 5, we present a summary and conclusions.

2. Mixed axion/neutralino abundance from coupled Boltzmann equations

Here, we present a brief description of our procedure to calculate the relic abundance of

mixed a?Z1CDM in the PQMSSM. A more detailed discussion is left to Appendix A.

2.1 Boltzmann equations

The general Boltzmann equation for the number density of a particle species can be gener-

ically written as[32]:

˙ ni+ 3Hni= Si−1

γiΓini

(2.1)

where Sirepresents a source term, Γiis the decay width and γiis the relativistic dilation

factor to take into account the suppressed decays of relativistic particles. To describe the

thermal production of a particle species i as well as its decoupling from the radiation fluid

and the non-thermal production coming from other particles decays, we include in Sithe

following terms:

Si= −[n2

i− (neq

i(T))2]?σv?i(T) +

?

j

BR(j,i)Γjnj

γj

(2.2)

3Here, we assume standard rates for thermal axino production as calculated in the literature[21, 29, 30].

In Ref. [28], it has been shown that if PQ-charged matter multipletsˆΦ exist well below the PQ breaking

scale fa, then axino production is suppressed by factors of mΦ/TR.

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where ?σv? is the (temperature dependent) thermally averaged annihilation cross section

times velocity for the particle species i, neq

iis its equilibrium number density and BR(j,i)

is the branching fraction for particle j to decay to particle i.4

The Boltzmann equation then becomes:

˙ ni+ 3Hni= −Γimin2

i

ρi

+ [(neq

i(T))2− n2

i]?σv?i+

?

j

BR(j,i)Γjmj

n2

ρj

j

,(2.3)

where we have used γi= ρi/mini. As discussed in Appendix A, the above equation is also

valid for coherent oscillating fields once we take BR(j,i) = 0 and ?σv?i= 0.

It is also convenient to write an equation for the evolution of entropy:

˙S =

?2π2

45g∗(T)1

S

?1/3

R4?

i

BR(i,X)1

γiΓiρi

or

˙S =R3

T

?

i

BR(i,X)Γimini

(2.4)

where BR(i,X) is the fraction of energy injected in the thermal bath from i decays.

Along with Friedmann’s equation,

H =1

R

dR

dt

=

?ρT

3M2

P

, with ρT≡

?

i

ρi+π2

30g∗(T)T4, (2.5)

the set of coupled differential equations, Eq’s. 2.3, 2.4 and 2.5, can be solved as a function

of time. More details on the solution of the above equations and the expressions used for

?σv?i, BR(i,j) and BR(i,X) are presented in Appendix A.

2.2 Present day abundances and constraints from BBN

To compute the relic density of neutralinos and axions we evolve the various particle and

sparticle abundances from T = TRuntil the final temperature TF is reached at which all

unstable particles (save the axion itself) have decayed. The relic densities of the various

dark matter species labeled by i are then given by:

Ωih2=ρi(TF)

s(TF)×s(TCMB)

ρc/h2

. (2.6)

In our calculations, a critical constraint comes from maintaining the success of the

standard picture of Big Bang nucleosynthesis. Constraints from BBN on late decaying

neutral particles (labeled X) have been calculated recently by several groups[34, 35, 36]

(we explicitly use the results of Ref. [36]) and are presented as functions of 1. the decaying

4In this paper, i is summed over 1. neutralinos?Z1, 2. TP axinos ˜ a, 3. and 4. CO- and TP-produced

saxions s(x), 5. and 6. CO- and TP-axions a, 7. TP gravitinos?G and radiation. We allow for axino

decay to g˜ g, γ? Zi and Z? Zi states (i = 1 − 4), and saxion decay to gg and ˜ g˜ g. Additional model-dependent

saxion decays e.g. to aa and/or hh are possible and would modify our results. We assume?G decay to all

particle-sparticle pairs, and include 3-body gravitino modes as well[33].

– 5 –

Page 7

neutral particle’s hadronic branching fraction Bh, 2. the decaying particle’s lifetime τX,

and 3. the decaying particle’s relic abundance ΩXh2had it not decayed. The constraints

also depend on 4. the decaying particle’s mass mX. We have constructed digitized fits

to the constraints given in Ref. [36], and apply these to late decaying gravitinos, axinos

and saxions. Typically, unstable neutrals with decay temperature below 5 MeV (decaying

during or after BBN) and/or large abundances will be more likely to destroy the predicted

light element abundances.

2.3 Example: calculation from a generic mSUGRA point

As an example calculation, we adopt a benchmark point from the paradigm minimal super-

gravity model (mSUGRA), with parameters (m0, m1/2, A0, tanβ, sign(µ)) = (400 GeV,

400 GeV, 0, 10, +). The sparticle mass spectrum is generated by Isasugra[37], and has

a bino-like neutralino with mass m? Z1= 162.9 GeV and a standard relic abundance from

IsaReD[38] of Ωstd

the standard neutralino freeze-out calculation). We assume a gravitino mass m? G= 1 TeV.

Here, we work in the PQMSSM framework, and take TR= 1010GeV with PQ param-

eters as m˜ a= 1 TeV, ms= 5 TeV, θi= 0.5 and fa= 1012GeV. We also take θs= 1,

where θsfais the initial field amplitude for coherent oscillating saxions. The various energy

densities ρiare shown in Fig. 1 for i = R (radiation),?Z1(neutralinos), aTP(thermally

sCO(coherent oscillating saxions), ˜ aTP(thermally produced axinos) and?GTP(thermally

where R0 is the scale factor at T = TR. We also plot the temperature T of radiation

(green-dashed curve).

We see that, at R/R0< 1010, the universe is indeed radiation-dominated. At T ≫ 1

TeV, the TP axions, saxions and axinos all have similar abundances. At these temperatures,

the saxion coherent abundance as well as the gravitino thermal abundance are far below the

other components. As the universe expands and cools, most components are relativistic,

and decrease with the same slope as radiation: ρi ∼ T−4. The exception is the CO-

produced saxions, which are non-relativistic, and fall-off as ρCO

temperature T ∼ 1 TeV, and the thermally-produced axinos, saxions and gravitinos become

non-relativistic, so now ρTP

neutralinos begin to freeze-out, and their abundance falls steeply. At T ∼ m? Z1/20, they do

freeze-out, and normally their density would fall as ρ? Z1∼ T−3, as indicated by the blue dot-

dashed curve, which shows neutralino abundance in the MSSM, without PQ-augmentation.

In the PQMSSM however, saxions– and later still axinos– begin decaying in earnest, and

feed into the neutralino abundance, preventing its usual fall as T−3. At T ∼ 0.5 GeV,

the energy density of axinos surpass the radiation component and the universe becomes

axino-dominated until the axino decays at T ∼ 10 MeV. Also, around R/R0∼ 3×1010with

T ∼ 1 GeV, CO production of axions begins, and by R/R0∼ 4×1011, with T ? ΛQCD, its

abundance begins to fall as T−3. For even lower temperatures (T < 10 MeV), the axinos

have essentially all decayed, feeding back into the neutralino abundance, and also increasing

the entropy per co-moving volume, which would otherwise be conserved. At R/R0∼ 1014,

? Z1h2= 1.9 (it would thus be excluded by WMAP7 measurements assuming

produced axions), aCO(coherent oscillating axions), sTP(thermally produced saxions),

produced gravitinos). The energy densities are plotted against scale factor ratio R/R0,

s

∼ T−3. At R/R0∼ 107, the

˜ a,s,? G∼ T−3. For even lower temperatures with R/R0 ∼ 109,

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0

R/R

6

10

7

10

8

10

9

10

10

10

11

10

12

10

13

10

14

10

15

10

16

10

17

10

18

10

19

10

20

10

)

4

(GeV

ρ

-30

10

-23

10

-16

10

-9

10

-2

10

5

10

12

10

19

10

22

10

T = 0.5 GeV

T = 10 MeV

-30

10

-23

10

-16

10

-9

10

-2

10

5

10

12

10

19

10

22

10

R

Z~

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

T (GeV)

(MSSM)

1

1

(TP)

(CO)

(TP)

(CO)

(TP)

a~

(TP)

G~

a

a

s

s

Z~

ρ

= 1

s θ

= 0.5,

iθ

= 5 TeV

s

m

= 1 TeV

G~

= m

a~

m

GeV

12

=10

a

f

GeV

10

=10

R

T

Figure 1:

densities versus scale factor R.

(m0,m1/2,A0,tanβ,sign(µ)) = (400 GeV,400 GeV,0,10,+). We also take m? G= 1 TeV and

TR= 1010GeV and PQ parameters m˜ a= 1 TeV, ms= 5 TeV, θi= 0.5, θs= 1 with fa= 1012

GeV.

Evolution of radiation, neutralino, axion, saxion, axino and gravitino energy

We adopt an mSUGRA SUSY model with parameters

the universe moves from radiation domination to matter (neutralino) domination, while

at even lower temperatures, the gravitinos decay away. In this case, the final neutralino

abundance is Ω? Z1h2∼ 90017– far beyond its standard value. This is mainly due to its

abundance being augmented by thermal axino and saxion production and cascade decay

to neutralinos. In the standard axion cosmology, the axion abundance would have been

Ωstd

gravitino decays has diluted its abundance to just Ωah2∼ 0.004.

As an example of the relevance of using the full set of Boltzmann equations instead of

the semi-analytical approach of Refs. [24] and [26], we compare in Fig. 2 the neutralino and

axion relic densities as a function of the axino mass using the Boltzmann equation formalism

and the semi-analytical approach. The other PQMSSM parameters are the same as used

in Fig. 1, but to compare with the semi-analytical results of Refs. [24] and [26] we neglect

the saxion component. For these choices of PQ parameters and for m˜ a? 50 TeV, the axino

decays after neutralino freeze-out (as seen on Fig. 1), significantly enhancing its final relic

abundance. Furthermore, the axino decay injects entropy, diluting the axion abundance.

As we can see from Fig. 2, the axion relic density obtained using the analytical expressions

derived in Ref. [26] agree extremely well with the solution of the Boltzmann equations. On

the other hand, the analytic neutralino abundance disagrees with the Boltzmann solution

by almost an order of magnitude for m˜ a? 50 TeV. The primary reason for this is the fact

ah2∼ 0.06[39]. In the case illustrated here, entropy injection from saxion, axino and

– 7 –

Page 9

(GeV)

a~

m

3

10

4

10

5

10

2

h

Ω

-5

10

-3

10

-1

10

10

3

10

5

10

7

10

8

10

2

h

1

h

Z~

Ω

2

a

Ω

Analytical

Boltzmann

Figure 2: Neutralino and axion relic densities as a function of the axino mass for θi = 0.5,

TR = 1010GeV, fa = 1012GeV and the mSUGRA point (m0,m1/2,A0,tanβ,sign(µ)) =

(400 GeV,400 GeV,0,10,+). The solid lines correspond to the solution of the Boltzmann equations

while the dashed lines correspond to the results obtained using the analytical expressions derived

in Ref. [26].

that, for this mSUGRA point, the neutralino is bino-like and ?σv?? Z1is no longer constant,

but strongly depends on the temperature. This dependence has not been included in the

semi-analytical approach. Also, the sharp transition seen in the semi-analytical result

at m˜ a ≃ 18 TeV, where T˜ a

sudden decay approximation. As shown by the Boltzmann solution, the enhancement of

the neutralino relic abundance smoothly decreases, going up to m˜ a≃ 50 TeV.

Dbecomes bigger than Tfr, is artificially introduced by the

3. Neutralino abundance in the PQMSSM

3.1 Neutralino Abundance in several PQMSSM models

In this section, we adopt four SUSY benchmark models listed in Table 1. The first two

points, labeled BM1 and BM2, are generic mSUGRA points with a bino-like?Z1 which

sign(µ)) = (400 GeV, 400 GeV, 0, 10, +) we have m? Z1= 162.8 GeV with a standard abun-

dance Ωstd

(3000 GeV, 1000 GeV, 0, 10, +) with m? Z1= 436.3 GeV and Ωstd

point, BM3, is a mSUGRA point with an apparent underabundance of neutralino dark

matter, with m? Z1= 163.8 GeV, mA = 367.5 GeV, lying in the A-funnel region[40], so

give rise as expected to an apparent excess of CDM. For BM1, with (m0, m1/2, A0, tanβ,

?Z1h2= 1.9, while the second point (BM2) has (m0, m1/2, A0, tanβ, sign(µ)) =

? Z1h2= 49.6. The next

– 8 –

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BM1

400

400

103

10

162.9

1.9

BM2

3000

1000

103

10

436.3

49.6

BM3

400

400

103

55

163.8

0.019

2.1 × 10−8

BM4

0

AMSB

5 × 104

10

142.1

0.0016

4.3 × 10−9

m0

m1/2

m3/2

tanβ

m? Z1

Ωstd

σSI(?Z1p) pb

?Z1h2

8.1 × 10−10

1.1 × 10−10

Table 1: Masses and parameters in GeV units for several benchmark points computed with

Isajet7.81 using A0= 0 and mt= 173.3 GeV.

Ωstd

and TR, in order to see if the relic density of mixed a?Z1CDM can lie in the WMAP-allowed

neutralino with Ωstd

In order to keep our results as general as possible, we will not assume particular PQ

parameters, but instead we scan over the following parameter values:

? Z1h2= 0.019. For all cases, we take m? G= 1 TeV, but now will vary the PQ parameters

region. The last point is taken from the gaugino AMSB model[41, 42] and has a wino-like

?Z1h2= 0.0016, but with m3/2≡ m? G= 50 TeV.

109GeV < fa < 1016GeV,

500 GeV < m˜ a< 104GeV,

103GeV < ms< 105GeV,

(3.1)

(3.2)

(3.3)

0.1 < θs < 10,

105GeV < TR < 1012GeV.

(3.4)

(3.5)

Since we will be mostly concerned with the neutralino relic abundance, we leave the axion

mis-alignment angle θiundetermined for now.

3.2 Benchmark BM1

Our results are shown as the resultant relic density of neutralinos Ω? Z1h2in the PQMSSM,

where we plot each model versus fain Fig. 3. The blue points are labeled as BBN-allowed,

while red points violate the BBN bounds as described in Sec. 2.2.

From Fig. 3, we see that at low values of PQ breaking scale fa∼ 109− 1011GeV, the

value of Ω? Z1h2is always bounded from below by its standard value Ωstd

points with Ω? Z1h2≃ 1.9 are typically those for which axinos and saxions decay before Tfr,

or those for which axino/saxion production is suppressed by low TRso that axinos/saxions

decays do not significantly contribute to Ω? Z1h2.

In Fig. 3, frequently the neutralino abundance is enhanced beyond 1.9, making these

points even more excluded. The reason why points only have enhanced relic densities at

the lower farange is because the axino-matter coupling is proportional to 1/fa, and so

thermal axino production is enhanced compared to higher favalues. In addition, ˜ a → ˜ gg

decays may be phase space suppressed, so that axino decay takes place at temperature

T˜ a

D< Tfr, thereby augmenting the neutralino abundance. Saxion decay is never phase

? Z1h2∼ 1.9. Those

– 9 –

Page 11

(GeV)

a f

9

10

10

10

11

10

12

10

13

10

14

10

15

10

16

10

17

10

2

h

Z~

Ω

1

-9

10

-7

10

-5

10

-3

10

-1

10

10

3

10

5

10

7

10

9

10

BBN Allowed

BBN Excluded

) = (400 GeV, 400 GeV, 0, 10, > 0)

µ

,

β

, tan

0

, A

1/2

, m

0

(m

= 0.1123

2

h

1

Z~

Ω

Figure 3: Calculated neutralino relic abundance from mSUGRA model BM1 versus fa. We take

m? G= 1 TeV. The spread in dots is due to a scan over PQ parameters fa, TR, m˜ a, msand θs.

space suppressed, since s → gg is always possible, so at the lower range of fa, saxion decay

typically takes place at Ts

no longer decay before neutralino freeze-out, and so the neutralino abundance is always

enhanced. The value of fawhere the neutralino abundance is always enhanced is somewhat

an artifact of our scanning range, since if we allow m˜ a> 104GeV, axinos could become

shorter-lived for a higher value of fasince Γ˜ a∼ m3

At even higher values of fa? 1012GeV, axino/saxion thermal production becomes

increasingly suppressed, while saxion production via CO becomes enhanced: entropy dilu-

tion by saxions starts winning over neutralino production from thermal axino production

and decay. Also, both saxion and axino become even longer-lived, so more points become

BBN-disallowed. Although the entropy injection from s → gg decays grows with fa, the

BBN-allowed blue points are never pushed below Ω? Z1h2∼ 1.9, since s → ˜ g˜ g also injects

additional neutralinos into the thermal bath.5

To understand why the neutralino injection from saxion decays always wins over the

entropy dilution of the neutralino abundance, we must look at the neutralino Yield from

saxion decays. For simplicity, we will neglect the axino component as well as neutralino

re-annihilation at Ts

Dand assume that the PQ parameters are chosen so the universe has

a saxion-dominated era, since this is the only scenario with significant entropy injection.

D> Tfr. For values of fa∼ 1011GeV and beyond, axinos can

˜ a/f2

a.

5We checked the effect of artificially turning off s → ˜ g˜ g decays in Fig. 3. In this case, at high fa ? 1014

GeV, the enhanced saxion production via COs produces only entropy dilution of the neutralino abundance,

and some BBN-allowed points remain with a highly suppressed neutralino abundance at high fa. This effect

has lead to claims that large fa ∼ MGUT values may be allowed in SUSY models due to entropy injection

by saxions[43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. By properly including s → ˜ g˜ g decay and the concommitant

neutralino re-annihilation at Ts, the pure entropy injection effect is counter-balanced in this case, and the

highly diluted cases become BBN-forbidden.

– 10 –

Page 12

Under these assumptions, the Yield of neutralinos emitted from saxion decays is simply

given by[52]:

Y? Z1=1

rYs× 2BR(s → ˜ g˜ g)

where the factor 2 above takes care of the multiplicity of neutralinos from each saxion

decay and r is the entropy injection factor, which can be approximated by

(3.6)

r =Te

Ts

D

, (3.7)

where Ts

Dis the saxion decay temperature and Te= 4msYs/3 (see Refs. [52, 53]). Therefore:

Y? Z1=3

2

Ts

msBR(s → ˜ g˜ g) ⇒

D

Ωs

? Z1h2≃ 4 × 108GeV−1m? Z1

Ts

msBR(s → ˜ g˜ g).

D

(3.8)

The above expression shows that the relic density can be suppressed for large ms, small Ts

and/or small BR. However, as seen in Fig. 3, such suppression never seems to drive Ω? Z1h2

below its standard value, except in the BBN excluded region. To see why this happens,

using Eq. 3.8 we show in Fig. 4 contours of Ωs

with TR= 106GeV and θs= 1. We also show the BBN excluded region (Ts

and the region with Ts

D> Te(r < 1), where there is no saxion dominated era and Eq. 3.8

is no longer valid. As we can see, the allowed region (white) can only satisfy the WMAP

constraints at very large ms and fa values. The main reason for the low Ω? Z1h2values

obtained in this region is due to the suppression of BR(s → ˜ g˜ g). This can be seen in Fig.

5, where we show the branching ratio as a function of msfor the same benchmark point.

We can see that– for the region where the s → ˜ g˜ g decay mode is closed– the saxion lifetime

falls into the BBN-forbidden zone. This can also be seen in Fig. 3, where all the low Ω? Z1h2

points at large faare BBN excluded.

As seen from the above results, the neutralino relic abundance can indeed be diluted

by including the saxion field, but only at the expense of going to extremely high msand

favalues. However, since msis expected to be of order the soft SUSY masses (or ∼ m? Gin

e.g. AMSB models), we consider such high values extremely unnatural. Furthermore, the

PQMSSM is most likely not the correct effective theory at fa> 1016− 1019GeV, where

we expect a Grand Unified theory and/or large supergravity corrections. Nonetheless, to

confirm the approximate results obtained from Eq. 3.8, we extend our previous scan over

to

fa∈ [1015, 1022] GeV ,

and use the full set of Boltzmann equations to compute the neutralino relic abundance.

The results are shown in Fig. 6, where we plot in the msvs. faplane all solutions satisfying

Ω? Z1h2< 0.11. As we can see, the numerical results agree very well with the analytical

results in Fig. 4. The only discrepancy is in the region near Te = Ts

present viable solutions in the scan. This is simply due to the fact that in our estimate of

Eq. 3.8, we neglected the neutralino freeze-out component, which becomes dominant when

r ≃ 1 or Te≃ Ts

D

?Z1h2in the msvs. faplane for the BM1 point,

D< 5 MeV)

ms∈ [104, 109] GeV(3.9)

D, which does not

D, increasing the value of Ω? Z1h2in this region.

– 11 –

Page 13

(GeV)

s

m

3

10

4

10

5

10

6

10

7

10

(GeV)

a

f

14

10

15

10

16

10

17

10

18

10

19

10

20

10

21

10

< 5 MeV

s

D

T

s

D

< T

e

T

= 0.01

2

h

1

Z~

Ω

= 0.11

2

h

1

Z~

Ω

= 1

2

h

1

Z~

Ω

= 1

s θ

GeV,

6

= 10

R

= 951 GeV, T

g~

= 163 GeV, m

1

Z~

m

Figure 4: Regions in the msvs. faplane where Ts

of constant Ωs

D< 5 MeV and Ts

D> Te. We also show contours

? Z1h2as estimated using Eq. 3.8.

From the results presented above, we see that for reasonable values of fa, msand θs,

the result illustrated for point BM1 in Fig. 3 seems to generalize to all SUSY model points

with a standard neutralino overabundance: SUSY models with a standard overabundance of

neutralino dark matter are still at least as excluded when augmented by the PQ mechanism.

In Fig. 7, we show the evolution of various energy densities versus the scale factor

for a large fa value. In this case, we see that the universe is radiation-dominated out

to R/R0∼ 108, whereupon it becomes saxion dominated. If only s → gg is considered,

the saxion entropy injection would cause a large dilution of neutralinos. But by including

s → ˜ g˜ g decays, we see the neutralino enhancement during 107? R/R0? 109. We also

show by the dash-dotted line the expected neutralino energy density by neglecting s → ˜ g˜ g

decays: in this case, the neutralino abundance is highly suppressed compared to the case

where s → ˜ g˜ g is accounted for.

3.3 Benchmark BM2

To emphasize some of the generality of our previous results, we show a further point with

a standard overabundance of neutralinos in Fig. 8, with (m0,m1/2,A0,tanβ,sign(µ)) =

(3000 GeV,1000 GeV,0,10,+), for which Ωstd

again we find that for low fa, Ω? Z1h2either remains at its standard value (if axinos/saxions

decay before freeze-out), or are enhanced (if axinos/saxions decay after freeze-out). At

high fa, entropy dilution from CO-produced saxions again can suppress Ω? Z1h2, but the

?Z1h2∼ 50. By scanning over PQ parameters,

– 12 –

Page 14

(GeV)

s

m

3

10

4

10

5

10

6

10

7

10

)

g~

g~

→

BR(s

-10

10

-9

10

-8

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

-1

10

1

Figure 5: Branching ratio of saxion decays into gluino pairs as a function of ms, for m˜ g = 951

GeV.

suppression is counterbalanced by s → ˜ g˜ g decays: only BBN-forbidden points where msis

so light that s → ˜ g˜ g is kinematically closed yield points with Ω? Z1h2< 0.11.

3.4 Benchmark BM3: A-resonance region

In Fig. 9, we show the neutralino abundance in the case of an mSUGRA point lying in the

A-resonance annihilation region[40] where 2m? Z1∼ mA. We adopt mSUGRA parameters

(m0,m1/2,A0,tanβ,sign(µ)) = (400 GeV,400 GeV,0,55,+), for which Ωstd

a standard underabundance.6In this case, a scan over PQ parameters yields many points

at low fawith Ω? Z1h2∼ 0.02 − 10. Thus, the standard neutralino underabundance may

be enhanced up to the WMAP-allowed value, or even beyond. As we push to higher fa

values, the axino becomes so long-lived that it only decays after neutralino freeze-out, and

hence the neutralino abundance is always enhanced. Above fa∼ 1012GeV, the neutralino

abundance is enhanced into the WMAP-forbidden region, with Ω? Z1h2always larger than

0.11. As we push even higher in fa, then axino production is suppressed, but CO-production

of saxions becomes large. Entropy dilution turns the range of Ω? Z1h2back down again, and

at fa∼ 1014GeV, some BBN-allowed points again reach Ω? Z1h2∼ 0.11. In this case, rather

large favalues approaching MGUT are allowed.

For the points with Ω? Z1h2< 0.11, the remaining dark matter abundance can be

accommodated by axions via a suitable adjustment of the initial axion mis-alignment angle

?Z1h2∼ 0.02, i.e.

6We have also scanned a benchmark point in the hyperbolic branch/focus point region[55] of mSUGRA,

again with a standard underabundance of neutralino dark matter. The Ω?

tively much like Fig. 9. We do not present these results here in the interests of brevity.

Z1h2vs. fa results look qualita-

– 13 –

Page 15

(GeV)

s

m

3

10

4

10

5

10

6

10

7

10

8

10

9

10

10

10

(GeV)

a

f

14

10

15

10

16

10

17

10

18

10

19

10

20

10

21

10

22

10

23

10

24

10

BBN Allowed

BBN Excluded

< 0.11

2

h

1

Z~

Ω

< 5 MeV

s

D

T

s

D

< T

e

T

= 0.11

2

h

1

Z~

Ω

Figure 6: Points with Ω? Z1h2< 0.11 obtained through the random scan described in the text. For

comparison, we also show the curves for Ts

from Eq. 3.8.

D< 5 MeV and Ts

D> Teand Ωs

? Z1h2= 0.11 obtained

θi. In Fig. 10, we show the required value of θineeded to enforce Ω? Z1h2+ Ωah2= 0.11.

At low fa, the points satisfying Ω? Z1h2< 0.11 have axinos and saxions decaying before the

neutralino freezes out and consequently before axions start to oscillate. Hence the axion

relic density is not affected by the entropy injection of axinos/saxions and is given by the

standard expression[39]:

?

From the above equation, we see that as faincreases, θimust decrease in order to maintain

Ω? Z1h2+ Ωah2= 0.11. This behavior is clearly seen in Fig. 10 for fa< 1012GeV. Once

fabecomes sufficiently large so axinos and saxions start to decay after the axion starts to

oscillate, the entropy injected from saxions and axinos considerably dilute the axion relic

density, thus allowing for larger θivalues. However, as seen in Fig. 10, this only happens

for the BBN-forbidden solutions at fa? 1014GeV. The only BBN-allowed points at large

fawith Ω? Z1h2< 0.11 are the ones where the saxion production is either suppressed or

where it decays before neutralino freeze-out. In this case there is no significant entropy

injection and the axion relic density is once again given by Eq. 3.10. Thus, extremely

small values of θiare required in order to suppress the axion relic density at large fa, as

seen in Fig. 10. Therefore these points tend to have neutralino domination of the dark

matter density, rather than axion domination. For these points, the large neutralino halo-

annihilation rates, enhanced by the A-resonance, may lead to visible production rates of

Ωah2≃ 0.23θ2

i

fa

1012GeV

?7/6

.(3.10)

– 14 –

Page 16

0

R/R

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

10

10

10

11

10

12

10

13

10

14

10

15

10

)

4

(GeV

ρ

-30

10

-23

10

-16

10

-9

10

-2

10

5

10

12

10

19

10

25

10

R

Z~

ρ

ρ

ρ

ρ

ρ

ρ

ρ

ρ

T (GeV)

(no s

Z~

1

(TP)

(CO)

(TP)

(CO)

(TP)

a~

(TP)

G~

a

a

s

s

) g~ g~

→

1

ρ

= 1

s θ

= 0.5,

iθ

= 10 TeV

s

m

= 1 TeV

G~

= m

a~

m

GeV

15

=10

a

f

GeV

6

=10

R

T

Figure 7: Evolution of radiation, neutralino, axion, saxion, axino and gravitino energy densities

versus scale factor R starting at T = TR. We adopt an mSUGRA SUSY model with parameters

(m0,m1/2,A0,tanβ,sign(µ)) = (400 GeV,400 GeV,0,10,+). The PQ parameters are listed on the

plot.

γs, e+s and ¯ ps in cosmic ray detectors[56], while corresponding direct neutralino detection

rates may remain low.

3.5 Benchmark BM4: AMSB with wino-like neutralino

In Fig. 11, we plot Ω? Z1h2for an anomaly-mediated SUSY breaking model (AMSB) with

a wino-like neutralino[41]. We choose the gaugino-AMSB model with m0∼ A0∼ 0, since

this model avoids tachyonic sleptons without introduction of an additional scalar mass

parameter[42]. Model parameters are (m3/2, tanβ, sign(µ)) = (50 TeV,10,+), with a

standard abundance Ωstd

see that for fa∼ 109− 1015GeV, the neutralino abundance can be enhanced and brought

into accord with measured values. For low fa, axino production and decay augments the

abundance, while for high fa, saxion production and decay both augments and dilutes

the abundance. In this case, as with BM3, the standard underabundance of DM can be

augmented and brought into accord with cosmological measurements. Unlike BM3, there

exists no intermediate range of fawhich is always excluded by the production of too much

neutralino DM. The PQ scale can be as large as fa∼ 1015GeV, in accord with expectations

from string theory.

Originally, Moroi and Randall had proposed augmenting the relic wino abundance

from AMSB via moduli production and decay[57, 58, 59]. Here, we see that an alternative

?Z1h2≃ 0.0016, far below the measured value. From the figure, we

– 15 –

Page 17

(GeV)

a f

9

10

10

10

11

10

12

10

13

10

14

10

15

10

16

10

17

10

2

h

Z~

Ω

1

-9

10

-7

10

-5

10

-3

10

-1

10

10

3

10

5

10

7

10

9

10

BBN Allowed

BBN Excluded

) = (3000 GeV, 1000 GeV, 0, 10, > 0)

µ

,

β

, tan

0

, A

1/2

, m

0

(m

= 0.1123

2

h

1

Z~

Ω

Figure 8: Calculated neutralino relic abundance versus fafrom mSUGRA SUSY model BM2. The

spread in dots is due to a scan over PQ parameters fa, TR, m˜ a, ms, θs.

(GeV)

a f

9

10

10

10

11

10

12

10

13

10

14

10

15

10

16

10

17

10

2

h

Z~

Ω

1

-9

10

-7

10

-5

10

-3

10

-1

10

10

3

10

5

10

7

10

9

10

BBN Allowed

BBN Excluded

) = (400 GeV, 400 GeV, 0, 55, > 0)

µ

,

β

, tan

0

, A

1/2

, m

0

(m

= 0.1123

2

h

1

Z~

Ω

Figure 9: Calculated neutralino relic abundance versus fafrom mSUGRA SUSY model BM3. The

spread in dots is due to a scan over PQ parameters fa, TR, m˜ a, ms, θs.

mechanism introducing the several PQMSSM fields can also do the job. Direct and indirect

detection rates for wino-like neutralinos have been presented in Ref. [60].

4. The case of very large θsand ms< 2m˜ g

From the results presented in Secs. 3.2-3.5, it seems difficult to suppress the neutralino

– 16 –

Page 18

(GeV)

a f

9

10

10

10

11

10

12

10

13

10

14

10

15

10

16

10

17

10

iθ

-2

10

-1

10

1

10

BBN Allowed

BBN Excluded

< 0.11

2

h

1

Z~

Ω

Figure 10: Values of the axion mis-alignment angle θifor the points in Fig. 9 with Ω? Z1h2< 0.11.

The parameter θiis chosen such as Ω? Z1h2+ Ωah2= 0.11.

(GeV)

a f

9

10

10

10

11

10

12

10

13

10

14

10

15

10

16

10

17

10

2

h

Z~

Ω

1

-9

10

-7

10

-5

10

-3

10

-1

10

10

3

10

5

10

7

10

9

10

BBN Allowed

BBN Excluded

) = (0 GeV, 161.4 GeV, 0, 10, > 0)

µ

,

β

, tan

0

, A

1/2

, m

0

inoAMSB (m

= 0.1123

2

h

1

Z~

Ω

Figure 11: Calculated neutralino relic abundance versus fa from inoAMSB model BM4. The

spread in dots is due to a scan over PQ parameters fa, TR, m˜ a, ms, θs.

CDM abundance below the standard neutralino abundance. This conclusion relies on the

fact that the (CO) saxion production and decay are correlated through the value of the PQ

scale, since the saxion field strenght (s(x) = θsfa)– which sets the amplitude of the coherent

oscillations– is assumed to be of order fa(θs∼ O(0.1 − 10)). Hence large (CO) saxion

production only happens at large fa values and leads to late decaying saxions, usually

– 17 –

Page 19

violating the BBN bounds. As also discussed above, the BBN bounds can be avoided if

saxions have masses in the multi-TeV range, but then the s → ˜ g˜ g decay is kinematically

allowed and the injection of neutralinos enhances the CDM abundance. However, if the

saxion field strength (s(x)) is not set by the PQ breaking scale, but by a much larger

scale, such as the reduced Planck mass (as suggested in some models[54]), it is possible

to envision a large production of coherent oscillating saxions even at small fa values.

In this scenario, small fa easily satisfies the BBN bounds, allowing for sub-TeV saxion

masses, such as ms< 2m˜ g. Thus, assuming s(x) ≫ fa(θs≫ 1), it is possible to have

large saxion production via coherent oscillations, small favalues and small saxion masses

without violating the BBN bounds. In this case, if ms< 2m˜ g, saxion decay leads to large

entropy production, but does not inject neutralinos.

To illustrate the large θs (≫ 1) scenario, in Fig. 12 we fix the initial saxion field

strength to s(x) = θsfa= 5 × 1017GeV, but allow fato vary and compute the neutralino

and axion CDM abundances assuming m? G= ms= m˜ a= 1 TeV, TR= 106GeV, θi= 0.5

and the BM1 benchmark point. In this case, ms< 2m˜ gso that if saxions can dominate

the energy density of the universe, they only lead to entropy dilution, and not CDM

production. From the plot, we see that for low fathe neutralino abundance is enhanced

due to large thermal production of axinos and their decay to neutralinos. As faincreases,

thermal production of axinos and saxions becomes suppressed, while the saxion decay

temperature decreases, leading to increased entropy dilution of the neutralino abundance.

At fa∼ 1012GeV, Ω? Z1h2drops below 0.1, and the BM1 point becomes allowed in the

PQMSSM. Meanwhile, the axion abundance will also suffer entropy dilution as faincreases,

but this is counterbalanced by an increasing axion field strength, which leads to greater

axion production via COs: the net result is an almost flat value of Ωah2as favaries. Once

faincreases past ∼ 1013GeV, the saxion becomes sufficiently long-lived that the model

begins to violate BBN bounds. While this scenario does provide a strong dilution of dark

matter relics, we note here that the values of θsneeded are in the range θs∼ 105− 106so

that the saxion field strength is far beyond the value of faand must be given by another

physics scale.

5. Conclusions

In this paper we have presented the results of a calculation of mixed axion/neutralino CDM

abundance using a set of eight coupled Boltzmann equations. The calculation improves

upon previous results in several respects: 1. it allows for non-constant values of ?σv?,

as occurs for bino-like neutralinos, where s-wave annihilation is suppressed, 2. it allows

for interplay between neutralino enhancement via axino production and decay, while si-

multaneously allowing for neutralino production and dilution via saxion production and

decay, 3. it includes the effect of gravitino production and decay (not a big effect for the

parameters presented here) and 4. it moves out of the “sudden decay” approximation and

allows for continuous axino, saxion and gravitino decay. Our calculation allows for the

accurate estimate of mixed axion/neutralino abundance for general choices of PQMSSM

parameters.

– 18 –

Page 20

(GeV)

a f

11

10

12

10

13

10

2

h

Ω

-4

10

-3

10

-2

10

-1

10

1

10

2

10

3

10

2

h

a

Ω

2

h

Z~

Ω

GeV

17

10

×

= 5

a

f

s θ

= 0.5,

iθ

GeV,

6

= 10

R

= 1 TeV, T

a~

= m

s

= m

G~

BM1: m

BBN excluded

Figure 12: Calculated neutralino and axion abundance versus fafor SUSY model BM1 with θsfa

fixed at 5 × 1017GeV, and with m? G= ms= m˜ a= 1 TeV, TR= 106GeV and θi= 0.5.

In most gravity-mediated SUSY breaking models with gaugino mass unification, it is

typically the case that the lightest SUSY particle is a bino-like neutralino. Over most of pa-

rameter space of models such as mSUGRA, bino-like neutralinos give rise to a dark matter

abundance far above WMAP limits[19], and hence vast regions of parameter space are con-

sidered as excluded due to overproduction of neutralino dark matter. In this paper, we have

shown that if the MSSM is extended to the PQMSSM– including an axion/saxion/axino

supermultiplet– then SUSY models with a standard overabundance of neutralinos are typ-

ically still excluded, even for very large values of fa? 1014− 1015GeV, where it might be

expected that a high rate of entropy production from saxion decay would dilute the DM

abundance. Here, we find that s → ˜ g˜ g compensates against entropy dilution, and prevents

the neutralino abundance from dropping into the measured range, unless the saxion decays

are in violation of BBN bounds on late-decaying neutral particles. As noted earlier, our

conclusion depends on at least three assumptions. First, we implemented the standard

thermal axino production rates as calculated in the Ref’s[21, 29, 30]. These rates should

apply in supersymmetric versions of the KSVZ model where PQ-charged matter multiplets

ˆΦ exist at or around the PQ breaking scale fa. In a recent publication[61], it has been

shown that in the SUSY DFSZ model, thermal axino production rates can be enhanced or

diminished compared to their KSVZ values depending on PQMSSM parameters. Secondly,

we assumed that saxion decay is dominated by two-body modes into gluon and gluino pairs.

In the DFSZ model, decays into Higgs pairs or aa may also contribute, and even dominate

the saxion decay modes. Thirdly, we have assumed saxion field strength s(x) ≡ θsfais of

– 19 –

Page 21

order the PQ-breaking scale fa, i.e. that θs∼ 1. We have also shown in Sec. 4 that if

θs≫ 1 and ms< 2m˜ g, then CO-produced saxions can dominate the universe and dilute

all thermal relics while avoiding BBN constraints.

In the case of a standard underabundance of neutralino CDM, a wide range of fa

values are permitted, and can augment the neutralino DM into the measured range. In

cases where the neutralinos still maintain an underabundance, the remaining abundance

can be accommodated by axions. In these cases of a standard underabundance of neutralino

DM, the PQ scale facan be pushed into the 1014− 1015GeV range, which is closer to

expectations from string theory. For the case of very high fa, then we typically expect the

DM to be neutralino rather than axion dominated, since the neutralino abundance cannot

be suppressed too much without violating BBN constraints.

Acknowledgments

This research was supported in part by the U.S. Department of Energy, by the Fulbright

Program, CAPES and FAPESP.

A. Boltzmann Equations for the PQMSSM

As discussed in Sec. 2, we assume the following set of coupled differential equations:

˙ ni= −3Hni− Γimin2

i

ρi

+ [(neq

i(T))2− n2

i]?σv?i+

?

j

BR(j,i)Γjmj

n2

ρj

j

,

˙S =R3

T

?

i

BR(i,X)Γimini, (A.1)

with H given by:

H =1

R

dR

dt

=

?ρT

3M2

P

, (A.2)

where ρT is the total energy density.

In order to simplify the above equations we define:

x = ln(R/R0), Ni= ln(ni/s0), and NS= ln(S/S0)(A.3)

so we can write Eq’s. A.1 as:

N′

S=

1

HT

?

i

BR(i,X)Γimiexp[Ni+ 3x − NS] (A.4)

N′

i= −3 −Γi

H

mi

ρi/ni

+

?

j?=i

BR(j,i)Γj

H

mj

ρj/nj

nj

ni

+?σv?i

H

ni[

?neq

i

ni

?2

− 1](A.5)

where′= d/dx and niis given by ni= s0eNi.

The above equation for Nialso applies for coherent oscillating fields, if we define:

Ni= ln(ni/s0), and ni≡ ρi/mi

(A.6)

– 20 –

Page 22

so

N′

i= −3 −Γi

H

(A.7)

where we assume that the coherent oscillating component does not couple to any of the

other fields.

Since H depends on the energy densities, to solve the above equations we must compute

ρifrom ni. However, even for particles following a thermal distribution, the energy density

for each component cannot be directly obtained from ni, unless the chemical potential

(µi) is also given. Nonetheless, µi(T) is usually small in the relativistic regime, while in

the non-relativistic regime we always have ρi= mini. Therefore, assuming that the fields

follow a thermal distribution, a good approximation for ρias a function of niis given by:

where the modified Bessel functions, K1 and K2, are necessary to describe a smooth

relativistic/non-relativistic transition and NF= 1(7/6) for bosons (fermions).

The only remaining piece of information necessary for computing ρiand H and solv-

ing the Boltzmann equations is the definition of temperature for each component. The

radiation temperature can be directly obtained from NSand x:

?g∗(TR)

For thermal fluids in equilibrium we always have Ti= T, but once they decouple, this is

no longer true. However, the temperature of relativistic fluids scales as T ∝ R−1, while

non-relativistic fluids have T ∝ R−2. Thus, we approximate Tiby

where Tdec

i

, Rdec

i

and RNR

i

are the decoupling (freeze-out) temperature, the scale factor at

freeze-out and the scale factor at the non-relativistic transition (Ti= 3mi/2), respectively.

If the fluid was never in thermal equilibrium, we take Tdec

fluids we always have Ti= 07.

Eq’s. A.4 and A.5, with the auxiliary equations for H (Eq. A.2), ρi(Eq. A.8) and Ti

(Eq. A.10) form a set of closed equations, which can be solved once the initial conditions

for the number densities (ni) and entropy (S) are given. The initial entropy S0is trivially

obtained, once we assume a radiation dominated universe at T = TR:

ρi= ni×

mi

mi

NF

, if Ti< mi/10

K1(mi/Ti)

K2(mi/Ti)+ 3Ti, if mi/10 < Ti< 3mi/2

π4

ξ(3)

30

, if 3mi/2 < Ti

Ti

(A.8)

T =

g∗(T)

?1/3

TRexp[NS/3 − x]. (A.9)

Ti= ×

T

Tdec

i

, if coupled

Rdec

i

R

?RNR

, if Ti> 3mi/2 and decoupled

3

2mi

i

R

?2

, if Ti< 3mi/2 and decoupled

(A.10)

i

= TR. For coherent oscillating

S(TR) =2π2

45g∗(TR)T3

RR3

0.(A.11)

7In principle, the approximations in Eq’s. A.8 and A.10 can be avoided if we include equations for the

chemical potentials µi(T). However, for simplicity, we use Eq’s. A.8 and A.10 instead.

– 21 –

Page 23

For thermal fluids we take the initial number density as

ni(TR) =

?

0

neq

, if ?σv?ineq

i/H|T=TR< 10

i/H|T=TR> 10,

i(TR) , if ?σv?ineq

(A.12)

while for coherent oscillating fluids the initial condition is set at the beginning of oscilla-

tions:

ni(Tosc

i

) =

ρ0

i

mi(Tosc

i

)

(A.13)

where Tosc

energy density for oscillations. For the oscillating saxion and axion[39] fields the initial

energy densities are given by:

i

is the oscillation temperature, given by 3H(Tosc

i

) = mi(Tosc

i

) and ρ0

ithe initial

ρ0

a= 1.44ma(T)2(fa)2θ2

?

i

2

f(θi)7/6

?2π2g∗(TR)T3

ρ0

s= min2.1 × 10−9

R

45

??TR

105

??θs(fa)

1012

?2

,m2

sθ2

s(fa)2

2

?

where f(θi) = ln[e/(1 − θ2

amplitudes. The definition of ρ0

during inflation (if TR< Tosc).

In order to compute the source term in Eq. A.1, we must specify the annihilation

cross-sections ?σv?i, the branching ratios BR(i,j) and BR(i,X) and the the decay widths

Γi. The annihilation cross-sections for axions, saxions, axinos and gravitinos are given by

the expressions [29, 30, 62]

i/π2)] and θifaand θsfaare the initial axion and saxion field

saccounts for the possibility of saxion oscillations beginning

?σv?a= 10−4g6

s

(fa)2

?4.19ln(1.5/g2

3 + 3.87 × θ(T − m˜ a)

s) + 1.68 × θ(T − ma(T))?

23.863g−0.7

4.47 + 31ln(1.4/gs) − 0.784 ,if gs< 0.35

sln(1.271/gs)(1 +M2

3m2

? G

+ 27g2ln(1.312/g)(1 +M2

3m2

? G

?σv?˜ a= 10−5g6

s

(fa)2

??

s

− 0.784,if gs> 0.35

?

?σv?? G=1.37

M2

P

×

?

72g23

)

2

) + 11g′2ln(1.266/g′)(1 +M2

1

3m2

? G

)

?

,

while ?σv?? Z1(T) is extracted from IsaReD[38]. The second term in the expressions for ?σv?a

and ?σv?˜ a represent contributions from 1 → 2 decays of particles with thermal masses.

Therefore, these terms should not be included unless T > ma,˜ a, as indicated by the θ

functions above. The expression for the axino effective cross-section is set to reproduce the

numerical results in [30]. Since the saxion thermal production has not been computed, we

approximate it by the axion expression:

?σv?s= ?σv?a

(A.14)

with ma→ ms.

– 22 –

Page 24

For obtaining the various unstable particle widths, we calculate Γ˜ afrom the ˜ a → ˜ gg,

?Ziγ and?ZiZ partial widths as presented in Ref. [26]. For gravitino decays, we adopt the

s → gg and s → ˜ g˜ g decays as presented in Ref. [53]. We note here that in the DFSZ

model, it is also possible to have s → hh decays and possibly s → aa decays. We neglect

these latter two cases, so that our results apply to the supersymmetrized KSVZ model,

where the gg and ˜ g˜ g final states should dominate.

Once the total and partial widths are known, we can easily compute the required

branching ratios:

gravitino widths as presented in Ref. [33]. For the saxion width, we include Γsfrom the

BR(˜ a,?Z1) = 1, BR(s,?Z1) = 2 ×Γ(s → ˜ g˜ g)

The factor 2 in BR(s,?Z1) takes care of the multiplicity of neutralinos for each saxion

that s → ˜ g˜ g plays a crucial role in the PQMSSM dark matter cosmology.

Finally, we assume that the branching ratios for computing the energy injection into

the thermal bath from unstable particle decays are given by:

Γs

, BR(?G,?Z1) = 1(A.15)

cascade decay. While the s → gg decay width is always dominant, we showed in Sec. 3

BR(˜ a,X) = BR(s,X) = BR(?G,X) = 1.(A.16)

Although some of the decay energy is lost into neutralinos (except for s → gg decays),

we assume that in the final product of the cascade decay of axinos, saxions and gravitinos

most of the initial energy has been converted into radiation, so Eq. A.16 consists in a good

approximation.

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