Ran Huo- Decaying dark matter and the CMB and LSS power spectrum

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Decaying dark matter and the CMB and LSS power spectrum

Ran Huo

March 16, 2010

Abstract

Gravitino as dark matter is usually overproduced after reheating in leptogenesis,especially in gauge mediated susy breaking where it is the LSP. Yanagida [1] haveproposed a way to introduce a still lighter susy particle, say axino, and the gravitinoeventually decay to axino to avoid overclosure. However, it failed to consider the

impact it brings to large scale structure. We modified the open code cmbfast, andperformed calculations for CMB angular power spectrum and transfer functions. Ourresults should apply to any models with relatively late dark matter decay, with twocollisionless produced particles.

1 Motivation

Gravitino is a promising SuperWIMP candidate for dark matter. In gauge mediated susybreaking, gravitino is usually the LSP and the mass has a large available region belowTeV scale. In fact the mass of gravitino is m 3

2

= <F>√3M P

, while M P = (8πGN )− 1

2 =

2.4

×1018GeV is the reduced Planck mass and

F

is the supersymmetry-breaking scale

squared; compared to the soft susy breaking mass scale msoft F S , usually taken as TeV

scale, and S is the messenger scale. So m 3

2

S M P

msoft and it can be several orders lower

than the mass of susy particle.Gravitino can be produced in many ways. In literature gravitino is widely considered

to be produced as the decay product of other susy particle, especially the NLSP. However,the way of direct thermal production may also be important, or in some cases disastrous.Because it is so weakly coupled to other particles, even in the very early stage of the uni-verse when the temperature is extremely high, namely during the reheating after inflation,it is still very difficult for gravitino to get thermal equilibrium. Exception exists that if the mass of gravitino is below several tens of eV, the goldstino component of gravitino will

be significant and the interaction is enhanced, gravitino may achieve thermal equilibriumin a T R = 1010GeV reheating temperature. However, apparently there is still a largeparameter window that the non-equilibrium thermal production applies.

A high reheating temperature of, say 1010GeV, is usually required for leptogenesis. Thereheating temperature should be higher than the mass of the lightest right hand Majorananeutrino, which is determined by see-saw mechanism to be roughly that order, for the righthand neutrino to be at some sizable equilibrium abundance and the the observed matteranti-matter asymmetry is created through a small CP-violation. Then several calculation

1



[2] [3] shows that gravitino is easily overproduced in this scenario

Y 32

1.4 × 10−10

T R

1010GeV

100GeV

m 3

2

2mg(µ)

1TeV

3

, (1)

Ω3

2

h2 0.28

T R

1010GeV

100GeV

m 3

2

mg(µ)

1TeV

3. (2)

where mg(µ) is the mass of the gluino at scale µ, where µ 100GeV. Given an O(GeV)or O(MeV) gravitino mass the Ω 3

2

h2 will be many orders larger than the observed CDM

contribution ΩCDMh2 0.105, or even the overclosure condition. This is the gravitino

problem in leptogenesis.One way to avoid this problem is to lower the reheating temperature. For leptogenesis

to still produce the observed matter anti-matter asymmetry, we require a much enhancedCP-violation for Majorana neutrino decay. This can be achieved in a similar way as the

resonant decay of the K 0

−¯K

0

system, where the near degeneracy of two mass eigenstatesmixed together to enhance the CP-violation. This way is sorted as “resonant leptogenesis”[4]. However, this requires a near degeneracy of the Majorana neutrino mass, which arenaturally hierarchy. One should introduce additional symmetry like A4 to achieve this.

In this sense it seems that ordinary leptogenesis is directly conflicted with gauge medi-ated susy breaking, in the cross of gravitino problem. Is there a way to remedy? Yanagida[1] proposed a way that the gravitino is actually not the LSP, but there still is some otherlighter susy particles. As the super partner of axion in strong CP problem, axino has avery model dependent mass and in some models it can have a mass still lower than themass of gravitino. Then gravitino finally decays to axion and axino g −→ a + a, theyare both collisionless so that the BBN constraint does not apply. The gravitino density

is reduced by the mass ratio of axino and gravitino, which is taken to be like O(keV)axino mass to O(100GeV) gravitino mass, so the density is reduced by 8 orders and theoverclosure problem is avoided.

However, [1] failed to consider the impact of the decay to structure formation. Becauseof the suppression of Planck scale for interaction, the decay is characterized by a large lifetime, say O(109s). It is not too far before the epoch of structure formation. The producedaxino and axion is relativistic and there is no time for them to cool down by redshift. Itbehaves like hot dark matter and the structure will be spoiled.

Cosmic Microwave Background (CMB) forms as the decoupling of relic photon, it isthe best observed imprint for early universe. it happens at around O(1013s) in standardcosmology, so we expect the hypothetical gravitino decay will be seen from CMB. We

modified the open code cmbfast to calculate the effect. We generalize it to universalscenario. The cmb power spectrum can directly be compared toHere we want to point out that our dark matter decay calculation also applies to

other physical scenario. If gravitino is really the LSP, and we don’t consider the thermaloverproduction but just the scenario that other susy particles like sneutrino decay to it,we have a similar effect [5]. Or if the gravitino is the LSP but the R parity is slightlybroken, gravitino will eventually decay to ordinary particle (but should be collisionlessand a large number of photon is not allowed). All the decay widths are suppressed by thePlanck scale squared and the decay life times are relatively large.

In section 2 we will briefly go through our physical picture of gravitino decay. In section3 we introduce the cmbfast code itself and our modification without details. Section 4 and

5 contain sample of our calculations. In section 6 we conclude.

2



2 Axino and Gravitino

We work in KSVZ axion model. Axion is the Nambu-Goldstone boson for the Peccei-Quinnsymmetry, with a mass estimation of ma f πmπ

f PQ. In supersymmetric models the axion

will have a super partner, the axino. Axion is CP odd, it also has a CP even partner, calledsaxion σ, with largely unknown mass. So the super multiplet is Φ = σ+ ia+√2θa+θ2F Φ.However, saxion is unrelated to the interaction.

When axion is referred in the literature of dark matter, it is through coherent pro-duction of vacua misalignment. It is produced at the long-wave state, so although it isextremely light, it is still cold. However, if we treat the decay produced axino as all darkmatter, we don’t have a significant contribution of axion to the cold dark matter. We as-sume some mechanism that prevents a large v.e.v of Θ angle, so the coherent production isnegligible. Note that the current upper bound of PQ scale f PQ 1012GeV comes from therequirement that the axion does not to exceed the dark matter density with an arbitraryΘ angle v.e.v, if Θ angle v.e.v is fine tuned to zero this upper bound is invalidated.

For axino, there are two production ways. One is from thermal production in thereheating, the other is the late time decay production from gravitino. We will omit thedirect decay of other susy particles (except gravitino) to axino, which may have a differentlife time, because other susy particles are supposed to be much less than the dominantgravitino.

In our real calculation through modified cmbfast we will use arbitrary parameter of decaying dark matter and nondecay dark matter.

Finally the decay time for gravitino is

T D =192πM 2 p

m33

2

1 − m2

a

m23

2

−4∼ 2.35 × 109

100GeV

m 3

2

3

1 − m2a

m23

2

−4s. (3)

3 Modified cmbfast with decay

3.1 Original code

Cmbfast [6] is based on worldline integration of photon, instead of solving Boltzmannequation explicitly. It works in synchronous gauge. The effective Boltzmann equation forphoton in synchronous gauge is

∆T + ikµ∆T = η − αµ2k2 + κ

∆T 0 + µv − 1

2P 2(µ)(∆T 2 + ∆P 0 + ∆P 2)

. (4)

Where ∆T ( k, n) =δT ( k,n)

T is the observation temperature fluctuation. Overdot ˙ meansderivative to conformal time τ . k is the perturbation wave number, n is the light of sightdirection of the observer and µ = k · n. κ = anexeσT is the differential optical depth,where a is the scale factor normalized to unity today, ne is the electron number density (incoordinate space), xe is the electron ionization fraction, σT is the Thomson cross section.

α ≡ h+6η2k2

and h and η are two gauge perturbation modes of spatial part of FRW metricin synchronous gauge. The subindex 0 or 2 of the last several terms are multipole index,T or P denotes different Stokes parameters.

Formally we can integrate it out till present τ 0

∆T ( k, n) = τ 0

0

dτeikµ(τ −τ 0)eκ(τ −τ 0)(η−αµ2k2)+κ∆T 0+µv

−1

2P 2(µ)(∆T 2+∆P 0+∆P 2).

(5)

3



Then we expand it in spherical harmonic functions

∆T ( k, n) =

(−i)(2 + 1)P (µ)∆T (k),

∆T( k) = τ 00

dτj(kτ 0−kτ ) eκ(η + α) + eκκ

14δγ 0 + α + 1

k2θb + 1

16Π + 3

16k2Π

+d

dτ (eκκ)

α +

1

k2θb +

3

8k2Π

+d2

dτ 2(eκκ)

3

16k2Π

,

after some manipulation. Here j is the spherical Bessel function, δ ≡ δρρ

and δ ∼ 4∆ forphoton is the quantity directly calculated in code. Π = 4(∆T 2 + ∆P 0 + ∆P 2). Finally theusually measured angular anisotropy is

C =

∞0

d ln k∆2T(k), (6)

and the plot is usually in T 2CMB( + 1) C 2π .

The detailed calculation of the quantity in integration depends on the evolution of allparticle species and metric perturbation h and η. Baryon photon plasma is directly relatedto CMB, but other species like cold dark matter, massless and massive neutrino are alsoimportant because they talk in gravity. For a review, see [7]. Here we just take massiveneutrinos for example, the evolution equations of perturbation are

Ψ0 = −qk

Ψ1 +

1

6hd ln f

d ln q(7)

Ψ1 =

qk

Ψ0 − 2

3Ψ2 (8)

Ψ2 =

qk

2

5Ψ1 − 3

5Ψ3

−

1

15h +

2

5η

d ln f

d ln q(9)

Ψ =

qk

1

2 + 1

Ψ−1 − ( + 1)Ψ+1

≥ 3 (10)

Ψ =

qk

Ψ−1 +

+ 1

τ Ψ As Truncation (11)

where Ψ is the fractional perturbation to equilibrium distribution f + δf = f (1 + Ψ) =1

eE/T +1(1 + Ψ).

In the way of perturbation growth cmbfast also calculate the transfer function for

different species of cold dark matter, baryon, photon and neutrino. Transfer function isby definition the ratio of perturbation growth factor from an very early stage to certainlater stage, between certain interesting scale and very large scale which is outside horizonin the whole evolution

T (k) ≡δ(k, tf )

δ(k, ti)

δ(k = 0, tf )

δ(k = 0, ti)

(12)

Sky survey provide some large scale structure information, in the form of spatial powerspectrum P (k). It is related to dimensionless perturbation δ by

δ2(k) = 8k

3

π2 P (k) (13)

4



3.2 Our modification

We introduced free parameter of Ωd to be the decay produced energy density today , whilestill keeping Ωc. That means we can treat any combination of decaying and nondecay darkmatter. For decaying dark matter, we introduced the mass ratio md1

mo

and md2

mo

where md1

and md2 are separately the masses of two daughter particles and mo is the mass of parentparticle. No exact mass scale is needed here, what matters is only the ratio. Essentiallythey are free parameters1, and massless case will be covered by setting one to zero. Thelast free parameter is the decay time T d.

Before decay the parent particle is taken to be at rest, so for a two body decay thetransverse momentum p is fixed. After decay, the comoving momentum q = ap is conservedin expansion for collisionless particle. We discretize the produced particles into like 30channels, each channel corresponds to particles which are produced around certain scalefactor, or equivalently, having certain comoving momentum q. Then perturbation for eachchannel evolvers in a way like (7), the only difference is that the unperturbed distributionis now decay distribution. A fraction of dΩ is distributed into q space d3q = 4πq2dq =4πp3a2da, so partition function from decay is

f ≡ dn

d3q=Ce

− tT d

dtT d

4πp3a2da=Ce

− tT d

T d p3aa(14)

And the quantity of d ln f d ln q

in (7) can be calculated as

d ln f

d ln q= − 1

T daa2

− 3

2+

3¯ p

2ρ(15)

The evolution mode for each channel is determined uniquely by its velocity at arbitrary

scale factor, independent of whether it is really “filled” with decay produced particles. Theinitial condition for each channel is set at a very early stage with a very small scale factor(a = 10−8), in the same way massive neutrino.

Right now we have only completed the modification for flat universe (Ωk = 0) sector,and only for scalar mode (not for tensor mode), but it suffice for our purpose. Themodification has not been extensively tested for its accuracy.

4 CMB Calculations

The best fit cosmological parameter set that we are using for fiducial is Ωb = 0.046,

Ωc = 0.224, Ωv = 0.73, Ωn = 0 for massive neutrino, H 0 = 71km/s/Mpc, T CMB = 2.725K,nν = 3.04 for massless neutrino, and effective massless d.o.f is 10.75 after BBN. Our “Nodecay” case always refer to this, without decay effect.

The effect of decay is always to introduce some lighter particles, they will behave in away somehow like radiation, depends on the exact mass ratio, so that the velocity. Rela-tivistic energy density during recombination will contribute to δT 0i terms and δT i j terms,while cold dark matter’s contribution will be suppressed because they are proportional tov or v2. Then the metric term in (5) will give a lifted peak. A similar effect is known forless cold dark matter in literature, which also has a smaller mass to radiation ratio.

1Some small error may exist relating to the initial conditions. And usually for the second particle we

calculate higher component, so it’s better to use the second one as massless, where higher multipole are

not suppressed by velocity.

5



0

1000

2000

3000

4000

5000

6000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

T 0

2 l ( l +

1 ) / 2 π

i n

( µ K ) 2

l

WMAP 7 year dataACBAR data

BOOMERANG dataCBI data

No decay25% decay, md1 /mo=0.1, md2 /mo=0, td=109

50% decay, md1 /mo=0.1, md2 /mo=0, td=109

75% decay, md1 /mo=0.1, md2 /mo=0, td=109

all decay, md1 /mo=0.1, md2 /mo=0, td=109

Figure 1: CMB calculation. Varying the fraction of DM from decay, with a total Ωc+Ωd =0.224. In following figures “all decay” means the same.

0

2000

4000

6000

8000

10000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

T 0

2 l (

l + 1 ) / 2 π

i n

( µ K ) 2

l


BOOMERANG dataCBI dataNo decay






Figure 2: CMB calculation. Varying the life time. As decay time goes smaller, it ap-proaches the non decay case. On the other hand, Larger decay time gives more relativisticenergy density during recombination, because there are less time for it to get redshifted,so the peak is further lifted.

6



0

2000

4000

6000

8000

10000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

T 0

2 l ( l +

1 ) / 2 π

i n

( µ K ) 2

l



No decayall decay, md1 /mo=0.9, md2 /mo=0, td=10

9







Figure 3: CMB calculation. Varying the mass ratio. We fix the decay produced energydensity to be Ωd = 0.224 today , so small mass ratio means more relativistic energy densityright after decay and higher peaks.

0

1000

2000

3000

4000

5000

6000

7000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

T 0

2 l ( l +

1 ) / 2 π

i n

( µ K )

2

l



No decay10

-5decay, md1 /mo=10

-5, md2 /mo=0, td=10

9

3*10-5

decay, md1 /mo=10-5

, md2 /mo=0, td=109

10-4

decay, md1 /mo=10-5

, md2 /mo=0, td=109

Figure 4: CMB calculation. A tiny Ωd with a similarly hierarchial mass ratio, and Ωc =0.224. We have checked that even for this small mass ratio the produced massive particleenergy density today is nearly all from mass, so the two tiny ratio combines an order onecontribution to Ω before decay. Actually it corrects the relativistic energy by O(10−1),and it is still discernible from CMB spectrum.

7



1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 Small

T r a n s f e r F u n c t i o n

k in h/Mpc

SDSS dataNo decay

25% decay, md1 /mo=0.1, td=109

50% decay, md1 /mo=0.1, td=109

75% decay, md1 /mo=0.1, td=109

All decay, md1 /mo=0.1, td=109

Figure 5: CDM Transfer function calculation. Varying the fraction of DM from decay,with a total Ωc + Ωd = 0.224. We set the second decay product to be massless in all cases.

We can see that CMB is able to trace relativistic energy density. If there is a sizableadditional relativistic energy density from decay, although it is very small today by red-shift, it still imprint on the CMB spectrum. On the other hand, it is not so sensitive tomatter, a near degenerate decay is relatively hard to tell from mere CMB data.

5 Transfer Function Calculations

A similar modification is [8], but it only calculate the transfer function. We found a matchin transfer function calculation, given that the decay time O(107)s and md1

mo 0.5, md2

mo 0.

We do transfer function for cold dark matter. On small scales the perturbation will beoscillating, so there may be several calculated wave number points with minus perturbationvalue, and we don’t get continuous fit line. That’s more severe for baryon, so althoughbaryon trace the measured spatial power spectrum better than cold dark matter, we plotfor cold dark matter.

Transfer function, or the spatial power spectrum, is complementary to CMB analysis.It tells sensitively on the matter, even on decay with close mass between parent particle

and daughter particle, given that a significant part of dark matter comes from decay. Ithave less power in telling the radiation. Another drawback is that the spatial spectrum isnot so well measured compared with CMB.

6 Conclusion and Outlook

With our calculation we can see that the hierarchy late decaying dark matter scenariois impossible, which is represented by gravitino decaying into axino in our case. If thegravitino is part of dark matter, it should either be stable, then we still have the gravitinoover production problem; or whatever decay is between near degenerate mass states, thenit doesn’t help to solve the gravitino problem; or decay happens at very early stage of theuniverse, but it seems unlikely because of the Planck scale suppressed interaction. If late

8



1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 Small


k in h/Mpc

SDSS dataNo decay


6





Figure 6: CDM Transfer function calculation. Varying the life time. As decay time goeslarger, it affect larger scale.

1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 Small


k in h/Mpc

SDSS dataNo decay






Figure 7: CDM Transfer function calculation. Varying the mass ratio. If the masses arecloser, decay will produce smaller effect.

9



1e-007

1e-006

1e-005

0.0001

0.001

0.01

0.1

1

0.001 0.01 0.1 1 10 Small


k in h/Mpc

SDSS dataNo decay

10-5

decay, md1 /mo=10-5

, td=109

3*10-5

decay, md1 /mo=10-5

, td=109

10-4

decay, md1 /mo=10-5

, td=109

Figure 8: CDM Transfer function calculation. A tiny Ωd with a similarly hierarchy massratio, and Ωc = 0.224.

and hierarchy decay does happen, this decaying part is constraint to be small compared toall dark matter. So even if it is still possible at some parameter region (by a conspiracy of reheating temperature, near degeneracy of mass states, and relic density of NLSP in [5]),the light gravitino dark matter scenario as predicted by gauge mediated susy breaking isat least unnatural.

We haven’t performed a global χ2

analysis at this stage, that will be the future objec-tive. As for observation, we call for more measurement on spatial power spectrum on asmaller scale.

References

[1] T. Asaka and T. Yanagida, “Solving the gravitino problem by axino,” Phys. Lett. B494, 297 (2000) [arXiv:hep-ph/0006211].

[2] M. Bolz, A. Brandenburg and W. Buchmuller, “Thermal Production of Graviti-nos,” Nucl. Phys. B 606, 518 (2001) [Erratum-ibid. B 790, 336 (2008)] [arXiv:hep-

ph/0012052].

[3] V. S. Rychkov and A. Strumia, “Thermal production of gravitinos,” Phys. Rev. D75, 075011 (2007) [arXiv:hep-ph/0701104].

[4] A. Pilaftsis and T. E. J. Underwood, “Resonant leptogenesis,” Nucl. Phys. B 692,303 (2004) [arXiv:hep-ph/0309342].

[5] J. L. Feng, S. f. Su and F. Takayama, “SuperWIMP gravitino dark matter from slep-ton and sneutrino decays,” Phys. Rev. D 70, 063514 (2004) [arXiv:hep-ph/0404198].

[6] U. Seljak and M. Zaldarriaga, “A Line of Sight Approach to Cosmic Microwave Back-ground Anisotropies,” Astrophys. J. 469, 437 (1996) [arXiv:astro-ph/9603033].

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[7] C. P. Ma and E. Bertschinger, “Cosmological perturbation theory in the syn-chronous and conformal Newtonian gauges,” Astrophys. J. 455, 7 (1995) [arXiv:astro-ph/9506072].

[8] M. Kaplinghat, “Dark matter from early decays,” Phys. Rev. D 72, 063510 (2005)[arXiv:astro-ph/0507300].

[9] E. Komatsu et al., arXiv:1001.4538 [astro-ph.CO].

http://lambda.gsfc.nasa.gov/product/map/dr4/pow tt spec get.cfm

C. L. Reichardt et al., Astrophys. J. 694, 1200 (2009) [arXiv:0801.1491 [astro-ph]].

http://cosmology.berkeley.edu/group/swlh/acbar/data 2008/index.html

W. C. Jones et al., Astrophys. J. 647, 823 (2006) [arXiv:astro-ph/0507494].

http://cmb.phys.cwru.edu/boomerang/data/2005 July/

J. L. Sievers et al., arXiv:0901.4540 [astro-ph.CO].

http://www.astro.caltech.edu/ tjp/CBI/data2009/index.html

[10] M. Tegmark et al. [SDSS Collaboration], Phys. Rev. D 74, 123507 (2006) [arXiv:astro-ph/0608632].

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Ran Huo- Decaying dark matter and the CMB and LSS power spectrum