# Constraining Light Gravitino Mass from Cosmic Microwave Background

**ABSTRACT** We investigate the possibilities of constraining the light gravitino mass m_{3/2} from future cosmic microwave background (CMB) surveys. A model with light gravitino with the mass m_{3/2}<O(10) eV is of great interest since it is free from the cosmological gravitino problem and, in addition, can be compatible with many baryogenesis/leptogenesis scenarios such as the thermal leptogenesis. We show that the lensing of CMB anisotropies can be a good probe for m_{3/2} and obtain an expected constraint on m_{3/2} from precise measurements of lensing potential in the future CMB surveys, such as the PolarBeaR and CMBpol experiments. If the gravitino mass is m_{3/2}=1 eV, we will obtain the constraint for the gravitino mass as m_{3/2} < 3.2 eV (95% C.L.) for the case with Planck+PolarBeaR combined and m_{3/2}=1.04^{+0.22}_{-0.26} eV (68% C.L.) for CMBpol. The issue of Bayesian model selection is also discussed. Comment: 22 pages, 6 figures, 7 tables, references are added, accepted for publication in JCAP

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**ABSTRACT:**We propose an extended version of the gauge-mediated SUSY breaking models where extra SUL(2) doublets and singlet field are introduced. These fields are assumed to be parity-odd under an additional matter parity. In this model, the lightest parity-odd particle among them would be dark matter in the Universe. In this Letter, we discuss direct detection of the dark matter and the collider signatures of the model.Physics Letters B 03/2010; · 6.02 Impact Factor - SourceAvailable from: ArXiv[Show abstract] [Hide abstract]

**ABSTRACT:**Recent cosmological observations, such as the measurement of the primordial 4He abundance, CMB, and large scale structure, give preference to the existence of extra radiation component, ΔNν>0. The extra radiation may be accounted for by particles which were in thermal equilibrium and decoupled before the big bang nucleosynthesis. Broadly speaking, there are two possibilities: (1) there are about 10 particles which have very weak couplings to the standard model particles and decoupled much before the QCD phase transition; (2) there is one or a few light particles with a reasonably strong coupling to the plasma and it decouples after the QCD phase transition. Focusing on the latter case, we find that a light chiral fermion is a suitable candidate, which evades astrophysical constraints. Interestingly, our scenario predicts a new gauge symmetry at TeV scale, and therefore may be confirmed at the LHC. As a concrete example, we show that such a light fermion naturally appears in the E6-inspired GUT.Physics Letters B 10/2010; · 6.02 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We study a U(1)R-symmetric extension of the supersymmetric standard model with supersymmetry breaking in the visible as well as hidden sectors. Specifically, we study U(1)R-breaking effects parameterized by the gravitino mass. A special R-charge assignment of right-handed neutrinos allows us to have neutrino Yukawa couplings with the R-charged Higgs field, which develops a tiny vacuum expectation value after the inclusion of U(1)R-symmetry breaking. Even with O(1) Yukawa couplings, a suitable size of Dirac neutrino mass can be generated if the gravitino mass is very small, m3/2 = 1-10 eV. Our flipped R-charge assignment also allows a new type of dimension-five operator that can induce proton decay. It turns out that the proton stability mildly constrains the allowed range of the gravitino mass: A gravitino heavier than 10 keV can evade the proton decay constraint as well as cosmological ones. In this case, the largest neutrino Yukawa coupling is comparable to the electron Yukawa. We also calculate the mass of the pseudo goldstino and its mixing with neutralinos, and briefly discuss its implications in cosmology and Higgs phenomenology.Progress of Theoretical and Experimental Physics. 05/2013; 2013(5).

Page 1

arXiv:0905.2237v2 [astro-ph.CO] 11 Aug 2009

IPMU 09-0042

ICRR-Report-540

Constraining Light Gravitino Mass from

Cosmic Microwave Background

Kazuhide Ichikawa1, Masahiro Kawasaki2,3, Kazunori Nakayama2,

Toyokazu Sekiguchi2and Tomo Takahashi4

1Department of Micro Engineering, Kyoto University, Kyoto 606-8501, Japan

2Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan

3Institute for the Physics and Mathematics of the Universe, University of Tokyo,

Kashiwa, Chiba, 277-8568, Japan

4Department of Physics, Saga University, Saga 840-8502, Japan

Abstract

We investigate the possibilities of constraining the light gravitino mass m3/2

from future cosmic microwave background (CMB) surveys.

gravitino with the mass m3/2< O(10) eV is of great interest since it is free from

the cosmological gravitino problem and, in addition, can be compatible with many

baryogenesis/leptogenesis scenarios such as the thermal leptogenesis. We show that

the lensing of CMB anisotropies can be a good probe for m3/2and obtain an ex-

pected constraint on m3/2from precise measurements of lensing potential in the

future CMB surveys, such as the PolarBeaR and CMBpol experiments. If the grav-

itino mass is m3/2= 1 eV, we will obtain the constraint for the gravitino mass

as m3/2≤ 3.2 eV (95% C.L.) for the case with Planck+PolarBeaR combined and

m3/2= 1.04+0.22

is also discussed.

A model with light

−0.26eV (68% C.L.) for CMBpol. The issue of Bayesian model selection

Page 2

1Introduction

One of the most important prediction of local supersymmetry (SUSY), or supergravity,

is the existence of gravitino, the spin-3/2 superpartner of the graviton. Although the

range of the gravitino mass m3/2can vary from a fraction of eV up to of an order of TeV,

depending on the scales of SUSY breaking, a light gravitino with m3/2< O(10) eV is of

great interest since it is free from the cosmological gravitino problem. Furthermore, for

some baryogenesis scenario to work such as thermal leptgenesis, high cosmic temperature

is required, which favors the range of light mass for the gravitino. Thus, in this respect,

the light gravitino would be attractive.

The determination of the gravitino mass is one of the most important issues to under-

stand how supersymmetry is broken. Some authors have discussed this issue, in particular,

focusing on probing the mass of the light gravitino with LHC experiment [1]. Since grav-

itino with very light mass of m3/2< 100 eV can play a role of warm dark matter (WDM)

in the universe, cosmology would be a powerful tool as well. Some authors have obtained

a constraint on the light gravitino mass from Lyman-α forest data in combination with

WMAP [2,3] and its bound is m3/2< 16 eV at 2σ level [2]#1. Although Lyman-α forest

can be very useful as a cosmological probe, it may suffer from some systematic uncer-

tainties [5,6]. Notice that the above mentioned limit crucially depends on the usage of

Lyman-α forest data, thus in this respect, another independent cosmological probe of the

light gravitino would be of great importance. Since light gravitino with the mass of in-

terest here acts as WDM, it affects cosmic density fluctuations through two major effects.

One is the change of radiation-matter equality due to the fact that light gravitions behave

as relativistic component at earlier times. The second one is the effect of free-streaming

which erases density contrast on the scale under which it can free-stream. Since these

two effects can make great influence on cosmic microwave background (CMB) anisotropy,

CMB can be a powerful probe of its mass. But in fact, when the gravitino is very light as

m3/2< O(10) eV, which is the range of the mass of interest here, its energy density do not

have a large fraction of the total one in general to affect the CMB. Thus the mass of light

gravitino can be mainly probed through the latter effect, free-streaming. The ultralight

gravitino is almost relativistic at the time of the recombination, its effects imprinted on

temperature anisotropy is not significant enough. However, by looking at the gravitational

lensing of CMB photons, we can well probe the change of the gravitational potential driven

during the intermediate redshift after the recombination when light gravitino comes to act

as a non-relativistic component. Thus the lensed CMB is a very useful cosmological tool

for investigating the mass of light gravitino#2, which we discuss in this paper. Since much

#1Recently the authors of [3] also conducted a similar analysis using Lyman-α data combined with

updated WMAP5 [4]. Although they did not report the constraint on the light gravitino mass, the SDSS

Lyman-α dataset would be more effective in constraining m3/2than the dataset adopted in [2]. Thus an

analysis optimized for the light gravitino model could give a severer constraint on m3/2.

#2A similar argument has been made for massive neutrinos on the use of CMB lensing to constrain its

mass Ref. [7–9].

1

Page 3

more precise CMB experiments will be available in the near future, it is an interesting

subject to investigate possible limits with such a future probe. Therefore, we in this paper

study the possibilities of constraining its mass with future CMB observations without us-

ing other cosmological data. In particular, we focus on the lensing potential which can be

reconstracted from CMB maps. As future CMB surveys, we consider Planck, PolarBeaR

and CMBpol to discuss a possible constraint on the light gravitino mass.

The structure of this paper is as follows. We first briefly review a model which predicts

light gravitino and its phenomenology in the early universe in Section 2. In Section 3 we

discuss the effects of the light gravitino on CMB anisotropy, paying particular attention

to the lensing of CMB. We then present forecasts for constraints on the mass of light

gravitino with future CMB surveys such as Planck, PolarBeaR and CMBpol in Section 4.

In addition to the parameter estimation, we also discuss Bayesian model selection analysis

for light gravitino model with future CMB surveys in Section 5. The final section is

devoted to summary of this paper.

2Light gravitino: A model and its phenomenology

in the early universe

A light gravitino scenario is realized in the framework of gauge-mediated SUSY breaking

(GMSB) models [10]. In GMSB models, the SUSY breaking effect in the hidden sector

is transmitted to the minimal supersymmetric standard model (MSSM) sector through

gauge-interactions, giving superparticles TeV scale masses. As an example, let us consider

a model where the SUSY breaking field S couples to N pairs of messenger particles ψ and

¯ψ, which transform as fundamental and anti-fundamental representations of SU(5), having

a superpotential W = λSψ¯ψ with coupling constant λ (in the following we set λ = 1 for

simplicity). The superfield S has a vacuum expectation value as ?S? = M + FSθ2. Here

FS gives SUSY breaking scale, which is related to the garvitino mass m3/2through the

relation FS=√3m3/2MP, with MP being the reduced Planck energy scale, for vanishing

cosmological constant. In this model, gaugino masses Ma(a = 1,2,3 are gauge indices)

and sfermion masses squared m2

˜fiat the messenger scale are given by

Ma= N

?αa

?αa

4π

?

?2

Λmess,(1)

m2

˜fi= 2N

?

a

4π

C(i)

aΛ2

mess,(2)

where αadenotes the gauge coupling constants, C(i)

˜fiand the messenger scale is given by Λmess= FS/M. In order to obtain TeV scale masses,

Λmess∼ 100 TeV is required, but still the SUSY breaking scale FS, or gravitino mass m3/2

can take wide range of values as 1 eV ? m3/2? 10 GeV. The upper bound comes from the

requirement that the gravity-mediation effect does not dominate. On the other hand, there

a are Casimir operators for the sfermion

2

Page 4

also exists a lower bound on the gravitino mass in order not to destabilize the messenger

scalar and lead to the unwanted vacuum. This requires m3/2? O(1) eV.

However, if cosmological effects of the gravitino are taken into account, not all of its

mass range is favored. This is because gravitinos are efficiently produced at the reheat-

ing era and it can easily exceed the present dark matter abundance unless the reheat-

ing temperature TR is very low [12]. This is problematic since many known leptogene-

sis/baryogenesis scenarios require high enough reheating temperature which may conflict

with the upper bound coming from the gravitino problem. In particular, thermal lepto-

genesis scenario [13], which requires TR ? 109GeV, seems to be inconsistent with the

gravitino problem except for the very light gravitino mass range m3/2? 100 eV. As we

will see, gravitinos with such a small mass are thermalized in the early Universe. Thus

their abundance does not depend on the reheating temperature and also it is smaller than

the dark matter density for m3/2 ? 100 eV. This is the reason why we pay particular

attention to a light gravitino scenario.

Having described that the light gravitino scenario is appealing from the view point

of cosmological gravitino problem, next we briefly discuss thermal evolution of the light

gravitino in the early universe. Gravitinos are relativistic well before the recombination.

In such a case, the energy density of the gravitino is parameterized by the effective number

of neutrino species, and it is given by

N3/2=ρ3/2

ρν

=

?T3/2

Tν

?4

=

?g∗ν

g∗3/2

?4/3

, (3)

where ρν(ρ3/2) and g∗ν(g∗3/2) are, respectively, the energy density and the effective degrees

of freedom of neutrinos (gravitinos) evaluated at the epoch when neutrinos (gravitinos)

have decoupled from thermal plasma while they are still relativistic. In the standard

cosmology, g∗ν = 10.75. Temperatures of neutrino and gravitino are represented by Tν

and T3/2. From Eq. (3) we can calculate the temperature of gravitino at present:

T3/2= (N3/2)1/4Tν= 1.95(N3/2)1/4[K],(4)

where we have adopted the temperature of neutrino in the standard cosmology at the

second equality. Eventually the gravitino loses its energy and becomes non-relativistic

due to the Hubble expansion. Its present energy density is given by

ω3/2≡ Ω3/2h2= 0.13

?m3/2

100 eV

??

90

g∗3/2

?

.(5)

For later convenience, we also define the fraction of gravitino in the total dark matter

density ωdmas

f3/2≡ω3/2

In the following, we assume that dark matter consists two components: light gravitino,

which acts as warm dark matter, and CDM. As CDM component, the Peccei-Quinn axion,

ωdm. (6)

3

Page 5

a messenger baryon proposed in [11] and so on can be well-fitted into the framework of

light gravitino.

Thus in order to evaluate the relic abundance of light gravitino, we must know the value

of the effective degrees of freedom of relativistic particles at the freeeze-out epoch, g∗3/2.

Since the production and/or destruction of the light gravitino due to scattering processes

are known to be inefficient for the low temperature regime, in which we are interested, the

gravitino maintains equilibrium with thermal bath through the decay and inverse-decay

processes [12,14], schematically represented by a ↔ b +˜G, where b is the standard model

(SM) particle and a is its superpartner. As the temperature decreases, particles in thermal

bath (b and˜G) lose an ability to create a heavy particle (a). Then gravitinos decouple

from thermal plasma after the time when a decays into b and˜G without inverse creation

processes.

In order to see these processes in detail, we must solve the Boltzmann equation which

governs time evolution of the system. The Boltzmann equation for the gravitino number

density n3/2is given by

˙ n3/2+ 3Hn3/2=

?

a,b

Γ(a → b˜G)

?ma

Ea

?

na

?

1 −n3/2

n(eq)

3/2

?

,(7)

where H is the Hubble expansion rate and ?ma/Ea? represents thermally averaged Lorentz

factor with maand Eabeing the mass and energy of the particle a, respectively. Γ(a → b˜G)

is the decay width of a into b and˜G and superscript (eq) denotes its equilibrium value.

As an example, the decay rate of the stau (˜ τ) into tau (τ) and gravitino is given by

Γ(˜ τ → τ˜G) =

1

48π

m5

3/2M2

˜ τ

m2

P

,(8)

and similar expressions hold for other particles. By solving this equation, one obtains the

final gravitino abundance which can be represented in terms of the gravitino-to-entropy

ratio, Y3/2= (n3/2/s)(t → ∞) and this translates into g∗3/2through the relation Y3/2=

0.417/g∗3/2. In Fig. 1, the value of g∗3/2is shown as a function of m3/2for several values

of Λmess. We have adopted Λmess= 50,100,200 TeV and N=1 and ignored running of the

masses from the messenger scale down to the weak scale for simplicity.

We can understand these results intuitively. As the gravitino mass increases, the decay

width becomes smaller, and hence the equilibrium lasts for rather shorter duration. This

leads to higher freeze-out temperature of the gravitino, which corresponds to large g∗3/2.

On the other hand, larger Λmessleads to heavier sparticle masses, which obviously makes

the time of freeze-out of the gravitino earlier, and hence higher g∗3/2. However, as seen

in the figure, the dependence on these parameters are not so strong and we can safely set

g∗3/2≃ 90 even if we take into account model uncertainties.

4

Page 6

86

88

90

92

94

96

98

100

1 10 100

g*3/2

m3/2 [eV]

Λmess=200TeV

Λmess=100TeV

Λmess=50TeV

Figure 1: g∗3/2as a function of m3/2for several values of Λmess.

3 Effects of light gravitino on CMB

In this section we discuss how light gravitino affects the structure formation and the

CMB anisotropies. Since light gravitino basically acts as WDM, it is characterized by

two quantities, its mass and number density. The number density is determined by the

effective number of degrees of freedom at the time of the decoupling, i.e. g∗3/2. Since, as

we have seen in the previous section, g∗3/2has only mild dependence on m3/2, we can take

g∗3/2= 90 as the representative value, which corresponds to

N3/2= 0.059, (9)

from Eq. (3). We assume Eq. (9) as a fiducial value throughout this section, except for

the last paragraph. We also assume that the universe is flat, dark energy is a cosmological

constant and the primordial fluctuations are adiabatic and its power spectrum obeys a

power-law without tensor perturbations. The fiducial values for cosmological parameters

are adopted from the the recent result of WMAP5 [4], except that we consider mixed dark

matter scenarios ωdm= ωc+ω3/2= 0.1099, instead of ωc= 0.1099 where ωcis the density

parameter for cold dark matter (CDM). By varying m3/2or f3/2, we can see the effects of

light gravitino on structure formation and CMB anisotropies. Moreover, we assume that

neutrinos are massless in the most part of this paper. We will make some comments on

the case where massive neutrinos are also included in Section 6.

As briefly discussed in the introduction, the effects of WDM on structure formation

can be understood by considering following two main aspects: (i) the change of the energy

contents of the universe, or the epoch of matter-radiation equality (unperturbed back-

ground evolutions), (ii) the erasure of perturbations on small scales via free-streaming

5

Page 7

(perturbation evolutions). The first point is due to the fact that WDM behaves as rela-

tivistic component at early times but non-relativistic one at late times. Thus it changes

the time of matter-radiation equality depending on the mass. It alters the evolution of

gravitational potential and drives the integrated Sachs-Wolfe (ISW) effect in the CMB

temperature anisotropy. However, in the case of light gravitino in which we are interested,

its abundance is so small N3/2 = 0.059 that the epoch of radiation-matter equality is

scarcely affected, even when we compare the two opposite limits, f3/2= 0 (m3/2= 0 eV)

and f3/2= 1 (m3/2= 86 eV), with fixed ωdm= 0.1099. Therefore it is almost impossible

to constrain the gravitino mass from unlensed CMB anisotropies, even when the ideal

observations (cosmic variance limeted survey) are available.

101

0.001

102

103

104

0.01 0.1 1

P(k) [h-3Mpc3]

k [h Mpc-1]

Figure 2: Matter power spectra P(k) for several values of m3/2. We have plotted the cases

with m3/2= 0 eV (solid red line), 1 eV (dashed green line), 10 eV (dotted blue line), 86

eV (dot-dashed magenta line). In all cases we fixed the total dark matter density and

energy density of gravitino as ωdm= 0.1099 and N3/2= 0.059, respectively. We adopt

HALOFIT [30] in calculating nonlinear corrections.

Possible constraints on the gravitino mass almost come from the second point. Light

gravitino free-streams to erase cosmic density fluctuations while it is relativistic.

scales smaller than the free streaming length of gravitino, fluctuations of matter and

hence gravitational potential are erased. To see the effect, we show matter power spectra

P(k) for several values of m3/2 with ωdm being fixed in Fig. 2. We take m3/2 = 0 eV

(solid red), m3/2= 1 eV (dashed green), m3/2= 10 eV (dotted blue) and m3/2= 86 eV

(dot-dashed magenta). As seen from the figure, as m3/2increases, the suppression of P(k)

becomes larger on small scales, while on large scales the amplitude of P(k) is unaffected

regardless of the value of m3/2. With more careful observation we can notice that when

m3/2is small, the suppression of the power is small, but the scale under which P(k) is

suppressed becomes large. On the other hand, when m3/2is large, the suppression is also

large, however the free-streaming scale becomes small. These can be simply understood

as follows. When the mass of gravitino is small, gravitino can erase cosmic structure up to

On

6

Page 8

large scales. However, the smallness of the mass in turn indicates that gravitino is minor

component in the contents of energy density and gravitationally irrelevant. Thus density

fluctuations are less suppressed. For the case of larger mass, the opposite argument holds.

10-8

10-7

10-6

10 100 1000

l4Cl

φφ

l

Planck

PolarBeaR

combine

CMBpol

10-7

10-6

10-5

10-4

10-3

10-2

10-1

10 100 1000

l3Cl

Tφ[µK]

l

Planck

PolarBeaR

combine

CMBpol

Figure 3: The angular power spectra of lensing potential Cφφ

temperature CTφ

ℓ

(Right). The values of m3/2are same as in Figure 2. The sensitivities of

future CMB surveys used in our analysis, Planck, PolarBeaR and CMBpol, are also shown

as points with error bars. Notice that the bins are linearly and logarithmically spaced in

ℓ for ℓ ≤ 100 and ℓ > 100, respectively. For visual reason, data points for Planck and

PolarBeaR are slightly displaced horizontally.

ℓ

(Left) and its correlation with

Now we move on to discuss how the suppression of matter fluctuations changes the

lensed CMB anisotropy. CMB photons last-scattered at the decoupling epoch, while trav-

eling to the present epoch, are deflected by the gravitational potential Φ(r,η) generated

by the matter fluctuations (For a recent review see e.g. [15]). The lensing potential φ is

given by

?χ∗

where χ is the comoving distance along the line of sight, χ∗is the comoving distance to the

last scattering surface and η(χ) is the conformal time corresponds to the comoving distance

of χ. Actually, the lensing potential is not a direct observable in CMB observations, but

we can reconstruct it with observed lensed CMB anisotropies. It contains much more

information than the lensed power spectrum [16,17], since the reconstruction is performed

by making use of off-diagonal components in correlation function of lensed anisotropies [28].

In Fig. 3, we show the angular power spectra of the lensing potential and its correlation

with the temperature anisotropy, Cφφ

ℓ

and CTφ

expected data of future CMB experiments: Planck, PolarBeaR and CMBpol. We can see

from the Cφφ

ℓ

in the Fig. 3 that the suppression of the power spectra depends on the mass

of gravitino, which can be probed with future observations of CMB. When we carefully

observe the power on small scales, the trend how the power is suppressed is similar to what

φ(ˆ n) = −2

0

dχχ∗− χ

χ∗χ

Φ(χˆ n,η(χ)), (10)

ℓ, respectively. For reference, we also show

7

Page 9

we have seen in the matter power spectra. Therefore we can expect the mass of gravitino

is constrained with reconstructed lensing potential, which would be obtained from future

CMB surveys. Although cross correlation of the lensing potential with CMB temperature

anisotropy CTφ

ℓ

is affected by the mass of gravitino, the effect is much small compared

with the expected errors for future CMB surveys, which can be seen from the right panel

in Fig. 3. Thus it is suggested that CTφ

ℓ

has little advantage for constraining the mass of

light gravitino.

Here it should be noted that some careful consideration must be given for the following

fact: the heavier the light gravitino mass is, the smaller the free-streaming scale would

be. Although, regarding the suppression of the power, the effects of the light gravitino is

more significant for a larger mass, when a survey cannot observe up to high multipoles

due to its limitation of the resolution, gravitino with lighter mass can be better probed.

This is because gravitino with lighter mass can erase cosmic structure up to larger scales

compared to the case with larger mass although the power suppression is milder. As a

simple example, let us compare the two cases, f3/2= 0 and f3/2= 1 while keeping ωdm

fixed. At small angular scales, the lensing potential φ for f3/2= 1 is more suppressed than

that for f3/2= 0. However, for the case with f3/2= 1 corresponding to m3/2≃ 86 eV,

the free streaming length is small. Therefore the suppression occurs only at limited small

angular scales. If the observed multipoles are limited to low ℓ’s, where the suppression

cannot be seen, gravitino with f3/2 = 1 cannot leave any imprint on such a measure-

ment, which means that we cannot differentiate models between f3/2= 1 and f3/2= 0.

This makes the likelihood surface multi-modal and highly-degenerate. To break up these

degeneracies, high-resolution measurement of lensing potential is needed, and currently

available observations cannot suffice this requirement. In the next section, we discuss how

future CMB surveys will constrain the light gravitino models.

4Constraints on light gravitino mass

Now in this section, we investigate the constraints on the light gravitino mass from future

CMB surveys. As discussed in the previous section, since the current CMB surveys are

not precise enough to measure the lensing potential, it is almost impossible to probe

m3/2. However, in future surveys of CMB, the measurement of lensing potential would

be significantly improved. To forecast constraints on light gravitino mass from future

CMB surveys, we make use of the following three surveys in this paper, the Planck [18],

PolarBeaR [19] and CMBpol [20]. The parameters for instrumental design for these surveys

are summarized in Table 1, where θFWHMis the size of Gaussian beam#3at FWHM and

σT (σP) is the temperature (polarization) noise.

In this paper, to generate samples from the Bayesian posterior distributions of cosmo-

logical parameters, we make use of the public code MultiNest [21] integrated in the vastly

#3We assume Gaussian beam and neglect any anisotropies in beam and distortion arising from the scan

strategy.

8

Page 10

surveys

Planck [18]

fsky

0.65

bands [GHz]

100

143

217

90

150

220

100

150

220

θFWHM[arcmin]

9.5

7.1

5.0

6.7

4.0

2.7

4.2

2.8

1.9

σT [µK]

6.8

6.0

13.1

1.13

1.70

8.00

0.84

1.26

1.84

σP [µK]

10.9

11.4

26.7

1.6

2.4

11.3

1.18

1.76

2.60

PolarBeaR [19]0.03

CMBpol [20]0.65

Table 1: Instrumental parameters for future CMB surveys used in our analysis. θFWHM

is Gaussian beam width at FWHM, σT and σP are temperature and polarization noise,

respectively. For the Planck and PolarBeaR surveys, we assume 1-year duration of obser-

vation and for the CMBpol survey, we assumed 4-year duration.

used Monte Carlo sampling code COSMOMC [22]. While COSMOMC samples the posterior dis-

tributions via the Markov chain Monte Carlo (MCMC) sampling method, MultiNest is

based on the different sampling called nested-sampling method [23]. Use of MultiNest

has several advantages in our analysis. One of the greatest advantages is that it enables

efficient exploring of multi-modal/highly-degenerate likelihood surface, which is indeed

the case for light gravitino models, as we have discussed in the previous section. Also it

provides Bayesian evidence of a model and hence enables us to employ Bayesian model

selection.

To obtain the limit for the mass of light gravitino, we can translate the constraint on

the parameter f3/2 = ω3/2/ωdmusing Eq. (5). Since light gravitino has almost definite

prediction of its abundance, we mainly report our results for the case with N3/2= 0.059

being fixed. However, in some scenario such as those with late-time entropy production,

this number may be altered. In this respect, we also make analysis with N3/2being varied.

Furthermore, regarding the treatment of the primordial abundance of4He (denoted as Yp),

we assume two cases: treating Ypas a free parameter and fixing Ypwith the derived value

via the big bang nucleosynthesis (BBN) relation [24]. In the BBN theory, Ypis determined

once baryon density ωband the effective number of neutrino Neff are given. Thus such

a fixing of the value of Ypwas adopted in some analysis [25–27]. Since, in the precise

measurement of future CMB survey, the prior on Ypcan also affect the determination of

other cosmological parameters [26,27], we consider the case with Ypfreely varied as well.

Thus the full parameter space that we explore for light gravitino models are basically

nine-dimensional:

(ωb,ωdm,θs,τ,N3/2,f3/2,Yp,ns,As),(11)

where θsis the acoustic peak scale, τ is the optical depth of reionization and Asand ns

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prior ranges

parameters

ωb

ωdm

θs

τ

N3/2

f3/2

Yp

ns

ln(1010As)

fiducial values

0.02273

0.1099

1.0377

0.087

0.059

0.013

0.248

0.963

3.063

Planck/PolarBeaR/combined

0.018 → 0.28

0.08 → 0.30

1.02 → 1.06

0.01 → 0.30

(0 → 5)

0 → 1

(0.1 → 0.5)

0.8 → 1.2

2.8 → 3.5

CMBpol

0.021 → 0.024

0.10 → 0.14

1.03 → 1.04

0.06 → 0.14

(0 → 2)

0 → 0.1

(0.2 → 0.3)

0.9 → 1

3.0 → 3.2

Table 2: The fiducial values and prior ranges for the parameters used in the analysis. Note

that priors shown with parenthesis are imposed only when the corresponding parameters

(N3/2 and Yp) are treated as free parameters and not imposed when they are fixed or

derived from other parameters. For CMBpol, we take narrower range for the top priors

since its accuracy is much higher than the former two surveys. Hence we do not need

broad range for the priors.

are the amplitude and spectral index of initial power spectrum of scalar perturbations at

a pivot scale k0= 0.05 Mpc−1. In the following, we investigate four different cases: (I)

fixing N3/2= 0.059 and deriving Ypvia the BBN relation, (II) fixing N3/2= 0.059 and

treating Ypas a free parameter, (III) treating N3/2as a free parameter and Ypas a derived

parameter via the BBN relation, and (IV) treating N3/2and Ypas free parameters. The

fiducial values and top-hat priors for parameter estimation are summarized in Table 2.

The likelihood function is adopted from Ref. [9].

spectra for correlation of CMB anisotropy and lensing potential up to ℓ ≤ 2500. We

assume lensing reconstruction is performed by adopting the method based on quadratic

estimator [28], and the expected noise in lensing potential is calculated by the publicly

available FuturCMB2 code developed by the authors of [9]. Angular power spectra are

calculated using the method in Ref. [29]. For corrections for lensing potential due to

nonlinear evolution of matter density perturbations, we adopt HALOFIT, which is based

on the N-body simulations of CDM models [30]. Though the light gravitino model is

not exactly the CDM models, we believe that the change of the nonlinear correction is

negligible. This is because that gravitino has small N3/2and regardless of the mass of m3/2,

dark matter can be approximated by CDM very well when it begins to evolve in nonlinear

regime. Furthermore, nonlinear correction changes the spectra of lensing potential by only

a few percent at ℓ ≤ 2500 [29], and hence the treatment here can be justified. In addition,

we also performed same analyses without including nonlinear corrections and checked that

resultant constraints do not significantly change by the treatment of nonlinearity.

Now we are going to present our results. In Tables 3-6 we summarize the constraints on

We include TT,TE,EE,φφ,Tφ

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the cosmological parameters from Planck alone, PolarBeaR alone, Planck and PolarBeaR

combined, and CMBpol alone, separately for different priors. First we discuss the case with

fixed N3/2= 0.059 and the BBN relation adopted for Yp. The 1d posterior distributions

of cosmological parameters are shown in Fig. 4. From the posterior distributions for f3/2

in Fig. 4, we can easily see that light gravitino models are not constrained very well with

Planck or PolarBeaR alone. The posterior distributions have decaying tails from the peak

near f3/2= 0 to f3/2= 1. Actually, they have very smooth second peaks at around f3/2= 1.

This multi-modal structure of posterior distributions comes from the degeneracies what

we have discussed in Section 3. Light gravitino with a relatively large mass (f3/2≃ 1) can

suppress the power via free-streaming only at very small scales where the Planck surveys

cannot sufficiently measure the CMB. Thus a model with such a gravitino mass can fit the

data from Planck alone. On the other hand, PolarBeaR has better resolution, gravitino

with large mass is much constrained from observation of lensing potential at small scales.

However, the sky coverage of the PolarBeaR survey is much smaller than Planck, thus

the observation is worse at large angular scales. In this case, other parameters than f3/2

are still not well-constrained from PolarBeaR alone. Therefore gravitino with relatively

large mass can fit the data to some extent by tuning other cosmological parameters in

this case too. To remove the degeneracy it is necessary to combine observations precise at

large and small angular scales or, ultimately, use measurements precise both at large and

small angular scales. With data from Planck and PolarBeaR combined, we can obtain a

constraint

f3/2≤ 0.036 (95% C.L.).

When we use CMBpol, whose measurement is precise both on large and small scales than

other two survey, this constraint can be improved as

(12)

f3/2= 0.0121 ± 0.0027 (68% C.L.). (13)

These constraints are translated into the limits on the mass of light gravitino. For the

case with Planck and PolarBeaR combined, the constraint is given as

m3/2≤ 3.2 eV (95% C.L.), (14)

and with CMBpol as

m3/2= 1.04+0.22

−0.26eV (68% C.L.). (15)

Since gravitino mass should be larger than 1eV not to destabilize the messenger scalar,

CMBpol would be expected to give (counter-)evidense for existence of gravitino if its mass

is (not) in the mass range considered here. In Section 5 we will discuss this point more

quantitatively using Bayesian models selection analysis.

So far we kept assuming the energy density of gravitino fixed as N3/2= 0.059. If we

loosen this assumption and take N3/2as a free parameter, the constraints are significantly

weakened. Notice that the free-streaming scale of light gravitino is determined by its

mass. Thus when N3/2is freely varied and hence the mass of gravitino can be large, the

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free-streaming scale can be shifted toward smaller scales over which PolarBeaR cannot

observe for a wide range of the mass and one cannot see the damping of the power there.

(On the other hand, since CMBpol is very precise on small scales, the constraint on m3/2

from CMBpol is not affected much.) Thus we cannot obtain a meaningful constraint on

f3/2even if we combine Planck and PolarBeaR when we take N3/2as a free parameter.

Furthermore, the change in N3/2renders the shift of the radiation-matter equality which

can be absorbed by tuning ωdm[31–33], significant degeneracies arise among ωdm, f3/2and

N3/2as shown in Fig. 5#4. For meaningful constraints we need sensitivities as good as those

of CMBpol-like survey. In Fig. 6 1d posterior distributions for parameters including N3/2

are shown. The 68 % limit of f3/2for this case is f3/2= 0.0118+0.0032

to the constraint on the gravitino mass m3/2= 1.19+0.16

−0.0031, which corresponds

−0.50eV.

parameters

100 ωb

ωdm

100 θs

τ

N3/2

f3/2

Yp

ns

ln(1010× As)

m3/2[eV]

Planck

2.276+0.013

0.1101+0.0013

103.783+0.024

0.0871+0.0043

—

—

—

0.9622+0.0045

3.0640+0.0073

—

PolarBeaR

2.274+0.021

0.1106+0.0022

103.777+0.035

0.090+0.009

—

—

—

0.9621+0.0081

3.077+0.015

—

combined

2.275+0.012

0.1100+0.0012

103.777+0.020

0.0873+0.0039

—

< 0.036 (95%)

—

0.9629+0.0036

3.0641+0.0063

< 3.2 (95%)

CMBpol

2.2739+0.0041

0.10993+0.00058

103.774+0.0056

0.0872+0.0022

—

0.0121+0.0027

—

0.9637+0.0017

3.0637+0.0040

1.04+0.22

−0.26

−0.012

−0.022

−0.009

−0.0033

−0.0010

−0.0022

−0.0010

−0.00059

−0.026

−0.031

−0.019

−0.0061

−0.0045

−0.013

−0.0043

−0.0026

−0.0027

−0.0036

−0.0083

−0.0033

−0.0017

−0.0094

−0.026

−0.0094

−0.0047

Table 3: Constraints on cosmological parameters for the case with fixing N3/2= 0.059

and adopting the BBN relation to fix the value of Yp(CASE I). We basically present the

mean values as well as 1σ errors. For parameters that are bounded only from one side we

present 95% credible intervals.

5Model selection analysis on light gravitino model

In the previous section, we have seen that future CMB surveys give rather tight constraints

on mass of light gravitino, so that we can expect they would give (counter-)evidence for

existence of gravitino to more or less extent. But then a question arises how strong the

evidence for gravitino is. This is a kind of model selection problem in statistics theory,

which has been often argued in cosmology [34–45]. In Bayesian statistics, the natural

#4In Figure 7 and 12 in [3], a similar degeneracy can also be seen as band-like allowed region along

FWDMaxis.

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Page 14

parameters

100 ωb

ωdm

100 θs

τ

N3/2

f3/2

Yp

ns

ln(1010× As)

m3/2[eV]

Planck

2.278+0.021

0.1100+0.0015

103.788+0.048

0.0873+0.0041

—

—

0.250+0.008

0.9627+0.0067

3.064+0.008

—

PolarBeaR

2.272+0.035

0.1107+0.0025

103.775+0.067

0.090+0.011

—

—

0.248+0.016

0.961+0.012

3.071+0.018

—

combined

2.274+0.016

0.1100+0.0011

103.775+0.032

0.0873+0.0040

—

< 0.035 (95%)

0.2486+0.0094

0.9625+0.0063

3.064+0.006

< 3.1 (95%)

CMBpol

2.2722+0.0055

0.10990+0.00061

103.771+0.011

0.0870+0.0020

—

0.0118+0.0030

0.2481+0.0031

0.9629+0.0026

3.0631+0.0043

1.02+0.26

−0.22

−0.017

−0.034

−0.014

−0.0047

−0.0010

−0.0021

−0.0011

−0.00056

−0.042

−0.058

−0.035

−0.010

−0.0050

−0.013

−0.0044

−0.0028

−0.0025

−0.011

−0.016

−0.0066

−0.0028

−0.0063

−0.014

−0.0052

−0.0025

−0.010

−0.028

−0.010

−0.0045

Table 4: Same tables as in Table 3 but for the cases with fixing N3/2= 0.059 and treating

Ypas a free parameter (CASE II).

measure for evidence of a model is Bayesian evidence E,

E(M) =

?

dθP(data|θ)π(θ|M),(16)

where θ represents a set of parameters included in a model M. P(data|θ) and π(θ|M)

are the likelihood and prior probability functions, respectively. Bayesian evidence can

be efficiently calculated by nested sampling method [23]. Predictiveness of a model M1

against another M2can be assessed by differencing the logarithm of Bayes factors of the

models, that is

B12= ln(E(M1)/E(M2)),(17)

which is called Bayes factor. If B12is positively (negatively) large, we can say the observed

data can be explained well by model M1 (M2) compared with M2 (M1). As a rule of

thumb the Jeffreys’ scale is often used to translate a Bayes factor into literal expression for

strength of an evidence: B12< 1 is not significant, 1 < B12< 2.5 significant, 2.5 < B12< 5

strong and 5 < B12 is decisive. For more details we refer to a recent review [45] and

references therein.

Now we are going to see how large the evidence is from future CMB surveys. In Table 7

we summarized values of obtained Bayes factor for light gravitino model with different sets

of data and priors against the conventional CDM model (f3/2= N3/2= 0). Here we have

assumed a same fiducial model (m3/2= 1eV and N3/2= 0.059), as in the previous section.

First of all, from Table 7 we can see that Planck or PolarBeaR alone and even Planck

and PolarBeaR combined give only negative Bayes factor for the gravitino model against

the CDM model, regardless of priors on Ypand N3/2. This is because for most of values

of added parameters f3/2 (and N3/2) from the CDM model, gravitino model can only

marginally improve fit to the data, even though it indeed improves the fit at some values

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parameters

100 ωb

ωdm

100 θs

τ

N3/2

f3/2

Yp

ns

ln(1010× As)

m3/2[eV]

Planck

2.276+0.015

0.1099+0.0022

103.782+0.035

0.0873+0.0040

< 0.24 (95%)

—

—

0.9638+0.0034

3.064+0.009

—

PolarBeaR

2.284+0.023

0.1125+0.0036

103.763+0.053

0.091+0.008

< 0.47 (95%)

—

—

0.968+0.009

3.076+0.021

—

combined

2.276+0.011

0.1097+0.0017

103.778+0.027

0.0874+0.0036

< 0.21 (95%)

—

—

0.9639+0.0038

3.064+0.008

—

CMBpol

2.2735+0.0043

0.1097+0.0011

103.776+0.0092

0.0871+0.0022

< 0.10 (95%)

0.0118+0.0032

—

0.9635+0.0020

3.0632+0.0051

1.19+0.16

−0.50

−0.015

−0.030

−0.014

−0.0048

−0.0034

−0.0060

−0.0033

−0.0011

−0.029

−0.038

−0.027

−0.0082

−0.0047

−0.015

−0.0046

−0.0027

−0.0031

−0.0065

−0.012

−0.0054

−0.0025

−0.010

−0.027

−0.010

−0.0049

Table 5: Same tables as in Table 3 but for the cases with treating N3/2as a free parameter

and adopting the BBN relation (CASE III).

around the fiducial ones, f3/2≃ 0.013 (and N3/2≃ 0.059). In other words, the complexity

of the gravitino model has little advantage in explaining the data.

The situation dramatically changes for the case of the CMBpol survey. From Table 7

we can see from CMBpol data we obtain Bayes factor for gravitino model against the

CDM model as lnB = 3.40 ± 0.16 for a case with fixing N3/2 = 0.059 and using the

BBN relation. This is interpreted as strong evidence in the Jeffreys’ scale, though there’s

always some disagreement in that how much Bayes factor can be regarded as giving enough

evidence. For the cases with treating N3/2as a free parameter and using the BBN relation,

we obtain lnB ? 1.08 ± 0.17. This can be regarded as giving only marginal evidence. So

it is difficult to obtain enough evidence for general WDM model whose number density

is not theoretically limited in some small range. Fortunately, since gravitino has small

model-dependence of N3/2we can take the former value of lnB.

So far we have discussed the case of fiducial gravitino mass m3/2. For larger gravitino

mass, as long as it is less than the current bound (m3/2? 16 eV [2,3]), the evidence surely

improves. This is because for this range of mass, the power spectra of lensing potential

differ more and more from those for the CDM model as the gravitino mass increases.

Since, in Section 2 we have seen that gravitino mass is expected to be O(1) eV or larger

theoretically, the fiducial model of m3/2 = 1 eV, which we used throughout this paper,

can be supposed as a rather pessimistic case. Since we have seen that even evidence for

gravitino with mass 1 eV can be probed by a CMBpol-like survey, we would expect such

a survey can probe most part of theoretically-motivated range of light gravitino mass. We

hope such a survey would be realized and probe light gravitino model in the near future.

14

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