arXiv:0905.2237v2 [astro-ph.CO] 11 Aug 2009
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
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
is also discussed.
A model with light
−0.26eV (68% C.L.) for CMBpol. The issue of Bayesian model selection
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 . 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 #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  also conducted a similar analysis using Lyman-α data combined with
updated WMAP5 . 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 . 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].
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.
2 Light 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 . 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
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
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 . 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 , 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
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
For later convenience, we also define the fraction of gravitino in the total dark matter
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,
a messenger baryon proposed in  and so on can be well-fitted into the framework of
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
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=
Γ(a → b˜G)
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) =
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.
1 10 100
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 , 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
(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.
0.01 0.1 1
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  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
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.
10 100 1000
10 100 1000
Figure 3: The angular power spectra of lensing potential Cφφ
(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.
(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. ). The lensing potential φ is
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 .
In Fig. 3, we show the angular power spectra of the lensing potential and its correlation
with the temperature anisotropy, Cφφ
expected data of future CMB experiments: Planck, PolarBeaR and CMBpol. We can see
from the Cφφ
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
Φ(χˆ n,η(χ)), (10)
ℓ, respectively. For reference, we also show
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
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
has little advantage for constraining the mass of
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.
4 Constraints 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 ,
PolarBeaR  and CMBpol . 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  integrated in the vastly
#3We assume Gaussian beam and neglect any anisotropies in beam and distortion arising from the scan
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 . 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 . 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
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 . 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
where θsis the acoustic peak scale, τ is the optical depth of reionization and Asand ns
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
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. .
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 , and the expected noise in lensing potential is calculated by the publicly
available FuturCMB2 code developed by the authors of . Angular power spectra are
calculated using the method in Ref. . 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 . 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 , 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
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
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
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
−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
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
< 0.036 (95%)
< 3.2 (95%)
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.
5 Model 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 , a similar degeneracy can also be seen as band-like allowed region along
< 0.035 (95%)
< 3.1 (95%)
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,
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 . 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  and
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
< 0.24 (95%)
< 0.47 (95%)
< 0.21 (95%)
< 0.10 (95%)
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.
< 0.47 (95%)
< 1.3 (95%)
< 0.34 (95%)
< 0.17 (95%)
Table 6: Same tables as in Table 3 but for the cases with treating both N3/2and Ypas
free parameters (CASE IV).
CASE ICASE II
−2.81 ± 0.17
−2.66 ± 0.18
−2.86 ± 0.18
3.63 ± 0.17
−5.76 ± 0.17
−5.02 ± 0.16
−5.80 ± 0.17
1.08 ± 0.17
−4.70 ± 0.18
−3.62 ± 0.18
−5.82 ± 0.18
1.44 ± 0.19
−3.36 ± 0.17
−3.39 ± 0.15
−3.03 ± 0.17
3.40 ± 0.16
Table 7: Bayes factors for the light gravitino model against the CDM model. Shown are
the mean and standard errors from two independent samplings.
6Summary and discussion
We investigated a possible constraint on the light gravitino mass with m3/2< 100 eV in
the light of future precise measurements of CMB. Although the effects of free-streaming
of light gravitino barely leave an imprint on CMB photons at the time of last scatter, they
can be deflected by the gravitational potential altered after recombination due to the free-
streaming effect, which can be probed with lensed CMB. Thus we in this paper discussed
the effects of the light gravitino, paying particular attention to the lensing potential, then
investigated the future constraint on its mass. For this purpose, we adopt the future CMB
surveys such as Planck, PolarBeaR and CMBpol and study the issue by generating poste-
rior distributions with nested-sampling method. For a simple (but physically motivated)
case, we obtained the limit on the light gravitino mass, assuming m3/2= 1 eV as a fiducial
value, as m3/2≤ 3.2 eV (95% C.L.) for the case with Planck+PolarBeaR combined and
can expect that the (counter-)evidence of the light gravitino can be found in cosmological
−0.26eV (68% C.L.) for CMBpol. Thus at the time of CMBpol experiment, we
0.0225 0.023 0.0235
0.06 0.08 0.1
0 0.2 0.4 0.6 0.8 1
0.94 0.95 0.96 0.97 0.98 0.99
3.05 3.1 3.15
Figure 4: The 1d posterior distributions for cosmological parameters with fixed N3/2=
0.059 and the BBN relation for Ypadopted. Shown are constraints from Planck alone (red
solid line), PolarBeaR alone (green dashed line), Planck and PolarBeaR combined (blue
dotted line), and CMBpol (magenta dash-dotted line).
In a simple case, the effective degrees of freedom at the time of the gravitino decoupling
is assumed to have definite value and fixed in the analysis. However in some scenarios,
this assumption may not hold, thus we have also made analysis by treating N3/2as a free
parameter. In addition, in the future precise CMB measurements, the primordial abun-
dance of4He, which is usually not assumed as a free parameter but fixed, can also affect
the determination of cosmological parameters, therefore we also performed the analysis
by varying Yptoo. When both or one of these parameters are varied, the constraint on
the mass becomes weak. The results are summarized in Tables 3-6. In addition, we also
discussed how strong future CMB surveys can find an evidence for light gravitino by em-
ploying Bayesian model selection analysis. Even if the mass of gravitino is around 1 eV, a
future CMBpol-like surveys is capable of providing some rather strong evidence for light
gravitino model, which can be even stronger for larger m3/2.
In principle the light gravitino mass of O(1) eV can be probed with the LHC experiment
0 0.1 0.2
Figure 5: The degeneracies arise among ωdm, f3/2and N3/2. Shown are the cases where we
adopted the BBN relation for Ypusing data from Planck and PolarBeaR combined (blue
dotted line) and CMBpol alone (magenta dash-dotted line). The thick and thin lines show
contours at 68 % and 95% C.L., respectively
with the method proposed in Ref. . But this requires some amount of tuning for the
sparticle mass spectrum together with the gravitino mass. Thus even if the LHC will fail
to determine the light gravitino mass, the future CMB experiments can do this job.
Finally we make some comment on the case where massive neutrinos are also included
in the analysis. From atmospheric, solar, reactor and accelerator neutrino experiments,
now we know that neutrinos have finite masses. Furthermore it has been discussed that
neutrino masses can be well probed with future CMB survey [7–9], which motivates us
to conduct the analysis assuming that neutrinos are massive. Thus we also investigated
the constraint on the light gravitino mass while the mass of neutrino is also varied. Since
the effects of massive neutrino and light gravitino on the lensing potential are essentially
the same, we found that strong degeneracies arise in particular, between their masses
and we could not obtain any meaningful constraints on those. However, neutrino masses
can also be constrained in future laboratory experiments such as tritium beta-decay and
neutrinoless double beta decay. Thus in the future, we will also have some inputs from
such neutrino experiments, which can remove the degeneracy in CMB survey. In light of
these considerations, we can expect that cosmology and particle physics experiments will
0.08 0.085 0.09 0.095
0 0.05 0.1 0.15
0 0.01 0.02
Figure 6: The 1d posterior distributions for cosmological parameters with freely varying
N3/2and the BBN relation for Ypadopted. The case for CMBpol data is only shown.
push us toward more severe constraint/precise determination of the light gravitino mass
in the near future.
The authors would like to thank Kiyotomo Ichiki and Shun Saito for useful discussions
and providing data to check our numerical calculations. The authors would also like to
thank Oleg Ruchayskiy for useful comments. This work is supported by Grant-in-Aid for
Scientific research from the Ministry of Education, Science, Sports, and Culture, Japan,
No. 14102004 (M.K.) and No. 19740145 (T.T.), and also by World Premier International
Research Center Initiative, MEXT, Japan. K.N. and T.S. would like to thank the Japan
Society for the Promotion of Science for financial support.
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