# Gravitino constraints on supergravity inflation

Shinsuke Kawai  Department of Physics, Sungkyunkwan University, Suwon 16419,
Republic of Korea  Nobuchika Okada  Department of Physics and Astronomy,
University of Alabama, Tuscaloosa, AL35487, USA

###### Abstract

Supergravity embedding of the Standard Model of particle physics provides
phenomenologically well-motivated and observationally viable inflationary
scenarios. We investigate a class of inflationary models based on the
superconformal framework of supergravity and discuss constraints from the
reheating temperature, with the particular focus on the gravitino problem
inherent in these scenarios. We point out that a large part of the parameter
space within the latest BICEP/Keck 95% confidence contour may have been
excluded by the gravitino constraints, depending on the mass scale of the
inflaton. Precision measurements of the scalar spectral index by a future
mission may rule out some of these scenarios conclusively.

Supersymmetric models, Supergravity, Inflation, Cosmic microwave background,
Dark matter

Introduction.— Understanding the origin of cosmic inflation is an important
goal of particle cosmology, and for that purpose, model building in a theory
beyond the Standard Model is a promising direction of research. In particular,
supergravity embedding of the Standard Model offers a well-motivated
framework; supersymmetry allows natural gauge unification, softens the
hierarchy problem and provides a natural candidate for the dark matter.
Realizing a realistic inflationary scenario within supergravity was once
considered challenging. The statement of this difficulty, known as the $\eta$
problem, is based on assumptions including the canonical form of the Kähler
potential. The avenues to circumvent the $\eta$ problem are now well known. In
this letter, we will be concerned with a class of supergravity inflationary
models obtained by relaxing the assumption of the canonical Kähler potential.
These are the direct supersymmetric analogue of the nonminimally coupled Higgs
inflation type model [1, 2], which has been a focus of much attention due to
its solid phenomenological origin and the excellent fit of the cosmological
parameters to the measurements by the WMAP and Planck satellites.

Figure 1:  The prediction of the primordial tilt $n_{s}$ and the tensor-to-
scalar ratio $r$ by the nonminimally coupled supergravity inflation model,
shown for the $e$-folding number $N_{e}=50$, $60$ and $70$. The points for
$y=10^{-6}$, $2\times 10^{-6}$, $5\times 10^{-6}$, $10^{-5}$, $2\times
10^{-5}$, $5\times 10^{-5}$, $1$ are marked with $\bullet$. The thick dashed
line $r=\frac{16}{3}(1-n_{s})$ corresponds to the minimally coupled ($\xi=0$)
model. The blue contours on the background are the Planck+BICEP/Keck 2018 1-
and 2-$\sigma$ constraints [3].

Basic structure of the supergravity inflation model.— The inflationary model
of our interest is constructed from the supergravity lagrangian

$\displaystyle{{\mathcal{L}}}\supset\int
d^{4}\theta\,\phi^{\dagger}\phi\,{{\mathcal{K}}}+\Big{\\{}\int
d^{2}\theta\phi^{3}W+{\rm h.c.}\Big{\\}},$ (1)

in which the superpotential is assumed to include the coupling of a singlet or
adjoint superfield $S$ and a vector-like pair $(\Phi,\overline{\Phi})$ under a
certain gauge symmetry

$\displaystyle W\supset yS\overline{\Phi}\Phi.$ (2)

This structure is common. Examples include the singlet $S$ and the Higgs
doublet superfields $(\Phi,\overline{\Phi})=(H_{u},H_{d})$ of the NMSSM [4, 5,
6], and $S={{\bm{2}}{\bm{4}}}_{H}$ and
$(\Phi,\overline{\Phi})=({{\bm{5}}}_{H},\overline{{\bm{5}}}_{H})$ of the
minimal $SU(5)$ grand unification model [7, 8]. See also [9] for the
construction in the Pati-Salam model, [10, 11, 12, 13, 14] for the type I and
type III seesaw models, [15] for the $B-L$ model, [16] for the $SO(10)$ grand
unified theory, [17] for the hybrid inflation model and [18] for the gauge
mediated supersymmetry breaking model. The Kähler potential in the
superconformal framework [19, 20, 21, 22, 23, 24, 25] is chosen in the form

$\displaystyle{{\mathcal{K}}}=$ $\displaystyle-3M_{\rm
P}^{2}+|\overline{\Phi}|^{2}+|\Phi|^{2}+|S|^{2}$ (3)
$\displaystyle\qquad-\frac{3}{2}\gamma\left(\overline{\Phi}\Phi+{\rm
h.c.}\right)-\frac{\zeta}{M_{\rm P}^{2}}|S|^{4},$ (4)

where $M_{\rm P}=2.44\times 10^{18}$ GeV is the reduced Planck mass, and
$\gamma$, $\zeta$ are real parameters. One may always adjust the parameter
$\zeta$ so that $S$ is stabilized at some constant value, which is assumed to
be small compared to the scale of inflation. Parametrizing the scalar
component of the vector-like fields along the D-flat direction as
$\overline{\Phi}=\Phi=\frac{1}{2}\varphi$, the standard supergravity
computation gives the scalar part of the action

$\displaystyle S_{\rm scalar}=\int d^{4}x\sqrt{-g}\left[\frac{M_{\rm
P}^{2}+\xi\varphi^{2}}{2}R-\frac{1}{2}(\partial\varphi)^{2}-\frac{y^{2}}{16}\varphi^{4}\right].$
(5)

Here, $\xi\equiv\frac{\gamma}{4}-\frac{1}{6}$ parametrizes the nonminimal
coupling between the scalar field $\varphi$ and the scalar curvature $R$. The
action (5) is recognized as that of the nonminimally coupled
$\lambda\varphi^{4}$ model [26] and the prediction for the cosmological
parameters is obtained in the standard slow roll paradigm, after transforming
it into the Einstein frame. The inflaton field $\widehat{\varphi}$ canonically
normalized in the Einstein frame is related to $\varphi$ by the relation

$\displaystyle d\widehat{\varphi}=\frac{M_{\rm P}\sqrt{M_{\rm
P}^{2}+\xi\varphi^{2}(1+6\xi)}}{M_{\rm P}^{2}+\xi\varphi^{2}}d\varphi.$ (6)

The scalar potential in the Einstein frame is deformed by the factor arising
from the Weyl transformation as

$\displaystyle V_{\rm E}(\varphi)=\frac{y^{2}}{16}\frac{M_{\rm
P}^{4}\varphi^{4}}{(M_{\rm P}^{2}+\xi\varphi^{2})^{2}}.$ (7)

This potential is concave for not too small $\xi$, giving the observationally
supported perturbation spectrum with the suppressed tensor mode at the CMB
scale. The model has two tunable parameters $\xi$ (or $\gamma$) and $y$, but
with the normalization of the scalar perturbation amplitude, there remains
only one parameter degree of freedom. As $\xi$ is increased from zero, the
coupling $y$ is also increased towards a larger value. The predicted
primordial tilt $n_{s}$ and tensor-to-scalar ratio $r$ are shown in Fig. 1 for
different values of $e$-folding number $N_{e}$. It can be seen that $y\gtrsim
10^{-6}\sim 10^{-5}$ is in good agreement with the recent cosmological data.
Note that $y\sim 10^{-6}$ is not unnaturally small from the phenomenological
perspective, as it is in the same order as the Standard Model electron Yukawa
coupling. The fact that the ‘self-coupling’ in the potential (7) appears as
$y^{2}$, and not as $y$, is a salient feature of this supergravity inflation
model which is in stark contrast to the nonsupersymmetric counterpart. For
example, the Higgs inflation model [1, 2] requires a large nonminimal coupling
$\xi\sim 10^{4}$ in order to accommodate the Standard Model Higgs self
coupling, which led some authors to worry about the unitarity issue [27, 28,
29, 30] (see however [31]). Since the self coupling is $y^{2}$ in
supergravity, this awkwardness, if it exists, may be easily avoided.

Figure 2:  The prediction for the primordial tilt $n_{s}$ and the tensor-to-
scalar ratio $r$, computed for the rescaled reheating temperature
$\delta^{-1}T_{\rm R}=1\;{\rm MeV}$, 1 GeV, 1 TeV and $10^{6}$, $10^{9}$,
$10^{12}$, $10^{15}$, $10^{18}$, $10^{21}$, $10^{24}$, $10^{27}$,
$10^{30}\;{\rm GeV}$ (red lines). The curves for $e$-foldings $N_{e}=45$, 50,
55, 60, 65, 70 are also indicated in green. The near-horizontal grey curve
$n_{s}=1-3r/8-\sqrt{r/3}$ is the prediction in the limit $\xi\gg 1$. The
contours on the background are the Planck+BICEP/Keck 2018 1- and 2-$\sigma$
constraints [3].

Gravitino problem.— Supergravity entails the gravitino, which is potentially
harmful in cosmological scenarios [32, 33, 34, 35] depending on its mass
$m_{3/2}=F/\sqrt{3}M_{\rm P}$, see e.g. [36]. A stable gravitino may be
produced by the decay of the inflaton, by the decay of a heavier
supersymmetric particle, or thermally produced via the freeze-in mechanism.
See [37] for the details of computations of the thermal production rate. The
stable gravitino in the mass range $4.7\,{\rm eV}\lesssim m_{3/2}\lesssim
0.24\,{\rm keV}$ becomes a hot or warm dark matter component, which is
severely constrained by the analysis of small scale structure formation [38,
39]. In the range $0.24\,{\rm keV}\lesssim m_{3/2}\lesssim 1\,{\rm GeV}$, the
gravitino behaves as cold dark matter. The condition that the Universe is not
overclosed by the gravitino sets an upper bound on the reheating temperature
$T_{R}\lesssim 10^{2}\sim 10^{7}$ GeV, depending on the mass $m_{3/2}$ [40].
The gravitino in the range $1\,{\rm GeV}\lesssim m_{3/2}\lesssim 1\,{\rm TeV}$
is restricted due to light element photodestruction. The overclosure bound for
the $m_{3/2}\gtrsim 1\,{\rm TeV}$ gravitino dark matter gives $T_{\rm
R}\lesssim 10^{9}$ GeV. The gravitino with $m_{3/2}\geq 1\,{\rm TeV}$ is
likely to be unstable. The condition that the successful big bang
nucleosynthesis is not jeopardized by the decay of the gravitino gives a bound
on the reheating temperature $T_{\rm R}\lesssim 10^{5}\sim 10^{9}$ GeV [41].
Extremely light, $m_{3/2}\lesssim$ eV, or extremely heavy, $m_{3/2}\gtrsim
10^{7}$ GeV [42], gravitinos are unconstrained. Although realizing such mass
spectra in a realistic supersymmetry breaking mechanism is challenging, there
exist possible scenarios, e.g. gravitino dark matter at $m_{3/2}\gtrsim$ EeV
discussed in [43, 44, 45].

Constraints from the reheating temperature.— Regardless of the details of the
particle physics model that is embedded in supergravity, the constraints from
the gravitino problem are always present. The constraints give an upper bound
on the reheating temperature. It is thus important to elucidate the relation
between the reheating temperature and the prediction for the cosmological
parameters, whenever the viability of an inflationary model is discussed
within supergravity.

Assuming the standard thermal history of the Universe, inflation (accelerated
cosmic expansion) ends111 We use the condition that one of the slow roll
parameters $\epsilon_{V}=(M_{\rm P}^{2}/2)(V_{{\rm
E},\widehat{\varphi}}/V_{\rm E})^{2}$ or $\eta_{V}=M_{\rm P}^{2}V_{{\rm
E},\widehat{\varphi}\widehat{\varphi}}/V_{\rm E}$ reaches unity, namely, $\rm
max(\epsilon_{V},\eta_{V})=1$ for the end of inflation. This is in good
agreement with the actual termination of accelerated cosmic expansion for the
models studied here. at time $t_{\rm end}$, followed by a period of
(p)reheating characterized by the equation of state parameter $w$. The
Universe then thermalizes at time $t_{\rm th}$ and becomes radiation
dominant222 Strictly speaking, the completion of thermalization and the start
of radiation dominance (the end of reheating) are different, as emphasized
e.g. in [46]. However, the distinction has little significance in our analysis
due to the logarithmic dependance in the equation (8). We thus assume in our
analysis that the Universe becomes radiation dominant immediately after
thermalization. until matter-radiation equality is reached at time $t_{\rm
eq}$. After that the Universe stays matter dominated, until today $t_{0}$. The
$e$-folding number $N_{k}$ between the horizon exit of the comoving wave
number $k$ and the end of inflation is then expressed as [47, 48]

$\displaystyle N_{k}\equiv$ $\displaystyle\ln\frac{a_{\rm
end}}{a_{k}}=66.5-\ln
h-\ln\frac{k}{a_{0}H_{0}}+\frac{1-3w}{12(1+w)}\ln\frac{\rho_{\rm
th}}{\rho_{\rm end}}$ (8) $\displaystyle+\frac{1}{4}\ln\frac{V_{k}}{\rho_{\rm
end}}+\frac{1}{4}\ln\frac{V_{k}}{M_{\rm P}^{4}}+\frac{1}{12}\left(\ln
g_{*}^{\rm eq}-\ln g_{*}^{\rm th}\right),$ (9)

where $H_{0}=100\,h\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$ with $h=0.674$
[49] is the Hubble parameter today, $V_{k}$ is the potential (7) evaluated at
the time of the horizon exit of the wave number $k$, and $a$, $\rho$, $g_{*}$
are the scale factor, the energy density and the number of relativistic
degrees of freedom evaluated at the time specified by the super/subscripts
($k$ for the horizon exit, end for the end of inflation, th for the completion
of thermalization (end of reheating), eq for the matter-radiation equality and
0 for the present time).

The equation of state parameter $w$ in (8) is understood to be the averaged
value over the time $t_{\rm end}<t<t_{\rm th}$. In the supergravity inflation
scenario we consider, the inflaton has mass $M$ which is much smaller than the
inflationary scale and is thus negligible during inflation. Including this
mass, the potential (7) after inflation becomes

$\displaystyle V_{\rm
E}(\varphi)\simeq\frac{y^{2}}{16}\varphi^{4}+\frac{1}{2}M^{2}\varphi^{2}.$
(10)

At the beginning of (p)reheating the quartic term dominates and the cosmic
expansion is radiation-like, $w\simeq w_{\rm r}=1/3$. As the amplitude of the
inflaton oscillations is diminished, the quartic and the quadratic terms
become comparable at time $t_{\star}$, when
$\varphi=\varphi_{\star}\simeq\sqrt{8}M/y$. Let us denote the energy density
at this moment as $\rho_{\star}(<\rho_{\rm end})$. After $t_{\star}$, the
quadratic term of the potential dominates and the cosmic expansion becomes
matter-like, $w\simeq w_{\rm m}=0$. Thus the (p)reheating of this model
proceeds stepwise, first with radiation-like equation of state, and then with
matter-like equation of state. Accordingly, the fourth term of (8) may be
written more concretely as

$\displaystyle\frac{1-3w}{12(1+w)}\ln\frac{\rho_{\rm th}}{\rho_{\rm end}}$
(11) $\displaystyle=\frac{1-3w_{\rm r}}{12(1+w_{\rm
r})}\ln\frac{\rho_{\star}}{\rho_{\rm end}}+\frac{1-3w_{\rm m}}{12(1+w_{\rm
m})}\ln\frac{\rho_{\rm th}}{\rho_{\star}}.$ (12)

Now using $w_{\rm r}=1/3$, $w_{\rm m}=0$ and introducing dimensionless
parameter $\delta$ ($0\leq\delta\leq 1$) to denote
$\rho_{\star}=\delta^{4}\rho_{\rm end}$, (11) becomes

$\displaystyle\frac{1}{12}\ln\frac{\rho_{\rm
th}}{\rho_{\star}}=\frac{1}{12}\ln\left[\frac{\pi^{2}g_{*}^{\rm
th}}{30\rho_{\rm end}}\left(\frac{T_{\rm R}}{\delta}\right)^{4}\right].$ (13)

Here, $T_{\rm R}$ is the reheating temperature and we have used $\rho_{\rm
th}=\pi^{2}g_{*}^{\rm th}T_{\rm R}^{4}/30$. The reheating temperature always
appears in the combination $T_{\rm R}/\delta$. The energy density at the end
of inflation may be evaluated as $\rho_{\rm end}\simeq 2\,V_{\rm end}$. The
parameter $\delta$ depends on the phenomenological model embedded in
supergravity; for example, in the messenger inflation model [18] we find
$\delta\sim 10^{-5}$ for the messenger mass $M=10^{8}$ GeV and Yukawa coupling
$y=5.735\times 10^{-6}$.

We solved the equations of motion for the supergravity inflation model to find
the primordial tilt $n_{s}$ and the tensor-to-scalar ratio $r$, for given
values of the reheating temperature $T_{\rm R}$. The results are shown in Fig.
2 as red curves, together with the 1- and 2-$\sigma$ contours from the Planck
+BICEP/Keck 2018 data [3]. The curves are found to be nearly straight lines
with fitting formula

$\displaystyle r=$ $\displaystyle
0.01-17.4\times(n_{s}-a_{0}-a_{1}x-a_{2}x^{2}-a_{3}x^{3}),$ (14)
$\displaystyle a_{0}=0.95935,\quad a_{1}=6.2000\times 10^{-4},$ (15)
$\displaystyle a_{2}=-8.7565\times 10^{-6},\quad a_{3}=7.3869\times 10^{-8},$
(16)

where $x\equiv\log_{10}(\delta^{-1}T_{\rm R}/\text{GeV})$. We used $g_{*}^{\rm
eq}=3.91$ and $g_{*}^{\rm th}=106.75$ of the Standard Model. Generically,
small inflaton mass is present over and above the quartic potential and the
prediction depends on $\delta$ which parametrizes the transition between the
radiation-like expansion and the matter-like expansion during (p)reheating.
The highest reheating temperature admissible in supergravity inflation is
$\sim 10^{9}$ GeV, and the lower bound of the reheating temperature compatible
with big bang nucleosynthesis is a few MeV. Since $\delta\leq 1$, the lower
bound on the reheating temperature constrains the model to lie to the right of
the leftmost red line of Fig. 2. When $\delta=1$, more than two thirds of the
1- and 2-$\sigma$ parameter regions on the $n_{s}$-$r$ plane are seen to be
excluded by the gravitino constraints. The 1-$\sigma$ bounds on the CMB
observables give $\delta^{-1}T_{\rm R}<10^{31}$ GeV, and combining this with
$T_{\rm R}\lesssim 10^{9}$ GeV we have a model-independent lower bound on the
parameter $\delta>10^{-22}$. The steep slope of the red lines indicates strong
correlation between the rescaled reheating temperature $\delta^{-1}T_{\rm R}$
and the primordial tilt $n_{s}$. Thus, measurements of $n_{s}$ are important
to test this class of inflationary scenarios. In future, precision
measurements of $n_{s}$ combined with the constraints from the gravitino
problem may well rule out this otherwise promising model of inflationary
cosmology.

Note Added.— After completing this work, we noticed a preprint [50] with
partially overlapping results appeared on the arXiv.

Acknowledgments.— This work was supported in part by the National Research
Foundation of Korea Grant-in-Aid for Scientific Research Grant No.
NRF-2018R1D1A1B07051127 (S.K) and by the United States Department of Energy
Grant No. DE-SC0012447 (N.O.).

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