Complexified Starobinsky Inflation in Supergravity
in the Light of Recent BICEP2 Result
Koichi Hamaguchi, Takeo Moroi, Takahiro Terada,
Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan
Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),
WPI, TODIAS, The University of Tokyo, Chiba 277-8568, Japan
Motivated by the recent observation of the -mode signal in the cosmic microwave background by BICEP2, we study the Starobinsky-type inflation model in the framework of old-minimal supergravity, where the inflaton field in the original (non-supersymmetric) Starobinsky inflation model is promoted to a complex field. We study how the inflaton evolves on the two-dimensional field space, varying the initial conditions. We show that (i) one of the scalar fields has a very steep potential once the trajectory is off from that of the original Starobinsky inflation, and that (ii) the -mode signal observed by BICEP2 is too large to be consistent with the prediction of the model irrespective of the initial conditions. Thus, the BICEP2 result strongly disfavors the complexified Starobinsky inflation in supergravity.
Recently, -mode polarization of cosmic microwave background (CMB) has been observed by BICEP2, which indicated a large tensor-to-scalar ratio of 
The observation of BICEP2 provides a significant constraint on inflationary models because the value of is directly related to the scale of inflation (i.e., the expansion rate during inflation). In particular, the BICEP2 result strongly disfavors one of the interesting possibilities, i.e., Starobinsky inflation model [2, 3] which utilizes a scalar degree of freedom in the gravitational sector as an inflaton. This is because the Starobinsky inflation predicts of the order of , which is significantly smaller than the BICEP2 result.
If one extends the model, this conclusion may change. The extension we consider in the present study is to supersymmetrize the model because supersymmetry is a prominent candidate of the physics beyond the Standard Model. In such a model, the inflation can still be realized solely by the gravitational sector, while new scalar degrees of freedom are automatically introduced, which may affect the dynamics of inflation.
The Starobinsky model is based on a modified theory of gravity, so we
need to consider a modified theory of supergravity. There are two
minimal off-shell formulations of supergravity: the
old-minimal [4, 5, 6] and
the new-minimal  supergravity. Supergravity
embedding of Starobinsky model has been studied both in the
old-minimal [8, 9, 10] and the
new-minimal [11, 12] supergravities.
These studies share the original philosophy of the Starobinsky model
in the sense that the supergravity generalizations of the model rely
solely on (super)geometrical or (super)gravitational
In this letter, we study the Starobinsky-type inflation model in the framework of old-minimal supergravity. We pay particular attention to the fact that there exist two scalar degrees of freedom originating from the gravity multiplet in such a model. We study the evolution of the inflaton on the two-dimensional field space. We will see that the potential of one of the scalar fields becomes very steep once the trajectory is off from that of the original Starobinsky inflation. We also show that the tensor-to-scalar ratio in the supergravity Starobinsky model is too small to be consistent with the BICEP2 result even though the field space is enlarged.
where and are the hermitian and the holomorphic functions of the scalar curvature chiral superfield , respectively. The superfield contains Ricci scalar curvature in its component and gravitino in its and components. It also contains a complex scalar and real vector . These are auxiliary fields in the case of the minimal action with and . For generic functions and , however, these become dynamical.
with the following no-scale type Kähler potential and superpotential:
Linearized analysis of the original picture (higher-curvature supergravity) for a simple function has been performed in Ref. . Bosonic Lagrangian of the original picture and comparisons of both pictures are described in Ref. . Note that any and functions lead to the unique Kähler and superpotentials for because the origin of is a Lagrange multiplier. In particular, canonically normalized field along the real axis () has the Starobinsky potential (cf. Eq. (9)). Roughly speaking, , , , and in this picture correspond to , , , and in the original geometrical picture, respectively. In this letter, we focus on the standard matter-coupled supergravity picture.
Consider a Kähler potential for ,
The first term (constant) is needed to reproduce Einstein supergravity. The second term leads to the kinetic term of the new degrees of freedom. However, this term produces the scalar potential unbounded below in the region of large . Instability for radial direction is stabilized by the third term proportional to (see e.g. Refs. [27, 28, 14, 10] and references therein). Small makes other local minima near the original minimum (). Because of these reasons, we take a sufficiently large value of . Note that, for sufficiently large , is stabilized for any value of . We also assume so that the potential value at the vacuum is zero. Thus, is set to the minimum , and the resultant effective theory has two fields and with only one parameter .
After stabilization of , the Lagrangian density is given by
Canonical normalization of both fields at the same time is impossible
in this case. We find it useful to define the semi-canonical basis
that does not have kinetic mixing and realizes canonical
normalization at the vacuum ():
Then, the Lagrangian density becomes
The third term is the Starobinsky potential. Looking at the second and fourth terms, one may naively guess that chaotic inflation  is possible neglecting the common factor . However, as we shall see, this exponential factor strongly drives to the positive direction in the large region.
Now let us investigate if the fields and/or play the role of inflaton which are responsible for the present density fluctuations of our universe. For this purpose, we first study the evolution of these fields. The evolution equations for and are given by
where the “dot” denotes the derivative with respect to time and (with being the scale factor) is the expansion rate of the universe. When the energy density of the universe is dominated by that of , we obtain
where is the total energy density:
By solving the above equations numerically, we follow the trajectories of and with various initial values.
In Fig. 1, we show the contours of the potential and the evolutions of the fields on plane. As representative initial conditions, we choose , , and . (The initial values of and are taken to be zero.) With such initial conditions, we can see that starts to move to the direction first, then it settles to the real axis (i.e., ). After reaching to the real axis, the motion of is well approximated by the single-field inflation with ; the situation is almost the same as the non-supersymmetric original Starobinsky inflation. As can be seen from the dashed lines, the trajectories are almost unchanged even if starts from .
On each contour, in particular for , we show several points which give rise to some specific values of the -folding numbers until the end of inflation. Here, the -folding number is defined as
where is the time at the end of inflation. In our analysis, we define it by , where the slow-roll parameter is given by
We have used the Einstein equation in the last equality. We can see that the change of the -folding value in the period of is small. Therefore, a large value of the -folding number during inflation, which is necessary to solve the horizon and flatness problems, should be accumulated when is on the real axis.
For and , the expansion of the universe is accelerating and decelerating, respectively. Thus, for inflation to happen, is necessary. To see when the expansion is accelerating, in Fig. 2, we plot as a function of , taking and . We can see that, just after the start of the motion, significantly increases and soon becomes larger than . In this period, the expansion of the universe is decelerating and not inflating. The drop of at (45) in the red (blue) line corresponds to the point at which becomes most negative and (cf. Fig. 1).
Thus, the universe transits from the decelerating epoch to the Starobinsky-type inflation. We call the period in between as “transition period,” and the period of the Starobinsky-like expansion as “Starobinsky-inflation period.” The important point is that the transition period is very short; during the transition period, changes or so. (For the case of , for example, the transition period corresponds to .) This is due to the fact that the motion of becomes suppressed soon after the condition is satisfied. If we require that the causal connection be realized for the scale much longer than (with being the wavenumber corresponding to the present Hubble scale), the mode with the wavenumber should leave the horizon in the Starobinsky-inflation period. Then, the tensor-to-scalar ratio becomes and is too small to be consistent with the value given in Eq. (1). Thus, in the light of the recent BICEP2 result, the Starobinsky inflation is disfavored even if the field space is complexified in the framework of old-minimal supergravity.
One of the possibilities to change this conclusion may be to consider the case where the mode with exits the horizon in the transition period. However, such a solution looks unlikely. Even though the density fluctuations with the wavenumber may be altered, fluctuations with the wavenumber larger than have almost the same property as those in the case of Starobinsky inflation. Consequently, for the angular scale of with , the density perturbations behave as those in the Starobinsky model. The BICEP experiment is sensitive to the -mode signal with , while the scalar-mode fluctuations for such an angular scale is well studied by using CMB and other observables. Thus, in the present model, it is difficult to enhance the tensor-mode fluctuations without conflicting observations.
This work was supported by JSPS KAKENHI Grant No. 22244021 (K.H., T.M.), No. 22540263 (T.M.), No. 23104008 (T.M.) and also by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The work of T.T. was supported by the Program for Leading Graduate Schools, MEXT, Japan.
- In this respect, see Ref.  for the inflationary scenario induced by gravitino condensation. Closely related works to Refs. [8, 9, 10, 11, 12] include Refs. [14, 15] in the old-minimal formulation and Refs.  (see also Refs. ) in both formulations. See also other recent related works [18, 19] in supergravity. These can reproduce the scalar potential of the dual theory of the Starobinsky model , but do not necessarily have pure (super)geometrical or (super)gravitational interpretation. Generalization of the duality  between higher-curvature supergravity and standard matter-coupled supergravity has recently been discussed in Ref.  which provides the higher-curvature supergravity representation of the attractor model .
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- Throughout this letter, we use the Planck unit , where is the reduced Planck scale.
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