Gravitational counterpart of the chiral magnetic effect, which is referred as
the chiral gravitational effect, can also be of interest in a cosmological
setup. In this study, we investigate this effect in the time-dependent chiral
asymmetric fermion background and in the expanding spacetime by formulating the
effective action of gravitational waves. We also analyze the anomaly equation
to see how the backreaction from gravitational waves to thermal chiral plasma
occurs. We find that the non-trivial time dependence of chiral chemical
potential, which can be induced in some scenarios of baryogenesis, is the key
ingredient of the chiral gravitational effect. It turns out that the "memory"
of the effect is imprinted on the high frequency gravitational waves
propagating in the plasma. Cosmological implications and potential effects on
the gravitational wave observation are briefly discussed.

We revisit gravitational particle production from the Stokes phenomenon
viewpoint, which helps us make a systematic way to understand asymptotic
behavior of mode functions in time-dependent background. One of our purposes of
this work is to make the method more practical for evaluation of
non-perturbative particle production rate. In particular, with several examples
of time-dependent backgrounds, we introduce some approximation methods that
make the analysis more practical. Specifically, we consider particle production
in simple expanding backgrounds, preheating after $R^2$ inflation, and a
transition model with smoothly changing mass. As we find several technical
issues in analyzing the Stokes phenomenon of each example, we discuss how to
simplify the problems while showing the accuracy of analytic estimation under
the approximations we make.

We study M-theory compactified on twisted 7-tori with $G_2$-holonomy. The
effective 4d supergravity has 7 chiral multiplets, each with a unit logarithmic
K\"ahler potential. We propose octonion, Fano plane based superpotentials,
codifying the error correcting Hamming (7,4) code. The corresponding 7-moduli
models have Minkowski vacua with one flat direction. We also propose
superpotentials based on octonions/error correcting codes for Minkowski vacua
models with two flat directions. We update phenomenological $\alpha$-attractor
models of inflation with $3\alpha=7,6,5,4,3,1$, based on inflation along these
flat directions. These inflationary models reproduce the benchmark targets for
detecting B-modes, predicting 7 different values of $r = 12\alpha/N_{e}^{2}$ in
the range $10^{-2}\gtrsim r \gtrsim 10^{-3}$, to be explored by future
cosmological observations.

We consider the renormalization in the pseudo-scalar inflation models with
the gravitational Chern-Simons term. In this model, lepton asymmetry is
generated from the chiral gravitational waves produced due to the Chern-Simons
term through the gravitational chiral anomaly. However, it is known that the
naive estimate of the expectation value of the gravitational Chern-Pontryagin
density as well as the resultant lepton number density depend on the UV-cutoff
scale, which raises a question on their validity. In this paper, we propose a
way to renormalize the expectation value of the Chern-Pontryagin density to
remove the UV-cutoff dependence. We also discuss the renormalized lepton number
density when we adopt the minimal subtraction scheme and the viability of the
gravitational leptogenesis scenario.

We discuss particle production associated with vacuum decay, which changes
the mass of a scalar field coupled to a background field which induces the
decay. By utilizing the Stokes phenomenon, we can optimally track the
time-evolution of mode function and hence calculate particle production
properly. In particular, we use real time formalisms for vacuum decay in
Minkowski and de Sitter spacetime together with the Stokes phenomenon method.
For each case, we consider the flyover vacuum decay model and stochastic
inflation, respectively. Within the real time formalism, the particle
production can be viewed as that caused by nontrivial external fields. This
gives us a novel perspective of the real time formalism of vacuum decay.

We revisit the gravitational leptogenesis scenario in which the inflaton is
coupled to gravity by the Chern-Simons term and the lepton asymmetry is
generated through the gravitational anomaly in the lepton number current during
inflation. We constrain the possible model parameter space by requiring the
absence of ghost modes below the Planck scale, which would suggest the
breakdown of the effective theory, and evaluate the net baryon asymmetry for
various reheating processes. We find that the mechanism with these requirements
is insufficient to explain the observed baryon asymmetry of the Universe for
standard reheating scenarios. We show that, however, with the kination scenario
realized in e.g. the k- and G-inflation models, a sufficient baryon asymmetry
can be generated within a feasible range of the model parameters.

We construct the Horndeski Lagrangian within non-linearly realized
supergravity. We will show that the bosonic part of the Horndeski Lagrangian
can be realized. Gravitino naturally couples to Horndeski sector in a
super-covariant way. Such gravitino couplings are also free from ghosts.

An anti-D3-brane plays a crucial role in the construction of semi-realistic
cosmological models in string theory. Part of its action provides an uplift
term that has been used to lift AdS solutions to phenomenologically viable dS
vacua in the KKLT and LVS setups. In the last few years it has been shown that
this uplift breaks supersymmetry spontaneously and can be described in the 4d
$\mathcal{N}=1$ supergravity language by using constrained supermultiplets.
Here we derive the complete 4d $\mathcal{N}=1$ supergravity action for an
anti-D3-brane coupled to all closed string background fields. In particular we
include the vector field, the scalar fields and all fermions that live on the
anti-D3-brane.

We discuss $N=2\to N=1$ reduction in four dimensional conformal supergravity.
In particular, we keep the off-shell structure of supermultiplets (except
hypermultiplets). As we will show, starting with (almost) off-shell conformal
supergravity makes the procedure simpler than that from $N=2$ Poincar\'e
supergravity, which makes it easier to show the correspondence to the standard
$N=1$ conformal supergravity. We find that the $N=1$ superconformal symmetry is
simply realized by truncating the gravitino multiplet. We also discuss the
consistency with the original $N=2$ system and show the reduced $N=1$ conformal
supergravity action.

We construct the simplest inflationary $\alpha$-attractor models in
supergravity: it has only one scalar, the inflaton. There is no sinflaton since
the inflaton belongs to an orthogonal nilpotent superfield where the sinflaton
depends on fermion bilinears. When the local supersymmetry is gauge-fixed,
these models have only one single real scalar (the inflaton), a graviton and a
massive gravitino. The sinflaton, sgoldstino and inflatino are all absent from
the physical spectrum in the unitary gauge. The orthogonality condition leads
to the simplest K\"ahler potential for the inflaton, while preserving the
Poincar\'e disk geometry of $\alpha$-attractors. The models are particularly
simple in the framework of the $\overline {D3}$ induced geometric inflation.

We revisit phenomenological as well as string-theoretical aspects of D-brane
inflation cosmological models. Phenomenologically these models stand out on par
with $\alpha$-attractors, as models with Planck-compatible values of $n_s$,
moving down to the sweet spot in the data with decreasing value of $r$. On the
formal side we present a new supersymmetric version of these models in the
context of de Sitter supergravity with a nilpotent multiplet and volume modulus
stabilization. The geometry of the nilpotent multiplet is evaluated in the
framework of string theory.

The UV finiteness found in calculations of the 4-point amplitude in $\mathcal
N=5$ supergravity at loop order $L=3, 4$ has not been explained, which
motivates our study of the relevant superspace invariants and on-shell
superamplitudes for both $\mathcal N=5$ and $\mathcal N=6$. The local 4-point
superinvariants for $L = 3,4$ are expected to have nonlinear completions whose
6-point amplitudes have non-vanishing SSL's (soft scalar limits), violating the
behavior required of Goldstone bosons. For $\mathcal N=5$, we find at $L=3$
that local 6-point superinvariant and superamplitudes, which might cancel these
SSL's, do not exist. This rules out the candidate 4-point counterterm and thus
gives a plausible explanation of the observed $L=3$ finiteness. However, at $L=
4$ we construct a local 6-point superinvariant with non-vanishing SSL's, so the
SSL argument does not explain the observed $L=4$ $\mathcal N=5$ UV finiteness.
For $\mathcal N=6$ supergravity there are no 6-point invariants at either $L=
3$ or 4, so the SSL argument predicts UV finiteness.

We constructed a model of natural inflation in the context of
$\alpha$-attractor supergravity, in which both the dilaton field and the axion
field are light during inflation, and the inflaton may be a combination of the
two. The T-model version of this theory is defined on the Poincare disk with
radius |Z| = 1. It describes a Mexican hat potential with the flat axion
direction corresponding to a circle of radius |Z| < 1. The axion decay constant
$f_{a}$ in this theory can be exponentially large because of the hyperbolic
geometry of the Poincare disk. Depending on initial conditions, this model may
describe $\alpha$-attractor inflation driven by the radial component of the
inflaton field, natural inflation driven by the axion field, or a sequence of
these two regimes. We also construct the E-model version of this theory, which
has similar properties. In addition, we describe generalized $\alpha$-attractor
models where the potential can be singular at the boundary of the moduli space,
and show that they can provide a simple solution for the problem of initial
conditions for the models with plateau potentials.

We derive the general supertrace formula for a system with $N$ chiral
superfields and one nilpotent chiral superfield in global and local
supersymmetry. The nilpotent multiplet is realized by taking the
scalar-decoupling limit of a chiral superfield breaking supersymmetry
spontaneously. As we show, however, the modified formula is not simply related
to the scalar-decoupling limit of the supertrace in linearly-realized
supersymmetry. We also show that the supertrace formula reduces to that of a
linearly realized supersymmetric theory with a decoupled sGoldstino if the
Goldstino is the fermion in the nilpotent multiplet.

We present a class of supergravity $\alpha$-attractors with an approximate
global U(1) symmetry corresponding to the axion direction. We also develop a
multi-field generalization of these models and show that the $\alpha$-attractor
models with U(1) symmetries have a dual description in terms of a two-form
superfield coupled to a three-form superfield.

In ${\cal N}=5, 6, 8$ supergravities there are hidden symmetries of equations
of motion, described by duality groups $SU(1,5), \, SO^*(12), \, E_{7(7)}$
respectively. UV divergences and known candidate counterterms violate the
deformed duality symmetry current conservation. Extra higher derivative terms
in the action are required to restore duality. We study the effect of a
two-vector part of the counterterm for ${\cal N}\geq 5$ supergravities using
the universality of the symplectic structure of extended supergravities. We
construct a compact form of a deformed action with infinite number of higher
derivative terms and restored duality symmetry with deformation parameter
$\lambda$. We find, in $\lambda^2$ approximation, that the $SU({\cal N})$
symmetry of the deformed theory is restored on shell.

$f(R)$ supergravity is known to contain a ghost mode associated with
higher-derivative terms if it contains $R^n$ with $n$ greater than two.We
remove the ghost in $f(R)$ supergravity by introducing auxiliary gauge field to
absorb the ghost. We dub this method as the ghostbuster
mechanism~\cite{Fujimori:2016udq}. We show that the mechanism removes the ghost
supermultiplet but also terms including $R^n$ with $n\geq3$, after integrating
out auxiliary degrees of freedom. For pure supergravity case, there appears an
instability in the resultant scalar potential. We then show that the
instability of the scalar potential can be cured by introducing matter
couplings in such a way that the system has a stable potential.

The dynamics of the Goldstino mode of spontaneously broken supersymmetry is
universal, being fully determined by the non-linearly realized symmetry. We
investigate the small-field limit of this theory. This model non-linearly
realizes an alternative supersymmetry algebra with vanishing anti-commutators
between the fermionic generators, much like an internal supersymmetry. This
Goldstino theory is akin to the Galilean scalar field theory that arises as the
small-field limit of Dirac-Born-Infeld theory and non-linearly realizes the
Galilean symmetry. Indeed, the small-field Goldstino is the partner of a
complex Galilean scalar field under conventional supersymmetry. We close with
the generalization to extended internal supersymmetry and a discussion of its
higher-dimensional origin.

We discuss the cosmological gravitino problem in inflation models in which
the inflaton potential is constructed from K\"ahler potential rather than
superpotential: a representative model is $\overline{\text{D}3}$-induced
geometric inflation. A critical ingredient in this type of models is the
coupling of the inflaton and Polonyi (supersymmetry-breaking) field in the
K\"ahler potential, which is needed to build the inflaton potential. We point
out the same coupling let the inflaton dominantly decay into a pair of
inflatino and gravitino causing the gravitino problem. We propose some possible
solutions to this problem.

We systematically construct ghost-free higher-derivative actions of Abelian
vector supermultiplets in four-dimensional ${\cal N}=1$ global supersymmetric
theories. After giving a simple example which illustrates that a naive
introduction of a higher-derivative term gives rise to a ghost, we discuss
possible building blocks for a ghost-free action and explicitly show that their
bosonic parts have no ghost mode and the auxiliary field $\boldsymbol{D}$ does
not propagate. Higher-derivative terms yield higher powers of the auxiliary
field $\boldsymbol{D}$ in the actions, and the D-term equations of motion
consequently admit multiple solutions in general. We confirm that the
well-known supersymmetric Dirac-Born-Infeld action falls into this class. We
further give another example in which the standard quadratic kinetic term
(Maxwell term) is corrected by a quartic term of the field strength. We also
discuss possible couplings to matter fields and a deformed D-term potential.

Fibre inflation is a specific string theory construction based on the Large
Volume Scenario that produces an inflationary plateau. We outline its relation
to $\alpha$-attractor models for inflation, with the cosmological sector
originating from certain string theory corrections leading to $\alpha=2$ and
$\alpha=1/2$. Above a certain field range, the steepening effect of
higher-order corrections leads first to the breakdown of single-field slow-roll
and after that to the onset of 2-field dynamics: the overall volume of the
extra dimensions starts to participate in the effective dynamics. Finally, we
propose effective supergravity models of fibre inflation based on an
${\overline {D3 } }$ uplift term with a nilpotent superfield. Specific moduli
dependent $\overline {D3}$ induced geometries lead to cosmological fibre models
but have in addition a de Sitter minimum exit. These supergravity models
motivated by fibre inflation are relatively simple, stabilize the axions and
disentangle the Hubble parameter from supersymmetry breaking.

Effective supergravity inflationary models induced by anti-D3 brane
interaction with the moduli fields in the bulk geometry have a geometric
description. The K\"ahler function carries the complete geometric information
on the theory. The non-vanishing bisectional curvature plays an important role
in the construction. The new geometric formalism, with the nilpotent superfield
representing the anti-D3 brane, allows a powerful generalization of the
existing inflationary models based on supergravity. They can easily incorporate
arbitrary values of the Hubble parameter, cosmological constant and gravitino
mass. We illustrate it by providing generalized versions of polynomial chaotic
inflation, T- and E-models of $\alpha$-attractor type, disk merger. We also
describe a multi-stage cosmological attractor regime, which we call cascade
inflation.

Extending the work of Ferrara and one of the authors, we present dynamical
cosmological models of $\alpha$-attractors with plateau potentials for
$3\alpha=1,2,3,4,5,6,7$. These models are motivated by geometric properties of
maximally supersymmetric theories: M-theory, superstring theory, and maximal $N
= 8$ supergravity. After a consistent truncation of maximal to minimal
supersymmetry in a seven-disk geometry, we perform a two-step procedure: 1) we
introduce a superpotential, which stabilizes the moduli of the seven-disk
geometry in a supersymmetric minimum, 2) we add a cosmological sector with a
nilpotent stabilizer, which breaks supersymmetry spontaneously and leads to a
desirable class of cosmological attractor models. These models with $n_s$
consistent with observational data, and with tensor-to-scalar ratio $r \approx
10^{-2}- 10^{-3}$, provide natural targets for future B-mode searches. We
relate the issue of stability of inflationary trajectories in these models to
tessellations of a hyperbolic geometry.

We list all potential candidates for U(1) anomalous non-local 1-loop 4-point
amplitudes and higher loop UV divergences in $\mathcal{N}\geq 5$
supergravities. The relevant chiral superinvariants are constructed from
linearized chiral superfields and define the corresponding superamplitudes. The
anomalous amplitudes, of the kind present in $\mathcal{N}=4$, are shown to be
absent in $\mathcal{N} \geq 5$. In $\mathcal{N}=6$ supergravity the result is
deduced from the double-copy $\mathcal{N}=4_{YM} \times (\mathcal{N}=2)_{YM}$
model, whereas in $\mathcal{N}=5,8$ the result on absence of anomalous
amplitudes is derived in supergravities as well as in the $(\mathcal{N}=4)_{YM}
\times (\mathcal{N}-4)_{YM}$ double-copy models.

We study non-thermal gravitino production in the minimal supergravity
inflation. In this minimal model utilizing orthogonal nilpotent superfields,
the particle spectrum includes only graviton, gravitino, inflaton, and
goldstino. We find that a substantial fraction of the cosmic energy density can
be transferred to the longitudinal gravitino due to non-trivial change of its
sound speed. This implies either a breakdown of the effective theory during
reheating or a serious gravitino problem.

We construct a Dirac-Born-Infeld (DBI) action coupled to a two-form field in
four dimensional $\mathcal{N}=1$ supergravity. Our superconformal formulation
of the action shows a universal way to construct it in various Poincar\'e
supergravity formulations. We generalize the DBI action to that coupled to
matter sector. We also discuss duality transformations of the DBI action, which
are useful for phenomenological and cosmological applications.

In the version 1 of this paper, we claimed that 4D ${\cal N}=1$ supergravity
formulations with different auxiliary fields can be unified using U(1) gauge
symmetry, whose gauge superfield does not have a kinetic term. However, after
submission, we found a critical error in our statement. This is the note on the
paper of the version 1.

We present for the first time a ghost-free higher-derivative chiral model
with a propagating auxiliary F-term field (highest component of the chiral
multiplet). We obtain this model by removing a ghost in a higher derivative
chiral model, with Higgsing it in terms of an auxiliary vector superfield.
Depending on the sign of the quadratic derivative term of the chiral
superfield, the model contains two ghost free branches of the parameter
regions. We find that supersymmetry is spontaneously broken in one branch while
it is preserved in the other branch. As a consequence of dynamical F-term
field, a conserved U(1) charge corresponding to the number density of $F$
appears, which can be regarded as a generalization of the R-symmetry.

We study whether the relaxion mechanism solves the Higgs hierarchy problem
against a high scale inflation or a high reheating temperature. To accomplish
the mechanism, we consider the scenario that the Higgs vacuum expectation value
is determined after inflation. We take into account the effects of the Hubble
induced mass and thermal one in the dynamics of the relaxion.

We consider Abelian tensor hierarchy in four-dimensional ${\cal N}=1$
supergravity in the conformal superspace formalism, where the so-called
covariant approach is used to antisymmetric tensor fields. We introduce
$p$-form gauge superfields as superforms in the conformal superspace. We solve
the Bianchi identities under the constraints for the superforms. As a result,
each of form fields is expressed by a single gauge invariant superfield. The
action of superforms is shown with the invariant superfields. We also show the
relation between the superspace formalism and the superconformal tensor
calculus.

We construct the Dirac-Born-Infeld (DBI) action of a real linear multiplet in
4D $\mathcal{N}=1$ supergravity. Based on conformal supergravity, we derive the
general condition under which the DBI action can be realized, and show that it
can be constructed in the new minimal supergravity. We also generalize it to
the matter coupled system.

We propose a supergravity model with a constrained curvature multiplet, which
realizes the Starobinsky inflation and the de Sitter vacuum in the present
universe simultaneously. Surprisingly, at the vacuum, the soft supersymmetry
breaking scale for minimal supersymmetric standard model sector becomes the TeV
scale, however, the gravitino mass scale becomes much higher than that of soft
supersymmetry breaking. Such a hierarchical structure, which naturally avoids
the gravitino problem, appears without introducing a new scale other than
inflation scale.

We express the action of six-dimensional supergravity in terms of
four-dimensional N=1 superfields, focusing on the moduli dependence of the
action. The gauge invariance of the action in the tensor-vector sector is
realized in a quite nontrivial manner, and it determines the moduli dependence
of the action. The resultant moduli dependence is intricate, especially on the
shape modulus. Our result is reduced to the known superfield actions of
six-dimensional global SUSY theories and of five-dimensional supergravity by
replacing the moduli superfields with their background values and by performing
the dimensional reduction, respectively.

We construct the component action of the system including an ordinary matter
and a nilpotent multiplet in global and local supersymmetric framework. The
higher dimensional operators of not only Goldstino but also matter and
gravitino fields are shown, which appear due to nonlinearly realized
supersymmetry.

We investigate the inflation model with a massive vector multiplet in a case
that the action of the vector multiplet is extended to the Dirac-Born-Infeld
(DBI) type one. We show the massive DBI action in 4 dimensional ${\cal N}=1$
supergravity, and find that the higher order corrections associated with the
DBI-extension make the scalar potential flat with a simple choice of the matter
couplings. We also discuss the DBI-extension of the new minimal Starobinsky
model, and find that it is dual to a special class of the massive DBI action.

We show the effects of supersymmetric higher derivative terms on inflation
models in supergravity. The results show that such terms generically modify the
effective kinetic coefficient of the inflaton during inflation if the cut off
scale of the higher derivative operators is sufficiently small. In such a case,
the $\eta$-problem in supergravity does not occur, and we find that the
effective potential of the inflaton generically becomes a power type potential
with a power smaller than two.

We construct the Dirac-Born-Infeld action in the context of N=1 conformal
supergravity and its possible extensions including matter couplings. We
especially focus on the Volkov-Akulov constraint, which is important to avoid
ghost modes from the higher derivative terms. In the case with matter
couplings, we find the modified D-term potential.

We express supersymmetric couplings among the vector and the tensor
multiplets in six dimensions (6D) in terms of N=1 superfields. The superfield
description is derived from the invariant action in the projective superspace.
The obtained expression is consistent with the known superfield actions of 6D
supersymmetric gauge theory and 5D Chern-Simons theory after the dimensional
reduction. Our result provides a crutial clue to the N=1 superfield description
of 6D supergravity.

We study reheating processes and its cosmological consequences in the
Starobinsky model embedded in the old-minimal supergravity. First, we consider
minimal coupling between the gravity and matter sectors in the higher curvature
theory, and transform it to the equivalent standard supergravity coupled to
additional matter superfields. We then discuss characteristic decay modes of
the inflaton and the reheating temperature $T_{\rm R}$. Considering a simple
model of supersymmetry breaking sector, we estimate gravitino abundance from
inflaton decay, and obtain limits on the masses of gravitino and supersymmetry
breaking field. We find $T_{\rm R}\simeq 1.0\times10^9$ GeV and the allowed
range of gravitino mass as $10^4$ GeV $\lesssim m_{3/2} \lesssim 10^5$ GeV,
assuming anomaly-induced decay into the gauge sector as the dominant decay
channel.

We propose a new class of inflation models in supergravity with higher
derivative terms. In those models, the K\"ahler potential does not contain the
inflaton multiplet, but a supersymmetric derivative term does. In the models,
inflation is effectively driven by a single scalar field with a standard
kinetic term and a scalar potential. Remarkably, the so-called $\eta$ problem
does not exist in our models.

We consider the supersymmetry breaking effects on typical inflation models
with different types of K\"ahler potential. The critical size of
supersymmetry-breaking scale, above which the flatness of the inflaton
potential is spoiled, drastically changes model by model. We present the
universal description of such effects in terms of a field-dependent scaling
factor by which gravity-mediated supersymmetry breaking terms are suppressed or
enhanced, based on the conformal supergravity framework. Such a description
would be useful when we estimate them in constructing supersymmetric models of
particle cosmology.

An exponentially large extra dimension can be naturally realized by the
Casimir energy and the gaugino condensation in 5D supersymmetric theory. The
model does not require any hierarchies among the 5D parameters. The key
ingredient is an additional modulus other than the radion, which generically
exists in 5D supergravity. SUSY is broken at the vacuum, which can be regarded
as the Scherk-Schwarz SUSY breaking. We also analyze the mass spectrum and
discuss some phenomenological aspects.

5D supergravity generically has non-geometric moduli other than the radion
that belong to 5D vector multiplets. We summarize the impacts of such moduli on
4D effective theory of 5D supergravity on S^1/Z_2. We mainly discuss the
structure of the effective Kahler potential including the one-loop quantum
corrections. As an illustrative example, we construct a model in which the size
of the extra dimension is stabilized at an exponentially large value compared
to the Planck length, which is similar to the LARGE volume scenario in string
theory.

The Hubble parameter during the inflationary era must be smaller than the
gravitino mass if the moduli are stabilized by the Kachru-Kallosh-Linde-Trivedi
mechanism. This condition represents the difficulty to combine the low scale
SUSY breaking and the high scale inflation. We propose a simple mechanism which
can naturally separate the inflation scale from the SUSY breaking scale today.

We perform a comprehensive and systematic analysis of the SUSY flavor
structure of generic 5D supergravity models on $S^1/Z_2$ with multiple
$Z_2$-odd vector multiplets that generate multiple moduli. The SUSY flavor
problem can be avoided due to contact terms in the 4D effective K\"ahler
potential peculiar to the multi-moduli case. A detailed phenomenological
analysis is provided based on an illustrative model.