We consider the classical energy conditions (strong, null and weak) in Starobinsky supergravity theories. We study in detail the simplest Starobinsky supergravity model in the "old-minimal" supergravity setup and find examples of violations of each energy condition for a certain range of the parameters due to extra scalar d.o.f. present in supergravity. We find the null and weak energy conditions to be preserved and the strong energy condition to be violated in the physically relevant examples such as inflation and dark energy.
Starobinsky inflation and Higgs inflation in gravity, and their equivalence based on the common inflationary potential are extended to supergravity in the proper framework, where the Starobinsky and Higgs descriptions of inflation arise in two different gauges of the same supergravity model.
The Starobinsky inflationary model of $(R+R^2)$ gravity is extended to the new-minimal (40+40) off-shell 4D, N=2 supergravity in N=2 chiral (curved) superspace. In its dual formulation, as a matter-coupled N=2 Einstein supergravity, inflaton belongs to a massive N=2 vector multiplet.
Polonyi-Starobinsky (PS) supergravity is the N=1 supergravity model of Starobinsky inflation with spontaneous supersymmetry breaking (after inflation) due to Polonyi superfield, and inflaton belonging to a massive vector supermultiplet. The PS model is used for an explicit realization of the (super-heavy) gravitino dark matter scenario in cosmology. We find a potential instability in this model, and offer a mechanism for its removal by adding a Fayet-Iliopoulos (FI) term.
In the new-minimal supergravity formulation we present the embedding of the $R+R^2$ Starobinsky model of inflation. Starting from the superspace action we perform the projection to component fields and identify the Starobinsky model in the bosonic sector. Since there exist no other scalar fields, this is by construction a single field model. This higher curvature supergravity also gives rise to a propagating massive vector. Finally we comment on the issues of higher order corrections and initial conditions.
We discuss how the higher-derivative Starobinsky model of inflation originates from N=1 supergravity. It is known that, in the old-minimal supergravity description written by employing a chiral compensator in the superconformal framework, the Starobinsky model is equivalent to a no-scale model with F-term potential. We show that the Starobinsky model can also be originated within the so-called new-minimal supergravity, where a linear compensator superfield is employed. In this formulation, the Starobinsky model is equivalent to standard supergravity coupled to a massive vector multiplet whose …
We find new realizations of Volkov-Akulov-Starobinsky supergravity, i.e. Starobinsky inflationary models in supergravity coupled to a nilpotent superfield describing Volkov-Akulov goldstino. Our constructions are based on the no-scale Kähler potential $K=-3\log(T+\bar{T})$ for the inflaton field, and can describe de Sitter vacuum after inflation where supersymmetry is broken by the goldstino auxiliary component. In fact, we show that a more general class of models with $K=-α\log(T+\bar{T})$ for $3\leqα\lesssim 6.37$ can accomodate Starobinsky-like inflation with the universal prediction $n_s\s…
The minimal Starobinsky supergravity with the inflaton (scalaron) and the goldstino in a massive vector supermultiplet is coupled to the dilaton-axion chiral superfield with the no-scale Kähler potential and a superpotential. The Kachru-Kallosh-Linde-Trivedi (KKLT)-type mechanism in the presence of a constant term in the superpotential is applied to stabilize the dilaton/axion during inflation, and it is shown to lead to an instability. The instability is cured by adding the alternative Fayet-Iliopoulos (FI) term that does not lead to the gauged $R$-symmetry. Other stabilization mechanisms, ba…
We provide a succinct way to construct the supersymmetric completion of $R^n$ $(n\ge3)$ in components using superconformal formulation of old minimal supergravity. As a consequence, we obtain the polynomial $f(R)$ supergravity extending the supersymmetric Starobinsky model to any higher power of $R$. The supersymmetric vacua in polynomial $f(R)$ supergravity are studied. We also present the $R^n$ extended Minimal Volkov-Akulov-Starobinsky Model.
We present a model for cosmological inflation based on a no-scale supergravity sector with an SU(2,1)/U(1) Kahler potential, a single modulus T and an inflaton superfield Phi described by a Wess-Zumino model with superpotential parameters (mu, lambda). This model yields a scalar spectral index n_s and a tensor-to-scalar ratio r that are compatible with the Planck measurements for values of lambda simeq mu/3M_P. For the specific choice lambda = mu/3M_P, the model is a no-scale supergravity realization of the R+R^2 Starobinsky model.
In the context of dynamical breaking of local supersymmetry (supergravity), including the Deser-Zumino super-Higgs effect, for the simple but quite representative cases of N=1, D=4 supergravity, we discuss the emergence of Starobinsky-type inflation, due to quantum corrections in the effective action arising from integrating out gravitino fields in their massive phase. This type of inflation may occur after a first-stage small-field inflation that characterises models near the origin of the one-loop effective potential, and it may occur at the non-trivial minima of the latter. Phenomenological…
Motivated by the recent observation of the B-mode signal in the cosmic microwave background by BICEP2, we stuty the Starobinsky-type inflation model in the framework of old-minimal supergravity, where the inflaton field in the original (non-supersymmetric) Starobinsky inflation model becomes a complex field. We study how the inflaton evolves on the two-dimensional field space, varying the initial condition. 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 B-mode signal observed by …
We propose a simple modification of the no-scale supergravity Wess-Zumino model of Starobinsky-like inflation to include a Polonyi term in the superpotential. The purpose of this term is to provide an explicit mechanism for supersymmetry breaking at the end of inflation. We show how successful inflation can be achieved for a gravitino mass satisfying the strict upper bound $m_{3/2}< 10^3$ TeV, with favoured values $m_{3/2}\lesssim\mathcal{O}(1)$ TeV. The model suggests that SUSY may be discovered in collider physics experiments such as the LHC or the FCC.
We propose the Starobinsky-type inflationary model in the matter-coupled $N=1$ four-dimensional supergravity with the massive vector multiplet that has inflaton (scalaron) and goldstino amongst its field components, whose action includes the Dirac-Born-Infeld-type kinetic term and the generalized (new) Fayet-Iliopoulos-type term, without gauging the R-symmetry. The $N=1$ chiral matter ("hidden sector") is described by the modified Polonyi model needed for spontaneous supersymmetry breaking after inflation. We compute the bosonic action and the scalar potential of the model, and show that it ca…
Models of cosmological inflation resembling the Starobinsky R + R^2 model emerge naturally among the effective potentials derived from no-scale SU(N,1)/SU(N) x U(1) supergravity when N > 1. We display several examples in the SU(2,1)/SU(2) x U(1) case, in which the inflaton may be identified with either a modulus field or a matter field. We discuss how the modulus field may be stabilized in models in which a matter field plays the role of the inflaton. We also discuss models that generalize the Starobinsky model but display different relations between the tilt in the spectrum of scalar density …
We review some recent trends in the inflationary model building, the supersymmetry (SUSY) breaking, the gravitino Dark Matter (DM) and the Primordial Black Holes (PBHs) production in supergravity. The Starobinsky inflation can be embedded into supergravity when the inflaton belongs to the massive vector multiplet associated with a (spontaneously broken) $U(1)$ gauge symmetry. The SUSY and R-symmetry can be also spontaneously broken after inflation by the (standard) Polonyi mechanism. Polonyi particles and gravitinos are super heavy and can be copiously produced during inflation via the Schwing…
Measurements of the cosmic microwave background (CMB) favour models of inflation with a small tensor-to-scalar ratio r, as predicted by the Starobinsky R + R^2 model. It has been shown previously that various models based on no-scale supergravity with different forms of superpotential make predictions similar to those of the Starobinsky model. In this paper we present a unified and general treatment of Starobinsky avatars of no-scale supergravity, using the underlying non-compact SU(2,1)/SU(2) x U(1) symmetry to demonstrate equivalences between different models, exhibiting 6 specific equivalen…
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…
We consider the ghost-free higher order corrections to the Starobinsky model in the old minimal supergravity. In general, higher order corrections cannot be forbidden by symmetries, which likely violate the flatness of the scalaron potential and makes inflation difficult to explain the present Universe. We find a severe constraint on the dimensionless coupling of the $R^4$ correction as $-5.5 \times 10^{-8}<s<9.1 \times 10^{-8}$ from the recent results of the Planck observation. If we start from the chaotic initial condition, the constraint becomes much severer. However, in the case where the …
We consider higher derivative supergravities that are dual to ghost-free $N=1$ supergravity theories in the Einstein frame. The duality is implemented by deforming the Kähler function, and/or the superpotential, to include nonlinear dependences on chiral fields that in other approaches play the role of the Lagrange multipliers employed to establish this duality. These models are of the no-scale type, and in the minimal case, require the presence of four chiral multiplets, with a Kähler potential having the structure of the $ SU(4,1)/SU(4) \times U(1) $ coset manifold. In the standard $N=1$ sup…
We construct a supergravity model whose scalar degrees of freedom arise from a chiral superfield and are solely a scalaron and an axion that is very heavy during the inflationary phase. The model includes a second chiral superfield $X$, which is subject however to the constraint $X^2=0$ so that it describes only a Volkov - Akulov goldstino and an auxiliary field. We also construct the dual higher - derivative model, which rests on a chiral scalar curvature superfield ${\cal R}$ subject to the constraint ${\cal R}^2=0$, where the goldstino dual arises from the gauge - invariant gravitino field …
We describe the primeval inflationary phase of the early Universe within a quantum field theoretical (QFT) framework that can be viewed as the effective action of vacuum decay in the early times. Interestingly enough, the model accounts for the "graceful exit" of the inflationary phase into the standard radiation regime. The underlying QFT framework considered here is Supergravity (SUGRA), more specifically an existing formulation in which the Starobinsky-type inflation (de-Sitter background) emerges from the quantum corrections to the effective action after integrating out the gravitino field…
Higgsino dark matter in the Starobinsky supergravity with the MSSM in light of the LUX-ZEPLIN event
Utrecht University（乌得勒支大学）；Dutch Institute for Emergent Phenomena（荷兰涌现现象研究所）；Tokyo Metropolitan University（东京都立大学）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study（理化学研究所 Kavli 宇宙粒子与宇宙线研究所（WPI），东京大学前沿研究综合中心）
We find a way to represent the Starobinsky model in terms of a simple conformally invariant theory with spontaneous symmetry breaking. We also present a superconformal theory, which, upon spontaneous breaking of the superconformal symmetry, provides a consistent supergravity generalization of the Starobinsky model.
A supergravity extension of the $(R+R^2)$ gravity with the additional (Born-Infeld) structure of a massive vector multiplet gives rise to the specific $F(R)$ gravity, whose structure is investigated in detail. The massive vector multiplet has inflaton (scalaron), goldstino and a massive vector field as its field components. The model describes Starobinsky inflation and allows us to extrapolate the $F(R)$ function beyond the inflationary scale (up to Planck scale). We observe some differences versus the $(R+R^2)$ gravity, and several breaking patterns of the well known correspondence between th…
We extend the Cecotti-Kallosh model of Starobinsky inflation in supergravity by adding a holomorphic function to the superpotential in order to generate a large peak in the power spectrum of scalar (curvature) perturbations. In our approach, the singular non-canonical kinetic terms are largely responsible for inflation (as an attractor solution), whereas the superpotential is engineered to generate a production of primordial black holes. We study the cases with (i) a linear holomorphic function, (ii) a quadratic holomorphic function, and (iii) an exponential holomorphic function, as regards th…
Institute for Advanced Study（高等研究院）；Sorbonne Université, CNRS（索邦大学，法国国家科学研究中心）；Scuola Normale Superiore and INFN（比萨高等师范学院和意大利国家核物理研究所）；Academy of Athens, Division of Natural Sciences（雅典科学院自然科学部）；Texas A&M University（德克萨斯农工大学）
Using a no-scale supergravity framework, we construct an SO(10) model that makes predictions for cosmic microwave background observables similar to those of the Starobinsky model of inflation, and incorporates a double-seesaw model for neutrino masses consistent with oscillation experiments and late-time cosmology. We pay particular attention to the behaviour of the scalar fields during inflation and the subsequent reheating.
As is well known, the gravitational degrees of freedom contained in $R+ζR^{2}$ (super)gravity lead to Starobinsky's potential, in a one-field setting for inflationary Cosmology that appears favored by Planck data. In this letter we discuss another interesting aspect of this model, related to gravitino production, with emphasis on the corresponding mass spectrum. Assuming that supersymmetry is broken at a very high scale, Super Heavy Gravitino Dark Matter (SHGDM) and Starobinsky's inflation can be coherently unified in a $R+ζR^{2}$ supergravity. Gravitinos are assumed to be the Lightest Supersy…
An N=1 Poincare supergravity action, suitable for describing the Starobinsky inflation, is proposed. It extends f(R) gravity to supergravity in its old-minimal version. The action is parametrized by a single holomorphic potential and a single non-holomorphic potential, and can be dualized into the standard matter-coupled supergravity action, with the "matter" given by two chiral superfields. The action extends the earlier proposals for embedding the Starobinsky inflation into supergravity, and can be further generalized to include quantum corrections to the inflation. Our approach assumes the …
Higgs boson mass and thermal wino dark matter from Starobinsky supergravity with the MSSM
Utrecht University（乌得勒支大学）；Dutch Institute for Emergent Phenomena（荷兰新兴现象研究所）；University of Southampton（南安普顿大学）；Rutherford Appleton Laboratory（卢瑟福·阿普尔顿实验室）；Tokyo Metropolitan University（东京都立大学）；International Institute of Physics, Natal（纳塔尔国际物理研究所）；Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes for Advanced Study（东京大学基础研究院宇宙物质宇宙学前沿研究中心）
We consider global issues in minimal supergravity models where a single field inflaton potential emerges. In a particular case we reproduce the Starobinsky model and its description dual to a certain formulation of R+R^2 supergravity. For definiteness we confine our analysis to spaces at constant curvature, either vanishing or negative. Five distinct models arise, two flat models with respectively a quadratic and a quartic potential and three based on the SU(1,1)/U(1) space where its distinct isometries, elliptic, hyperbolic and parabolic are gauged. Fayet-Iliopoulos terms are introduced in a …
A novel realization of the Starobinsky inflationary model within a moderate extension of the Minimal Supersymmetric Standard Model (MSSM) is presented. The proposed superpotential is uniquely determined by applying a continuous R and a Z_2 discrete symmetry, whereas the Kahler potential respects a no-scale-type SU(54,1)/SU(54)x U(1)_R x Z_2 symmetry too. The inflaton is identified with a Higgs-like modulus whose the vacuum expectation value controls the gravitational strength. Thanks to a strong enough coupling (with a parameter cT involved) between the inflaton and the Ricci scalar curvature,…
After reviewing the motivations for cosmological inflation formulated in the formalism of supersymmetry, we argue that the appropriate framework is that of no-scale supergravity. We then show how to construct within this framework inflationary models whose predictions for the tilt in the spectrum of scalar perturbations, $n_s$, and the ratio, $r$, of tensor and scalar perturbations coincide with those of the $R + R^2$ model of inflation proposed by Starobinsky. A more detailed study of no-scale supergravity reveals a structure that is closely related to that of $R^2$ modifications of the minim…
The Dark Side of the Universe, which includes the cosmological inflation in the early Universe, the current dark energy and dark matter, can be theoretically described by supergravity, though it is non-trivial. We recall the arguments pro and contra supersymmetry and supergravity, and define the viable supergravity models describing the Dark Side of the Universe in agreement with all current observations. Our approach to inflation is based on the Starobinsky model, the dark energy is identified with the positive cosmological constant (de Sitter vacuum), and the dark matter particle is given by…
We propose a four-dimensional N = 1 supergravity-based Starobinsky-type inflationary model in terms of a single massive vector multiplet, whose action includes the Dirac-Born-Infeld-type kinetic terms and a generalized (new) Fayet-Iliopolulos-type term without gauging the R-symmetry. The bosonic action and the scalar potential are computed. Inflaton is the superpartner of goldstino in our model, and supersymmetry is spontaneously broken after inflation by the D-type mechanism, whose scale is related to the value of the cosmological constant.
We show that a new class of helical phase inflation models can be simply realized in minimal supergravity, wherein the inflaton is the phase component of a complex field and its potential admits a deformed helicoid structure. We find a new unique complex-valued index $χ$ that characterizes almost the entire region of the $n_s-r$ plane favored by new Planck observations. Continuously varying the index $χ$, predictions interpolate from quadratic/natural inflation parameterized by a phase/axion decay constant to Starobinsky-like inflation parameterized by the $α$-parameter. We demonstrate that th…
We review the new theory of modified supergravity, dubbed F(\cal R) supergravity, and some of its recent applications to inflation and reheating in the early universe cosmology. The F(\cal R) supergravity is the N=1 locally supersymmetric extension of the f(R) gravity in four space-time dimensions. A manifestly supersymmetric formulation of the F(\cal R) supergravity exist in terms of N=1 superfields, by using the (old) minimal Poincar'e supergravity in curved superspace. We find the conditions for stability, the absence of ghosts and tachyons. Three models of the F(\cal R) supergravity are st…
We study the structure of neutron stars in $f(R)=R+αR^{2}$ theory of gravity (Starobinsky model), in an exact and non-perturbative approach. In this model, apart from the standard General Relativistic junction conditions, two extra conditions, namely the continuity of the curvature scalar and its first derivative needs to be satisfied. For an exterior Schwarzschild solution, the curvature scalar and its derivative has to be zero at the stellar surface. We show that for some equation of state (EoS) of matter, matching all conditions at the surface of the star is impossible. Hence the model brin…
We discuss the inflaton $ϕ$ in an interaction with an infinite number of fields treated as an environment (noise) with a friction $γ^{2}>0$. In a Markovian approximation we obtain a stochastic wave equation (appearing also in the warm inflation models). After the replacement of the environment by the white noise, the stochastic wave equation violates the energy conservation if $γ\neq 0$. We introduce a dark energy as a compensation of the inflaton energy-momentum. We add to the classical wave equation the Starobinsky-Vilenkin noise which in the slow-roll approximation describes the quantum flu…
We give a general analysis of AdS boundary conditions for spin-3/2 Rarita-Schwinger fields and investigate boundary conditions preserving supersymmetry for a graviton multiplet in AdS_4. Linear Rarita-Schwinger fields in AdS_d are shown to admit mixed Dirichlet-Neumann boundary conditions when their mass is in the range $0 \leq |m| < 1/2l_{AdS}$. We also demonstrate that mixed boundary conditions are allowed for larger masses when the inner product is "renormalized" accordingly with the action. We then use the results obtained for |m| = 1/l_{AdS} to explore supersymmetric boundary conditions f…
We consider minimal supergravity on (2+1)dimensional de-Sitter background. We fix the fall-off conditions for gravitini fields in order to fix the asymptotic phase space. Using the Chern-Simons formulation, we then derive the asymptotic symmetry algebra for this theory. The fall-off conditions impose constraints on the phase space which reduces the Chern Simons theory to a WZW model. Further constraints reduce it to a super-Liouville theory at the boundary. This can be treated as a classical dual for the supergravity theory in the bulk.
We obtain N = 3 chiral supergravity (SUGRA) compatible with the reality condition by applying the prescription of constructing the chiral Lagrangian density from the usual SUGRA. The $N = 3$ chiral Lagrangian density in first-order form, which leads to the Ashtekar's canonical formulation, is determined so that it reproduces the second-order Lagrangian density of the usual SUGRA especially by adding appropriate four-fermion contact terms. We show that the four-fermion contact terms added in the first-order chiral Lagrangian density are the non-minimal terms required from the invariance under f…
In this letter we discuss de Sitter vacua in maximal gauged supergravity in 4 dimensions. We show that, using the newly deformed theories introduced in arxiv:1209.0760, we can obtain de Sitter vacua with arbitrarily flat tachyonic directions in the SO(4,4)c models.
In this work, we build the most general boundary conditions for $\mathcal N=\left(1,1\right)$ and $\mathcal N=\left(2,2\right)$ extended supergravity. We show that for the loosest set of boundary conditions, their asymptotic symmetry algebras are two copies of the $\mathfrak{osp}\left(1\mid 2\right)_k$ and $\mathfrak{osp}\left(2\mid 2\right)_k$ algebra, respectively. Then, we restrict the gauge fields on the boundary conditions in order to obtain supersymmetric extensions of the Brown-Henneaux and Troessaert boundary conditions. Furthermore, for $\mathcal N=1$ and $\mathcal N=2$ (extended) sup…
We explain why it is necessary to use boundary conditions in the proof of supersymmetry of a supergravity action on a manifold with boundary. Working in both boundary (``downstairs'') and orbifold (``upstairs'') pictures, we present a bulk-plus-boundary/brane action for the five-dimensional (on-shell) supergravity which is supersymmetric with the use of fewer boundary conditions than were previously employed. The required Gibbons-Hawking-like Y-term and many other aspects of the boundary/orbifold picture correspondence are discussed.
We derive the complete orbit of boundary conditions for supergravity models which is closed under the action of all local symmetries of these models, and which eliminates spurious field equations on the boundary. We show that the Gibbons-Hawking boundary conditions break local supersymmetry if one imposes local boundary conditions on all fields. Nonlocal boundary conditions are not ruled out. We extend our analysis to BRST symmetry and to the Hamiltonian formulation of these models.
We consider the conditions for integrating out heavy chiral fields and moduli in N=1 supergravity, subject to two explicit requirements. First, the expectation values of the heavy fields should be unaffected by low energy phenomena. Second, the low energy effective action should be described by N=1 supergravity. This leads to a working definition of decoupling in N=1 supergravity that is different from the usual condition of gravitational strength couplings between sectors, and that is the relevant one for inflation with moduli stabilization, where some light fields (the inflaton) can have lon…
Boundary conditions in supergravity on a manifold with boundary relate the bulk gravitino to the boundary supercurrent, and the normal derivative of the bulk metric to the boundary energy-momentum tensor. In the 3D N=1 setting, we show that these boundary conditions can be stated in a manifestly supersymmetric form. We identify the Extrinsic Curvature Tensor Multiplet, and show that boundary conditions set it equal to (a conjugate of) the boundary supercurrent multiplet. Extension of our results to higher-dimensional models (including the Randall-Sundrum and Horava-Witten scenarios) is discuss…
We construct higher derivative supervertices in an effective theory of maximal supergravity in various dimensions, in the super spinor helicity formalism, and derive non-renormalization conditions on up to 14-derivative order couplings from supersymmetry. These non-renormalization conditions include Laplace type equations on the coefficients of $R^4$, $D^4R^4$, and $D^6R^4$ couplings. We also find additional constraining equations, which are consistent with previously known results in the effective action of toroidally compactified type II string theory, and elucidate many features thereof.
Bulk supergravity on a manifold with boundary must be supplemented by boundary conditions that preserve local supersymmetry. This "downstairs" picture has certain advantages over the equivalent "upstairs" picture, expressed in terms of orbifolds. In particular, Scherk-Schwarz supersymmetry breaking can be described much more simply in the downstairs picture. Nevertheless, physics on the fundamental domain can always be lifted upstairs, so long as fields are allowed to be discontinuous across the boundary. In this talk we apply these considerations to five-dimensional supergravity in a warped R…
In this paper, we present a candidate for $\mathcal{N}=(1,1)$ extended higher - spin $AdS_3$ supergravity with the most general boundary conditions discussed by Grumiller and Riegler recently. We show that the asymptotic symmetry algebra consists of two copies of the $\mathfrak{osp}(3|2)_k$ affine algebra in the presence of the most general boundary conditions.Furthermore, we impose some certain restrictions on gauge fields on the most general boundary conditions and that leads us to the supersymmetric extension of the Brown - Henneaux boundary conditions. We eventually see that the asymptotic…
This paper explains the Unimodular gauge fixing of gravity and supergravity in the framework of a perturbative BRST construction. The unphysical sector contains additional BRST-exact quartets to suppress possible ambiguities and impose both the Unimodular gauge fixing condition on the metric and a gauge condition for the reparametrization symmetry of the unimodular part of the metric. The Unimodular gauge choice of the metric must be completed by a $γ$-Traceless gauge condition for the Rarita--Schwinger field in the case of supergravity. This gives an interesting new class of gauges for gravit…
