The effects which quantum fields and an $α_0 R^2$ term in the gravitational Lagrangian have on future singularities are investigated. While all values of $α_0$ are considered, an emphasis is placed on those values which are compatible with the universe having undergone Starobinsky inflation in the past. These are also values which lead to stable solutions to the semiclassical backreaction equations in the present universe. The dark energy is modeled as a perfect fluid, and the focus is on type I-IV singularities and little rips which result when the classical Einstein equations are solved with various types of dark energy as a source. First, evidence is provided that the energy densities of massive conformally coupled scalar fields approach that of the conformally invariant scalar field as a type III singularity is approached. Then, solutions to the semiclassical backreaction equations are investigated when conformally invariant fields and the $ α_0 R^2$ term in the gravitational Lagrangian are present. General proofs regarding the behaviors of the solutions are given. The proofs are illustrated by analytic and numerical calculations in specific cases.
In this work we study numerically one kind of generalization of the Starobinsky inflationary model (power-law type), which is characterized by the parameter $p$. In order to find the parameter $p$ that fixes with observations, we compute the cosmological parameters $A_S$, $n_S$, and $r$ for several values of $p\simeq 1$. We have found that the value of $p=1.0004$ reproduces the value of $A_S$, $n_\sca$, and $r$ in agreement with current observational data.
In this work, we investigate the Higgs-Starobinsky (HS) model in the context of warm inflation scenario. The dissipative parameter as a linear form of temperature of warm inflation is considered with strong and weak regimes. We study the HS model in the Einstein frame with the slow-roll inflation framework. We compute the inflationary observables and then compare with the Plank 2018 data. With the sizeable number of e-folds and proper choices of parameters, we discover that the predictions of warm HS model present in this work are in very good agreement with the latest Planck 2018 results. Mor…
The lack of correlations on the large scale cosmic microwave background (CMB) anisotropy provides a potential window to probe beyond the standard inflationary scenario. In this paper, we investigate the primordial power spectrum based on the Hartle-Hawking (HH) no-boundary proposal for a homogeneous, isotropic, and spatially-closed universe that leads to a Starobinsky-type inflation after the classicalization. While we found that there is no suppression at large scales in the standard R + R^2 theory, we also found that it is possible to sufficiently suppress the large-scale power spectrum if a…
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.
We explore two saddle point inflationary scenarios in the context of higher order corrections related to different generalisations of general relativity. Firstly, we deal with Jordan frame Starobinsky potential, for which we identify a portion of a parameter space of inflection point inflation, which can accommodate all the experimental results. Secondly, we analyse Higgs inflation and more specifically the influence of non-renormalisible terms on the standard quartic potential. All results were verified with the PLANCK 2015 data.
The first inflationary model conceived was the one proposed by Starobinsky which includes an additional term quadratic in the Ricci-scalar R in the Einstein-Hilbert action. The model is now considered a target for several future cosmic microwave background experiments given its compatibility with current observational data. In this paper, we analyse the robustness of the Starobinsky inflation by inserting it into a generalized scenario based on a $β$-Starobinsky inflation potential, which is motivated through brane inflation. In the Einstein frame, the generalized model recovers the original m…
In the Starobinsky inflationary model inflation is driven by quantum corrections to the vacuum Einstein equation. We reduce the Wheeler-DeWitt equation corresponding to the Starobinsky model to a Schroedinger form containing time. The Schroedinger equation is solved with a Gaussian ansatz. Using the prescription for the normalization constant of the wavefunction given in our previous work, we show that the Gaussian ansatz demands Hawking type initial conditions for the wavefunction of the universe. The wormholes induce randomness in initial states suggesting a basis for time-contained descript…
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 consider the Starobinsky inflation with a set of higher order corrections parametrised by two real coefficients $λ_1, λ_2$. In the Einstein frame we have found a potential with the Starobinsky plateau, steep slope and possibly with an additional minimum, local maximum or a saddle point. We have identified three types of inflationary behaviour that may be generated in this model: i) inflation on the plateau, ii) at the local maximum (topological inflation), iii) at the saddle point. We have found limits on parameters $λ_i$ and initial conditions at the Planck scale which enable successful in…
We propose a one-parametric extension of the Starobinsky $R+R^2$ model by adding the $(R+m^2β^2)^{3/2}$ term. The parameter $m$ is the inflaton mass, which is determined in the same way as in the Starobinsky model, and $β$ is a dimensionless constant. Using the Einstein frame and the scalar field potential, we get the inflationary parameters of the model proposed. The value of the tensor-to-scalar ratio $r$ can be significantly larger than in the Starobinsky model. The considered inflationary model is in a good agreement with the current observational data. The corresponding scalar field poten…
We investigate the pre-inflationary dynamics of inflation with the Starobinsky potential, favored by recent data from the Planck mission, using techniques developed to study cosmological perturbations on quantum spacetimes in the framework of loop quantum cosmology. We find that for a large part of the initial data, inflation compatible with observations occurs. There exists a subset of this initial data that leads to quantum gravity signatures that are potentially observable. Interestingly, despite the different inflationary dynamics, these quantum gravity corrections to the powerspectra are …
In this paper we construct a bounce model that mimics the Starobinsky inflationary model. Our construction relies on Wands' duality, which shows that the Mukhanov-Sasaki equation has a symmetry transformation by changing appropriately its time-dependent mass term. One of the advantages of this constructive method is that one can control every contribution to the primordial power spectrum and check how far we can emulate a given primordial model with a different scenario. In particular, we show that mapping the Starobinsky inflation into a quasi-matter bounce gives the correct relation between …
We point out that the ability of some models of inflation, such as Higgs inflation and the universal attractor models, in reproducing the available data is due to their relation to the Starobinsky model of inflation. For large field values, where the inflationary phase takes place, all these classes of models are indeed identical to the Starobinsky model. Nevertheless, the inflaton is just an auxiliary field in the Jordan frame of the Starobinsky model and this leads to two important consequences: first, the inflationary predictions of the Starobinsky model and its descendants are slightly dif…
In this paper we point out that Starobinky inflation could be induced by quantum effects due to a large non-minimal coupling of the Higgs boson to the Ricci scalar. The Higgs Starobinsky mechanism provides a solution to issues attached to large Higgs field values in the early universe which in a metastable universe would not be a viable option. We verify explicitly that these large quantum corrections do not destabilize Starobinsky's potential.
An extension of the Starobinsky model is proposed. Besides the usual Starobinsky Lagrangian, a term proportional to the derivative of the scalar curvature, $\nabla_μR\nabla^μR$, is considered. The analyzis is done in the Einstein frame with the introduction of a scalar field and a vector field. We show that inflation is attainable in our model, allowing for a graceful exit. We also build the cosmological perturbations and obtain the leading-order curvature power spectrum, scalar and tensor tilts and tensor-to-scalar ratio. The tensor and curvature power spectrums are compared to the most recen…
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…
In this work we solved the equation of scalar and tensor perturbations for the generalized Starobinsky inflationary model using the improved uniform approximation method and the phase-integral method up to third-order in deviation. We compare our results with the numerical integration. We have obtained that both semiclassical methods reproduce the scalar power spectra $P_{S,T}$, the scalar spectral index $n_S$, and the tensor-to-scalar ratio $r$. Also we present our results in the $(n_S,r)$ plane.
Quantum geometry gives a regularization scheme-independent effective action, whoes equation of motion for the conformal mode has a stable de Sitter solution at the high-energy region where the coupling of the self-interactions of the traceless mode can be neglected because of the asymptotic freedom. However, the dynamics of the traceless mode suggests that inflation ends at the low-energy region.
We consider an extension of the Starobinsky model, whose parameters are functions of an extra scalar field. Our motivation is to test the robustness (or sensitivity) of the Starobinsky inflation against mixing scalaron with another (matter) scalar field. We find that the extended Starobinsky model is (classically) equivalent to the two-field inflation, with the scalar potential having a flat direction. For the sake of fully explicit calculations, we perform a numerical scan of the parameter space. Our numerical calculations support the viability of the Starobinsky-like two-field inflation for …
For zero spatial curvature, cosmological phase space of Starobinsky and extended Starobinsky inflationary model show three apparent attractors; the fixed angle attractor in the large field limit, the final attractor representing reheating phase in the small field region, and the apparent attractor corresponding to the slow-roll condition connecting between the large-field and small-field region. To consider the total $e$-folding likelihood of the model, Remmen-Carroll conserved measure is constructed and normalized. Using the measure, the total e-folding number $N$ and its expectation value $\…
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 consider the simplest extension to the Starobinsky model, by allowing an extra scalar field to help drive inflation. We perform our analysis in the Einstein frame and calculate the power spectra at the end of inflation to second order in the slow--roll parameters. We find that the model gives predictions in great agreement with the current Planck data without the need for fine-tuning. Our results encourage current efforts to embed the model in a supergravity setting.
The $R+R^2$ model of gravity with the corresponding shallow potential in the Einstein frame is consistent with the observations. Recently, many efforts have been made to generalize the $R+R^2$ (Starobinsky) model of inflation or use other shallow potentials to construct a model for the early Universe. We revise the question about the shallow potential. We propose a model in which the Starobinsky model can emerge through a dynamical mechanism. We show that the absence of ghost modes results to constraints on the parameters of the Starobinsky model. We obtain the scalar spectral index and the te…
We prove that the field equations of the Starobinsky model for inflation in a Friedmann-Lema\^ıtre-Robertson-Walker constitute an integrable system as the field equations pass the singularity test. The analytical solution in terms of a Painlevé Series for the Starobinsky model is presented for the case of zero and nonzero spatial curvature. In both cases the leading-order term describes the radiation era provided by the corresponding higher-order theory.
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.
In this work, we study unified picture of inflation and dark matter in the Higgs-Starobinsky (HS) model. As pointed out in the literature, Starobinsky $R^2$ inflation is induced by quantum correction effect from the large Higgs-curvature (graviton) coupling. We start with non-minimal coupling HS action in Jordan frame. We then transform the Jordan frame action into the Einstein one using the conformal transformation. The inflation potential is derived from the gravitational action of non-minimal-Higgs coupling and Starobinsky term in Einstein frame where the $R^2$ term is dominated in the infl…
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 …
Starobinsky Inflation in k-Essence Framework: Attractor Dynamics, Reheating, and Consistency with ACT DR6
Hunan Institute of Technology（湖南工学院）；SGT University（SGT大学）；Centre for Cosmology and Science Popularization (CCSP), SGT University（SGT大学宇宙学与科学普及中心）；Eurasian International Centre for Theoretical Physics（欧亚国际理论物理中心）
It is believed that soon after the Planck era, space time should have a semi-classical nature. According to this, the escape from General Relativity theory is unavoidable. Two geometric counter-terms are needed to regularize the divergences which come from the expected value. These counter-terms are responsible for a higher derivative metric gravitation. Starobinsky idea was that these higher derivatives could mimic a cosmological constant. In this work it is considered numerical solutions for general Bianchi I anisotropic space-times in this higher derivative theory. The approach is ``experim…
Tokyo Metropolitan University（东京都立大学）；Tomsk State University（托木斯克国立大学）；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 宇宙物理学与数学研究所（WPI），东京大学高等研究院）
We investigate inflationary attractor points by analyzing non-minimally coupled single field inflation models in two opposite limits; the `flat' limit in which the first derivative of the conformal factor is small and the `steep' limit, in which the first derivative of the conformal factor is large. We consider a subset of models that yield Starobinsky inflation in the steep conformal factor, strong coupling, limit and demonstrate that they result in chaotic inflation in the opposite flat, weak coupling, limit. The suppression of higher order powers of the inflaton field in the potential is sh…
We investigate the production of primordial black holes and their contribution to the presently observed dark matter in a dilaton two-field extension of Starobinsky's quadratic $f(R)$ model of inflation. The model features a multi-field amplification mechanism which leads to the generation of a sharp peak in the inflationary power spectrum at small wavelengths responsible for the production of primordial black holes. This mechanism is significantly different from single-field models and requires a stochastic treatment during an intermediate phase of the inflationary dynamics. We find that the …
The Starobinsky inflation model is one of the simplest inflation models that is consistent with the cosmic microwave background observations. In order to explain dark matter of the universe, we consider a minimal extension of the Starobinsky inflation model with introducing the dark sector which communicates with the visible sector only via the gravitational interaction. In Starobinsky inflation model, a sizable amount of dark-sector particle may be produced by the inflaton decay. Thus, a scalar, a fermion or a vector boson in the dark sector may become dark matter. We pay particular attention…
We consider a non-canonic field in the context of Starobinsky inflation. We work in Einstein-frame. In this frame, the gravitational part of the action is equivalent to the Hilbert-Einstein action, plus a scalar field called scalaron. We investigate a model with a heavy scalaron trapped at the effective potential minimum, where its fluctuations are negligible. To be more explicit, we consider a Dirac-Born-Infeld (DBI) field, which is usually considered within the brane inflation context, as the non-canonic field. Although, the DBI field governs inflation through implicit dependence on Scalaron…
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 point out that in the large field regime, the recently proposed superconformal D-term inflation model coincides with the Starobinsky model. In this regime, the inflaton field dominates over the Planck mass in the gravitational kinetic term in the Jordan frame. Slow-roll inflation is realized in the large field regime for sufficiently large gauge couplings. The Starobinsky model generally emerges as an effective description of slow-roll inflation if a Jordan frame exists where, for large inflaton field values, the action is scale invariant and the ratio \hat λ of the inflaton self-coupling a…
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…
We single out the Starobinsky model and its extensions among generic $f(R)$ gravity as attractors at large field values for chaotic inflation. Treating a $R^3$ curvature term as a perturbation of the Starobinsky model, we impose the phenomenological bounds on the additional term satisfying the successful inflationary predictions. We find that the scalar spectral index can vary in both the red or blue tilted direction, depending on the sign of the coefficient of the $R^3$ term, whereas the tensor-to-scalar ratio is less affected in the Planck-compatible region. We also discuss the role of highe…
In this work, we studied the slow-roll approximation of cosmic inflation within the context of $f(R,T)$ gravity, where $R$ is the scalar curvature, and $T$ is the trace of the energy-momentum tensor. By choosing a minimal coupling between matter and gravity, we obtained the modified slow-roll parameters, the scalar spectral index ($n_s$), the tensor spectral index ($n_{\textrm{T}}$), and the tensor-to-scalar ratio ($r$). We computed these quantities for a general power-law potential, Natural & Quartic Hilltop inflation, and the Starobinsky model, plotting the trajectories on the $(n_s,r)$ plan…
Zero temperature black hole solutions to the semiclassical backreaction equations are investigated. Evidence is provided that certain components of the stress-energy tensors for free quantum fields at the horizon only depend on the local geometry near the horizon. This allows the semiclassical backreaction equations to be solved near the horizon. It is found that macroscopic uncharged zero temperature black hole solutions to the equations may exist if the coefficient of one of the higher derivative terms in the gravitational Lagrangian is large enough and of the right sign for Starobinsky infl…
Cosmological solutions derived by Nariai and Tomita (1971) and by Starobinsky (1980) are compared, and it is shown that the former derived de Sitter expansion in the R^2 modified gravity (without cosmological constant) at the earliest stage, and nine years later the latter derived the well-known inflationary solution. Next the property of their simplified models is described using the method of conformal transformations, and how the inflation arises and the singularity is avoided is shown. Finally the initial and final states of the inflation are discussed.
The $R^2$ term in the Starobinsky inflationary model can be regarded as a leading quantum correction to the gravitational effective action. We assume that parity-preserving and parity-violating (axial) non-minimal couplings between curvature and electromagnetic field are also present in the effective action. In the Einstein frame, they turn into non-trivial couplings of the scalaron and curvature to the electromagnetic field. We make an assessment of inflationary magnetogenesis in this model. In the case of parity-preserving couplings, amplification of magnetic field is negligibly small. In th…
We study several extensions of the Starobinsky model of inflation, which obey all observational constraints on the inflationary parameters, by demanding that both the inflaton scalar potential in the Einstein frame and the $F(R)$ gravity function in the Jordan frame have the explicit dependence upon fields {\it and} parameters in terms of elementary functions. Our models are continuously connected to the original Starobinsky model via changing the parameters. We modify the Starobinsky $(R+R^2)$ model by adding an $R^3$-term, an $R^4$-term, and an $R^{3/2}$-term, respectively, and calculate the…
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.
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…
Testing the Starobinsky model of inflation with resonant cavities
Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Indian Institute of Science Education and Research Bhopal（博帕尔印度科学教育与研究机构）；Indian Institute of Technology Bombay（孟买印度理工学院）
