In the landscape, if there is to be any prospect of scientific prediction, it is crucial that there be states which are distinguished in some way. The obvious candidates are states which exhibit symmetries. Here we focus on states which exhibit discrete symmetries. Such states are rare, but one can speculate that they are cosmological attractors. We investigate the problem in model landscapes and cosmologies which capture some of the features of candidate flux landscapes. In non-supersymmetric theories we find no evidence that such states might be cosmologically favored. In supersymmetric theories, simple arguments suggest that states which exhibit $R$ symmetries might be. Our considerations lead us to raise questions about some popular models of eternal inflation.
The recent progress in the understanding of the landscape of string theory vacua hints that the hierarchy problem might be the problem of a super-selection rule. The attractor mechanism gives a possibility to explain the choice of a vacuum. We consider a toy model of self-interacting membranes and show that for a very generic interaction there are attractor solutions.
A 4D-brane realization of $q$-theory has been proposed a few years ago. The present paper studies the corresponding late-time cosmology and establishes the dynamic cancellation of an initial cosmological constant and an attractor behavior towards Minkowski spacetime.
We investigate the attractor mechanism for spherically symmetric extremal black holes in Einstein-Born-Infeld-dilaton theory of gravity in four-dimensions, in the presence of a cosmological constant. We look for solutions analytic near the horizon by using perturbation method. It is shown that the values of the scalar fields at the horizon are only dependent on the charges carried by the black hole and are irrelevant in their asymptotic values. This analysis supports the validity of non-supersymmetric attractors in the presence of higher derivative interactions in the gauge fields part and in …
We provide a unified description of cosmological $α$-attractors and late-time acceleration, in excellent agreement with the latest Planck data. Our construction involves two superfields playing distinctive roles: one is the dynamical field and its evolution determines inflation and dark energy, the other is nilpotent and responsible for a landscape of vacua and supersymmetry breaking. We prove that the attractor nature of the theory is enhanced when combining the two sectors: cosmological attractors are very stable with respect to any possible value of the cosmological constant and, interestin…
We study Noether symmetries in two-field cosmological $α$-attractors, investigating the case when the scalar manifold is an elementary hyperbolic surface. This encompasses and generalizes the case of the Poincare disk. We solve the conditions for the existence of a `separated' Noether symmetry and find the form of the scalar potential compatible with such, for any elementary hyperbolic surface. For this class of symmetries, we find that the $α$-parameter must have a fixed value. Using those Noether symmetries, we also obtain many exact solutions of the equations of motion of these models, whic…
As a test for the non-supersymmetric attractor mechanism, we consider extremal Reissner-Nordström-(anti-)de Sitter black holes. Based on the simple observation that the near-horizon geometry of a generic extremal black hole contains two-dimensional anti-de Sitter factor even in the presence of the positive cosmological constant, we apply Ashoke Sen's entropy function method to compute the entropy of these black holes. We find the results which exactly agree with the Bekenstein-Hawking entropy. We also obtain the constant higher-order correction to the entropy due to the Gauss-Bonnet term.
We investigate cosmological dynamics of multiple tachyon fields with inverse square potentials. A phase-space analysis of the spatially flat FRW models shows that there exists power-law cosmological scaling solutions. We study the stability of the solutions and find that the potential-kinetic-scaling solution is a global attractor. However, in the presence of a barotropic fluid the solution is an attractor only in one region of the parameter space and the tracking solution is an attractor in the other region. We briefly discuss the physical consequences of these results.
In [1], KKLT give a mechanism to generate de Sitter vacua in string theory. And the scenario, \emph{Landscape}, is suggested to explain the problem of the cosmological constant. In this paper, adopting a simple potential describing the \emph{landscape}, we investigate the decay of the vacuum and the evolution of the universe after the decay. We find that the big crunch of the universe is inevitable. But, according to the modified Friedmann equation in [11], the singularity of the big crunch is avoided. Furthermore, we find that this gives a cyclic cosmological model.
We study Lorentz-violating models of massive gravity which preserve rotations and are invariant under time-dependent shifts of the spatial coordinates. In the linear approximation the Newtonian potential in these models has an extra ``confining'' term proportional to the distance from the source. We argue that during cosmological expansion the Universe may be driven to an attractor point with larger symmetry which includes particular simultaneous dilatations of time and space coordinates. The confining term in the potential vanishes as one approaches the attractor. In the vicinity of the attra…
Applying the anholonomic frame deformation method, we construct various classes of cosmological solutions for effective Einstein -- Yang-Mills -- Higgs, and two measure theories. The types of models considered are Freedman-Lemaître-Robertson-Walker, Bianchi, Kasner and models with attractor configurations. The various regimes pertaining to plateau--type inflation, quadratic inflation, Starobinsky type and Higgs type inflation are presented.
We argue that the vast majority of flux vacua with small cosmological constant are unstable to rapid decay to a big crunch. Exceptions are states with large compactification volume and supersymmetric and approximately supersymmetric states. Neither weak string coupling, warping, or the existence of very light particles are, by themselves, enough to render states reasonably metastable. We speculate, as well, about states which might be cosmological attractors.
Theories of the cosmological constant fall into two classes, those in which the vacuum energy is fixed by the fundamental theory and those in which it is adjustable in some way. For each class we discuss key challenges. The string theory landscape is an example of an adjustment mechanism. We discuss the status of this idea, and future directions.
In this note we give multiple examples of the recently proposed New Attractors describing supersymmetric flux vacua and non-supersymmetric extremal black holes in IIB string theory. Examples of non-supersymmetric extremal black hole attractors arise on a hypersurface in $WP^{4}_{1,1,1,1,2}$. For flux vacua on the orientifold of the same hypersurface existence of multiple basins of attraction is established. It is explained that certain fluxes may give rise to multiple supersymmetric flux vacua in a finite region on moduli space, say at the Landau-Ginzburg point and close to conifold point. Thi…
I show that the problem of realizing inflation in theories with random potentials of a limited number of fields can be solved, and agreement with the observational data can be naturally achieved if at least one of these fields has a non-minimal kinetic term of the type used in the theory of cosmological $α$-attractors.
We suggest a novel picture of the quantum Universe -- its creation is described by the {\em density matrix} defined by the Euclidean path integral. This yields an ensemble of universes -- a cosmological landscape -- in a mixed state which is shown to be dynamically more preferable than the pure quantum state of the Hartle-Hawking type. The latter is dynamically suppressed by the infinitely large positive action of its instanton, generated by the conformal anomaly of quantum fields within the cosmological bootstrap (the self-consistent back reaction of hot matter). This bootstrap suggests a sol…
Quantum creation of the universe is described by the {\em density matrix} defined by the Euclidean path integral. This yields an ensemble of universes -- a cosmological landscape -- in a mixed quasi-thermal state which is shown to be dynamically more preferable than the pure quantum state of the Hartle-Hawking type. The latter is suppressed by the infinitely large positive action of its instanton, generated by the conformal anomaly of quantum matter. The Hartle-Hawking instantons can be regarded as posing initial conditions for Starobinsky solutions of the anomaly driven deSitter expansion, wh…
Recently, several broad classes of inflationary models have been discovered whose cosmological predictions are stable with respect to significant modifications of the inflaton potential. Some classes of models are based on a non-minimal coupling to gravity. These models, which we will call $ξ$-attractors, describe universal cosmological attractors (including Higgs inflation) and induced inflation models. Another class describes conformal attractors (including Starobinsky inflation and T-models) and their generalization to $α$-attractors. The aim of this paper is to elucidate the common denomin…
In this paper, the dynamical heteroclinic orbit and attractor have been employed to make the late-time behaviors of the model insensitive to the initial condition and thus alleviates the fine tuning problem in cosmological dynamical system of barotropic fluid and quintessence with a double exponential potential. The late-time asymptotic behavior of the double exponential model does not always correspond to the single case. The heteroclinic orbits are the non-stationary solutions and in general they will interpolate between the critical points. Although they can not be shown analytically, yet a…
We show that the inflaton coupling to all other fields is exponentially suppressed during inflation in the cosmological $α$-attractor models. In the context of supergravity, this feature is a consequence of the underlying hyperbolic geometry of the moduli space which has a flat direction corresponding to the inflaton field. A combination of these factors protects the asymptotic flatness of the inflaton potential.
Results of a general study on the dynamics of cosmological scalar fields with arbitrary potentials are presented. Exact and approximate attractor solutions are found, with applications to quintessence, moduli stabilization and inflation.
Recent developments in string theory have reinforced the notion that the space of stable supersymmetric and non-supersymmetric string vacua fills out a ``landscape'' whose features are largely unknown. It is then hoped that progress in extracting phenomenological predictions from string theory -- such as correlations between gauge groups, matter representations, potential values of the cosmological constant, and so forth -- can be achieved through statistical studies of these vacua. To date, most of the efforts in these directions have focused on Type I vacua. In this note, we present the firs…
The large class of inflationary models known as $α$- and $ξ$-attractors gives identical cosmological predictions at tree level (at leading order in inverse power of the number of efolds). Working with the renormalization group improved action, we show that these predictions are robust under quantum corrections. This means that for all the models considered the inflationary parameters $(n_s,r)$ are (nearly) independent on the Renormalization Group flow. The result follows once the field dependence of the renormalization scale, fixed by demanding the leading log correction to vanish, satisfies a…
We study cosmological $α$-attractors in superconformal/supergravity models, where $α$ is related to the geometry of the moduli space. For $α=1$ attractors \cite{Kallosh:2013hoa} we present a generalization of the previously known manifestly superconformal higher curvature supergravity model \cite{Cecotti:1987sa}. The relevant standard 2-derivative supergravity with a minimum of two chiral multiplets is shown to be dual to a 4-derivative higher curvature supergravity, where in general one of the chiral superfields is traded for a curvature superfield. There is a degenerate case when both matter…
We present a phase-space analysis of cosmology containing multiple scalar fields with positive and negative exponential potentials. We show that there exist power-law multi-kinetic-potential scaling solutions for sufficiently flat positive potentials or steep negative potentials. The former is the unique late-time attractor and the well-known assisted inflationary solution, but the later is never unstable in an expanding universe. Moreover, for steep negative potentials there exist a kinetic-dominated regime in which each solution is a late-time attractor. We briefly discuss the physical conse…
We propose a new scenario for generating a relic density of non-relativistic dark matter in the context of heterotic string theory. Contrary to standard thermal freeze-out scenarios, dark-matter particles are abundantly produced while still relativistic, and then decouple from the thermal bath due to the sudden increase of their mass above the universe temperature. This mass variation is sourced by the condensation of an order-parameter modulus, which is triggered when the temperature T(t) drops below the supersymmetry breaking scale M(t), which are both time-dependent. A cosmological attracto…
We discuss implications of the latest BICEP/Keck data release for inflationary models, with special emphasis on the cosmological attractors which can describe all presently available inflation-related observational data. These models are compatible with any value of the tensor to scalar ratio $r$, all the way down to $r = 0$. Some of the string theory motivated models of this class predict $10^{-3} \leq r \leq 10^{-2}$. The upper part of this range can be explored by the ongoing BICEP/Keck observations.
Cosmological alpha-attractors give a natural explanation for the spectral index n_s of inflation as measured by Planck while predicting a range for the tensor-to-scalar ratio r, consistent with all observations, to be measured more precisely in future detection of gravity waves. Their embedding into supergravity exploits the hyperbolic geometry of the Poincare disk or half-plane. These geometries are isometric under Mobius transformations, which include the shift symmetry of the inflaton field. We introduce a new Kahler potential frame that explicitly preserves this symmetry, enabling the infl…
We construct $α$-attractor versions of hybrid inflation models. In these models, the potential of the inflaton field $φ$ is uplifted by the potential of the second field $χ$. This uplifting ends due to a tachyonic instability with respect to the field $χ$, which appears when $φ$ becomes smaller than some critical value $φ_{c}$. In the large $N$ limit, these models have the standard universal $α$-attractor predictions. In particular, $n_{s }= 1- {2 \over N}$ for the exponential attractors. However, in some special cases the large $N$ limit is reached only beyond the horizon, for $N \gtrsim 60$.…
In an earlier paper, it is proposed that, due to resonance tunneling effect, tunneling from a large cosmological constant $Λ$ site in the stringy comic landscape can be fast, while tunneling from a small $Λ$ site may take exponentially long time. Borrowing the renormalization group analysis of the conductance in the Anderson localization transition, we treat the landscape as a multi-dimensional random potential and find that the vastness of the landscape leads to a sharp transition at a small critical value $Λ_{c}$ from fast tunneling for $Λ> Λ_{c} $ to suppressed tunneling for $Λ_{c} > Λ>0$. …
The Planck value of the spectral index can be interpreted as $n_s = 1 - 2/N$ in terms of the number of e-foldings $N$. An appealing explanation for this phenomenological observation is provided by $α$-attractors: the inflationary predictions of these supergravity models are fully determined by the curvature of the Kahler manifold. We provide a novel formulation of $α$-attractors which only involves a single chiral superfield. Our construction involves a natural deformation of no-scale models, and employs these to construct a De Sitter plateau with an exponential fall-off. Finally, we show how …
An earlier paper points out that a quantum treatment of the string landscape is necessary. It suggests that the wavefunction of the universe is mobile in the landscape until the universe reaches a meta-stable site with its cosmological constant $Λ_0$ smaller than the critical value $Λ_c$, where $Λ_c$ is estimated to be exponentially small compared to the Planck scale. Since this site has an exponentially long lifetime, it may well be today's universe. We investigate specific scenarios based on this quantum diffusion property of the cosmic landscape and find a plausible scenario for the early u…
We consider a generic class of the so-called inflationary $-α$ attractor models and compute the cosmological observables in the Einstein and Jordan frames, of the corresponding $F(R)-$gravity theory. We find that the two sets coincide (to within errors from the use of the slow-roll approximation) for moderate and large values of the number of e-foldings $N$, which is the novel result of this paper, generalizing previous results on the subject (see e.g. Ref. (\citen{oik1})). We briefly comment on the possible generalizations of these results.
We present a phase-space analysis of cosmology containing multiple scalar fields with a positive or negative cross-coupling exponential potential. We show that there exist power-law kinetic-potential-scaling solutions for a sufficiently flat positive potential or for a steep negative potential. The former is the unique late-time attractor, but it is difficult to yield assisted inflation. The later is never stable in an expanding universe. Moreover, for a steep negative potential there exists a kinetic-dominated regime in which each solution is a late-time attractor. In the presence of ordinary…
The possibility to construct an inflationary scenario for renormalization-group improved potentials corresponding to the Higgs sector of finite gauge models is investigated. Taking into account quantum corrections to the renormalization-group potential which sums all leading logs of perturbation theory is essential for a successful realization of the inflationary scenario, with very reasonable parameter values. The inflationary models thus obtained are seen to be in good agreement with the most recent and accurate observational data. More specifically, the values of the relevant inflationary p…
In a previous paper we found that in the context of the string theory ``discretuum'' proposed by Bousso and Polchinski, the cosmological constant probability distribution varies wildly. However, the successful anthropic predictions of the cosmological constant depend crucially on the assumption of a flat prior distribution. We conjectured that the staggered character of our Bousso-Polchinski distribution will arise in any landscape model which generates a dense spectrum of low-energy constants from a wide distribution of states in the parameter space of the fundamental theory. Here we calculat…
We show that the geometry of cutoffs on eternal inflation strongly constrains predictions for the timescales of vacuum domination, curvature domination, and observation. We consider three measure proposals: the causal patch, the fat geodesic, and the apparent horizon cutoff, which is introduced here for the first time. We impose neither anthropic requirements nor restrictions on landscape vacua. For vacua with positive cosmological constant, all three measures predict the double coincidence that most observers live at the onset of vacuum domination and just before the onset of curvature domina…
We have carried out detailed $k$-mode analysis of single-inflaton slow-roll inflationary perturbations including quantum fluctuations by setting up non-linear coupled differential equations of inflaton field ($ϕ$), its perturbation ($δϕ$) and the metric perturbation (the Bardeen potential, $Φ_B$), and calculated the number of e-folds ($N$), scalar spectral index ($n_s$), tensor spectral index ($n_h$), scalar power spectrum ($Δ_s$), tensor power spectrum ($Δ_h$), tensor-to-scalar ratio ($r$) and the Hubble parameter ($H$) for different $k$ values at the horizon crossing. In these calculations w…
In cosmological framework, Noether symmetry technique has revealed a useful tool in order to examine exact solutions. In this work, we first introduce the Jordan-frame Lagrangian and apply the conformal transformation in order to obtain the Lagrangian equivalent to Einstein-frame form. We then analyse the dynamics of the field in the cosmological alpha-attractors using the Noether sysmetry approach by focusing on the single field scenario in the Einstein-frame form. We show that with a Noether symmetry the coresponding dynamical system can be completely integrated and the potential exhibited b…
In this paper, we study the cosmological dynamics of scalar fields with O(N) symmetry in general potentials. We compare the phase space of the dynamical systems of the quintessence and phantom and give the conditions for the existence of various attractors as well as their cosmological implications. We also show that the existence of tracking attractor in O(N) phantom models require the potential with $Γ<1/2$, which makes the models with exponential potential possess no tracking attractor.
We show that, in quadratic lagrangian theories of gravity, isotropic cosmological singularities are stable to the presence of small scalar, vector and tensor inhomogeneities. Unlike in general relativity, a particular exact isotropic solution is shown to be the stable attractor on approach to the initial cosmological singularity. This solution is also known to act as an attractor in Bianchi universes of types I, II and IX, and the results of this paper reinforce the hypothesis that small inhomogeneous and anisotropic perturbations of this attractor form part of the general cosmological solutio…
We describe a broad class of multi-field inflationary models with spontaneously broken conformal invariance. It generalizes the recently discovered class of cosmological attractors with a single inflaton field. In the new multi-field theories, just as in the previously studied single-field models, the moduli space has a boundary (Kahler cone) in terms of the original homogeneous conformal variables. Upon spontaneous breaking of the conformal invariance and switching to the Einstein frame, this boundary moves to infinity in terms of the canonically normalized inflaton field. This results in the…
In the scalar-tensor theory of gravitation it seems nontrivial to establish if solutions of the cosmological equations in the presence of a cosmological constant behave as attractors independently of the initial values. We develop a general formulation in terms of two-dimensional phase space. We show that there are two kinds of fixed points, one of which is an attractor depending on the coupling constant and equation of state. In the case with a power-law potential in the Jordan frame, we also find new type of inflation caused by the coupling to the matter fluid.
We discuss a mechanism through which the multi-vacua theories, such as String Theory, could solve the Hierarchy Problem, without any UV-regulating physics at low energies. Because of symmetry the number density of vacua with a certain hierarchically-small Higgs mass diverges, and is an attractor on the vacuum landscape.The hierarchy problem is solved in two steps. It is first promoted into a problem of the super-selection rule among the infinite number of vacua (analogous to theta-vacua in QCD), that are finely scanned by the Higgs mass. This rule is lifted by heavy branes, which effectively c…
The creation of a quantum Universe is described by a {\em density matrix} which yields an ensemble of universes with the cosmological constant limited to a bounded range $Λ_{\rm min}\leq Λ\leq Λ_{\rm max}$. The domain $Λ<Λ_{\rm min}$ is ruled out by a cosmological bootstrap requirement (the self-consistent back reaction of hot matter). The upper cutoff results from the quantum effects of vacuum energy and the conformal anomaly mediated by a special ghost-avoidance renormalization. The cutoff $Λ_{\rm max}$ establishes a new quantum scale -- the accumulation point of an infinite sequence of garl…
Inflationary scenarios motivated by the Minimal Supersymmetric Standard Model (MSSM) where five scalar fields are non-minimally coupled to gravity are considered. The potential of the model and the function of non-minimal coupling are polynomials of two Higgs doublet convolutions. We show that the use of the strong coupling approximation allows to obtain inflationary parameters in the case when a combination of the four scalar fields plays a role of inflaton. Numerical calculations show that the cosmological evolution leads to inflationary scenarios fully compatible with observational data for…
In BP models with hundreds of fluxes, we compute the effects of cosmological dynamics on the probability distribution of landscape vacua. Starting from generic initial conditions, we find that most fluxes are dynamically driven into a different and much narrower range of values than expected from landscape statistics alone. Hence, cosmological evolution will access only a tiny fraction of the vacua with small cosmological constant. This leads to a host of sharp predictions. Unlike other approaches to eternal inflation, the holographic measure employed here does not lead to "staggering", an exc…
The so-called Einstein-Aether theory is General Relativity coupled (at second derivative order) to a dynamical unit time-like vector field (the aether). It is a Lorentz-violating theory, and gained much attention in the recent years. In the present work, we study the cosmological evolution of Einstein-Aether models with power-law-like potential, by using the method of dynamical system. In the case without matter, there are two attractors which correspond to an inflationary universe in the early epoch, or a de Sitter universe in the late time. In the case with matter but there is no interaction…
The canonical scalar-tensor theory model which exhibits spontaneous scalarization in the strong-gravity regime of neutron stars has long been known to predict a cosmological evolution for the scalar field which generically results in severe violations of present-day Solar System constraints on deviations from general relativity. We study if this tension can be alleviated by generalizing this model to include a disformal coupling between the scalar field $φ$ and matter, where the Jordan frame metric ${\tilde g}_{μν}$ is related to the Einstein frame one $g_{μν}$ by ${\tilde g}_{μν}=A(φ)^2 (g_{μ…
It has been argued that the quantum (conformal) trace anomaly could potentially provide us with a dynamical explanation of the cosmological constant problem. In this paper, however, we show by means of a semiclassical analysis that the trace anomaly does not affect the cosmological constant. We construct the effective action of the conformal anomaly for flat FLRW spacetimes consisting of local quadratic geometric curvature invariants. Counterterms are thus expected to influence the numerical value of the coefficients in the trace anomaly and we must therefore allow these parameters to vary. We…
We suggest a new cosmological scenario which naturally guarantees the smallness of scalar masses and VEVs, without invoking supersymmetry or any other (non-gravitationaly coupled) new physics at low energies. In our framework, the scalar masses undergo discrete jumps due to nucleation of closed branes during (eternal) inflation. The crucial point is that the step size of variation decreases in the direction of decreasing scalar mass. This scenario yields exponentially large domains with a distribution of scalar masses, which is sharply peaked around a hierarchically small value of the mass. Th…
The cosmological constant and the Higgs mass seem unnaturally small and anthropically selected. We show that both can be efficiently scanned in Quantum Field Theories with a large enough number of vacua controllable thanks to approximated $\mathbb{Z}_2$ symmetries (even for Coleman-Weinberg potentials). We find that vacuum decay in a landscape implies weaker bounds than previously estimated. Special vacua where one light scalar is accidentally light avoid catastrophic vacuum decay if its self-cubic is absent. This is what happens for the Higgs doublet, thanks to gauge invariance. Yukawa coupli…
We discuss FRW universe with time-dependent EoS dark fluid which leads to multiple de Sitter space. This model (as well as its scalar-tensor version) may be considered as some classical analog of cosmological landscape. The universe expansion history may look as transitions between different deSitter eras what suggests the interesting solution for cosmological constant problem. For specific time-dependent EoS dark fluid the possibility of transitions between universe regions with positive and negative cosmological constant is also established.
It is suggested that the properties of the mass spectrum of elementary particles could be related with cosmology. Solutions of the Klein-Gordon equation on the Friedmann type manifold with the finite action are constructed. These solutions (actons) have a discrete mass spectrum. We suggest that such solutions could select a universe from cosmological landscape. In particular the solutions with the finite action on de Sitter space are investigated.
In this paper we considered the bosonic string action in the presence of metric $G_{μν}$, Kalb-Ramond field $B_{μν}$ and dilaton field $Φ$. The quantum conformal invariance is achieved if all three one-loop $β$-functions are zero. But, $β^Φ$, $β$-function related to the dilaton field, can also be equal to some nonzero constant $c$, which is actually the central charge of Virasoro algebra. The anomaly contained in the Schwinger term of Virasoro algebra is canceled by adding Liouville term to the sigma model action and quantum conformal invariance is restored. The space-time action, which Euler-…
