Cosmological Scaling Solutions Research Articles

Constraining cosmological scaling solutions of a Galileon field

We study a Lagrangian with a cubic Galileon term and a standard scalar-field kinetic contribution with two exponential potentials. In this model the Galileon field generates scaling solutions in which the density of the scalar field $\ensuremath{\phi}$ scales in the same manner as the matter density at early-time. These solutions are of high interest because the scalar field can then be compatible with the energy scale of particle physics and can alleviate the coincidence problem. The phenomenology of linear perturbations is thoroughly discussed, including all the relevant effects on the observables. Additionally, we use cosmic microwave background temperature-temperature and lensing power spectra by Planck 2018, the baryon acoustic oscillations measurements from the 6dF galaxy survey and SDSS and supernovae type Ia data from Pantheon in order to place constraints on the parameters of the model. We find that despite its interesting phenomenology, the model we investigate does not produce a better fit to data with respect to $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$, and it does not seem to be able to ease the tension between high and low redshift data.

The general form of the coupled Horndeski Lagrangian that allows cosmological scaling solutions

We consider the general scalar field Horndeski Lagrangian coupled to dark matter. Within this class of models, we present two results that are independent of the particular form of the model. First, we show that in a Friedmann-Robertson-Walker metric the Horndeski Lagrangian coincides with the pressure of the scalar field. Second, we employ the previous result to identify the most general form of the Lagrangian that allows for cosmological scaling solutions, i.e. solutions where the ratio of dark matter to field density and the equation of state remain constant. Scaling solutions of this kind may help solving the coincidence problem since in this case the presently observed ratio of matter to dark energy does not depend on initial conditions, but rather on the theoretical parameters.

Cosmological scaling solutions for multiple scalar fields

The general k-essence Lagrangian for the existence of cosmological scaling solutions is derived in the presence of multiple scalar fields coupled to a barotropic perfect fluid. In addition to the scaling fixed point associated with the dynamics during the radiation and matter eras, we also obtain a scalar-field dominated solution relevant to dark energy and discuss the stability of them in the two-field set-up. We apply our general results to a model of two canonical fields with coupled exponential potentials arising in string theory. Depending on model parameters and initial conditions, we show that the scaling matter-dominated epochs followed by an attractor with cosmic acceleration can be realized with/without the couplings to scalar fields. The different types of scaling solutions can be distinguished from each other by the evolution of the dark energy equation of state from high-redshifts to today.

Dynamics of cosmological scalar fields

The background dynamical evolution of a universe filled with matter and a cosmological scalar field is analyzed employing dynamical system techniques. After the phenomenology of a canonical scalar field with exponential potential is revised, square and square root kinetic corrections to the scalar field canonical Lagrangian are considered and the resulting dynamics at cosmological distances is obtained and studied. These noncanonical cosmological models imply new interesting phenomenology including early time matter dominated solutions, cosmological scaling solutions and late time phantom dominated solutions with dynamical crossing of the phantom barrier. Stability and viability issues for these scalar fields are presented and discussed.

Towards scaling cosmological solutions with full coupled Horndeski Lagrangian: the KGB model

We study a general scalar field Lagrangian coupled with matter andlinear in □ϕ (also called KGB model). Within this class ofmodels, we find the most general form of the Lagrangian that allowsfor cosmological scaling solutions, i.e. solutions where the ratioof matter to field density and the equation of state remain constant.Scaling solutions of this kind may help solving the coincidence problemsince in this case the presently observed ratio of matter to darkenergy does not depend on initial conditions, but rather on the theoreticalparameters. Extending previous results we find that it is impossibleto join in a single solution a matter era and the scaling attractor.This is an additional step towards finding the most general scalingLagrangian within the Horndeski class, i.e. general scalar-tensormodels with second order equations of motion.

Dynamical study of DBI-essence in loop quantum cosmology and brane world

We have studied homogeneous isotropic FRW model having dynamical dark energy DBI-essence with scalar field. The existence of cosmological scaling solutions restricts the Lagrangian of the scalar field $\phi$. Choosing $p=X g(X e^{\lambda \phi})$, where $X=-g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi /2$ with $g$ is any function of $X e^{\lambda \phi}$ and defining some suitable transformations, we have constructed the dynamical system in different gravity: (i) Loop Quantum Cosmology (LQC), (ii) DGP BraneWorld and (iii) RS-II Brane World. We have investigated the stability of this dynamical system around the critical point for three gravity models and investigated the scalar field dominated attractor solution in support of accelerated universe. The role of physical parameters have also been shown graphically during accelerating phase of the universe.

Interacting Kasner-type cosmologies

It is well known that Kasner-type cosmologies provide a useful framework for analyzing the three-dimensional anisotropic expansion because of the simplification of the anisotropic dynamics. In this paper relativistic multi-fluid Kasner-type scenarios are studied. We first consider the general case of a superposition of two ideal cosmic fluids, as well as the particular cases of non-interacting and interacting ones, by introducing a phenomenological coupling function $q(t)$. For two-fluid cosmological scenarios there exist only cosmological scaling solutions, while for three-fluid configurations there exist not only cosmological scaling ones, but also more general solutions. In the case of triply interacting cosmic fluids we can have energy transfer from two fluids to a third one, or energy transfer from one cosmic fluid to the other two. It is shown that by requiring the positivity of energy densities there always is a matter component which violates the dominant energy condition in this kind of anisotropic cosmological scenarios.

Assisted dark energy

Cosmological scaling solutions, which give rise to a scalar-field density proportional to a background fluid density during radiation and matter eras, are attractive to alleviate the energy scale problem of dark energy. In the presence of multiple scalar fields the scaling solution can exit to the epoch of cosmic acceleration through the so-called assisted inflation mechanism. We study cosmological dynamics of a multi-field system in details with a general Lagrangian density p=sum_{i=1}^n X_i g(X_i e^{lambda_i phi_i}), where X_i=-(nabla phi_i)^2/2 is the kinetic energy of the i-th field phi_i, lambda_i is a constant, and g is an arbitrary function in terms of Y_i=X_i e^{lambda_i phi_i}. This covers most of the scalar-field models of dark energy proposed in literature that possess scaling solutions. Using the bound coming from Big-Bang-Nucleosynthesis and the condition under which the each field cannot drive inflation as a single component of the universe, we find the following features: (i) a transient or eternal cosmic acceleration can be realized after the scaling matter era, (ii) a "thawing" property of assisting scalar fields is crucial to determine the evolution of the field equation of state w_{phi}, and (iii) the field equation of state today can be consistent with the observational bound w_{phi}<-0.8 in the presence of multiple scalar fields.

Cosmological scaling solutions in generalised Gauss–Bonnet gravity theories

The conditions for the existence and stability of cosmological power-law scaling solutions are established when the Einstein-Hilbert action is modified by the inclusion of a function of the Gauss-Bonnet curvature invariant. The general form of the action that leads to such solutions is determined for the case where the universe is sourced by a barotropic perfect fluid. It is shown by employing an equivalence between the Gauss-Bonnet action and a scalar-tensor theory of gravity that the cosmological field equations can be written as a plane autonomous system. It is found that stable scaling solutions exist when the parameters of the model take appropriate values.

Cosmological scaling solutions with tachyon: Modified gravity model

Modifying the Einstein's gravity at large distance scales is one of the interesting proposals to explain the late time acceleration of the universe. In this Letter, we analyse scaling solutions in modified gravity models where the universe is sourced by a background matter fluid together with a tachyon type scalar field. We describe a general prescription to calculate the scaling potential in such models. Later on we consider specific examples of modifications and apply our method to calculate the scaling potential and the scale factor. Our method can be applied to any modified gravity model, in presence of tachyon field.

Scaling cosmologies of gauged supergravity

We construct exact cosmological scaling solutions in gauged supergravity. We restrict to solutions for which the scalar fields trace out geodesic curves on the scalar manifold. Under these restrictions it is shown that the axionic scalars are necessarily constant. The potential is then a sum of exponentials and has a very specific form that allows for scaling solutions. The scaling solutions describe eternal accelerating and decelerating power-law universes, which are unstable. An uplift of the solutions to 11-dimensional supergravity is carried out and the resulting time-dependent geometries are discussed. In the discussion we briefly comment on the fact that gauged supergravity allows accelerating stable scaling solutions.

String-inspired cosmology: a late time transition from a scaling matter era to a dark energyuniverse caused by a Gauss–Bonnet coupling

The Gauss–Bonnet (GB) curvature invariant coupled to a scalar fieldϕ can lead to an exit from a scaling matter dominated epoch to a late time acceleratedexpansion, which is attractive for alleviating the coincidence problem of dark energy.We derive the condition for the existence of cosmological scaling solutions inthe presence of the GB coupling for a general scalar field Lagrangian densityp(ϕ,X), where X = −(1/2)(∇ϕ)2 is a kinetic term of the scalar field. The GB coupling and the Lagrangian densityare restricted to being in the forms f(ϕ) ∝ eλϕ and p = Xg(Xeλϕ), respectively,where λ is aconstant and g is an arbitrary function. We also derive fixed points for such a scaling Lagrangian with aGB coupling f(ϕ) ∝ eμϕ and clarify the conditions under which the scaling matter era is followed by a de Sittersolution which can appear in the presence of the GB coupling. Among the scaling modelsproposed in the current literature, we find that the models which allow such a cosmologicalevolution are an ordinary scalar field with an exponential potential and a tachyon field withan inverse square potential, although the latter requires a coupling between dark energyand dark matter.

DYNAMICS OF DARK ENERGY

We review in detail a number of approaches that have been adopted to try and explain the remarkable observation of our accelerating universe. In particular we discuss the arguments for and recent progress made towards understanding the nature of dark energy. We review the observational evidence for the current accelerated expansion of the universe and present a number of dark energy models in addition to the conventional cosmological constant, paying particular attention to scalar field models such as quintessence, K-essence, tachyon, phantom and dilatonic models. The importance of cosmological scaling solutions is emphasized when studying the dynamical system of scalar fields including coupled dark energy. We study the evolution of cosmological perturbations allowing us to confront them with the observation of the Cosmic Microwave Background and Large Scale Structure and demonstrate how it is possible in principle to reconstruct the equation of state of dark energy by also using Supernovae Ia observational data. We also discuss in detail the nature of tracking solutions in cosmology, particle physics and braneworld models of dark energy, the nature of possible future singularities, the effect of higher order curvature terms to avoid a Big Rip singularity, and approaches to modifying gravity which leads to a late-time accelerated expansion without recourse to a new form of dark energy.

Scaling solutions and geodesics in moduli space

In this paper we consider cosmological scaling solutions in general relativity coupled to scalar fields with a non-trivial moduli space metric. We discover that the scaling property of the cosmology is synonymous with the scalar fields tracing out a particular class of geodesics in moduli space—those which are constructed as integral curves of the gradient of the log of the potential. Given a generic scalar potential we explicitly construct a moduli metric that allows scaling solutions, and we show the converse—how one can construct a potential that allows scaling once the moduli metric is known.

General analytic formulas for attractor solutions of scalar-field dark energy models and their multifield generalizations

We study general properties of attractors for scalar-field dark energy scenarios which possess cosmological scaling solutions. In all such models there exists a scalar-field dominant solution with an energy fraction ${\ensuremath{\Omega}}_{\ensuremath{\phi}}=1$ together with a scaling solution. A general analytic formula is given to derive fixed points relevant to dark energy coupled to dark matter. We investigate the stability of fixed points without specifying the models of dark energy in the presence of nonrelativistic dark matter and provide a general proof that a nonphantom scalar-field dominant solution is unstable when a stable scaling solution exists in the region ${\ensuremath{\Omega}}_{\ensuremath{\phi}}&lt;1$. A phantom scalar-field dominant fixed point is found to be classically stable. We also generalize the analysis to the case of multiple scalar fields and show that for a nonphantom scalar-field assisted acceleration always occurs for all scalar-field models which have scaling solutions. For a phantom field the equation of state approaches that of cosmological constant as we add more scalar fields.

Cosmological scaling solutions (and their applications to string scenarios)

AbstractIn this review we will study the phase space of cosmologies consisting of a scalar field with an exponential potential and a barotropic fluid. This is a very simple system which, however, gives rise to many interesting solutions that can be classified according to the values of the parameters in the game, namely those that define the type of fluid and the steepness of the potential. We shall then pay particular attention to scaling solutions, where the scalar energy density tracks that of the fluid.In the second part of this review we shall turn to specific models in order to apply the previous results. The goal is to study the dynamics of scalar fields in the Early Universe, and the framework to do so is different types of string compactifications. The corresponding scalar fields are then the so‐called moduli for which non perturbative potentials, of an exponential type, usually arise. We shall see how, contrary to previous expectations, the existence of scaling in this context allows these fields to dynamically stabilise at their tiny minima for a large fraction of initial conditions. This represents a further step in the quest for finding a suitable Particle Physics model compatible with the existing cosmology.

Cosmological scaling solutions (and their applications to string scenarios)

Cosmology has undergone a significant advance in recent years both observationally and theoretically. New high quality data from a host of Cosmic Microwave Background and high redshift surveys has provided strong support for the inflationary paradigm, which postulates that the universe expanded exponentially fast when it was a fraction of a second old. A key question that has arisen is the nature of the underlying physics that drove such a phase of accelerated expansion. An exciting theoretical development has been the realization that the standard model interactions may be confined to a four–dimensional brane, whereas gravitational interactions may also propagate through the extra dimensions. The radical proposal of the ‘braneworld scenario’ is that our observable universe is a brane embedded in a higher–dimensional bulk. We will review some of the recent developments in this field.

Phantom damping of matter perturbations

Cosmological scaling solutions are particularly important in solving the coincidence problem of dark energy. We derive the equations of sub-Hubble linear matter perturbations for a general scalar-field Lagrangian—including quintessence, tachyon, dilatonic ghost condensate and k-essence—and solve them analytically for scaling solutions. We find that matter perturbations are always damped if a phantom field is coupled to dark matter and identify the cases in which the gravitational potential is constant. This provides an interesting possibility to place stringent observational constraints on scaling dark energy models.

Coupled dark energy: towards a general description of the dynamics

In dark energy models of scalar-field coupled to a barotropic perfect fluid, theexistence of cosmological scaling solutions restricts the Lagrangian of the fieldφ top = Xg(Xeλφ), where , λ is aconstant and g is an arbitrary function. We derive general evolution equations in an autonomous form forthis Lagrangian and investigate the stability of fixed points for several different darkenergy models: (i) ordinary (phantom) field, (ii) dilatonic ghost condensate, and(iii) (phantom) tachyon. We find the existence of scalar-field dominant fixed points(Ωφ = 1) with an accelerated expansion in all models irrespective of the presence of the couplingQ between dark energy and dark matter. These fixed points are always classically stable for aphantom field, implying that the universe is eventually dominated by the energy density ofa scalar field if a phantom is responsible for dark energy. When the equation of statewφ for thefield φ islarger than −1, we find that scaling solutions are stable if the scalar-field dominant solution is unstable, andvice versa. Therefore in this case the final attractor is either a scaling solution with constantΩφ satisfying0 < Ωφ < 1 or a scalar-fielddominant solution with Ωφ = 1.

Generalized cosmological scaling solutions

Motivated by the recent interest in cosmologies arising from energy density modifications to the Friedmann equation, we analyze the scaling behavior for a broad class of these cosmologies comprised of scalar fields and background barotropic fluid sources. In particular, we determine the corresponding scalar field potentials which lead to attractor scaling solutions in a wide class of braneworld and dark energy scenarios. We show how a number of recent proposals for modifying the Friedmann equation can be thought of as being dual to one another, and determine the conditions under which such dualities arise.