The growth of DM and DE perturbations in DBI non-canonical scalar field scenario

doi:10.1016/j.aop.2020.168299

Annals of Physics

Volume 422, November 2020, 168299

https://doi.org/10.1016/j.aop.2020.168299 Get rights and content

Highlights

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In contrast to other works, we do not set the effective sound speed to zero or unity. The effective and adiabatic sound speed is set to what the theory anticipates.
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In the present work, we will further apply the results of the linear regime of perturbations for growth factor to examine the variation of the Integrated Sachs–Wolfe (ISW). We study the consequences of the clustering DBI dark energy model in the ISW effect and compare the results with the $Λ$ CDM model.

Abstract

We study the effect of varying sound speed on clustering dark energy in the Dirac–Born–Infeld (DBI) scenario. The DBI action is included in the class of $k$ -essence models, and it has an important role in describing the effective degrees of freedom of D-branes in the string theory. In the DBI setup, we take the anti-de Sitter (AdS) warp factor $f (ϕ) = f_{0} ϕ^{- 4}$ , and investigate the self-interacting quartic potential $V (ϕ) = λ ϕ^{4} ∕ 4$ . We calculate the full expression of the effective sound speed for our model, and show that it can evolve with time during the cosmological evolution. Besides, the adiabatic sound speed evolves with time here, and this influences the background dynamics to some extent. We show that the effective sound speed is very close to the adiabatic sound speed. We examine the effect of the variable sound speed on growth of the perturbations in both the linear and non-linear regimes. In the linear regime, we apply the Pseudo-Newtonian formalism, and show that dark energy suppresses the growth of perturbations at low redshifts. From study of the Integrated Sachs–Wolfe (ISW) effect in our setup, we see that the model manifests some deviation from the concordance $Λ$ CDM model. In the non-linear regime, we follow the approach of spherical collapse model, and calculate the linear overdensity $δ_{c} (z_{c})$ , the virial overdensity $Δ_{vir} (z_{c})$ , overdensity at the turn around $ζ (z_{c})$ and the rate of expansion of collapsed region $h_{ta} (z)$ . Our results imply that the provided values of $δ_{c} (z_{c})$ , $Δ_{vir} (z_{c})$ , $ζ (z_{c})$ and $h_{ta} (z)$ in our clustering DBI dark energy model approach the fiducial value in the EdS universe at high enough redshifts. We further compute relative number density of halo objects above a given mass in our setting, and show that the number of structures with respect to the $Λ$ CDM model is reduced more in the high mass tail at high redshifts.

Introduction

Cosmological data from different instruments indicate that the present-day Universe experiences an accelerating expansion [1], [2], [3], [4], [5], [6], [7]. This acceleration is usually attributed to the dark energy (DE) whose nature is still unknown for us. The most familiar candidate for the DE is the Einstein cosmological constant $Λ$ which gives rise to a constant energy density for the Universe and this in turn leads to the negative pressure required for having an accelerating expansion. The cosmological model based on the cosmological constant is called the $Λ$ CDM model, and it matches very well with observational data from the cosmic microwave background radiation and large-scale scale structure formation. In spite of its achievements, the cosmological constant suffers from some catastrophic drawbacks such as the fine-tuning and cosmic coincidence problems [8], [9], [10], [11], [12]. To overcome these problems, cosmologists consider the dynamical DE models. One important class of the dynamical DE models is based on a scalar field describing the evolution of vacuum energy content. In these kinds of models, the equation of state of the scalar field tends to the vacuum energy behavior just after the pressureless matter constituent which has dominated over radiation contribution. A cosmological solution which shows such a trend is called the tracker solution.

The most famous scalar field DE model is the quintessence model [13], [14], in which a scalar field with a canonical kinetic energy is coupled minimally to the Einstein gravity. The quintessence model can provide the tracker solutions required to solve the coincidence problem, but to do so, the parameters of the scalar potential should be adjusted very carefully. Therefore, the model requires high level of fine-tuning which is at odds with the primitive intentions of the model. An important alternative to the quintessence is the $k$ -essence model [15], [16], [17] of DE, in which a generalized form is taken for the kinetic term of the scalar field. In contrast to the quintessence model, the tracker solutions of the $k$ -essence model are general solutions of it, and hence the $k$ -essence does not require high level of fine-tuning to overcome the coincidence problem. One privileged $k$ -essence model which has robust theoretical motivations is the Dirac–Born–Infeld (DBI) dark energy model. The DBI action was formerly offered as a substitute for the standard action of electrodynamics. In the recent theoretical developments in the string theory, the DBI action has a prominence role in implying the effective D-brane degrees of freedom [18], [19], [20].

So far, the DBI scalar field has been considered in different contexts of cosmology in the literature. Some of these articles have considered DBI as a source for inflation [21], [22], [23], [24], [25] or dark energy [26], [27], [28], [29], [30], [31], [32], [33], [34]. It has been shown in [28] that the DBI model provides several new classes of dark energy behavior beyond quintessence due to its relativistic kinematics. In that paper, the authors discussed that the dark energy dynamics demonstrates attractor solutions which include the cosmological constant behavior. The authors also argued that the novel signature of DBI attractors is that the sound speed can be driven to zero, unlike for quintessence where it is the speed of light. In [29], the dynamics of the DBI field is analyzed in a cosmological setup which includes a perfect fluid. There, the authors supposed arbitrary power law or exponential functions for the potential and the brane tension of the DBI field, and concluded that scaling solutions can exist if powers of the field in the potential and warp factor fulfill specific relations. The scaling solutions of the DBI scalar field also have been regarded in [30], [31]. The DBI dark energy field interacting with dark matter in terms of late-time scaling solutions was studied in [32]. In [34] for unified DBI dark energy model, linear growth of perturbation was studied by fixing some of its degrees of freedom, and also a Bayesian analysis was performed to set observational constraints on the parameters of the model.

An attractive property of the DBI dark energy model is that the sound speed of the scalar field perturbations in this model can be different from the light speed. Therefore, it is expected that this feature has a noticeable implications in study of the large-scale structure formation in the Universe. In the process of structure formation, the gravitational instability causes the primordial density perturbation collapse [35], [36], [37], [38], [39], [40]. The primordial density perturbations are produced during the inflationary era [41], [42]. During inflation the Hubble horizon shrinks and so the wavelength of the cosmological perturbations exceeds the horizon size. In the subsequent stages of the Universe history, the horizon expands, and the perturbations enter the horizon again. At the early times after their reentry, the overdensities are small so that the linear theory of perturbations is applicable. In this period, the interesting scales in cosmology are much smaller than the size of the Hubble horizon, and also the velocities are non-relativistic. So, the linear regime of perturbations holds, and we can apply the Pseudo-Newtonian formalism to analyze the evolution of the overdensities. Since in the Pseudo-Newtonian formalism, the relativistic effects lead to the appearance of pressure terms in the Poisson equation, the Newtonian hydrodynamical equations can be used in the expanding Universe [43], [44]. However, at late times the perturbations grow and the overdensities enter the non-linear regime. From investigating the dynamics of the overdensities in the non-linear regime, we can obtain valuable cosmological predictions which can be assessed by observations. A simple analytical manner in study of the non-linear perturbations is the spherical collapse model (SCM) [35]. A feasibility of SCM is that in this scenario DE can behave like a fluid with clustering features same as those of DM [43]. This property of DE originates from the naive idea that when fluctuations in the fluid pressure grow, the effective equation of state of the collapsed sphere becomes different from that of the unperturbed background [43]. As discussed in [44], the clustering properties of DE can manifest signals which are observable on the cosmological data.

In the present paper, we study the cosmological structure formation within the framework of DBI clustering DE. In contrast to other works (see e.g. [45], [46], [47], [48], [49], [50], [51]), we do not set the effective sound speed to zero or unity. The effective and adiabatic sound speed are set to what the theory anticipates. The effective sound speed is a key quantity which has a crucial role in the equations of the large-scale structure formation and it determines the amount of clustering DE. We aim to provide a method to derive the full expression of the effective sound speed, which is applicable for all the $k$ -essence DE models. Besides, in the present work, we will further apply the results of the linear regime of perturbations for growth factor to examine the variation of the Integrated Sachs–Wolfe (ISW). We study the consequences of the clustering DBI dark energy model in the ISW effect and compare the results with the $Λ$ CDM model. Furthermore, we study consequences of our model in spherical-collapse scenario and compare the results with $Λ$ CDM and EdS models.

In the following, we examine the growth of the perturbations in both the linear and non-linear regimes. In our examination, we assume that the dynamics of the DBI scalar field is determined by the self-interacting quartic potential $V (ϕ) = \frac{λ}{4} ϕ^{4}$ where $λ$ is a constant. We also take the DBI warp factor in the anti-de sitter form $f (ϕ) = f_{0} ∕ ϕ^{4}$ with constant $f_{0}$ , and study the cosmological implications of our model. For this purpose, we first investigate the consequences of our model in the background cosmology in Section 2. Then, in Section 3, we examine the growth of the perturbations at the linear level. Subsequently, in Section 4, we proceed to study the non-linear dynamics of DM and DE overdensities by using of SCM. Finally, in Section 5, we present our concluding remarks.

Section snippets

DBI dark energy model

The action of DBI dark energy in the context of Einstein gravity is in the form [52] $S = \int d^{4} x \sqrt{- g} [\frac{M_{P}^{2}}{2} R + L (X, ϕ)] + S_{m},$ where $M_{P} = {(8 π G)}^{- 1 ∕ 2}$ is the reduced Planck mass. Also $g$ and $R$ are respectively the determinant of the metric $g_{μ ν}$ and the Ricci scalar. Furthermore, $S_{m}$ is the matter field action and $L (X, ϕ)$ is the DBI non-canonical scalar field Lagrangian given by $L (X, ϕ) \equiv f^{- 1} (ϕ) [1 - \sqrt{1 - 2 f (ϕ) X}] - V (ϕ) .$ In the above relations, $f (ϕ)$ , $X \equiv - g^{μ ν} \partial_{μ} ϕ \partial_{ν} ϕ ∕ 2$ and $V (ϕ)$ , are respectively the warp factor, the canonical kinetic term,

Linear perturbation theory

In this section, we study the linear regime of structure formation in the DBI dark energy scenario. For this purpose, it is suitable to start with the perturbations of the DBI scalar field. Our analysis is concise here, and for more details on the perturbations of scalar fields with non-canonical Lagrangian, one can see [60], [61], [62], [63], [64], [65]. Perturbing the background metric leads to a perturbation for the scalar field which is denoted here by $φ \equiv δ ϕ$ . Like the scalar field $ϕ$ , we

Spherical collapse in the DBI model

Here, we study evolution of the non-linear overdensities of DM and DE in the context of clustering DBI non-canonical scalar field. To this aim, we use the spherical collapse model (SCM) which is the simplest analytical approach to investigate structure formation in non-linear regime [35], [74], [75]. In the spherical collapse scenario, spherically symmetric regions with different peculiar expansion rates are separated from the homogeneous background. The spherical overdense regions in the

Conclusions

Within the framework of DBI non-canonical scalar field models of DE, we investigated the perturbations of DM and DE in both the linear and non-linear regimes. The setup of our DBI model is characterized by the AdS warp factor $f (ϕ) = f_{0} ∕ ϕ^{4}$ and the quartic potential $V (ϕ) = λ ϕ^{4} ∕ 4$ . At the background level and in a flat FRW universe, we obtained the evolutionary behaviors of the normalized Hubble parameter $\tilde{H}$ , normalized DBI scalar field $\tilde{ϕ}$ , density parameters ( $Ω_{m}$ , $Ω_{d}$ ), deceleration parameter $q$ , EoS

CRediT authorship contribution statement

K. Rezazadeh: Conceptualization, Methodology, Software, Data curation, Writing - original draft. S. Asadzadeh: Conceptualization, Methodology, Software, Data curation, Writing - original draft. K. Fahimi: Conceptualization, Methodology, Software, Data curation, Writing - original draft. K. Karami: Supervision. A. Mehrabi: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors thank the anonymous referee for very valuable comments.

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The growth of DM and DE perturbations in DBI non-canonical scalar field scenario

Highlights

Abstract

Introduction

Section snippets

DBI dark energy model

Linear perturbation theory

Spherical collapse in the DBI model

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgment

Phys. Rep.

Phys. Lett. B

Nuclear Phys. B

Phys. Lett. B

Phys. Lett. B

Phys. Rep.

Phys. Lett. B

Phys. Rep.

Astron. J.

Astrophys. J.

Astrophys. J.

Phys. Rev. D

Phys. Rev. D

Mon. Not. R. Astron. Soc.

Planck 2018 results. VI. Cosmological parameters

Rev. Modern Phys.

Internat. J. Modern Phys. D

Living Rev. Relativ.

Internat. J. Modern Phys. D

Phys. Rev. Lett.

Phys. Rev. Lett.

Phys. Rev. D

Phys. Rev. Lett.

Phys. Rev. D

J. High Energy Phys.

Phys. Rev. D

Phys. Rev. D

Gen. Relativity Gravitation

Astrophys. J.

Eur. Phys. J. C

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

J. Cosmol. Astropart. Phys.

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Gen. Relativity Gravitation

Astrophys. J.

Astrophys. J.

Mon. Not. R. Astron. Soc.

Rev. Modern Phys.

Space Sci. Rev.

Annu. Rev. Astron. Astrophys.

Phys. Rev. D

Phys. Rev. D

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