# Growth rate in the dynamical dark energy models

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## Abstract

Dark energy models with a slowly rolling cosmological scalar field provide a popular alternative to the standard, time-independent cosmological constant model. We study the simultaneous evolution of background expansion and growth in the scalar field model with the Ratra–Peebles self-interaction potential. We use recent measurements of the linear growth rate and the baryon acoustic oscillation peak positions to constrain the model parameter \(\alpha \) that describes the steepness of the scalar field potential.

### Keywords

Dark Energy Scalar Field Dark Energy Model Linear Growth Rate Background Dynamic## 1 Introduction

Cosmological observations now convincingly show that the expansion of the Universe is accelerating [1, 2, 3, 4]. One of the possible explanations of this empirical fact is that the energy density of the Universe is dominated by the so-called *dark energy* (DE) [5, 6], a component with effective negative pressure.

The simplest DE candidate is a time-independent cosmological constant \(\Lambda \), and the corresponding cosmological model, the so-called \(\Lambda \)CDM model, is considered to be a *concordance* model. This simple model, however, suffers from fine tuning and coincidence problems [7, 8]. In the attempt of constructing a more natural model of DE many alternative scenarios have been proposed [9, 10, 11, 12, 13, 14, 15].

One of the alternatives to a cosmological constant are the models of a dynamical scalar field. In these models a spatially uniform cosmological scalar field, slowly rolling down its almost flat self-interaction potential, plays the role of a time-dependent cosmological constant. This family of models avoid the fine tuning problem, having a more natural explanation for the observed low energy scale of DE [17, 18, 39, 40, 41]. For the scalar field models (the so-called \(\phi \)CDM model) the equation of state \(P_{\phi }= w\rho _{\phi }\) (with \(P_\phi \) and \(\rho _\phi \) the pressure and energy density of the scalar field) is time dependent, \(w= w(t)\), and unlike the cosmological constant, \( w (t) \ne -1\), although at late-times it approaches \(-1\). When the scalar field energy density starts to dominate the energy budget of the Universe, the Universe expansion starts to accelerate [19, 20]. Even though at low redshifts the predictions of the model are very close to the ones of the cosmological constant, the two models (\(\Lambda \)CDM and the dynamical DE model) predict different observables over a wide range of redshifts.

The scalar field models can be classified via their effective equation of state parameter. The models with \( -1<w<-{1}/{3}\) are referred to as quintessence models, while the models with \(w<-1\) are referred to as phantom models. The quintessence models can be divided in two broad classes: tracking quintessence, in which the evolution of the scalar field is slow, and thawing quintessence, in which the evolution is fast compared to the Hubble expansion [21, 22, 23, 24].

In tracking models the scalar field exhibits tracking solutions in which the energy density of the scalar field scales as the dominant component at the time; therefore the DE is subdominant but closely tracks first the radiation and then matter for most of the cosmic evolution. At some point in the matter domination epoch the scalar field becomes dominant, which results in its effective negative pressure and accelerated expansion [25, 26]. The simplest example of such a model is provided by a scalar field with an inverse-power-law potential energy density \(V_{\phi } \propto \phi ^{-\alpha }\), \(\alpha >0\) [27], the so-called *Ratra–Peebles* model.

The scalar field models predict a different background expansion history and a growth rate compared to the cosmological constant model ones. Thus the scalar field model can be distinguished from the \(\Lambda \)CDM model through high precision measurements of distances and growth rates over a wide redshift range [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

In this paper we study generic predictions of slowly rolling scalar field models by taking the Ratra–Peebles model as a representative example. We present a self-consistent and effective way of solving the joint equations for the background expansion and the growth rate. We use a compilation of recent growth rate and baryon acoustic oscillation (BAO) peak measurements to put constraints on the parameter \(\alpha \) describing the steepness of the scalar field’s potential.

This paper is organized as follows. In Sect. 2 we investigate in detail the dynamics and the energy of the \(\phi \)CDM models. In Sect. 3 we study the influence of the \(\phi \)CDM models on the growth factor of matter density perturbations. In Sect. 4 a comparison is presented of the obtained theoretical results with observational data. We discuss our results and conclude in Sect. 5. We use the natural units with \(c= {\hbar }=1\) throughout this paper.

## 2 Background dynamics in \(\phi \)CDM models

### 2.1 Background equations

We assume the flat and isotropic Universe that is described by the standard Friedmann–Lemaître–Robertson–Walker homogeneous cosmological spacetime model (FLRW) \( \mathrm{d}s^2=-\mathrm{d}t^2+a(t)^2\mathrm{d}\mathbf{x}^2\), and we normalize the scale factor to be equal to 1 at present time, \(a_\mathrm{today}=a_0=1\), i.e. \(a=1/(1+z)\), where *z* is the redshift.

The flatness of the Universe requires that the total energy density of the Universe is equal to the critical energy density, i.e. \(\rho _\mathrm{tot}\) = \(\rho _\mathrm{cr} \) = \(3H_0^2 M_\mathrm{pl}^2/ (8\pi )\). We also introduce the energy density parameters for each component as \(\Omega _i = \rho _i/\rho _\mathrm{cr}\) (where the index \(i\) denotes the individual components, such as radiation, matter or the scalar field).

#### 2.1.1 Initial conditions

#### 2.1.2 The results of computations of the dynamics and the energy of the \(\phi \)CDM model.

The evolution of the equation of state \(w(a)\) is presented on Fig. 3. We find that for all values of the \(\alpha \) parameter, the Chevallier–Polarsky–Linder (CPL) parametrization of the DE equation of state \(w(a)=w_0+w_a(1-a)\) near \(a=1\) (where \(w_0=w(a=1)\) and \(w_a=(-\mathrm{d}w/\mathrm{d}a)|_{a=1}\)) [44, 45, 46] provides a good approximation in the range of the scale factor \(a = [0.98\)–\(1]\).

The evolution of \(E(a)\) for different values the \(\alpha \) parameters is shown on Fig. 3. As we can expect the expansion of the Universe occurs more rapidly with increasing value of the \(\alpha \) parameter, the \(\Lambda \)CDM limit corresponding to the slowest rate of the expansion. The value of the \(\alpha \) parameters affects also the redshift of the equality between matter and scalar field energy densities (see Fig. 4); with larger values of \(\alpha \) the scalar field domination begins earlier and vice versa.

## 3 Growth factor of matter density perturbations in dark energy models

Following [47] we use the initial conditions \(\delta (a_\mathrm{in})=\delta ^{'}(a_\mathrm{in})=5\times 10^{-5}\), with \(a_\mathrm{in}=5\times 10^{-5}\) as defined above.

### 3.1 The results of computations of the growth factor of matter density perturbations in \(\phi \)CDM dark energy model

## 4 Comparison with observations

The \(\phi \)CDM models generically predict a faster expansion rate and a slower rate of growth at low redshifts. Tight measurements of the expansion rate, distance–redshift relationship and the growth rate at multiple redshift ranges can be used to simultaneously constrain the background dynamics and the growth of structure and discriminate between \(\phi \)CDM and \(\Lambda \)CDM models.

## 5 Discussion and conclusions

We explored observable predictions of the scalar field DE model. We showed that the model differs from \(\Lambda \)CDM in a number of ways that are generic and do not depend on the specific values of model parameters. For example, in scalar field models the expansion rate of the Universe is always faster and the DE dominated epoch sets in earlier than in \(\Lambda \)CDM model when other cosmological parameters are kept fixed. The two models also differ in their predictions for the growth rate, where the scalar field model generically predicts a slower growth rate than \(\Lambda \)CDM.

We used a compilation of BAO, growth rate, and the distance prior from the CMB to constrain the model parameters of the scalar field model. We find that if only the growth rate data is used there is a strong degeneracy between \(\Omega _{m}\) and \(\alpha \), where higher values of \(\alpha \) are allowed as long as the \(\Omega _{m}\) parameter is large as well. When combining these constraints with the constraints coming from a distance–redshift relationship (BAO data and the distance prior from CMB) the degeneracy is broken and we get \(\Omega _{m} = 0.30 \pm 0.04\) and \(\alpha < 1.30\) with a best-fit value of \(\alpha = 0.00\).

## Notes

### Acknowledgments

We appreciate useful comments from Leonardo Campanelli. We thank Gennady Chitov, Omer Farooq, Vasil Kukhianidze, Anatoly Pavlov, Bharat Ratra, and Alexander Tevzadze for discussions. We acknowledge partial support from the Swiss NSF grant SCOPES IZ7370-152581, the CMU Berkman foundation, the NSF grants AST-1109180, and the NASA Astrophysics Theory Program grant NNXlOAC85G. N. A. and T.K. acknowledge hospitality of International Center for Theoretical Physics (ICTP, Italy) where this work has been designed.

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