Reconstruction of Dynamical Dark Energy Potentials: Quintessence, Tachyon and interacting models

TL;DR

This paper develops methods to reconstruct dark energy potentials for various dynamical models, enabling differentiation of models with identical expansion histories by analyzing perturbations beyond smooth cosmology.

Contribution

It provides closed-form and quadrature-based reconstruction techniques for dark energy potentials in tachyon, quintessence, and interacting models, applicable to arbitrary w(z).

Findings

Reconstruction formulas for dark energy potentials derived.

Applicable to constant w and CPL parameterization.

Method generalizes to any w(z) form.

Abstract

Dynamical models for dark energy are an alternative to the cosmological constant. It is important to investigate properties of perturbations in these models and go beyond the smooth FRLW cosmology. This allows us to distinguish different dark energy models with the same expansion history. For this, one often needs the potential for a particular expansion history. We study how such potentials can be reconstructed obtaining closed formulae for potential or reducing the problem to quadrature. We consider three classes of models here: tachyons, quintessence, and, interacting dark energy. We present results for constant w and the CPL parameterization. The method given here can be generalized to any arbitrary form of w(z).

Equations

$$w=p+q(a-a_i)$$

$$\frac{d{\rho }_{de}}{dt}=-3(1+p+q(a-a_i))\frac{\dot{a}}{a}{\rho }_{de}$$

$${\rho }_{de}={\rho }_{de}^i{\left(\frac{a_i}{a}\right)}^{3(1+p-q{a}_i)}\exp [-3q(a-a_i)]$$

$$\frac{{\dot{a}}^2}{a^2}=\frac{\alpha }{a^3}+\frac{\beta }{a^{3(1+p-q{a}_i)}{e}^{3qa}}$$

$$\alpha ={\Omega }_{mi}\beta =(1-{\Omega }_{mi})a_i^{3(1+p-q{a}_i)}{e}^{3q{a}_i}$$

$$I=\int d^{4}x\sqrt{-g}\left[-V(\varphi )\sqrt{1-{\partial }^{\mu }\varphi {\partial }_{\mu }\varphi }\right]$$

$${\rho }_{\varphi }$$

$$P_{\varphi }$$

$$\frac{d\varphi }{dt}=\sqrt{1+p+q(a-a_i)}$$

$$\varphi (a)=\int \frac{\sqrt{a(1+p+q(a-a_i))}}{\sqrt{\alpha +\frac{\beta }{a^{3p-q{a}_i}{e}^{3qa}}}}da$$

$$V(\varphi )=\sqrt{-w}{\rho }_{de}$$

$$\varphi (a)=\int \frac{\sqrt{a(1+w)}}{\sqrt{\alpha +\frac{\beta }{a^{3w}}}}da$$

$$x^2=\alpha +\frac{\beta }{a^{3w}}$$

$$\varphi (x)=\int \frac{\sigma }{(x^2-\alpha {)}^k}dx$$

$$\sigma =-\frac{2\sqrt{1+w}}{3w\beta }{\beta }^k,k=\frac{w+\frac{1}{2}}{w}$$

$$\varphi (a)=\sigma \sqrt{\alpha +\beta a^{3/2}}$$

$$\frac{V(a)}{H_i^2}=\frac{3\sqrt{-w}\beta }{8\pi G{a}^{3(1+w)}}$$

$$\frac{V(\varphi )}{H_i^2}=\frac{3\beta }{8\pi G\sqrt{2}\left[\frac{{\varphi }^2}{\beta {\sigma }^2}-\frac{\alpha }{\beta }\right]}$$

$$\begin{array}{rl}\varphi (a)=& \frac{2a}{3}{\left[\frac{a\left(1+w\right)\left(\beta a^{-3w}+\alpha \right)}{\alpha \left(\beta a^{-3w}+\alpha \right)}\right]}^{1/2}\\ & \times {}_2{F}_1\left[\frac{1}{2},-\frac{1}{2w};1-\frac{1}{2w};-\frac{a^{-3w}\beta }{\alpha }\right]\end{array}$$

$$\begin{array}{rl}{}_2{F}_1(a,b,c,g)=& \frac{1}{(1-g{)}^a}\frac{\Gamma (c)\Gamma (b-a)}{\Gamma (b)\Gamma (c-a)}\\ & {}_2{F}_1(a,c-b,a-b+1,\frac{1}{(1-g)})\\ & +\frac{1}{(1-g{)}^b}\frac{\Gamma (c)\Gamma (a-b)}{\Gamma (a)\Gamma (c-b)}\\ & {}_2{F}_1(b,c-a,b-a+1,\frac{1}{(1-g)})\end{array}$$

$$g=-\frac{a^{-3w}\beta }{\alpha }$$

$$g(1-g)\frac{d^2\varphi }{d{g}^2}+\left[\left(\frac{1}{2w}+1\right)-\left(\frac{3}{2}+\frac{1}{2w}\right)g\right]\frac{d\varphi }{dg}=0$$

$$\varphi (g)=C_{1}B(g;1-u,1+u+v)+C_2$$

$$\begin{array}{rl}u=& \left(\frac{1}{2w}+1\right)\\ v=& -\left(\frac{3}{2}+\frac{1}{2w}\right)\end{array}$$

$$B(g;a,b)=\frac{g^a}{a}{}_2{F}_1(a,1-b,a+1,g)$$

$$V(\varphi )=c{\varphi }^b$$

$$I=\int d^{4}x\sqrt{-g}\left[\frac{1}{2}{g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -V(\varphi )\right]$$

$${\rho }_{\varphi }=\frac{{\dot{\varphi }}^2}{2}+V(\varphi )$$

$$P_{\varphi }=\frac{{\dot{\varphi }}^2}{2}-V(\varphi )$$

$$w_{\varphi }=\frac{P_{\varphi }}{{\rho }_{\varphi }}=\frac{{\dot{\varphi }}^2-2V}{{\dot{\varphi }}^2+2V}$$

$$\frac{d\varphi }{dt}=\sqrt{(1+w){\rho }_{\varphi }}=\sqrt{(1+p+q(a-a_i)){\rho }_{\varphi }}$$

$$V(\varphi )=\frac{1}{2}(1-w){\rho }_{\varphi }=\frac{1}{2}(1-p-q(a-a_i)){\rho }_{\varphi }$$

Full text

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1 Introduction

Observations (Riess et al. 1998; Perlmutter et al. 1999) have indicated that the Universe is expanding at an increasing rate. This has led to Dark Energy (Efstathiou et al. 1990; Ostriker & Steinhardt 1995; Bagla et al. 1996; Amendola & Tsujikawa 2010), the component with unusual properties that causes the accelerated expansion of the Universe. Besides the simple and successful Cosmological Constant model ($\Lambda$), there are a number of competing theories (Amendola & Tsujikawa 2010; Durrer 2011; Bamba et al. 2012) that are consistent with observations. In these theories, dark energy is dynamical in that its properties are a function of space and time. In order to study the theoretical and observational implications for these theories, we have to solve the equations describing the dark energy. Analysis of some observations only requires the variation of scale factor with time, however other observations can have a dependence on spatial variations in dark energy and thus details of the model become relevant.

A number of models have been proposed for dark energy, e.g., tachyon dark energy (Padmanabhan 2002; Bagla et al. 2003) and quintessence (Tsujikawa 2013; Caldwell et al. 1998). In both of these a scalar field and its gradients give rise to dark energy densities, but the forms of the Lagrangian densities for these are very different.

It is well known that if two models have the same evolution of the scale factor, tests relying only on distance measurements cannot distinguish between such models. Therefore it is important to study growth of perturbations in matter for different models of dark energy with the same evolution of the scale factor. This opens up comparison based on CMB anisotropies (Jassal 2012; Wang et al. 2010; Mifsud & De Bruck 2017), weak lensing (Bernardis et al. 2011; Pettorino & Baccigalupi 2008; Yang et al. 2016), and growth of perturbations (Rajvanshi & Bagla 2018). In this context, it is useful to have a formalism for constructing potentials for different models of dark energy that lead to the same expansion history. In this article we compute the corresponding potentials in quintessence and tachyon models which can give same background evolution. We reconstruct potential $V(\phi)$ assuming a particular equation of state $w(z)$. We give analytical expressions wherever possible, in other cases we reduce the problem to quadrature for numerical reconstruction of $V(\phi)$.

There has been a lot of interest in recovering dark energy potential from the observed expansion history (Wu & Yu 2007; Sahni & Starobinsky 2006; Saini et al. 1999). For example Huterer and Turner (Huterer & Turner 1999), provide an early work on constructing potential from simulated data and inspired further research. Li et.al (Li et al. 2007) construct potential by approximating luminosity distances and also do a comparison for reconstruction using parameterization of equation of state $w(z)$. A number of other attempts for reconstruction using a parametric or a non-parametric approach have been made. For example, see (Gerke & Efstathiou 2002; Clarkson & Zunckel 2010; Huterer & Shafer 2017) for a review. We approach this problem by attempting to construct potential for a given redshift dependence of the equation of state parameter $w(z)$ for the dark energy component. We do this for both quintessence and tachyon models: while a number of solutions exist for quintessence models (Scherrrer 2015; Battye & Pace 2016), few solutions are available for tachyon models. In Scherrer (2015), a mapping between CPL parameters and potentials is explored while an analytic approximation for various scalar field models is obtained by Battye & Pace (2016).

In §2, we set up equations for tachyon and quintessence models. In §2.2 and §2.3, we do reconstruction of potential for $w(z)=constant$. In §3, we outline the numerical recipe for reconstruction for any general $w(z)$ and illustrate it with results for some simple cases.

2 Basic Equations

We are interested in late time evolution of the Universe. Given observations that indicate that the spatial curvature is consistent with zero, and that radiation does not contribute to the expansion history at $z\leq 100$, we choose to work with only matter and dark energy. The method we outline can be generalized without any modifications to include other cases. For illustration of the method, we work with the CPL parameterization (Chevallier & Polarski 2000; Linder 2002). The functional form for $w(z)$ is defined in terms of two constants, which we call $p$ and $q$:

[TABLE]

$p$ is the value of $w$ at some $t=t_{i}$ while $q$ gives rate of change of $w$ with scale factor. Symbols $w_{0}$ (for $p$) and $w_{1}$ (for $q$) are often used while using this parameterization, if $t_{i}$ is taken to be the present time $t_{0}$. Continuity equation for dark energy density ${{\rho}}_{de}$ is:

[TABLE]

Using this equation, we get:

[TABLE]

where ${\rho}_{de}^{i}$ is density at some initial time. From now on we use a scaled dimensionless variable for time: $t=tH_{i}$. Friedmann equation then takes the form:

[TABLE]

where $\alpha$ and $\beta$ are constants defined as:

[TABLE]

These are related to the density parameter for matter and dark energy at the initial time.

2.1 Tachyon field

Tachyon models for dark energy have an action of the following form:

[TABLE]

In these models the energy density and pressure can be written as:

[TABLE]

For these models the equation of state parameter is related to the time derivative of the field as $w=-1+{\dot{\phi}}^{2}$ for a homogeneous field. Thus we have:

[TABLE]

Combining eq.(4) and eq.(7)

[TABLE]

Using the relation between the energy density and the potential, we can write:

[TABLE]

Since we know $\rho_{de}$ as a function of $a$ from eq.(3), we can compute $V(a)$. The combination of Eqn.8 and Eqn.9 gives a parametric solution for the potential as a function of the field $\phi$, with the scale factor $a$ playing the role of the intermediate parameter.

2.2 Tachyon field: Constant $w$

We start by considering the special case of $w=$ constant, i.e., $q=0$. The integral in equation (8) takes following form for constant $w$:

[TABLE]

Defining:

[TABLE]

reduces the integral to form:

[TABLE]

where $\sigma$ and $k$ are:

[TABLE]

Integral in eq.(12) is trivial for $w=-\frac{1}{2}$ where we get:

[TABLE]

Potential $V(a)$ for constant $w$ case is:

[TABLE]

When $w=-\frac{1}{2}$, we get:

[TABLE]

For other values of $w$, integral in equation (10) does not have a closed form solution. The result can be expressed in the form of hypergeometric functions:

[TABLE]

From eq.(15), we have $V(a)$, we need to invert eq.(17) to get $a(\phi)$ and substitute it in equation (15) to get $V(\phi)$. Please note that for background calculations one does not really need $V(\phi)$, $V(a)$ contains the relevant information. However for a study of spatial perturbations we require $V(\phi)$ as $\phi$ can take on different values at different points at a given time. A number of numerical libraries provide routines for calculation of ${}_{2}F_{1}(a,b,c,g)$. GNU Scientific library has function gsl_sf_hyperg_2F1, which computes ${}_{2}F_{1}(a,b,c,g)$ for $|g|<1$. In case of eq.(17), $g<0$ and for extending to $g<-1$, there are standard transformations available in literature(see Pearson’s thesis (2009) for a detailed account of computation of hypergeometric functions, we use transformations mentioned in section 4.6 of Pearson’s thesis (2009)). For $g=-\frac{a^{-3w}\beta}{\alpha}<-1$, we use following formulae for computing ${}_{2}F_{1}(a,b,c,g)$:

[TABLE]

Equation (10) can be written in the form of a differential equation which makes its relationship with other functions clear. Let

[TABLE]

Then eq.(10) can be differentiated to obtain:

[TABLE]

It can be integrated twice to obtain $\phi(g)$ in terms of incomplete beta functions $B(g;a,b)$, which are related to ${}_{2}F_{1}(a,b,c,g)$:

[TABLE]

where $C_{1}$, $C_{2}$ are constants of integration and

[TABLE]

$B(g;a,b)$ is related to ${}_{2}F_{1}(a,b,c,g)$ (Weisstein webpage 2018) as follows:

[TABLE]

We can invert either eq.(17) or eq.(23) to obtain $a(\phi)$ and then us eq.(15) to obtain $V(\phi)$. We have used the Newton-Raphson method for inversion from $\phi(a)$ to $a(\phi)$ and then on to $V(\phi)$. This is useful in dynamically calculating $V(\phi)$ and the derivative $V_{\phi}(\phi)$ when $\phi$ has spatial variations in presence of perturbations.

2.2.1 Form of the potential for constant $w$

Here we plot(figure 1) the potential $V(\phi)$ for different values of $w$. We can see from the plot that the dependence of $V(\phi)$ is close to a power law. To get insight into this behaviour, we plot derivatives of log of potential with respect to log of field in figure 2. We see that in central part there is approximate flat curve indicating that in this region the potential can be approximated by power laws.

We can approximate potential in this flat region with form:

[TABLE]

For this form we have done fitting for different values of constant $w$ and then we find the relationship between constant $w$ and $b$ which is linear as shown in 3 . These fittings are crude given that evolution of $w$ and other quantities is very sensitive to form of potential.

This can potentially be used to constrain the potential for tachyon fields if one already has observational constraints on $w$. We are working on a detailed analysis of observational constraints to be presented in a forthcoming publication, here we present an example of such an exercise. We make use of existing studies of observational constrains on $wCDM$ models. In one such study, Tripathi et al. (2017) combined the results from 3 different data sets to obtain $3\sigma$ confidence intervals for constant $w$ and CPL $w(z)$ models. We use confidence intervals for the constant $w$ while working in the regime $w\geq-1$, i.e., we use the confidence interval ($-1.0,-0.95$) and reconstruct corresponding potential slope in lower panel of 2. This is shown as a shaded region and marks the allowed slope for the potential. This is a simplistic approach and we are working on a detailed analysis while accounting for possible variations of other cosmological parameters.

2.3 Quintessence

The action for quintessence field is:

[TABLE]

with effective pressure and density:

[TABLE]

For Quintessence models of dark energy, $w$ is related to time derivative of the field and the potential, and we have:

[TABLE]

where

[TABLE]

From equations (3) and (29), we obtain:

[TABLE]

Equation (31) can be combined with eq.(4) to obtain $\frac{d\phi}{da}$. For potential we have from (30) and (3):

[TABLE]

This system of equations specifies the solution.

2.4 Quintessence field: Constant w

For $w(a)=constant$, we obtain a closed formula for $V(\phi)$(see Sangwan et al. (2018) and references within for previous work on this). In this case, eq.(31) reduces to:

[TABLE]

and

[TABLE]

Defining:

[TABLE]

and

[TABLE]

We have,

[TABLE]

here $C_{1}$ is a constant of integration. The solution is :

[TABLE]

Inverting this we get:

[TABLE]

Defining:

[TABLE]

We rewrite eq.(39):

[TABLE]

And we get

[TABLE]

Substituting this in eq.(32),

[TABLE]

Equivalently,

[TABLE]

Derivations for constants $w$ case for quintessence and phantom models were done by Sangwan et al.(2018) and they obtain the same form for quintessence models as in eq.(44). The reconstruction approach can be used to constrain potentials from observations, as was done in Sangwan et al.(2018). They used the constrained ranges from Tripathi et al. (2017), to constrain the quintessence potentials for constant $w$. They also constrain the potentials for CPL and logarithmic $w(z)$ using some approximations. Their work can be numerically generalized, using formalism developed in this article, to various $w(z)$ for both tachyonic as well as quintessence models.

3 General case

For an arbitrary function $w(a)$, continuity equation for that component is:

[TABLE]

giving

[TABLE]

Equivalently

[TABLE]

Subscript i represent values at some initial time.

Using this evolution equation for energy density we can write differential equations for tachyon and quintessence fields:

[TABLE]

[TABLE]

where $\Omega_{\phi q}$ and $\Omega_{\phi tach}$ are quintessence and tachyon field density parameters scaled as shown in eq.(47) respectively. The potentials for two fields are:

[TABLE]

[TABLE]

One can numerically integrate equations (47) and (48)/(49) to get $\phi(a)$ and alongside use (50)/(51) to obtain $V(a)$. Hence one can obtain a numerical table of $V(\phi)$ vs $\phi$ in desired range. This table can be used for numerical fitting or interpolation functions. For example, cubic splines(see book by Antia (2012)) can be used for fitting to obtain spline coefficients which can be used for calculating $V(\phi)$ and its gradients given a value of $\phi$. Once we have spline coefficients and $\phi$, task is to find the interval in which the value of $\phi$ lies so that we can use coefficients corresponding to that interval. Evaluation of the function can be time consuming, but the fact, that for background values $\phi$ there is a correspondence between $\phi$ and $a$, comes to our rescue. Typically perturbations have a small amplitude and hence deviation from background in a particular simulation domain is small, and this can be used to guess spline interval in that region. For example, one might be simulating a spherical collapse in real space and perturbations may be really strong towards centre but they merge into background as one moves away from centre. In this case for large radii, interval can be guessed from background and then one can move toward smaller radii. In this way for each new inner point one has to only search in the adjacent intervals for interpolation if the field is continuous. As an example we show here CPL potentials for quintessence and tachyon field in 4. The form obtained is similar to that obtained by Scherrer(2015).

4 Coupled Quintessence mimicking $\Lambda CDM$

Minimally coupled quintessence models(Amendola 1999; Shahalam et al. 2015) can exactly mimic $\Lambda CDM$ only with a completely flat potential, that is no field dynamics is involved and equations just reduce to that in case of $\Lambda$. However if energy exchange is allowed between quintessence field and dark matter, a $\Lambda$ like evolution is possible even with field dynamics and a time varying $w$. In this section we consider a quintessence model with following type of coupling(Barros et al. 2018):

[TABLE]

[TABLE]

Please note a bit different notation in this section as described below. $Q$ is the coupling constant between matter and Quintessence. Subscript $lcdm$ denotes quantity corresponding to $\Lambda CDM$ and $cdm$ subscript is for corresponding quantities for cold dark matter in model with field, e.g. $\rho_{cdm}$ is density for cold dark matter in model with an interacting dark energy field while $\rho_{lcdm}$ is cold dark matter density as evolved within $\Lambda CDM$. Also $\Omega_{c}^{i}$ is density parameter for dark matter at initial time and $\Omega_{\Lambda}^{i}$ is $\Lambda$ counterpart. Basic equations for this type of coupled model mimicking $\Lambda CDM$ were derived in Barros et al. (2018). They write the potential $V(\phi)$ in terms of other variables, and do not specify exact formula for $V(\phi)$. Here we start from the equations derived in Barros et al. (2018) and then reconstruct the formula for potential that gives the required $\Lambda$ like behaviour. For a field model giving same $a(t)$ as that of $\Lambda CDM$, we have:

[TABLE]

Ignoring baryons and radiation we have:

[TABLE]

and

[TABLE]

Combining the two, we have:

[TABLE]

Continuity equation for matter is:

[TABLE]

giving:

[TABLE]

Using (57) and (59) along with standard Friedmann equation for $\Lambda CDM$, we get:

[TABLE]

Arranging and integrating equation we obtain:

[TABLE]

where $\omega=\pm\frac{2\sqrt{3}}{Q}$(- for negative $Q$)

Writing $a(\phi)$ as a function of $\phi$:

[TABLE]

with $C_{1}$ taking care of any constant of integration and

[TABLE]

Potential $V$ can be obtained from equations (56), (57) and (59):

[TABLE]

Where $a^{3}(\phi)$ has a functional form as mentioned in (62). Form of potential is illustrated in 5.

[TABLE]

Studies of perturbations in the coupled quintessence models can play an important role in distinguishing these from $\Lambda$CDM models. Our analysis of such models will be reported elsewhere.

5 Summary

In this work we have described basic equations for reconstructing potentials for quintessence and tachyon field. We have given results for $w=constant$ case. We show that analytical closed formulas are possible for quintessence potentials in these cases while for tachyon fields such formulae are obtained only for $w=-0.5$ case. For other values of constant $w$, we provide formulae for numerical reconstruction. We also find a rough approximation to these constant $w$ potentials for tachyon dark energy. We describe numerical methods for numerical construction of tachyon and quintessence potentials for arbitrary $w(a)$. From numerical calculation of potentials for CPL cases for quintessence and tachyon we show that the shape pf potential is same for both of these, but the field rolls much more in quintessence case than in tachyon case. This could motivate further investigations in context of String Swampland (Heisenberg et al. 2018; Agrawal et al. 2018; Akrami et al. 2018). We have also studied coupled quintessence models.

The results of this study can be used for analysis of perturbations in such models. In particular we can compare growth of perturbations in models of different types that have the same expansion history. We will report results on spherical collapse in perturbed dark energy models in a forthcoming publication.

Acknowledgements

Authors thank Dr. Ankan Mukherjee, Dr. H. K. Jassal, Dr. Varadharaj Srinivasan and Avinash Singh for useful discussions. This research has made use of NASA’s Astrophysics Data System Bibliographic Services.

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