Modeling interacting dark energy models with Chebyshev polynomials:Exploring their constraints and effects

Modeling interacting dark energy models with Chebyshev polynomials:
Exploring their constraints and effects

Freddy Cueva Solano Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo
Edificio C-3, Ciudad Universitaria, CP. 58040, Morelia, Michoacán, México.
freddy@ifm.umich.mx,  freddycuevasolano$2009$@gmail.com
Abstract

In this work, we examine the main cosmological effects derived from a time-varying coupling () between a dark matter (DM) fluid and a dark energy (DE) fluid with time-varying DE equation of state (EoS) parameter (), in two different coupled DE models. These scenarios were built in terms of Chebyschev polynomials. Our results show that such models within dark sector can suffer an instability in their perturbations at early times and a slight departure on the amplitude of the cosmic structure growth () from the standard background evolution of the matter. These effects depend on the form of both and . Here, we also perform a combined statistical analysis using current data to put tighter constraints on the parameters space. Finally, we use some selections criteria to distinguish our models.

pacs:
98.80.-k, 95.35.+d, 95.36.+x, 98.80.Es

I Introduction

A number of observations Conley2011 (); Jonsson2010 (); Betoule2014 (); Jackson1972 (); Kaiser1987 (); Mehrabi2015 (); Alcock1979 (); Seo2008 (); Battye2015 (); Samushia2014 (); Hudson2013 (); Beutler2012 (); Feix2015 (); Percival2004 (); Song2009 (); Tegmark2006 (); Guzzo2008 (); Samushia2012 (); Blake2011 (); Tojeiro2012 (); Reid2012 (); delaTorre2013 (); Planck2015 (); Hinshaw2013 (); Beutler2011 (); Ross2015 (); Percival2010 (); Kazin2010 (); Padmanabhan2012 (); Chuang2013a (); Chuang2013b (); Anderson2014a (); Kazin2014 (); Debulac2015 (); FontRibera2014 (); Eisenstein1998 (); Eisenstein2005 (); Hemantha2014 (); Bond-Tegmark1997 (); Hu-Sugiyama1996 (); Neveu2016 (); Sharov2015 (); Zhang2014 (); Simon2005 (); Moresco2012 (); Gastanaga2009 (); Oka2014 (); Blake2012 (); Stern2010 (); Moresco2015 (); Busca2013 () have indicated that the present universe is undergoing a phase of accelerated expansion, and driven probably by a new form of energy with negative EoS parameter, commonly so-called DE DES2006 (). This energy has been interpreted in various forms and extensely studied in OptionsDE (). However, within General Relativity (GR) the DE models can suffer the coincidence problem, namely why the DM and DE energy densities are of the same order today. This latter problem could be solved or even alleviated, by assuming the existence of a non-gravitational within the dark sector, which gives rise to a continuous energy exchange from DE to DM or vice-versa. Currently, there are not neither physical arguments nor recent observations to exclude Interacting (); Pavons (); Wangs (); Cueva-Nucamendi2012 (). Moreover, due to the absence of a fundamental theory to construct , different ansatzes have been widely discussed in Interacting (); Pavons (); Wangs (); Cueva-Nucamendi2012 (). So, It has been shown in some coupled DE scenarios that an appropriate choice of may cause serious instabilities in the dark sector perturbations at early times valiviita2008 (); Jackson2009 (); He2009 (); Xu2011 (); He2011 (); Clemson2012 (). Also, It has been argued in the coupled DE scenarios that can affect the background evolution of the DM density perturbations and the expansion history of the universe Mehrabi2015 (); Alcaniz2013 (); Yang2014 (). Thus, and could very possibly introduce new features on during the matter era.
On the other one, within dark sector we can propose new ansatzes for both and , which can be expanded in terms of the Chebyshev polynomials , defined in the interval and with a diverge-free at z Chevallier-Linder (); Li-Ma (). However, that polynomial base was particularly chosen due to its rapid convergence and better stability than others, by giving minimal errors Simon2005 (); Martinez2008 (). Besides, could also be proportional to the DM energy density and to the Hubble parameter . This new model will guarantee an accelerated scaling attractor and connect to a standard evolution of the matter. Here, will be restricted from the criteria exhibit in Campo-Herrera2015 ().
Therefore, our main motivation in the present work is to explore the effects of and on the cosmological variables in differents times and scales, including the search for new ways to alleviate the coincidence problem.
On the other hand, two distinct coupled DE models (XCPL and DR) are discussed, on which we have performed a global fitting using an analysis combined of Joint Light Curve Analysis (JLA) type Ia Supernovae (SNe Ia) data Conley2011 (); Jonsson2010 (); Betoule2014 (), including the growth rate of structure formation obtained from redshift space distortion (RSD) data Jackson1972 (); Kaiser1987 (); Mehrabi2015 (); Alcock1979 (); Seo2008 (); Battye2015 (); Samushia2014 (); Hudson2013 (); Beutler2012 (); Feix2015 (); Percival2004 (); Song2009 (); Tegmark2006 (); Guzzo2008 (); Samushia2012 (); Blake2011 (); Tojeiro2012 (); Reid2012 (); delaTorre2013 (); Planck2015 (), together with Baryon Acoustic Oscillation (BAO) data Hinshaw2013 (); Beutler2011 (); Ross2015 (); Percival2010 (); Kazin2010 (); Padmanabhan2012 (); Chuang2013a (); Chuang2013b (); Anderson2014a (); Kazin2014 (); Debulac2015 (); FontRibera2014 (); Eisenstein1998 (); Eisenstein2005 (); Hemantha2014 (), as well as the observations of anisotropies in the power spectrum of the Cosmic Microwave Background (CMB) data Planck2015 (); Bond-Tegmark1997 (); Hu-Sugiyama1996 (); Neveu2016 () and the Hubble parameter (H) data obtained from galaxy surveys Sharov2015 (); Zhang2014 (); Simon2005 (); Moresco2012 (); Gastanaga2009 (); Oka2014 (); Blake2012 (); Stern2010 (); Moresco2015 (); Busca2013 () to constrain the parameter space of such models and break the degeneracy of their parameters, putting tighter constraints on them.
In this paper, we will use the following criteria of selection (: degrees of freedom), Goodness of Fit (), Akaike Information Criterion () AIC_Criterion (), Bayesian Information Criterion () BIC_Criterion (); Kurek2014 (); Arevalo2016 () to distinguish our cosmological models from the number of their parameters that require to explain the data.
Finally, we organize this paper as follows: We describe the background equation of motions of the universe in Sec. II and the perturbed universe in Sec. III. The cu-rrent observational data and the priors considered are presented in Sec. IV. In Sec. V, we describe the different selection criteria. We discuss our results in Sec. VI and show our conclusions in Sec. VII.

Ii Background equations of motion

We consider here a flat Friedmann-Robertson-Walker (FRW) universe composed by radiation, baryons, DM and DE. Moreover, we postulate that the dark components can also interact through a non-gravitational coupling . For () implies that the energy flows from DE to DM (the energy flows from DM to DE). Also, to satisfy the requirements imposed by local gravity experiments Koyama2009-Brax2010 () we also assume that baryons and radiation are coupled to the dark components only through the gravity. Thus, the energy balance equations for these fluids may be described Cueva-Nucamendi2012 (),

(1)
(2)
(3)
(4)

where , , and are the energy densities of the baryon (b), radiation (r), DM and DE, respectively, and .
We also defined the critical density and the critical density today (here is the current value of the Hubble parameter). Considering that , the normalized densities are

(5)

and the first Friedmann equation is given by

(6)

The following relation is valid for all time

(7)

ii.1 Parameterizations of and

Due to the fact that the origin and nature of the dark fluids are unknown, it is not possible to derive from fundamental principles. However, we have the freedom of choosing any possible form of that satisfies Eqs. (3) and (4) simultaneously. Hence, we propose a new phenomenological form for a varying so that it can alleviate the coincidence problem. This coupling could be chosen proportional to , to and to . Therefore, in the dark sector can be written as

(8)

Here measures the strength of the coupling, and was modeled as a varying function of in terms of Chebyshev polynomials. This polynomial base was chosen because it converges rapidly, is more stable than others and behaves well in any polynomial expansion, giving minimal errors. The coefficients are free dimensionless parameters Cueva-Nucamendi2012 () and the first three Chebyshev polynomials are

(9)

Similarly, we propose here a new phenomenological ansatz for a time-varying and divergence-free at . Thus, we can write

(10)

where and are free dimensionless parameters. The polynomial and the parameter were included conveniently to simplify the calculations. Here, behaves nearly linear at low redshift and , while for , .

ii.2 Ratios between the DM and DE energy densities

From Eqs. (3) and (4), we define as the ratio of the energy densities of DM and DE. Then, we can rewrite as Campo-Herrera2015 (); Ratios ()

(11)

This leads to the evolution equation of

(12)

Now, we substitute Eq. (8) into Eq. (12), and then, we impose the following condition to guarantee the possibility that the coupling can solve the coincidence problem. This implies two solutions and ,

(13)

From Eq. (13), we note that and are not independent but its product can be approached to the order unity. So,

(14)

In the limiting cases , and , must be either constant or very slowly. Then, the quantities , and the value of today () must fulfill .

ii.3 DE models

ii.3.1 CDM model

Fixing both and into Eqs. (1)-(4) and solving Eq. (12), we can find and

(15)

ii.3.2 CPL model

Replacing both , where , are real parameters and into Eqs. (1)-(4), and solving Eq. (12), we obtain and ,

(16)

where is given by Eq. (15).

ii.3.3 XCPL model

Putting , where , are real free parameters and using Eq. (8) into Eqs. (1)-(4), then the solution of Eq. (12) gives

(17)

where is the maximum value of such that and and (see Cueva-Nucamendi2012 ()).

ii.3.4 DR model

This scenario can be modeled setting Eqs. (8) and (10) into Eqs. (1)-(4), and then solving Eq. (12), we get

(18)

Eqs. (8)-(12) show that from simple arguments based on the evolution of , one can find an appropriated restriction for the coupling between the dark components of the universe Campo-Herrera2015 ().
On the other hand, the coincidence problem can be alleviated, if we impose that in Eq. (11). From here, and using Eq. (12) at present time, we find

(19)

This result means that the slope of at is more gentle than that found in the CDM model Campo-Herrera2015 ().

ii.4 Crossing of line with a coupling.

From Eqs. (8) and (9), we note that exist real values of that leads to , which are called the redshift crossing points,

(20)

Then, the solution of Eq. (20) is given by

(21)

This result depends of the choice for . However, the only possibility for a crossing happends when

(22)

From Eq. (22), we impose the following restraint to gua-rantee real values in

(23)

In general, if and are both positive or both negative, then could be positive or negative. Moreover, may be zero when , , and are all zero (i.e. uncoupled DE models) or when . From here, we can describe the sign of ().

Iii Perturbed equations of motion.

Let us consider a spatially flat universe with scalar perturbations about the background. The perturbed line element in the Newtonian gauge is given by valiviita2008 (); Jackson2009 (); He2009 (); Xu2011 (); He2011 (); Clemson2012 ()

(24)

where is the conformal time, and are gravitational potentials, and “” is the scale factor. The four-velocity of fluid A is

(25)

where is the peculiar velocity potential, and is velocity perturbation in Fourier space. The energy-momentum tensor for each A is valiviita2008 (); Jackson2009 (); He2009 (); Xu2011 (); He2011 (); Clemson2012 ()

(26)

where the density and the pressure . Then, the total energy-momentum tensor is with and ).
Each fluid A satisfies the following energy-momentum balance equation valiviita2008 (); Jackson2009 (); He2009 (); Xu2011 (); He2011 (); Clemson2012 ()

(27)

where the four-vector governs the energy-momentum exchange between the dark components and satisfies valiviita2008 (); Clemson2012 ().
A general can be split relative to the total four-velocity as valiviita2008 (); Jackson2009 (); He2009 (); Xu2011 (); He2011 (); Clemson2012 ()

(28)

where is the energy density transfer relative to and is the momentum density transfer rate, relative to . Here, is a momentum transfer potential. We choose each and the total as Clemson2012 ()

(29)

Thus, the total energy-frame is defined as

(30)

where is the total energy-frame velocity potential. From Eqs. (25) and (28) obtain

(31)

The perturbed energy transfer includes a metric perturbation term and a perturbation . In addition, we stress that the perturbed momentum transfer is made up of two parts: the momentum transfer potential and .
On the other hand, the physical sound-speed of a fluid or scalar field is defined by in the rest-frame (), and the adiabatic sound-speed can be defined as valiviita2008 (). For the adiabatic DM fluid, . By contrast, the DE fluid is non-adiabatic and to avoid any unphysical instability, should be taken as a real and positive parameter. A common choice (and the one we make here) is to take valiviita2008 (); Clemson2012 ().
Defining the density contrast as , we can find equations for the density perturbations and the velocity perturbations valiviita2008 (); Clemson2012 (),

(32)
(33)

The curvature perturbations on constant- surfaces and the total curvature perturbation are given by

(34)

Then, we need to specify a covariant form of and in the dark sectors valiviita2008 (); Clemson2012 (). For , the simplest physical choice is that there is no momentum transfer in the rest-frame of either DM or DE valiviita2008 (). This leads to two cases

(35)
(36)

Using Eqs. (25), (35), (36) and (31), finds

(37)

According to valiviita2008 (); Clemson2012 (), can be

(38)
(39)

For both cases or cases, Eq. (III) does not change, but Eq. (III) is different in both cases.
In this article, we focus only on the case. Besides, assuming that depends only on the cosmic time through the global expansion rate, then a possible choice can be

(40)

From Eqs. (8), (28) and (40), we have

(41)

For convenience, we impose that , so

(42)

In a forthcoming article we will extend our study, by considering other relations between , and . It is beyond the scope of the present paper.
Considering that , and using the above Eqs into Eqs. (III) and (III), we find valiviita2008 (); Mas ():

iii.1 Dark sector

(43)
(44)
(45)
(46)

iii.2 Photon-baryon and Neutrino sectors