1 Introduction
Abstract

We perform a comprehensive analysis of recently released ‘Joint Light-curve Analysis’ SNe Ia data to investigate its implications for different dark energy models with varying equation of state parameter , is the redshift. Observational constraints on the variations of , in terms of two parameters () appearing in the chosen expressions for , are obtained for CPL, JBP, BA and Logarithmic models. We attempt to realise dark energy with varying equation in terms of a homogeneous scalar field , whose dynamics is driven by a essence Lagrangian with a constant potential and a dynamical term with . The form of the function has been reconstructed at the observational best-fit points () for the different varying dark energy models. In the context of essence model, the evolution of the adiabatic sound speed squared are investigated. For each of the varying dark energy models, the values of () are obtained which are maximally favoured from the observational data and for which lies within its physical bound imposed by stability and causality for all values of accessible in SNe Ia data. At this point we also obtain the form of the function and nature of dependence of .

Implications of JLA data for essence model of dark energy with given equation of state

Abhijit Bandyopadhyay111Email: abhijit@rkmvu.ac.in and Anirban Chatterjee222Email: anirban.chatterjee@rkmvu.ac.in

Department of Physics

Ramakrishna Mission Vivekananda Educational and Research Institute

Belur Math, Howrah 711202, India

1 Introduction

Observations of Type Ia Supernova (SNe Ia) events and measurement of their redshift and luminosity distances [1], [2] over last few decades are instrumental in establishing the fact that the universe has undergone a transition from a phase of decelerated expansion to accelerated expansion during its late time phase of evolution. Other independent evidences in support of this fact come from the observations of Baryon Acoustic Oscillation ([3],[4]), Cosmic Microwave Background radiations [5, 6], measurement of differential ages of the galaxies in GDDS, SPICES and VDSS surveys [7, 8, 9, 10] and studies of power spectrum of matter distributions of the universe. ‘Dark Energy (DE)’ is a general label attributed to the origin of this late time cosmic acceleration. Besides, study of rotation curves of spiral galaxies [11], Bullet cluster [12], gravitational lensing [13], established existence of non-luminous matter in present universe. Such ‘matter’ labeled as ‘Dark Matter’ (DM) manifest their existence only through gravitational interactions. At present epoch, dark energy and dark matter comprise around 96% of total energy density of the universe (69% dark energy and 27% dark matter). Rest 4% is contributed by baryonic matter with negligible contribution from radiations. This has been established by measurements in satellite borne experiments - WMAP [14] and Planck [15].

There exist diverse theoretical approaches aiming construction of models for dark energy to explain the present day cosmic acceleration. These include the CDM model [16] which provides best agreement with the cosmological data. Here ‘CDM’ refers to Cold Dark Matter content of the universe and , the cosmological constant, denotes vacuum energy density. Though the model provides a simple phenomenological solution, it is plagued with the problem of large disagreement between vacuum expectation value of energy momentum tensor and observed value of dark energy density (fine tuning problem). This motivates investigation of alternative models of dark energy. One of the key features of a class of such models, called varying dark energy models, is the variation of dark energy equation of state (EOS) ( is the energy density and the pressure of dark energy) with redshift (in CDM model , constant). The redshift dependence of the EOS parameter , in the context of the varying DE models may be constrained from the observational data. To access the entire range of variations of EOS parameter , the methodology adopted involves consideration of diverse functional forms of each involving a small number of parameters (denoted in the text by symbols ). The constraints on the dependence of imposed by the observational data may be expressed in terms of constraints on the parameters () for each different functional forms of considered. In this work we have performed a comprehensive analysis of recently released ‘Joint Light-curve Analysis’ (JLA) data ([17],[18],[19]) to obtain constraints on different forms of often used in literature in the context of varying dark energy, viz. CPL[20], JBP[21], BA[22] and Logarithmic model [23]. We have presented the best-fit values of the parameters ( ) and the regions of this parameter space at different confidence limits allowed from the analysis of observational data.

Dark energy with varying equation of state may be realised in terms of dynamics of a scalar field (). One class of such scalar field models, called ‘Quintessence’, is described in terms of standard canonical Lagrangian of the form where is the kinetic term. There also exist alternative class of models involving Lagrangians with non-canonical kinetic terms as , where is a function of . Such models, called essence models, have interesting phenomenological consequences different from that of quintessence models, in cosmological context. Another motivation for considering essence scalar fields is that they appear naturally in low energy effective string theory. Such theories with non-canonical kinetic terms was first proposed by Born and Infeld to get rid of the infinite self-energy of the electrons [24] and were also investigated by Heisenberg in the context of cosmic ray physics [25] and meson production [26]. In this work, we consider dark energy represented in terms of a homogeneous scalar field with its dynamics driven by a essence Lagrangian with a constant potential . Assumed constancy of the potential ensures existence of a scaling relation ( constant) in a Friedmann-Lemaitre-Robertson-Walker (FLRW) background space-time with scale factor . We exploited the scaling relation to reconstruct the forms of the function using the constraints on the parameters of the dark energy equation of state parameter obtained from the analysis of JLA data for the different varying dark energy models.

In the context of essence model, we also investigate the adiabatic sound speed squared () - the quantity relevant for the growth of small fluctuations in the background energy density. Imaginary value of the sound speed () implies instability of density perturbations. Also from causality, it requires the speed of propagation of density perturbations not to exceed the speed of light (). For each of the varying dark energy models considered here, we find -dependence of at best-fit values of parameters obtained from analysis of JLA data. This has been found over the entire redshift range accessible in SNe Ia observations corresponding to the JLA data. We note that the physical bound imposed by stability and causality is not satisfied at the best-fit for the all values in the above mentioned range. It is satisfied only for a narrow range of values of . For each of the varying DE models, we have found the regions in parameter space for which lies between 0 and 1 for the entire range of values of in JLA data. The best-fit values of parameters () for each models obtained from analysis of the observational data are found to lie outside this domain corresponding to implying that observational data allow values of parameters (), for which the physical bound on is respected, only at higher confidence limits. For example, for BA and Logarithmic models the values of parameters () corresponding to for all is allowed from observational data only beyond confidence limits. It is allowed only at and beyond for CPL model and even at larger confidence limits for JBP model. For each of the models we have found the point in parameter space which belongs to the domain for which for all and is maximally favoured from observational data. The form of the essence Lagrangian density are also reconstructed at these points.

The paper is organised as follows. In Sec. 2 we describe the methodology of analysis of the observed data and provide a brief description of the different data sets used in our analysis. In Sec. 3 we discussed different models of dark energy with varying equation of state and their realisations in terms of k−essence scalar field models. In this context we also discussed relevance of investigating variations of the adiabatic sound speed squared. The methodology of obtaining form of the Lagrangian density for essence models are also described. In Sec. 4 we discussed the results on the variation of , form of the function obtained from the analysis of the data.The conclusions are presented in Sec. 5.

2 Methodology of Analysis of Observational Data

Measurement of luminosity distances and redshift of type Ia supernovae (SNe Ia) are instrumental in probing nature of dark energy. There exist several systematic and dedicated measurements of SNe Ia. There are different supernova surveys in different domains of redshift (). High redshift projects () include Supernova Legacy Survey (SNLS) ([27],[28]), the ESSENCE project [29], the Pan-STARRS survey ([30],[31],[32]). The SDSS-II supernova surveys ([33],[34],[35], [36],[37]) probe the redshift regime . The surveys in the small redshift domain are the Harvard-Smithsonian Center for Astrophysics survey [38], the Carnegie Supernova Project ([39],[40],[41]) the Lick Observatory Supernova Search [42] and the Nearby Supernova Factory [43]. Nearly one thousand of SNe Ia events were discovered in all these surveys.

The recently released “Joint Light-curve Analysis” (JLA) data ([17],[18],[19]) is a compilation of several low, intermediate and high redshift samples including data from the full three years of the SDSS survey, first three seasons of the five-year SNLS survey and 14 very high redshift SNe Ia from space-based observations with the HST [7]. This data set contains 740 spectroscopically confirmed SNe IA events with high-quality light curves.

In this section we describe the methodology of analysis of JLA data to obtain bounds on equation of state parameter of dark energy. We take the function corresponding to JLA data as [18, 19]

(1)

where values of the dummy indices run from 1 to 740 corresponding to the 740 SNe IA events contained in the JLA data set [18]. stands for the theoretical expression for distance modulus in a flat FRW spacetime background for the entry of the JLA data set and is given by

(2)

where

(3)

is the luminosity distance, is the comoving distance to an object corresponding to a redshift . and are SNe IA redshifts in CMB rest frame and in heliocentric frame respectively. is the speed of light and is the value of Hubble parameter at present epoch. The function in Eq. (3) is the reduced Hubble parameter given by

(4)

where , and are the values of fractional energy density contributions from radiation, matter and dark energy respectively at present epoch. denotes the equation of state of dark energy.

is the observed value of distance modulus at a redshift corresponding to entry of the JLA data set . This is expressed through the following empirical relation as

(5)

where is the observed value of peak magnitude, denotes time stretching of the light-curve and is the supernova ‘color’ at maximum brightness. is the absolute magnitude which we take fixed at for our work and , are nuisance parameters. is the total covariant matrix given in terms of statistical and systematic uncertainties as

(6)

The terms in the first two lines of Eq. (6) represent the diagonal part of the covariance matrix. These include Statistical uncertainties in redshifts (), in SNe IA magnitudes (owing to intrinsic variation () and gravitational lensing (), in , and Color () and covariances between them () in each bin. The terms in last line of Eq. (6) involving matrices () correspond to the off-diagonal part of the covariance matrix originating from statistical and systematic uncertainties. All these matrices are given by JLA group and are extensively discussed in [18, 19].

We note from Eqs. (2), (3) and (4) that evaluation of requires values of parameters , and and knowledge of functional form of equation of state (EOS) of dark energy. Since in a spatially flat universe , neglecting the value of fractional density contribution of radiation at present epoch with respect to those from other components, we have . We may also choose different functional form of dark energy EOS which we denote by a general symbol , where denote the parameters of the chosen functional dependence. On the other hand, the nuisance parameters and enter in the expression for (Eq. (5)) and the covariance matrix as well (Eq. (6)). The function, in Eq. (1), are minimised with respect to the parameters , , , , . The (best-fit) values of these parameters corresponding to the minimum value of for different chosen models of dark energy EOS are presented in Sec 4.

Besides SNe Ia data, compilation of measurements of differential ages of the galaxies in GDDS, SPICES and VDSS surveys gives measured values of Hubble parameter at 15 different redshift values [7, 8, 9, 10] The function for the analysis of this observational Hubble data (OHD) may be defined as

(7)

where is the Observed value of the Hubble parameter at redshift with 1 uncertainty and is its theoretical value evaluated by multiplying in Eq. (4) by . Also, observation of Baryon Acoustic Oscillations (BAO) in Slogan Digital Sky Survey (SDSS) provide measurement of correlation function of the large sample of luminous red galaxies. Using the detected acoustic peak value of a dimensionless standard ruler corresponding to a typical redshift may be determined. The theoretical expression for the quantity is given by

(8)

where the parameters enter in the above expression though the function (Eq. (4)). The observed value of the standard ruler is and the -function for the BAO data is taken as

(9)

To illustrate the impact of the Observational Hubble data and BAO data we have performed a combined analysis of SNe IA, OHD and BAO data by minimising the () with respect to the parameter set (, , , , ). Results of the analysis are presented in Sec. 4.

3 essence and Varying Dark energy

Figure 1: Pictorial representations of the conditions for and in the parameter space spanned by and (See Eqs. (17) and (18)).

In this work we investigate realisation of dark energy with varying equation of state in terms of essence scalar field models. We assume dark energy represented in terms of a homogeneous scalar field whose dynamics is driven by a -essence Lagrangian with constant potential. In this context, we give below a brief outline of basic equations of essence model i.e.

(10)
(11)

where is the -essence Lagrangian, and respectively represent energy density and pressure of dark energy. is a function of , where , and represents the potential. In a flat FLRW spacetime background, and are related by the continuity equation

(12)

where is the Hubble constant and is the scale factor. For a homogeneous scalar field , in a flat FLRW spacetime background, we have . We consider essence models with constant potential which ensures existence of scaling relation [44, 45]

(13)

where is a constant. The equation of state of dark energy represented by essence field is given by

(14)

As mentioned in Sec. 1, for the classical solutions of the scaling relation Eq. (13) to be stable against small perturbations of the background energy density, the square of adiabatic sound speed (), should be positive. On the other hand, causality arguments require that this speed of propagation small perturbations of the background should not exceed the speed of light, implying [46, 47, 48]. In the context of essence model, we have

(15)

Using (where is the redshift and value of scale factor at present epoch is normalised to unity) in we have . Exploiting this result and transforming time dependences of , and to their dependences in Eq. (12) we may also express the sound speed squared as a function of redshift as

(16)

We note from Eq. (16) that the bound corresponds to

either
or (17)

and the bound corresponds to

either
or (18)

Whether the physical bound imposed by stability and causality is realised for any chosen functional form of equation of state may be verified by checking the conditions given in Eqs. (17) and (18). These conditions are pictorially demonstrated in Fig. 1 where we have studied effect of the conditions on the plane spanned by quantities and . The shaded region marked in the figure is bounded by two curves (dashed line) and (solid line). We have also shown the line in the plane. From Eq. (17) and (18) we see that corresponds to the region in the plane which is below the dashed line for and above the dashed line for . The condition , on the other hand, corresponds to the region lying above the solid line for and below the solid line for . The shaded region, bounded between these two lines in the plane, therefore corresponds for the entire range of values of accessible in the observations considered here.

Model
CPL [20]
JBP [21]
BA [22]
Logarithmic [23]
Table 1: Functional forms of equation of state of dark energy as used in different varying dark energy Models. Expressions for corresponding -dependences of dark energy density, expressed through the function (Eq. ((19))) are also given.
Figure 2: Regions of parameter space allowed at , and confidence limits from analysis of SNe Ia data (left panel) and SNe Ia + BAO + OHD (right panel). The results for 4 different types of parametrisations of are shown in different rows. The corresponding best-fit point and the point are also shown (see text for details).

Variation in the dark energy density may be expressed in terms of variation of dark energy equation of state . Expression for a general functional form of , in principle, involves infinite number of parameters. However, for a practical analysis its effective to express the functional dependence , in terms a small number of parameters and consider different forms of parametrizations of [21]. We consider here 4 different models, often used in literature in the context of varying dark energy, viz. CPL[20] , JBP[21], BA[22] and Logarithmic model [23]. Each of the models uses a specific functional form of expressed in terms of two parameters ( and ) and are listed in Table 1 where we have also given functional form of the quantity which gives the dependence of the corresponding dark energy density (see Eq. (19)). In the plane of Fig. 1, we have also shown the curves representing the different parametrisations of for the best-fit values of parameters and obtained from the analysis of observational data (see Sec. 4).

We finally exploit the equations of essence models to reconstruct the form the function . Using Eq. (12) we express energy density as a function of redshift as

(19)

where corresponds to value of dark energy density at present epoch . Using Eqs. (10), (11), (13) and (19) we obtain,

(20)

Writing in Eq. (10) and then using Eq. (19) we have,

(21)

For a given form of the equation of state , the right hand sides of Eqs. (20) and (21) may be evaluated numerically at each . Eliminating from both the equations one may obtain the -dependence of the function corresponding to a given form of . The dependences of on obtained from the analysis of observational data are shown and described in Sec. 4.

4 Results of analysis of observed data

In this section we present the results of analysis of the observational data using the methodology described in Sec. 2. We investigate implications of the observations in the context of the varying dark energy models listed in Table 1. For each of the models we obtain the best-fit values of the parameters (, ) along with their allowed domains at different confidence limits from the analysis. We perform analysis of SNe Ia data alone and also a combined analysis of data from SNe Ia, BAO and OHD (discussed in Sec. 2) where we freely vary the parameters , , , and to find their best-fit values (corresponding to minimum value of ). The obtained best-fit values of the above parameters for different models are presented in Table 2. In Fig. 2, we have shown regions of parameter space for the different parametrisations allowed at , and confidence limits from analysis of observed data. The corresponding best-fit () points are also shown.

In the context of essence model of dark energy, we investigate, to what extent the condition as imposed by causality and stability is favoured from observational data for different parametrisations of the dark energy equation of state . We used the different parametrisations of in Eqs. (17) and (18) to find the range of values of the parameters and for which the condition is satisfied for all within the domain of observations. For each of the parametrisations of mentioned in Table 1, this range is shown by a shaded region in plane in Fig. 2. We then observe that for all the models, the shaded region corresponding to has no overlap with the region bounded by contour allowed from analysis of SNe Ia data alone. For BA and Logarithmic models the overlap is seen when one considers allowed ranges at and beyond confidence limits. This implies that the physical bound for the entire range of values of probed by the observed data considered here, is favoured from observational data (SNe Ia only) at and above confidence level if we consider parametrisations of as in BA and Logarithmic models. For CPL parametrisation the physical bound is disfavoured below and with JBP its disfavoured upto even higher confidence limits. For all the different parametrisations of the physical bound is disfavoured to a larger extent from a combined analysis of SNe Ia , BAO and OHD. In the parameter space shown in Fig. 2, we have also marked a point in the shaded region corresponding to , at which the value of is closest to the value of for the corresponding model. Thus refers to the maximally favoured values of parameters and from observational data for which lies between 0 and 1 for all values. The values of () corresponding to are shown in the last column of table 2.

At the best-fit values of parameters and obtained from the combined analyses of observational data from Sne IA, BAO and OHD for different parametrisations we have shown the variation of sound speed squared with redshift in left panel of Fig. 3. We see that at the best-fit values of the parameters the calculated value of lies within its physical bound only for a vary narrow range of values of . These ranges are also shown in Table 2. The variation of at values of corresponding to the point are shown for different models in the middle panel of Fig. 3. These correspond to a monotonous variation of with within its physical bound imposed by causality and stability.

In Sec. 3 we discussed the methodology to reconstruct the form of function for different form of parametrisations of . We have shown in right panel of Fig. (3) the obtained dependences of on , for each of the models at the corresponding best-fit values of the parameters . The Figure shows that for JBP model has a monotonous dependence of whereas for the other models (CPL, BA and Logarithmic) the function is double valued in a certain domain of .

Data Range of
Set Model /DOF for which
CPL -0.63 -0.93 0.23 0.14 3.1 685.42/735 0.93 - 1.15 (-0.77,1.15)
JBP -0.59 -1.16 0.20 0.14 3.1 685.39/735 - (-0.49,2.22)
SNe Ia BA -0.65 -0.44 0.22 0.14 3.1 685.48/735 0.94 - 1.10 (-0.87,0.65)
Log. -0.65 -0.82 0.24 0.14 3.1 685.44/735 0.79 - 1.01 (-0.85,0.76)
CPL -0.71 -1.94 0.29 0.14 3.13 708.26/751
SNe Ia JBP -0.63 -3.14 0.28 0.14 3.13 707.65/751
+ BAO BA -0.77 -0.97 0.29 0.14 3.13 708.86/751
+ OHD Log. -0.74 -1.52 0.29 0.14 3.13 708.56/751
Table 2: Best-fit values of parameters for different models from analysis of SNe Ia data alone and SNe + BAO + OHD. The values of () at the best-fit point per DOF (degrees of freedom) are also shown. The range of for which the value of evaluated at best-fit lies between 0 and 1 are also shown. Last column shows the values of () corresponding to point (see text for details).
Figure 3: Plot of vs for different parametrisations of at best-fit (right panel) from SNe Ia data and at (middle panel) (see text for details). Corresponding plots of vs reconstructed at best-fit and are shown in right panel.

5 Conclusion

In this work we have performed a comprehensive analysis of recently released ‘Joint Light-curve Analysis’ (JLA) data to investigate its implications for models of dark energy with varying equation of state parameter . As a benchmark, we considered 4 different varying dark energy models, viz. CPL[20], JBP[21], BA[22] and Logarithmic model [23], each of which involves a specific functional form of dependence of the dark energy equation of state . The analytical expression for the function in each case involves two parameters, denoted by and . From the analysis of observational data we have obtained best-fit values of these parameters and also their ranges allowed at , and confidence level. Description of the data and methodology of analysis has been discussed in detail in Sec. 2. The results of the analysis are presented in table 2 and depicted in Fig. 2.

We make an attempt to realise the scenario of varying equation state of dark energy in terms of dynamics of a scalar field . We assume the scalar field to be homogeneous with its dynamics driven by a essence Lagrangian , with a constant potential and a dynamical term with . Consideration of constant potential ensures a scaling relation of the form ( constant) in FLRW spacetime background with scale factor . We have exploited this to reconstruct functional form of for the different varying dark energy models considered here (See Sec. 3 for details). The nature of are obtained for each kind of dependences corresponding to the best-fit parameters (). In this context, we also obtain the dependences of on . , as mentioned earlier, is the the speed with which small fluctuation in the background energy density grows. Stability of the density perturbations and causality requires to lie in the domain . The results show that at the best-fits, lies within its physical bound only for a small range of values of . For each of the varying DE models we have shown the region of the parameter space for which lies with its physical bound for all values of accessible in SNe Ia observations. We finally find the point in parameter space for which for all and which is maximally favoured from the observational data. The dependence of and form of are also obtained for this point () for all the varying DE models. These results are discussed in detail in Sec. 4 and depicted in Fig. 3.

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