Observational constraints on Quintessence models of dark energy
Abstract
Scalar fields aptly describe equation of state of dark energy. The scalar field models were initially proposed to circumvent the fine tuning problem of cosmological constant. However, the model parameters also need a fine tuning of their own and it is important to use different observations to determine these parameters. In this paper, we use a combination of low redshift data to constrain the low redshift evolution of canonical scalar field parameters. For this analysis, we use the Supernova Type Ia observations, the Baryon Acoustic Observations and the Hubble parameter measurement data. We consider scalar field models of the thawing type of two different functional forms of potentials. The constraints on the model parameters are more stringent than those from earlier observations although these datasets do not rule out the models entirely. The parameters which let dark energy dynamics closely emulate that of a cosmological constant are preferred. The constraints on the parameters are suitable priors to further quintessence dark energy studies.
I Introduction
The discovery of late time cosmic acceleration by the Supernova Type Ia observations has been one of the most important results in cosmology riess:1998cb (); perlmutter:1998np (). Nearly twothirds of the energy of the universe is due to the cosmological constant or an alternative description called the dark energycaldwell2004 (). The presence of dark energy has been further confirmed by many different observations such as Baryon Acoustic Oscillations bao1 () and Cosmic Microwave Background observations cmb1 (); cmb2 (). More recently, data from direct measurements of the Hubble parameter has also shown to be a useful probe of low redshift evolution of the universe hz1 (); hz2 (); hz3 ().
Dark energy can be modeled by invoking the presence of a cosmological constant model for which the value of the equation of state parameter is cc1 (); cc2 () and this model is consistent with observational data. Einstein’s cosmological constant is attributed to the zeropoint energy of the vacuum, with a constant energy density , a negative pressure with an equation of state given by . Observations do, however, allow a which is different from that of a cosmological constant and has a dynamical nature i.e. it varies with time. The variation with time is achieved by extending the description of barotropic fluid equation of state parameter to be a function of time or the scale factor. A few parameterizations which have been proposed are described in fluid (); reconst () and there are non baroptropic fluids such as the Chaplygin gas in chap1 ().
A slowly varying scalar field have been proposed to be viable substitute for the cosmological constant as the negative equation of state parameters arises naturally in these scenarios. The scalar field models include those based on canonical scalar field matter such as the quintessence quin (); quin2 (); quin3 (); quin4 (); quin5 (); quin6 (); quin7 (), kinetic energy driven kessence kess1 (); kess2 () and others like tachyon Padmanabhan:2002cp (); Bagla:2002yn (). There is at present no consensus as to which of these models better describe dark energy. Although proposed to do away with the fine tuning problem of the cosmological constant, the scalar field models have fine tuning of their own. The potential parameters need to be fine tuned such that the acceleration of the universe begins after a sufficiently long matter dominated era in order that the large scale structures form. Among the models listed above, the quintessence model is described by a canonical scalar field. For a slowly varying field, the scalar field potential dominated universe has a positive acceleration.
Since a large amount of data is now available and there is a large variety of observations, it is possible to constrain cosmological parameters to to better precision than before. This is especially true of observations which have constraints orthogonal to each other and hence the combined range is much smaller than those by individual datasets. Many observations such as supernovae data riess:1998cb (); perlmutter:1998np (); snia3 (); snia4 (); snia5 (); snia6 (); snia7 (); snia8 (); snia9 (), Baryonic Acoustic oscillation(BAO) data bao1 (); bao2 (); bao3 (); bao4 (); bao5 (); bao6 (); bao7 () and Hubble distance (H(z)) measurements hz1 (); hz2 (); hz3 () have been employed to determine these cosmological parameters.
In this work, we revisit the quintessence dynamics in the light of recent and more diverse cosmological observations. We consider different quintessence scenarios, with different scalar field potentials. These different scenarios have been broadly classified as thawing and freezing type review (); Pantazis (); linder2008 (); caldwell2005 (); huterer2006 (); linder2006 (). This broad classification is based on the whether the equation of state parameters is cosmological constant like in the past, or if this behavior is at later time. While it may be expected that the equation of state parameter can be effectively constrained by assuming dark energy to be a fluid, it is important to explicitly study different scalar field models. The equation of state parameter depends on the time evolution of the scalar field and the functional form of the scalar field potential. In this work, we determine constraints on the equation of state parameters for different thawing scalar field models. Earlier similarly motivated studies include watson2003 (); scherrer2007 (); dutta2011 (); chang2016 (); steinhardt1999 (); Macorra2000 (); Nunes2001 (); corasaniti2003 (); Slepian:2013ug (). In a recent study, it is reported that scalar fields provides better constraints than that of CDM model, but the difference is not significant Ooba:2018dzf ().
This paper is structured as follows. After introduction in section I, in section II, we discuss cosmological equations for quintessence scalar field model. In section III, we show the solutions of the equations for different potentials. The key results are discussed in section IV and we summarize and conclude in section V.
Potential  Parameter  Lower Limit  Upper Limit 

0.01  0.6  
1.0  1.0  
0.01  190.0  
0.01  0.6  
1.0  1.0  
1.0  10.0 
Ii Quintessence Dynamics
We consider a scalar field minimally coupled, i.e. experiencing gravity passively through the spacetime curvature and a selfinteraction described by the scalar field potential V() and with a canonical kinetic energy contribution. The action for a quintessence field is therefore given by
(1) 
where
(2) 
In a flat Friedmann background, the pressure and energy density of a homogeneous scalar field are given by
(3)  
The equation of state, which is in general is time varying, is defined as
(4) 
The equation of motion for the scalar field, the KleinGordon equation
(5) 
follows from functional variation of the Lagrangian and is interchangeable with the continuity equation.
Data set  confidence  Best Fit Model  

SNIa  1.0 0.63  =0.97  
0.01 0.31  563.42  =0.24  
0.01 1.0  =0.03  
BAO  1.0 0.85  =0.99  
0.26 0.31  2.35  =0.28  
0.01 1.0  =0.07  
H(z)  1.0 0.14  =1.0  
0.19 0.32  17.04  =0.26  
0.01 1.0  =1.0 
The evolution of a spatially flat universe is described by Friedmann equations,
(6) 
(7) 
where is the Hubble parameter, and denote the total energy density and pressure of all the components present in the universe at a given epoch. Using Eq. 3 yields
(8) 
(9) 
For an accelerating universe . This implies that one requires an almost flat potential for an accelerated expansion The equation of state for the scalar field is given by
(10) 
Depending on the evolution of , different quintessence models are classified into two broad categories scherrer2007 (); dutta2011 (); steinhardt1999 (); Gupta2014 (); Chiba2009 (); scherrer2005 (); schimd2006 (); Sahlen2006 (); Chiba2005 (). The first corresponds to thawing models, in which the field is nearly frozen by a Hubble damping during the early cosmological epoch and it starts to evolve at late times. Here the field is displaced from its frozen value recently, when it starts to roll down to the minimum. In this case, the evolution of is characterized by the growth from , at early times the equation of state is 1, but grows less negative with time. We analyze the following concave potentials for thawing behavior Ferreira1997 (); Ferreira1997b (); Kallosh2003 (); Linde1991 (); Linde1994 ().

Exponential potential quin5 (); Halliwell:1986ja (); Barreiro:1999zs (); Rubano:2001su (); Heard:2002dr ():
(11) 
Polynomial (concave) potential :
(12)
For the potential described by a polynomial, we consider . These different values correspond to potentials with different shapes. The other class of potentials consists of a field which was already rolling towards minimum of its potential, prior to the onset of acceleration, but slows down because of the shallowness of the potential at late times and comes to a halt as it begins to dominate the universe. For freezing models, the equation of state parameter approaches . For this work, we will focus on homogeneous scalar field belonging to thawing class.
Iii Solutions to cosmological equations
n  SnIa data  BAO data  H(z) data 

1  
2  
3  
n  SNIa data  BAO data  H(z) data 

1  1.0 0.92  1.0 0.995  1.0 0.1 
0.1 0.29  0.26 0.31  0.19 0.32  
1.0 10.0  1.0 10.0  1.0 10.0  
2  1.0 0.91  1.0 0.996  1.0 0.1 
0.1 0.29  0.26 0.31  0.18 0.32  
1.0 10.0  1.9 10.0  1.0 10.0  
3  1.0 0.91  1.0 0.997  1.0 0.08 
0.1 0.29  0.26 0.31  0.18 0.32  
1.0 10.0  2.8 10.0  1.0 10.0 
In this section, we discuss the background cosmology and numerical solutions for the different types of potentials that we have discussed in the previous section.
iii.1 The exponential potential
To study how the universe evolves in the presence of this potential, we solve the KleinGordon equation, Eq. 5, and Friedmann equations for the scalar field, Eq. 8. In order to solve the equations, we define the following dimensionless variable:
(13) 
The potential, then, takes the form
(14) 
In terms of the new variables, the cosmological equations can be written as
(15)  
where and is the present value of Hubble parameter. For , the initial conditions are given by
(16) 
(17) 
The variables , and are values of nonrelativistic matter density parameter, field and equation of state parameter at some initial time (and represents the present day value of ).
By solving these coupled equations numerically, we solve for and as a function of the scale factor. These values are, then, used to determine the value of equation of state parameter , which in terms of the dimensionless parameters is given by
(18) 
From the above equation we can see that, depending upon the form of potential , lies between and .
To study the evolution of the model, we evolve the system from early time to late time. We plot the results obtained for this potential in Fig. 1. The plot in the first row shows the variation of as a function of scale factor (past to future). The plot on the right shows the behavior of equation of state parameter or the potential as scale factor changes. In the second row, the left plot is for energy density of the field as a function of scale factor and the right figure is the phase plot obtained for the potential.
iii.2 The Polynomial (concave) potential
The second potential of thawing class that we analyzed is a power potential given by
(19) 
The background equations then take the following form:
(20)  
And equation of state becomes
(21) 
The value of for this potential is found to be
(22) 
and the initial value of field velocity, is given by
(23) 
As mentioned earlier, to study this potential, we consider , corresponding to different background evolution.
We plot the results obtained by evolving the system from past to present and then to future for this potential in Fig. 5. In the first row, the left plot shows the variation of as a function of redshift. The next plot shows the behavior of equation of state parameter for the potential with respect to redshift. In the second row, the left plot is for energy density of the field as a function of redshift and the right figure is the phase plot obtained for the polynomial potential.
Iv observational constraints on parameters
In this section, we discuss the results obtained by using the three different cosmological observations in the analysis. For the data analysis we use the minimization technique. The observational data consists of n points of observables(), such as luminosity distance for supernova data or angular diameter distance for the BAO data, at a particular redshift(), along with error associated with the observable (). In this technique, we calculate the same observable quantity () at the same redshift, with the equation state parameter obtained by solving cosmological equations in the presence of scalar field with a particular potential . The measures the goodness of fit i.e., by how much the observational value differs in comparison to theoretically expected value and is defined as
(24) 
We have listed the priors used for the analysis in Table 1.
For the exponential potential, , the free parameters are the dark energy equation of state parameter , matter density parameter and in combination with the present day value of , where is the Plank mass. In Fig. 2, we show the , and confidence contours in plane. Here, and denote the present day value of nonrelativistic matter density parameter and present day dark energy equation of state parameter. The plot on the left is from SNIa data, the plot in the middle is for BAO data and the plot on the right shows the results from H(z) data. To obtain the contours we have marginalized over the entire range of the third parameter . The minimum value of () and the constraints obtained for the parameters are listed in Table 2. BAO data provides the narrowest constraints on and on the upper limit of ; none of the data sets provide a lower limit on . The Hubble data constrains strongly but it allows the regions of within limits, which gives decelerated expansion. Supernovae data allows the maximum range in ; between to and the range of below , and it does not allow for a decelerated expansion of the universe.
In Fig. 3, we present the confidence contours corresponding to , and levels in and plane. Here, we show the results for the range of . We find that the most stringent constraints are provided for BAO data set, and the widest range is allowed for H(z) data and for SNIa data set the range lies between the range provided by other two datasets. In Fig. 4, we show the allowed range of and for different datasets in first row and in second row we show the constraints on versus . The first plot is obtained for SNIa, second plot is obtained for BAO and third plot is the result from H(z) data respectively. The results are consistent with the confidence contours of Fig. 2, this is because the value of depends upon both and .
We now discuss the results obtained for the exponential potential, . This gives us three different potentials, as n takes three values; and . The free parameters in the analysis for each of these potentials are , nonrelativistic matter density parameter and the initial value of the field . The Fig. 6 shows the , and confidence contours in plane. The contours in the first row are obtained from analysis of SNIa data, second row represents plots from BAO data and the third row shows results for H(z) dataset. The contours in first, second and third columns are for , and respectively. The contours in plane are obtained by marginalizing over the third parameter , the initial value of the field. The value of the minimum is listed in Table 3 and the constraints on the parameters are listed in Table 4. Again, the most stringent constraints are provided by BAO data followed by SNIa and H(z) data. The H(z) data also allows models with decelerated expansion for all three values of , within limit. We find that for a dataset, the tightest range is given by the potential corresponding to ; as the value of goes from to , the allowed range for increases for all three datasets. None of the data sets provide a lower limit on and as the value of increases the contours move towards , the cosmological constant model. All the three datasets constrain well, with SNIa giving maximum allowed range for this parameter.
In Fig. 7, we show the allowed range of corresponding to , and confidence regions as a function of field . The scheme of plots is same as in Fig. 6. As the value of increases, the allowed range of values of decreases. This trend is same for the three datasets for all values of . The maximum value is required for a smaller value of . The maximum range is allowed by H(z) data and the narrowest range is provided by BAO data for all values of . The results are consistent with the confidence contours in plane of Fig. 6 as the value of depends upon those parameters (see Eq. (22)). The solid blue region in the middle is the allowed region at level, the hatched lines (red) and the slanted lines (blue) regions represent and regions respectively.
V Summary
In this paper, we present current constraints on thawing models of canonical scalar field models of dark energy. Restricting ourselves to thawing models, we present these results for an exponential potential and polynomial potentials with different exponents. To constrain model parameters, we have taken three different types of observational datasets into account, the most widely used supernova observations, the baryon acoustic oscillation data and data from direct measurements of Hubble parameter. The observations considered here are sensitive to different cosmological quantities and therefore the datasets allow for a small range in variation of scalar field parameters.
For this analysis, we have considered two widely used phenomenological scenarios, the exponential potential and the power law potential where we have considered three integer exponents. The exponential potential has two scalar field parameters, while after fixing the power exponent, the polynomial model reduces to a one parameter potential. In all the models considered in this paper, the most stringent constraints are due to the Baryon Acoustic Oscillation data. While it allows for a moderately large range in the equation of state parameter, the allowed range of the value of the matter density parameter is very strongly limited and hence enables ruling out a large range of parameters. The supernova data shows a correlation between the matter density parameter and the equation of state parameter. The Hubble parameter determination data allows for the largest range in the equation of state parameter and does not rule out nonaccelerating solutions. The observations do not entirely rule out any of the models considered here although the allowed range of parameters is narrow. In general, the models which closely emulate the background evolution of a cosmological constant are preferred by observations. This result is consistent with constraints on fluid models of dark energy and other studies on scalar field dark energy models. While the datasets do limit the range of parameters, it is important to confirm and further tighten the constraints with forthcoming observations. The constraints obtained from purely distance measurement observations can be further used as priors for dark energy studies, especially in studies of structure formation.
Vi acknowledgments
The numerical work in this paper was done on the High Performance Computing facility at IISER Mohali.
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