Dark Energy Parametrization motivated by Scalar Field Dynamics
We propose a new Dark Energy parametrization based on the dynamics of a scalar field. We use an equation of state , with , the ratio of kinetic energy and potential . The eq. of motion gives and has a solution where and . The resulting EoS is . Since the universe is accelerating at present time we use the slow roll approximation in which case we have and . However, the derivation of is exact and has no approximation. By choosing an appropriate ansatz for we obtain a wide class of behavior for the evolution of Dark Energy without the need to specify the potential . The EoS can either grow and later decrease, or other way around, as a function of redshift and it is constraint between as for any canonical scalar field with only gravitational interaction. To determine the dynamics of Dark Energy we calculate the background evolution and its perturbations, since they are important to discriminate between different DE models. Our parametrization follows closely the dynamics of a scalar field scalar fields and the function allow us to connect it with the potential of the scalar field .
In the last years the study of our universe has received a great deal of attention since on the one hand fundamental theoretical questions remain unanswered and on the other hand we have now the opportunity to measure the cosmological parameters with an extraordinary precision. Existing observational experiments involve measurement on CMB planck (); wmap9 () or large scale structure ”LSS” LSS () or supernovae SNIa SN (), and new proposals are carried out new ().
Taking a flat universe dominated at present time by matter and Dark Energy ”DE”, and using a constant equation of state for DE, one finds , with from WMAP9 results wmap9 () and , with from Planck planck (), SNLS SN () and BAO LSS () measurements. The constraint on curvature is for WMAP9 wmap9 () and for Planck planck () using a model, i.e. for DE. At present time, the equation of state ”EoS” of DE depends on the priors, choice of parameters and on the data used for the analysis as can be seen from the results obtained by either WMAP or Planck collaboration groups planck (); wmap9 (). A more precise determination of the EoS of DE will be carried out in new () which together with precise measurements of CMB such as planck (); wmap9 () will yield a better understanding of the dynamics of Dark Energy. With better data we should be able to study in more detail the nature of Dark Energy, a topic of major interest in the field DE.rev (). Since the properties of Dark Energy are still under investigation, different DE parametrization have been proposed to help discern on the dynamics of DE DEparam ()-quint.ax (). Some of these DE parametrization have the advantage of having a reduced number of parameters, but they may lack a physical motivation and may also be too restrictive. The evolution of DE background may not be enough to distinguish between different DE models and therefore the perturbations of DE may be fundamental to differentiate between them.
Perhaps the best physically motivated candidates for Dark Energy are scalar fields, which can interact only via gravity SF.Peebles (); tracker (); quint.ax () or interact weakly with other fluids, e.g. Interacting Dark Energy ”IDE” modelsIDE (); IDE.ax (). In this work we will concentrate on canonically normalized scalar fields minimally coupled to gravity. Scalar fields have been widely studied in the literature SF.Peebles (); tracker (); quint.ax () and special interest was devoted to tracker fields tracker (), since in this case the behavior of the scalar field is weakly dependent on the initial conditions set at an early epoch, well before matter-radiation equality. In this class of models a fundamental question is why DE is relevant now, also called the coincidence problem, and this can be understand by the insensitivity of the late time dynamics on the initial conditions of . However, tracker fields may not give the correct phenomenology since the have a large value of at present time. We are more interested at this stage to work from present day redshift to larger values of , in the region where DE and its perturbations are most relevant. Interesting models for DE and DM have been proposed using gauge groups, similar to QCD in particle physics, and have been studied to understand the nature of Dark Energy GDE.ax () and also Dark Matter GDM.ax ().
Here we propose a new DE parametrization based on scalar fields dynamics, but the parametrization of can be used without the connection to scalar fields. This parametrization has a rich structure that allows to have different evolutions, it may grow and later decrease or other way around. We also determine the perturbations of DE which together with the evolution of the homogenous part can single out the nature of DE. With the underlying connection between the evolution of and the dynamics of scalar field we could determine the potential . The same motivation of parameterizing the evolution of scalar field was presented in an interesting paper huang (). We share the same motivation but we follow a different path. We have the same number of parameters but a richer structure and it is easier to obtain information of the scalar potential .
We organize the work as follows: in Sec.I.1 we give a brief overview of our DE parametrization. In Sec.II we present the scalar field and we define the variables used in this work. In Sec.III we present the dynamics of a scalar field and the set up for our DE parametrization given in Sec.IV. We calculate the DE perturbations in Sec.V and finally we conclude in Sec.VI.
We present here an overview of our parametrization. The EoS is
with the ratio of kinetic energy and potential . This corresponds to a canonically normalized scalar field. The equation of motion of the scalar field gives (c.f. eq.(21)),
where the ratio of matter density and , with . Eq.(2) is an exact equation and is valid for any fluid evolution and/or for an arbitrary potential . In terms of and the EoS is
which we consider our master equation and it is valid for any value of and and not only in the slow roll regime.
The aim of our proposed parametrization for is to cover a wide range of DE behavior. Of course other interesting parameterizations are possible. The dynamics of scalar fields with canonical kinetic terms has an EoS constrained between and gives an accelerating universe only if or to a constant with quint.ax () (for an EoS one needs ). From the dynamics of scalar fields we know that the evolution of close to present time is model dependent. For example, in the case of , used as a model of DE derived from gauge theory GDE.ax (), the shape of close to present time depends on the initial conditions and it may grow or decrease as a function of redshift . We also know that tracker fields are attractor solutions but in most cases they do not give a negative enough tracker (). In this class of models the EoS, regardless of its initial value, goes to a period of kinetic domination where and later has a steep transition to , which may be close to present time, and finally it grows to in a very model and initial condition dependent way. Furthermore, if instead of having a single potential term we have two competing terms close to present time, the evolution of would even be more complicated. Therefore, instead of deriving the potential from theoretical models as in GDE.ax () we propose to use an ansatz for the functions and which on one hand should cover a wide range of DE behavior with the least number of parameters but without sacrificing generality, and on the other hand we would like to have this ansatz as close as possible to the known scalar field dynamics. We believe that using our model will greatly simplify the extraction of DE dynamics from the future observational data. We propose therefore the ansatz (c.f. eqs.(75) and (77) )
where and are free parameters giving at present time (a subscript represents present time quantities) and at early times, is a function that goes from at to at large . The parameter sets the transition redshift between and while sets the steepness of the transition and takes only two values or (see sect.(IV)).
Since the universe is accelerating at present time we may take the slow roll approximation where and . In this case one has . However, the derivation of in eq.(2) is exact and has no approximation. We will show in section IV that can have a wide range of behavior and in particular can decrease and later increase as a function of redshift and vice versa, therefore the shape and steepness are not predetermined by the choice of our parametrization. Of course we could use other parametrizations for since the evolution of and in eqs.(1) and (2) are fully valid. There is also no need to have any reference to the underlying scalar field dynamics, i.e. our parametrization is not constrained to scalar field dynamics. However, it is when we interpret and as: , and , that we connect the evolution of to the scalar potential .
Finally DE perturbations are important in distinguishing between different DE models hu ()-DE.p () and we will show that a steep transition of has a bump in the adiabatic sound speed which could be detected in large scale structure DE.p.sf (); DE.p (). At an epoch where the universe is dominated by DM and DE the total perturbation and if is not much smaller than , i.e. for low redshift , then the evolution of may have an important contribution to , as discussed in Sec.V.
Ii Scalar Field Dynamics
We are interested in obtaining a new DE parametrization inferred from scalar fields SF.Peebles ()-IDE.ax (). Since it is derived from the dynamics of a scalar field we can also determine its perturbations which are relevant in large scale structure formation. We start with a FRW metric with a line element
and a canonically normalized scalar field with a potential and minimally coupled to gravity. The homogenous part of has an equation of motion
where , is the Hubble constant, is the scale factor and a dot represents derivative with respect to time . Since we are interested in the epoch for small redshift z, with , we only need to consider matter and DE and we have
in natural units . The energy density and pressure for the scalar field are
We define the ratio of kinetic energy and potential energy as
and the equation of state parameter ”EoS” becomes
The value of determines or inverting eq.(11) we have . Since the Dark Energy EoS is in the range . For growing the EoS becomes larger and at one has while a decreasing has for . In terms of we have
where we defined the ratio of and scalar potential as
Finally we can express and in terms of and as
The equation of motion for is
and we can write
which has a simple solution
If we multiply in eq.(22) the numerator and denominator by we obtain an alternative and useful expression for the EoS
which we consider our master expression for . It is a relatively simple expression but more important is that eqs.(22) and (23) are exact and therefore valid for any values of and and not only at a slow roll regime. We will see in the next section some physically motivated limits of eq.(23). Finally, inverting the expression of we can obtain as a function of and
and as a function of and
Dynamics and Limits for
Using eq.(20) we can express
and it is an exact equation, however depends on through eq.(21). If we take , valid in the slow roll approximation, one has for arbitrary values of and
and as we have and . Notice that
and we then expect that increases as a function of redshift z when matter dominates over Dark Energy and with at present time. We expect then that will be satisfied beyond the slow roll approximation and eq.(28) is valid for a wide range of the parameter’s values.
If we take with constant in eq.(23) we get the limit
and for constant with we have and also constants given by
Clearly depending on the value of we can have a decreasing or increasing and as a function of redshift. For example for one has while for one requires at large .
Since is only a function of we have or as a function of redshift
The sign of depends then only on the sign of given by
Clearly is positive definite while . In general we can assume that DE redshifts slower than matter, at least for small z , since DE has and matter , so is a growing function of z, i.e. and . Therefore if is negative we have and a decreasing and as a function of redshift. However, if is positive then the sign of is positive and may grow or decrease depending on the magnitude of compared to .
Slow Roll Approximation
The evolution of as a function of and is given in eq.(21) and here we will show some phenomenological interesting limits. Since the field should be responsible for accelerating the universe we know that must be close to at present time, and the field must satisfy the slow roll approximation. In the slow roll approximation one has and we then have
with and the function becomes
We name a full slow roll approximation when (i.e. and ). This full slow roll approximation is more suitable in inflation where one has a long period of inflation, however Dark Energy does dominate only recently and so we do not expect this approximation to hold for a long period of time and should not be taken as a working hypothesis for DE parametrization.
Late time attractor Solution
The evolution of scalar fields has been studied in quint.ax () and a late time attractor for an accelerating universe requires with . In the limit with one has
Notice that eq.(28) reduces to eq.(39) in the limit with , and therefore eq.(28) generalizes eq.(39). For large redhsift z we expect to increase and eq.(39) would not longer be valid and we should take instead eq.(28).
Iii Dynamical Evolution
Differentiating and w.r.t. time or equivalently as a function , where is the scale factor, we get the evolution of and . Using the definition of in eq.(18), , and for any function , we get
We see that eqs.(41) and (42) are uniquely determined by a single function . The critical points for , i.e. , have or while is satisfied for and . The case, has and which gives a solution , a constant and the limits and . In the case the EoS becomes and it will take different constant values. Setting constant in eq.(42) we get with constant one has a solution
For we have , and . Finally, we can satisfy for or giving and constant (c.f. eq.(43)).
Therefore, having an increasing or decreasing depends on the sign of and it can vary as a function on time depending on the values of and , i.e. on the choice of the potential . If we take what we call a full slow roll defined by and then eq.(41) becomes
which is positive definite, i.e. . Using we can express eq.(44) giving a solution
Therefore if the condition or is satisfied, then from eq.(45) we have a decreasing function for as a function of redshift z and therefore also decreases. However, we do not expect the universe to be in a full slow roll regime and when is small, e.g. one has , the slow roll condition does not imply that . Therefore, the sign of can be positive or negative depending on the sign and size of compared to and can either grow or decrease. In the region where we have while for we have . The value of parameterizes then the amount of slow roll of the potential and a full slow roll has but we expect to be only in an approximate slow roll regime with and . We discuss the dynamics of in section V. In the present work we do not want to study the critical points of the dynamical equations but the evolution of close to present time when the universe is accelerating with close to zero ( close to -1) but not exactly zero with and
iii.1 Dynamical evolution of and
From eqs.(III.1) we clearly see that the behavior of and is completely determined by a single function and we define
In the case where the scalar field rolls down its minimum we will have negative giving a . In the simple case of having constant the solution to eqs.(III.1) are simply
with initial conditions at . The asymptotic values of and depend on the value of . The critical points of the system in eqs.(III.1) give for a solution
and if we invert eq.(51) we simply get
and an EoS
which is valid for constant. We clearly see that the value of determines the asymptotic behavior of and of the scalar field energy density . If the scalar field will dominate since and we will have and , while for we have and constant given in eq.(59). Finally for we get and with . Of course an accelerated universe requires and therefore close to present time. However, in general the value of will be time (or ) dependent.
as a function of and
The function is a function of and , as we can see form eq.(47) and we can express in terms of as
(for a rolling scalar field but we have kept the term for completeness),
and the eqs.(III.1) read now
The amount of can be determined at the critical point giving (i.e. the scalar field redshifts as the barotropic fluid which in our case is matter ) and
This solution is valid for and has a . The critical point for has
iii.2 Dynamical System for and
As a matter of completeness we present here also the dynamical evolution equations in term of the variables and which have been widely used in the literature for studying the dynamics of scalar fields quint.ax (). In this case the dynamical equations are given by n
we can write
and therefore with as in eq.(52). On the other hand if then one finds
with and we recover as in eq.(52). An universe with a late time acceleration has then which corresponds to the slow roll approximation.
iii.3 Summary on , and
We have defined three different quantities , and given in eqs.(19) and (47). Both and determine uniquely the set of differential equations eqs.(III.1) and (III.2), respectively. The solution to these equations then depend on the initial conditions and on the functional form of or as a function of . These three quantities are related by eqs.(19) and (48)
and in the slow roll approximation they coincide since . Therefore all three quantities are equivalent in the full slow roll approximation and differ slightly once the evolution of does not obey it any more. All of them have advantages and can be used to determine the evolution of uniquely.
If we determine and as a function of the scale factor , then we can extract as a function of the scale factor using and . We get
with . From eq.(71) we have and as a function of the scale factor and we can then determine as a function of .
We have studied the critical points for or for constant or , respectively, which give the asymptotic behavior of the system. However, we do not expect in general to have (or ) constant and we would need to obtain the dynamical equation of motion for (). This can be easily done by taking for example the derivative , i.e. , and we can use the equation of motion of the scalar field to determine the rhs of the equation of . This is not a closed system and we could take further derivatives of and have an infinite series of equation in which the functions are given in terms of with . In order to have an exact solution for the scalar field we would need to solve the equation of motion of given a potential , but then we would have an exact solution only for the specific potential used.
In principle the quantity may have many free parameters and may be a complicated function since it depends on the potential via and the kinetic term . Instead of using as our free function we prefer to work with since it is only a function of the potential and not of . If we take the derivative of we get
and we will distinguish two different cases for . If we have a rolling scalar field in such a way that only vanishes at infinite value of , i.e. there is no local minimum, then the term
and this ansatz corresponds to a wide class of potentials tracker ()-IDE.ax (). We show in Figs.1a, 1b and 1c the evolution of for an inverse power IPL , an exponential and a sugra potentials, respectively, for different initial conditions and with . Notice that for the same potential we have different evolution of and it may grow or decrease as a function of z and we have a steep transition which also depends on the initial conditions. We show in figure 2a the evolution of for different initial conditions and same , where is the initial condition of . It is clear that and the EoS have an attractor solution but what figures 2b,c show is that the model may not have reached the attractor solution by the time one has and the evolution of and as a function of z depends strongly not only on the value of but also on the initial conditions and . We have used a with (green,red,yellow and blue respectively and with .
Iv Scalar Field DE parametrization
The aim here is to test a wide class of DE models in order to constrain the dynamics of Dark Energy form the observational data. We could parameterize or with a single function in terms of the scale factor (c.f. Sec.IV.3) but as we have seen in eq.(21) we get a better understanding of the evolution of if we parameterize the functions and .
In order to have the evolution of Dark Energy we need to either choose a potential and solve eq.(III.1) or parameterize the EoS or as a function of the scale factor . If we choose to parameterize or then we need to solve the differential equations eq.(III.1) or (III.2). On the other hand gives the evolution of and without needing to solve any differential equations and reduces to (or ) in the slow roll approximation. Therefore we propose here to use the exact equation for given in eq.(23) and parameterize the function and .
A priori it is impossible to know how many free parameters has the potential and since the evolution of different scalar field model requires the solve the eq.(III.1) and one has a difficult task to test a wide range of potentials . The potential may involve a single term, as in a runaway potential as in or , or it may have different terms in the potential with the same order of magnitude as for example or that may lead to a local minimum with .
In all these cases the evolution of would be different but still constrained to , and in the slow roll approximation (