Consistency relation in power law G-inflation

Consistency relation in power law G-inflation

Sanil Unnikrishnan111Current address: Department of Physics, The LNM Institute of Information Technology, Jaipur-302031, India. Email: sanil@lnmiit.ac.in    and S. Shankaranarayanan
July 13, 2019
Abstract

In the standard inflationary scenario based on a minimally coupled scalar field, canonical or non-canonical, the subluminal propagation of speed of scalar perturbations ensures the following consistency relation: , where is the tensor-to-scalar-ratio and is the spectral index for tensor perturbations. However, recently, it has been demonstrated that this consistency relation could be violated in Galilean inflation models even in the absence of superluminal propagation of scalar perturbations. It is therefore interesting to investigate whether the subluminal propagation of scalar field perturbations impose any bound on the ratio in G-inflation models. In this paper, we derive the consistency relation for a class of G-inflation models that lead to power law inflation. Within these class of models, it turns out that one can have or depending on the model parameters. However, the subluminal propagation of speed of scalar field perturbations, as required by causality, restricts .

Prepared for submission to JCAP

Consistency relation in power law G-inflation


  • School of Physics, Indian Institute of Science Education and Research, Thiruvananthapuram 695016, India

E-mail: sanil@iisertvm.ac.in, shanki@iisertvm.ac.in

Keywords: Inflation, physics of early universe, primordial gravitational waves (theory)

ArXiv ePrint: 1311.0177

 

 

1 Introduction

The inflationary paradigm not only heals the Big Bang theory afflicted with the horizon, flatness and monopole problems [1, 2], its prediction of a nearly scale invariant cosmological perturbations [3] has remarkably been verified by numerous cosmological observations with the recent one being the Cosmic Microwave Background (CMB) observation from the Planck mission [4, 5]. In spite of this, one is yet to identify the primary source of matter field that caused inflation, although numerous viable models have been proposed [6, 7, 8, 9, 10, 11, 12, 13]. In most of these proposed models, inflation is driven by a minimally coupled scalar field. Broadly speaking, all minimally coupled single scalar field models of inflation can be divided into the following three classes:

  1. Canonical scalar field models whose Lagrangian is of the form .

  2. Non-canonical scalar field models [14, 15, 16] in which the Lagrangian is a generic function of the field and the kinetic term .

  3. Galilean models of inflation or G-inflation, also known as kinetic gravity braiding models [18, 19], described by the Lagrangian , where and can be an arbitrary function of and the kinetic term .

In fact, the third case above is the most general class of models describing inflation and contains the other two cases. It should be noted that unlike the first two cases, the Lagrangian of the Galilean field contains the second order derivative of the field . However, it turns out that the resultant equation of motion for the scalar field still remains at the second order as the higher order derivative terms cancel away. It is one of the possible scalar field models in curved space time that contains higher order terms in the Lagrangian, but still maintains a second order equation for both metric and the field [20], similar to the Gauss-Bonnet term in the gravity action [21].

It should be noted that a Galilean field corresponds to those class of scalar field models which are invariant in a Minkowski space time under the Galilean type field transformation, viz, where is a constant and is a constant vector [22, 23]. Note that the transformation, , corresponds to shifting the field derivative by a constant vector similar to the standard Galilean transformation in particle mechanics. One of the type of scalar field model admitting this type of invariance has Lagrangian of the form  [24]. The Lagrangian of the G-inflation field contains the term which can be viewed as the generalization of this type of Galilean interaction, although a generic may not admit invariance under . Nevertheless, these scalar fields are dubbed as Galilean fields.

The three class of models of inflation discussed above can be characterized by observables such as , and , where are the spectral indices for scalar and tensor perturbations, respectively, and is the tensor-to-scalar ratio [25, 26]. For the first two classes of inflation models discussed above, namely, the canonical and the non-canonical scalar field models, it turns out that and satisfy the following consistency relation  [27]. This is, in fact, the consequence of the subluminal propagation111It is also possible to violate the consistency relation in standard canonical inflation models, even in the absence of superluminal propagations, if additional fields generates perturbations [31]. of the scalar field perturbations. In models which lead to superluminal speed of sound222The question of whether or not superluminal propagation of scalar field perturbations violates causality is debated in the literature [29]., one gets , see for instance [28]. However, it is recently demonstrated that for the G-inflation models [19, 30], this consistency relation can be violated even after ensuring that the speed of sound of scalar field perturbations is subluminal. It is therefore interesting to investigate whether the subluminal propagation of scalar field perturbations in G-inflation models put any upper limit on the ratio . For the case of canonical and non-canonical scalar field models of inflation, it is known that the upper limit of the above mentioned ratio is unity. However, it is not known whether such an upper limit exists for G-inflation models.

To address this issue, we consider a restricted class of analytically solvable G-inflation model in which the term only contributes the potential and and it is independent of the field . This will ensure that the contribution of the k-inflation term is minimal and any violation of the consistency relation can therefore be attributed to the G-inflation term . For this class of model, we obtain the form of the potential which can lead to power law inflation. For power law solution, it is possible to arrive at an exact inflationary consistency relation between and without imposing the slow roll condition. We therefore derive such a consistency relation for the class of G-inflation model driving power law inflation. The limit on the ratio can thus be found by imposing the restriction that the speed of sound for scalar field perturbations is subluminal.

This paper is organised as follows. In Sec. 2, all the basic equations describing the field dynamics of a generic G-inflation model in a spatially flat universe is discussed. In Sec. 3, we introduce a specific class of G-inflation model which can drive power law inflation. The inflationary consistency relation for such a power law model is derived Sec. 4. In Sec. 5, the observationally viable Galilean inflationary scenario is compared with those based on non-canonical scalar field settings. Finally, the main results of this paper are highlighted in Sec. 6. The derivation of the expression for the speed of sound in G-inflation models is described in Appendix A. Throughout this paper, we shall adopt the metric signature of and we express every equations in natural units thereby setting . In such units the reduced Planck mass is defined as .

2 G-Inflation Preliminaries

We consider the following Einstein-Hilbert action with a Galilean scalar field:

(2.1)

where

(2.2)

is the Lagrangian density of the G-inflation field [19, 18]. In the above Lagrangian the function and can, in general, be an arbitrary function of the field and the kinetic term . The form of the Lagrangian (2.2) takes care of almost all minimally coupled scalar field models of inflation. When , the model represent non-canonical scalar field inflation also known as k-inflation333The class of models with are known as k-inflation or kinetic inflation since in some of these models, first introduced in Ref. [14, 15], it is the kinetic term in the Lagrangian which drives inflation. Therefore, in the k-inflation models described in Ref. [14, 15], as . However, not all non-canonical models satisfies this criteria, see for instance Refs. [40, 41, 16]. Nevertheless, a generic model with , may be referred as either k-inflation or as non-canonical model of inflation. [14, 15] and in addition if it reduces to the standard canonical scalar field model of inflation.

The Lagrangian (2.2) contains the second derivative of the field . After removing the boundary term one gets the following equivalent Lagrangian density [18]:

(2.3)

where the notations such as denotes . Note that when , the above Lagrangian is a function only of and , and hence in this scenario it is equivalent to a k-inflation model. However, when , the Lagrangian (2.2) contains the second order derivative in after removing the boundary term and hence, in this case the model is phenomenologically distinct from the k-inflation models. We will be considering such a case where .

From the action (2.1), the field equation for is given by

(2.4)

On substituting the Lagrangian density (2.2) in the above equation, we get

(2.5)

Note that one gets the same equation of motion if instead of from Eq. (2.2) one substitutes the equivalent Lagrangian (2.3) in Eq. (2.4). It is also important to note that although we started with the action (2.1) in which the field is minimally coupled to gravity, the resulting field equation contains a term indicating a coupling between the Ricci tensor and the kinetic term. It is for this reason, these class of models are also known as kinetic gravity braiding models [18].

On varying the action (2.1) with respect to the metric gives the Einstein’s equation , where the energy momentum tensor is defined as

(2.6)

Substituting from Eq. (2.2) or from Eq. (2.3) in the above equation gives

(2.7)

Considering a spatially flat Friedmann-Robertson-Walker (FRW) universe described by the line element

(2.8)

the above expression (2.7) for the energy momentum tensor takes the diagonal form:

(2.9)

where the energy density and the pressure are given by

(2.10)
(2.11)

In the above two equations, , where is the scale factor. The Einstein’s equation implies that the scale factor satisfies the following Friedmann equations:

(2.12)
(2.13)

Note that the expression for the energy density , as described in Eq. (2.10), contains . Therefore, the first Friedmann equation (2.12) describes a quadratic equation for unlike the usual case when one considers canonical or non-canonical scalar field models. To ensure that is real and positive definite, it is necessary that the following conditions are satisfied:

(2.14)
(2.15)

With the above condition, the Friedmann equation (2.12) becomes

(2.16)

Moving on to the field equation for as described in Eq. (2.5), notice that because of the term , the equation of motion for contains terms proportional to . This can be eliminated using the two Friedmann equations (2.12) and (2.13). The equation of motion for can then be expressed as

(2.17)

where

(2.18)

Eqs. (2.16) and (2.17) forms the two closed set of equations describing the evolution of and . For the case of canonical scalar field which corresponds to choosing and , the field equation (2.17) reduces to the standard Klein-Gordon equation viz. .

3 Power law G-Inflation

Our aim in this paper is to derive an exact consistency relation in G-inflation models without assuming slow roll. It is possible to do so in the case of power law inflation for which one can obtain an exact analytical expression for and . For the case of kinetic power law inflation, such an exact consistency relation is derived in Ref. [14]. It follows from the analysis of Ref. [14] that when the speed of sound is subluminal.

Although, G-inflation corresponds to a wider class of models with generic and in the Lagrangian (2.2), to understand the exact reason for the violation of the consistency relation as noted in Ref. [30], it is necessary to minimize the contribution from the k-inflation term . If we eliminate the contribution of the altogether, and chose , then it lead to thereby making the system violently unstable. This also happens when and with an integer value for . For this reason we restrict ourself to a subclass of these models with and . The Lagrangian of this restricted class of G-inflation model considered in this paper is therefore given by:

(3.1)

where and are parameters of the model with being dimensionless and has dimensions of mass. We will now obtain the form of the potential which can drive power law inflation wherein the scale factor evolves as

For model (3.1), the energy density and the pressure turns out to be:

(3.2)
(3.3)

When , the Friedmann equations (2.12) and (2.13) implies that

(3.4)
(3.5)

where . From Eqs. (3.2) to (3.5), it follows that

(3.6)
(3.7)

It can be verified that Eq. (3.6) admit solution of the form:

(3.8)

and

(3.9)

Substituting the solution (3.8) in Eq. (3.7), we get the following form of the potential:

(3.10)

and

(3.11)

In the model (3.1) with the above form of the potential, although tedious, it is straight forward to verify that the solution (3.8) with satisfy both the Friedmann’s equations (2.12), (2.13) and the scalar field equation (2.17). Hence, in the Lagrangian (3.1), an inverse power law potential of the form (3.10) can drive power law inflation with . It is interesting to note that such inverse power law potentials also drive power law inflation in non-canonical scalar field models, see for instance power law models described in Refs. [14, 15, 32]. However, in the case canonical scalar field driven inflation, an inverse power law potential leads to intermediate inflation [33, 34].

4 Consistency relation in Galilean Power law inflation

In this section we shall derive the consistency relation for the Galilean power law inflation model described in the preceding section. However, before moving on the specific model (3.1), let us first consider the generic G-inflation scenario. To obtain the scalar and tensor perturbations generated by the inflation field, we consider the following FRW line element with metric perturbations [35, 36, 37]

where , , and are scalar degree of metric perturbation and is the tensor perturbations. The vector perturbations are ignored as it is known that scalar fields do not lead to such perturbations. The perturbation in the scalar field is defined as

(4.1)

where is the background field which, for the G-inflation case, satisfies Eq. (2.17). The perturbation being a gauge dependent quantity, one generally introduce the following gauge invariant quantity known as curvature perturbation:

(4.2)

In the generic model with the Lagrangian (2.2), the second order action for the curvature perturbation turns out to be [19, 18, 38]

(4.3)

where is the conformal time and is the speed of sound for the G-inflation field whose square is given by

(4.4)

In the action (4.3), the function is defined as

(4.5)

where

(4.6)

From the action (4.3), one gets the following equation of motion for the curvature perturbation

(4.7)

where is the amplitude of the curvature perturbation in the Fourier space and is the wavenumber. Note that just like in the case of canonical or non-canonical scalar field driven inflation, the curvature perturbation is also conserved at the superhorizon scales in G-inflation models, for a proof see Ref. [39].

In terms of the Mukhanov-Sasaki variable , Eq. (4.7) becomes:

(4.8)

This is exactly identical to the corresponding equation for the k-inflation field [14], the difference being and in the above equation are different from those that appear in the Mukhanov-Sasaki equation in k-inflation models (see Eq. (28) in Ref. [14]).

Similarly, for tensor perturbations one gets the following equation:

(4.9)

where , with being the Fourier amplitude of tensor perturbations. Unlike the case of Mukhanov-Sasaki equation for the scalar variable , the above equation is identically valid for all minimally coupled scalar field models of inflation since tensor perturbations evolves independent of the scalar perturbations at the linear order.

The scalar and tensor power spectrum are defined as

(4.10)
(4.11)

Furthermore, one defines the scalar and tensor spectral index as:

(4.12)
(4.13)

And finally the tensor-to-scalar ratio is defined as

(4.14)

It is clear from Eq. (4.8) that it is the function which can lead to a different evolution for the mode function in a G-inflation model from those in a k-inflation model which has the same value for the speed of sound for scalar perturbations. It is therefore illustrative to express the function defined in Eq. (4.5) as

(4.15)

where , defined as

(4.16)

is the one that appears in the Mukhanov-Sassaki equation for the k-inflation field [14] and is defined as

(4.17)

where is defined in Eq. (4.6) and is given by

(4.18)

Note that in the case of k-inflation which corresponds to setting , in the Lagrangian (2.2), one gets . Hence, it is because of this factor in Eq. (4.15) that makes the scalar power spectrum in G-inflation different from those in an equivalent k-inflation model which leads to the same background evolution and has the same value for . Since Eq. (4.9) is valid for all minimally coupled scalar field models of inflation, the tensor power spectrum in G-inflation is exactly the same as those in an equivalent k-inflation model which leads to the same background evolution. It is for this reason the tensor-to-scalar ratio in G-inflation will be different from those in an equivalent k-inflation model. This point will be illustrated in detail for the power law model considered in the preceding section

For the restricted class of G-inflation model (3.1) with an inverse power law potential (3.10) driving power law inflation with , it follows from Eqs. (3.8) and (4.4) that

(4.19)

where is the equation of state parameter which is related to parameter in the power law solution as . It is clear from Eq. (4.19) that the speed of sound is identically constant for the Galilean power law inflation model (3.1) and in the slow roll limit which corresponds to or equivalently , one gets . When , the slow roll value of is and this is consistent with Ref. [30] which considered Higgs G-inflation. In the left panel of Fig. 1, is plotted as a function of . Note that the speed of sound for the Galilean model is subluminal [51]. Furthermore, the solution (3.8) also implies that the parameter defined in Eq. (4.17) is a constant and is given by

(4.20)

The parameter is plotted as a function of in the right panel of Fig. 1. In the slow roll limit which corresponds to , the parameter irrespective of the value of in the Lagrangian (3.1).

Figure 1: In the left panel, for the Galilean power law inflation model, as given in Eq. (4.19), is plotted as a function of equation of state parameter for different values of in the Lagrangian (3.1). In the right panel given in Eq. (4.20) is plotted as a function of . Note that both and are positive definite and they are less than unity. In this figure, the range of is from to , although for inflation one only need to consider the regime where .

For the power law solution , the function defined in Eq. (4.15) becomes

(4.21)

With given by the above expression, the Mukhanov-Sasaki equation (4.8) can be expressed as

(4.22)

where

The general solution of the above equation can be expressed as

where and are constants of integration, and are Hankel functions of first and second kind, respectively. On imposing the Bunch-Davis initial condition that at the sub-horizon scales leads to and the above solution for the mode function becomes

(4.23)

Similarly for the tensor perturbations, the solution of the Eq. (4.9), satisfying the Bunch-Davis initial condition turns out to be identical to , except for tensor perturbations . Therefore,

(4.24)

From the above solutions for mode functions and , the scalar and tensor power spectrum defined in Eqs. (4.10) and (4.11), respectively, at the super horizon scales turns out to be

(4.25)
(4.26)

where

(4.27)
(4.28)

In the above equations is Gamma function and is a constant that appears in the equation describing the evolution of scalar factor in conformal time, viz. The value of this parameter in Eqs. (4.27) and (4.28) can be fixed using the CMB normalization, namely, at the pivot scale  [4]. For and , it turns out that . Since implies that , it turns out that . Let be the time at which the pivot scale exit the cosmological horizon (), which in turn implies that

Therefore, the CMB normalized value of can be used to determined the value of the Hubble parameter when the pivot scale exit the cosmological horizon. For and , we find that .

From Eqs. (4.25) and (4.26), the scalar and tensor spectral index, defined in Eqs. (4.12) and (4.13), respectively, turns out to be

(4.29)

This is exactly the same for the case of standard power law inflation driven by a canonical scalar field with an exponential potential [44, 45, 46] and one also gets the same and in the power law scenario in some non-canonical scalar field models such as those discussed in Refs. [14, 15, 32]. In fact, Eq. (4.29) is valid for any model of power law inflation based on the Lagrangian (2.2) but for which the parameters and defined in Eqs. (4.4) and (4.17), respectively, are identically constant. This simply follows from the fact that whenever and are constant, the solution (4.23) for the mode function satisfy the Mukhanov Sasaki equation (4.8) during power law expansion and the resultant (and ) is the consequence of this solution for (and ).

Eqs. (4.25) and (4.26) lead to the following tensor-to-scalar ratio

(4.30)

It is evident from the above expression that unlike the spectral indices and which depends only on the value of in the power law solution , the tensor-to-scalar ratio also depends on the model parameters and defined in Eqs. (4.4) and (4.17), respectively. Therefore, contains the details of the dynamics of inflation and it can play an important role in distinguishing models of inflation [47].

Eqs. (4.29) and (4.30) imply the following consistency relation

(4.31)

where the expression for and for the Galilean power law inflation model (3.1) are given by Eqs. (4.19) and (4.20), respectively. Note that the above consistency relation is an exact result since no slow roll approximation is imposed.

In Fig. 2, the ratios and are plotted as a function of slow roll parameter for different models of power law inflation. In this figure, relation (4.31) is used for the Galilean power law model while for the power law scenario in k-inflation [15] and in the non-canonical scalar field model [32], the same relation (4.31) is used but with . This is justified from the analysis of Ref. [14]. For the power law inflation driven by a canonical scalar field  [48] which leads to the following exact consistency relation .

Figure 2: In the left panel, the ratio is plotted as function of the slow roll parameter for the following four power law inflation models: Galilean PLI: Power law inflation in Galilean model (3.1) with and , C PLI: Canonical scalar field driven power law inflation, NC PLI: A non-canonical scalar field model of power law inflation describe by the Lagrangian (5.1) with and Kinetic PLI: A kinetic driven power law inflation model [14, 15] where and . In the right panel, the ratio is plotted for the same set of power law inflation models. Note that, in the slow roll limit, the ratio approaches unity for canonical, kinetic and non-canonical models whereas, for the Galilean model of power law inflation, this ratio approaches irrespective of the value of in the Lagrangian (3.1).

In the slow roll limit which corresponds to , the consistency relation (4.31) reduces to

(4.32)

In comparison, for the case of k-inflation, one gets . Therefore, as mentioned earlier, it is the parameter defined in Eq. (4.17) which alters the consistency relation in G-inflation. Note that, although, the consistency relation (4.32) was derived for the power law inflation, it is approximately valid, in the slow roll limit, for a generic G-inflation model with Lagrangian (2.2). For the power law inflation driven by a Galilean field, the slow roll regime of the consistency relation (4.32) can be re-expressed as

(4.33)

When , the above relation becomes consistent with those derived in Ref. [30]. For any value of , Eq. (4.33) can also be expressed as , where is the slow roll parameter, which for the power law solution turns out to be . Although, we considered the Galilean power law inflation model (3.1) for which , the expression is also valid in case of Higgs G-inflation model where as described in Ref. [49]. However, recall that the expression is slow roll limit of the exact relation (4.31).

When , Eq. (4.33) implies that even though the speed of sound is subluminal . This does not arise in the case of canonical and non-canonical scalar field models of inflation for which when  [14]. This leads us to ask the following important question: What is the physical reason behind the violation of the standard consistency relation in G-inflation models ? To go about answering this question let us re-write the scalar power spectrum (4.25) as

(4.34)

where is the scalar power spectrum that one gets in an equivalent non-canonical model of inflation which leads to the same background evolution, viz and for which the value of is the same as given in Eq.(4.19). Note that the expression for the tensor power spectrum (4.26) remains unchanged for an equivalent non-canonical scalar model, as tensor perturbations do not directly couple with scalar perturbations. In the slow roll limit the expression (4.20) implies that . therefore in the slow roll limit Eq. (4.34) implies that . This means that the scalar power spectrum in G-inflation models is suppressed by a factor as compared to the same in an equivalent k-inflation scenario and consequently this enhances the tensor-to-scalar ratio. It is because of this enhancement of tensor-to-scalar ratio, one can get in G-inflation models even when .

Note that for integer value of , it follows from Eq. (4.33) that only when . For one gets back the standard consistency relation as in the case of inflation driven by a canonical or non-canonical scalar field.

From Eqs. (4.19), (4.20) and (4.31), it follows that, if we restrict , then

(4.35)

Therefore, the upper bound on the ratio for the Galilean power law model (3.1) is , when the speed of sound for the scalar field perturbations is restricted to be subluminal. This is the main result of this paper. In comparison, for all canonical and non-canonical scalar field models of inflation, the upper bound on the ratio is unity when .

Although the expression (4.35) was obtained for a restricted class of G-inflation model (3.1) driving power law inflation, it may be worthwhile to investigate whether the inequality holds for a generic G-inflation model with the Lagrangian (2.2) but for which