Advances in High Energy Physics

Volume 2015 (2015), Article ID 926807, 11 pages

http://dx.doi.org/10.1155/2015/926807

## Light of Planck-2015 on Noncanonical Inflation

^{1}Islamic Azad University, Sanandaj Branch, Pasdaran Street, P.O. Box 618, Sanandaj, Iran^{2}Young Researchers and Elites Club, Sanandaj Branch, Islamic Azad University, Sanandaj, Iran^{3}Department of Physics, Faculty of Science, University of Kurdistan, Pasdaran Street, Sanandaj 66177-15175, Iran

Received 6 May 2015; Accepted 24 June 2015

Academic Editor: Rong-Gen Cai

Copyright © 2015 Kh. Saaidi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

Slow-roll inflationary scenario is considered in noncanonical scalar field model supposing a power-law function for kinetic term and using two formalisms. In the first approach, the potential is picked out as a power-law function, that is, the most common approach in studying inflation. Hamilton-Jacobi approach is selected as the second formalism, so that the Hubble parameter is introduced as a function of scalar field instead of the potential. Employing the last observational data, the free parameters of the model are constrained, and the predicted form of the potential and attractor behavior of the model are studied in detail.

#### 1. Introduction

Inflationary scenario is the best candidate for describing the very early times evolution of the universe, which could simply solve the problems of the standard hot big bang model. There are lots of observational data supporting this scenario [1]. The scenario was first proposed by American physicist, Guth, in order to solve the problem of standard hot big bang model [2]. So far, different model of inflationary cosmology has been introduced; however it might be said that the most interesting one is chaotic inflation, introduced by Linde in 1983 [3]. Based on this scenario, the universe in the very early times is dominated by a slowly varying homogenous scalar field and undergoes an exponential expansion in very short time.

The scenario has been constructed based on canonical scalar field; however recently the cosmological models of scalar field including noncanonical kinetic term have attracted scientists attention. The general form of its action is expressed by , where [4]. The case with comes into a well-known model as -essence. The main idea of -essence comes from Born-Infeld action of string theory [5, 6]. The model is able to give some interesting results about dark energy [7–12]. In [13, 14], the model is applied as a possible way for inflation and describing early time evolution of the universe.

In the present work, we are going to take and as scalar field potential. In other works, we take a pure kinetic -essence plus a potential term. This kind of model is known as noncanonical scalar field [15]. This case is another class of the general form which could be as important and interesting as -essence model. The cosmological solution of the model has been studied in [15], where it was shown that it is possible to construct a unified model of dark matter and dark energy for a simple form of noncanonical kinetic term . The same case has been considered in [16] as well, in which they found that producing a unified model of dark matter and dark energy for a pure kinetic -essence is very difficult. It seems that the noncanonical scalar field model has this ability to be an appropriate model of the universe evolution and has merit of consideration in more detail. Then, we were motivated to use noncanonical scalar field as a possible model for describing one example of the earliest universe evolution, namely, inflation.

There are two formalisms that cosmologists apply to study inflation, and in the present work, we are going to utilize both these approaches. The first formalism, and the most common and well-known one, relies on the potential. In this approach, we need to propose a specific form of the potential to be able to investigate the model in detail. In this regard, during this work, a familiar form of potential, as power-law, is picked out. The power-law potential, , has a huge contribution in inflationary scenario studies, which includes chaotic inflation. The two most known cases are and . Recent observational results have shown that the stays outside of the joint CL region in the plane. The quadratic potential model, , lies outside the joint CL region for Planck + WP + high- for -folds. The potential models with and lie on the boundary of the joint CL region [17–19]. Another interesting formalism is known as Hamilton-Jacobi formalism, which relies on a specific form of the Hubble parameter in terms of scalar field, instead of introducing a potential [20–24]. In comparison with the first formalism, Hamilton-Jacobi formalism includes some fascinating feature such as deducing a form of the potential, obtaining an exact solution for the scalar field. Although the formalism has got less attention than first formalism, its abilities in inflationary studies sound undeniable. Based on the presented argument, the later formalism stands in the center of our attention in this work.

The main goal of the work is to constrain the free parameters of the model utilizing the latest observational data released by Planck-2015 [19, 25]. Prediction of quantum perturbation is one of interesting features of inflationary scenario. The most important ones of these perturbations are scalar perturbation, which is seeds for large scale structure of the universe, and tensor perturbation, which is known as gravitational waves too [26–28]. The prediction is supported by huge amount of observational data. From Planck data, the amplitude of scalar perturbation is about , and the scalar spectra index, which is equal to one for a scale-invariant spectrum, is measured about [19]. In contrast with scalar perturbation, Planck does not give an exact value for tensor-to-scalar ratio ; it just specifies an upper bound for this parameter as [19].

The paper is organized as the following: the general form of noncanonical scalar field equations, slow-rolling parameters, and the perturbation parameters are derived in Section 2. In Section 3, the problem is studied by using the first formalism, and appropriate values for the free parameters are presented. The second formalism is applied in Section 4, where, as well as constraining the parameters, the predicted potential and attractor behavior of the model are considered.

#### 2. Noncanonical Scalar Field

To begin, we introduced the Lagrangian which is read aswhere is Ricci scalar constructed from the metric and is Planck mass. is the Lagrangian of noncanonical scalar field which is defined as . The kinetic term of noncanonical scalar field is expressed by an arbitrary function , in which . Potential of scalar field is denoted by .

The observation shows that the universe is spatially flat. The common metric to describe such a universe is spatially flat FLRW metric. The dynamical equations are derived by taking variation of action with respect to independent variable as and . Substituting the metric in the field equations leads to well-known Friedmann equations asin which and are, respectively, the energy density and pressure of noncanonical scalar field, expressed byDerivative of kinetic function with respect to is denoted by . And for the acceleration equation we haveTo have a positive acceleration phase for the universe, one must have . On the other hand, the equation of motion of noncanonical scalar field is obtainedwhich is another expression of familiar conservation equation , where and have been introduced in (3).

From now on, we take the kinetic term as a power-law function of ; namely, . The constant is dimensionless, and is a constant whose dimension is fitted in a way to give for kinetic energy density . It could be easily checked that the case and comes to usual canonical scalar field model.

##### 2.1. Noncanonical Inflation

Inflation is an era of the universe evolution where it stays in a positive accelerated phase and undergoes an extreme expansion. It is supposed that, in this era, the universe is dominated by an isotropic and homogeneous scalar field which causes quasi-de Sitter expansion. During this work, we assume that inflation happens due to a noncanonical inflation, and the general form of parameters is derived.

###### 2.1.1. Slow-Roll Approximation

In order to have a quasi-de Sitter expansion, the rate of the Hubble parameter during a Hubble time should be much smaller than unity; in other words [29]. The same situation is assumed for , which states that the rate of time derivative of scalar field during a Hubble time should be much smaller than unity: [29]. These two conditions are known as slow-roll approximations. The first condition allows us to ignore the kinetic energy density of scalar field against the potential part in the Friedmann equation, and the second condition lets one ignore the term against the term in the wave equation. Corresponding to each slow-roll approximation there is a slow-roll parameter, given by [29]Smallness of these two parameters during inflation shows that the scalar field slowly rolls down its potential and lets enough amount of inflation happen.

###### 2.1.2. Perturbation

Inflationary models predict three kinds of perturbations as scalar, vector, and tensor perturbation. One of the main metric perturbations is scalar perturbation. Scalar fluctuations become seeds for cosmic microwave background (CMB) anisotropies or for large scale structure (LSS) formation. Therefore, by measuring the spectra of the CMB anisotropies and density distribution, the corresponding primordial perturbation could be determined. First, let us have a brief look at the scalar perturbation.

Consider only an arbitrary scalar perturbation to the background FLRW metric, which is expressed by [26–28, 30, 31] is background spatial metric and is the covariant derivative with respect to this metric. The intrinsic curvature of the spatial hypersurface is expressed by the perturbation parameter as , where is usually named the curvature perturbation [31].

Inserting this metric in the main field equations leads one to the scalar perturbation, which are expressed in [26–28, 30–33]. In this order, we follow [30], which has considered the perturbation of generalized gravity, including our model. After some routine algebraic calculation, the results for scalar perturbation show the amplitude of scalar perturbation given by [30]where is sound speed and for our model is a constant and equal to (reader could refer to [30] for more detail).

Besides scalar fluctuation, the inflationary scenario predicts tensor fluctuation, which is known as a gravitational wave, too. The produced tensor fluctuations induce a curved polarization in the CMB radiation and increase the overall amplitude of their anisotropies at a large scale. The physics of the early universe could be specified by fitting the analytical results of CMB and density spectra to corresponding observational data. At first, it was thought that the possible effects of primordial gravitational waves are not important and might be ignored. However, a few years ago, it was found out that the tensor fluctuations have an important role, and they should be taken into consideration to determine the best-fit values of the cosmological parameters from the CMB and LSS spectra [34–36]. Contribution of tensor perturbation in metric is expressed asInserting it into field equations comes to tensor perturbation equations. In contrast with scalar and vector perturbation, energy-momentum perturbation has no role in tensor perturbation equation. After doing some algebraic analysis, the amplitude of tensor perturbation is obtained as [30]The imprint of tensor fluctuation on the CMB brings this idea to indirectly determine its contribution to power spectra by measuring CMB polarization [35]. Such a contribution could be expressed by the quantity, which is known as tensor-to-scalar ratio and represents the relative amplitude of tensor-to-scalar fluctuation: . Therefore, constraining is one of the main goals of the modern CMB survey. According to the current accuracy of observations, it is only possible to place a constant upper bound on the allowed range of [37–42]. Recent data from nine years of results of WMAP9 and South Pole Telescope (SPT) give the latest constraints of and at confidence level (CL) [43–46]. Combining Planck’s temperature anisotropy measurements with the WMAP large-angle polarization to constrain inflation gives an upper limit in CL [17, 46]. The latest data about the quantity comes from Planck collaboration on February 2015. Planck full mission data for model resulted in a new constraint on the quantity as (Planck TT, TE, EE + lowP), < (Planck TT + lowP + lensing) at CL, which indicates a slight improvement in comparison with the previous result of Planck-2013 [19, 25].

#### 3. Chaotic Inflation

The general form of dynamical equations and main parameters of the model have been derived, which brings us some crude results. In order to have more clear insight, a specific kind of potential is necessary. In this section, we are going to consider the model for a familiar kind of potential, which has received lots of attention, namely, power-law potential.

Using slow-roll approximations, the dynamical equation of the model is rewritten asand the acceleration equation is expressed as which shows a positive acceleration, a desirable situation for inflation. Also, for the wave equation, there isBy utilizing the definition of slow-roll parameters, and the reorganized form of dynamical equations, the slow-roll parameters could be obtained asThis form of slow-roll parameters is known as potential slow-roll parameters, whose smallness displays the flatness of potential during inflation. Inflation period lasts until the slow-roll parameter arrives at one, which corresponds to . Amount of expansion during this era is measured by number of -folds’ parameter, indicated by and defined asIt is expressed that, to overcome on standard cosmology problems, there should be about numbers of -folds.

Let us turn our attention to power-law potential . Since the general form of main equations has been acquired in the previous section, we ignored repeating it, and we only expressed the final results. Substituting this potential in slow-roll parameters (13), one arrives atin whichAt the end of inflation era, the acceleration parameter vanishes, and the slow-roll parameter reaches to one. Therefore, at this time, the scalar field could be read from (15) asUsing the number of -folds (14), one could easily derive the scalar field at the beginning of inflationin which is a constant parameter, defined by .

As it was mentioned, at the beginning of inflation, in order to have a quasi-de Sitter expansion, the slow-roll parameters should be much smaller than unity. Therefore, the constants should be in a way to satisfy this condition.

Using the definition of the potential in perturbation amplitude (8) and (10), the scalar and tensor perturbation amplitudes, respectively, are obtained asThe tensor perturbation is observed indirectly, by using the parameter which is the ratio of tensor perturbation amplitude to scalar perturbation amplitude: . These perturbations are used to constrain another free parameter of the model; however first we need to compute them at initial of inflation, namely, for .

On the other hand, the scalar and tensor spectra indices are expressed in terms of these slow-roll parameters

Utilizing the scalar spectra index (20), and tensor-to-scalar ratio equations (19), one could determine the behavior of the parameter in terms of and . Based on latest Planck data, there are a range for scalar spectra index and an upper bound for tensor-to-scalar ratio [19]: TT + lowP → , . TT + lowP + lensing → , . TT + lowP + BAO → = 0.9673 ± 0.0045, . TT, TE, EE + lowP → = 0.9645 ± 0.0049, .

Note that, as we concentrate on Planck-2015 data about the quantity , we realize that the previous mentioned constraint could rise in some cases. For instance, according to [19], for model, there is (Planck TT + lowP + lensing).

In order to specify a proper value for the constant parameter , the parameter is depicted versus for four different values of in Figure 1(a). The shadow covers the area which stays in the range ; the common interval for scalar spectra index resulted from Planck-2015 data. The curves cross the shadow line, which in turn indicates interval values of which are in perfect match with observational data. Figure 1(b) displays the parameter in terms of for four different values of tensor-to-scalar ratio. In return to the interval values of that comes from Figure 1(a), one could demonstrate appropriate values for the potential power . The final result has been prepared in Table 1. The existence of a modified kinetic term for scalar field is clear from the result, in which the power of kinetic term runs from to . In addition, the appropriate potential, to be in agreement with observational data, has a power of .