Evaluation of the Cosmological Constant in Inflation with a Massive Nonminimal Scalar Field

Abstract

In Schrödinger picture we study the possible effects of trans-Planckian physics on the quantum evolution of massive nonminimally coupled scalar field in de Sitter space. For the nonlinear Corley-Jacobson type dispersion relations with quartic or sextic correction, we obtain the time evolution of the vacuum state wave functional during slow-roll inflation and calculate explicitly the corresponding expectation value of vacuum energy density. We find that the vacuum energy density is finite. For the usual dispersion parameter choice, the vacuum energy density for quartic correction to the dispersion relation is larger than for sextic correction, while for some other parameter choices, the vacuum energy density for quartic correction is smaller than for sextic correction. We also use the backreaction to constrain the magnitude of parameters in nonlinear dispersion relation and show how the cosmological constant depends on the parameters and the energy scale during the inflation at the grand unification phase transition.

Academic Editor:Filipe R. Joaquim

1, Department of Mechanical Engineering, Physics Division, Ming Chi University of Technology, Taishan District, New Taipei City 24301, Taiwan

Received 31 July 2014; Revised 5 January 2015; Accepted 24 January 2015; 15 February 2015

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 SCOAP³ .

1. Introduction

In the standard inflationary scenario, usual realization of inflation is associated with a slow rolling inflation minimally coupled to gravity [1]. Nevertheless, it is well known that the extension to the nonminimal coupling with the Ricci scalar curvature can soften the problem related to the small value of the self-coupling in the quartic potential of chaotic inflation [2]. Further, nonminimal coupling terms also can lead to corrections on power spectrum of primordial perturbations [3], a tiny tensor-to-scalar ratio [4, 5], and non-Gaussianities [6]. A broad class of models of chaotic inflation in supergravity with an arbitrary inflation potential was also proposed. In these models the inflation field is nonminimally coupled to gravity [7, 8]. Recently, the viability of simple nonminimally coupled inflationary models is assessed through observational constraints on the magnitude of the nonminimal coupling from the BICEP2 experiment [9].

Moreover, the standard inflationary scenario has two possible extensions. The first extension is associated with the ambiguity of initial quantum vacuum state, and the choice of initial vacuum state affects the predictions of inflation [10, 11]. The second extension concerns with the trans-Planckian problem [12, 13] of whether the predictions of standard cosmology are insensitive to the effects of trans-Planckian physics. In fact, nonlinear dispersion relations such as the Corley-Jacobson (CJ) type were used to mimic the trans-Planckian effects on cosmological perturbations [12-14]. These CJ type dispersion relations can be obtained naturally from quantum gravity models such as Horava gravity [15, 16]. Recently, in several approaches to quantum gravity, the phenomenon of running spectral dimension of spacetime from the standard value of 4 in the infrared to a smaller value in the ultraviolet is associated with modified dispersion relations, which also include the CJ type dispersion relations [17, 18].

In the previous work [19-23] we used the lattice Schrödinger picture to study the free scalar field theory in de Sitter space, derived the wave functionals for the Bunch-Davies (BD) vacuum state and its excited states, and found the trans-Planckian effects on the quantum evolution of massless minimally coupled scalar field for the CJ type dispersion relations with sextic correction. In this paper we extend the study to the case of massive nonminimally coupled scalar field.

The paper is organized as follows. In Section 2, the theory of a generically coupled scalar field in de Sitter space is briefly reviewed in the lattice Schrödinger picture. In Section 3, we consider the massive nonminimally coupled scalar field during slow-roll inflation and use the CJ type dispersion relations with quartic or sextic correction to obtain the time evolution of the vacuum state wave functional. In Section 4, using the results of Section 3, we calculate the finite vacuum energy density, use the backreaction to constraint the parameters in nonlinear dispersion, and evaluate the cosmological constant. Finally, conclusions and discussion are presented in Section 5. Throughout this paper we will set [figure omitted; refer to PDF] .

2. De Sitter Scalar Field Theory in Schrödinger Picture

In this section, we begin by briefly reviewing the theory of a generically coupled scalar field in de Sitter space in the lattice Schrödinger picture (for the details of some derivations in this section see [23]). The Lagrangian density for the scalar field we consider is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a real scalar field, [figure omitted; refer to PDF] is the potential, [figure omitted; refer to PDF] is the mass of the scalar quanta, [figure omitted; refer to PDF] is the Ricci scalar curvature, [figure omitted; refer to PDF] is the coupling parameter, and [figure omitted; refer to PDF] = [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . For a spatially flat ( [figure omitted; refer to PDF] )-dimensional Robertson-Walker spacetime with scale factor [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] In the ( [figure omitted; refer to PDF] )-dimensional de Sitter space we have [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the Hubble parameter which is a constant.

For [figure omitted; refer to PDF] , in the lattice Schrödinger picture, we obtain from (2) the time-dependent functional Schrödinger equation in momentum space [23] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Here [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] is the overall comoving spatial size of lattice, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the conjugate momentum for [figure omitted; refer to PDF] , and the subscripts 1 and 2 denote the real and imaginary parts, respectively.

For each real mode [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Note that (8) arises from the field quantization of the Hamiltonian (5) through the functional Schrödinger representation [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where operators [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy the equal time commutation relations [figure omitted; refer to PDF] , and setting [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] . Thus (8) governs the time evolution of the state wave functional [figure omitted; refer to PDF] of the Hamiltonian operator [figure omitted; refer to PDF] in the [figure omitted; refer to PDF] representation. In terms of the conformal time [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] the normalized wave functionals of vacuum and its excited states are [figure omitted; refer to PDF] with the amplitude [figure omitted; refer to PDF] and phase [figure omitted; refer to PDF] [figure omitted; refer to PDF] Here [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the [figure omitted; refer to PDF] th-order Hermite polynomial, [figure omitted; refer to PDF] is the Hankel function of the first kind of order [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the prime in (12) denotes the derivative with respect to [figure omitted; refer to PDF] . The complete wave functionals can be written as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] means that mode [figure omitted; refer to PDF] is in the [figure omitted; refer to PDF] excited state, mode [figure omitted; refer to PDF] is in the [figure omitted; refer to PDF] excited state, and so forth. For [figure omitted; refer to PDF] , the ground state wave functional corresponds to the BD vacuum.

For [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and the mode index [figure omitted; refer to PDF] in [figure omitted; refer to PDF] carries labels [figure omitted; refer to PDF] which will be suppressed below. Furthermore, from (3)-(8) we get in the continuum limit ( [figure omitted; refer to PDF] ) [figure omitted; refer to PDF]

3. Trans-Planckian Effects on Vacuum Wave Functional

For the inflationary potential [figure omitted; refer to PDF] , the bounds on [figure omitted; refer to PDF] derived from the joint data analysis of Planck + WP + BAO + high- [figure omitted; refer to PDF] for the number of [figure omitted; refer to PDF] -foldings [figure omitted; refer to PDF] are [figure omitted; refer to PDF] (68% CL), [figure omitted; refer to PDF] (95% CL) [24].

For the mass in the tree-level potential, we have [figure omitted; refer to PDF] GeV [25]. Moreover, the recent BICEP2 experiment suggests that [figure omitted; refer to PDF] GeV [26-28]. Therefore, from [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , which will be used below. To study further the effects of trans-Planckian physics, we use the CJ type dispersion relation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a cutoff scale, [figure omitted; refer to PDF] is an integer, and [figure omitted; refer to PDF] is an arbitrary coefficient [12-14].

3.1. CJ Type Dispersion Relations with Quartic Correction

First, we use the CJ type dispersion relation (14) with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to obtain the time evolution of the vacuum state wave functional. Recall that these CJ type dispersion relations can be obtained from theories based on quantum gravity models [15-18].

Using [figure omitted; refer to PDF] which is the ratio of physical wave number [figure omitted; refer to PDF] to the inverse of Hubble radius, (13) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , and the ground state solution of (15) becomes [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy [figure omitted; refer to PDF] In region I where [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] , the dispersion relations can be approximated by [figure omitted; refer to PDF] , and the corresponding wave functional for the initial BD vacuum state is [23, 29] [figure omitted; refer to PDF] where the prime in (20) denotes the derivative with respect to [figure omitted; refer to PDF] .

On the other hand, in region II where [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] , linear relations recover [figure omitted; refer to PDF] , and the corresponding wave functional for the non-BD vacuum state is [23, 29] [figure omitted; refer to PDF] where the prime in (22) denotes the derivative with respect to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the time when the modified dispersion relations take the standard linear form. Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . The constants [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be obtained by the following matching conditions at [figure omitted; refer to PDF] for the two wave functionals (19) and (21): [figure omitted; refer to PDF] which can also be rewritten, respectively, as [figure omitted; refer to PDF] by requiring [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] when [figure omitted; refer to PDF] .

Using [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have from (20), (22), and (25) [figure omitted; refer to PDF] where we choose [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is a relative phase parameter. Then from (27) and [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Substituting (20) and (22) into (26) and keeping terms up to order [figure omitted; refer to PDF] on the right-hand side of (26), we obtain [figure omitted; refer to PDF] Using (28) in (29) gives [figure omitted; refer to PDF] Here we choose [figure omitted; refer to PDF] , so that [figure omitted; refer to PDF] is small for [figure omitted; refer to PDF] to avoid an unacceptably large backreaction on the background geometry. Then we have [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

3.2. CJ Type Dispersion Relations with Sextic Correction

In this subsection, we use the CJ type dispersion relation (14) with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to obtain the time evolution of the vacuum state wave functional. For this case, only (15), (18), and (20) are changed into [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , and the prime in (35) denotes the derivative with respect to [figure omitted; refer to PDF] . Using [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , we obtain from (35), (22), (25), and [figure omitted; refer to PDF] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Substituting (35) and (22) into (26) and keeping terms up to order [figure omitted; refer to PDF] on the right-hand side of (26), we find [figure omitted; refer to PDF] Using (37) in (38) gives [figure omitted; refer to PDF] Here we choose [figure omitted; refer to PDF] , so that [figure omitted; refer to PDF] is small for [figure omitted; refer to PDF] to avoid an unacceptably large backreaction on the background geometry. Then we have [figure omitted; refer to PDF] or [figure omitted; refer to PDF]

4. Vacuum Energy, Backreaction, and Cosmological Constant

Using the results of Section 3, we proceed to calculate the finite vacuum energy density and use the backreaction constraint to address the cosmological constant problem. Note that, in the slow-roll approximation, the energy density of the scalar field is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Therefore the relation between the expectation value of the vacuum energy density [figure omitted; refer to PDF] and the vacuum wave functional [figure omitted; refer to PDF] in (16) is [figure omitted; refer to PDF] where we use a field redefinition [figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF] denotes the real part of [figure omitted; refer to PDF] , and the factor [figure omitted; refer to PDF] in (43) appears through the normalization condition [figure omitted; refer to PDF]

For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , in region I, we have [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . Then, using [figure omitted; refer to PDF] and (20) in (42), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] GeV is the Planck mass) are the boundaries of the interval of integration. On the other hand, in region II, (22) can be expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is defined as [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . From (27), (31), and (32), we note that [figure omitted; refer to PDF] can be approximated by [figure omitted; refer to PDF] as [figure omitted; refer to PDF] decreases from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] (horizon exit). Then, using [figure omitted; refer to PDF] and (22) in (42), we obtain [figure omitted; refer to PDF] From (45) and (48) we have [figure omitted; refer to PDF] For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , (49) becomes [figure omitted; refer to PDF] From (50) we see that there is no backreaction problem if the energy density due to the quantum fluctuations of the inflation field is smaller than that due to the inflation potential; that is, [figure omitted; refer to PDF] In the slow-roll approximation, using [figure omitted; refer to PDF] and (50) in (51) gives the constraint on the parameter [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] GeV (the energy scale during inflation implied by the BICEP2 experiment [26, 27]), we have [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , in region I, we have [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . Then, using [figure omitted; refer to PDF] and (35) in (42), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are the boundaries of the interval of integration. On the other hand, in region II, (22) can be again expressed as (46) with [figure omitted; refer to PDF] defined by (47). From (36), (40), and (41), we also note that [figure omitted; refer to PDF] can be approximated by [figure omitted; refer to PDF] as [figure omitted; refer to PDF] decreases from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] . Then, using [figure omitted; refer to PDF] and (22) in (42), we obtain [figure omitted; refer to PDF] From (52) and (53) we have [figure omitted; refer to PDF] For [figure omitted; refer to PDF] and [figure omitted; refer to PDF] which are satisfied if [figure omitted; refer to PDF] , (54) becomes [figure omitted; refer to PDF] Moreover, we notice that there is no backreaction problem if [figure omitted; refer to PDF] Using [figure omitted; refer to PDF] and (55) in (56) gives the constraint on the parameter [figure omitted; refer to PDF] as [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] GeV, we have [figure omitted; refer to PDF] .

Comparing (50) with (54), we find that [figure omitted; refer to PDF] if the inequality [figure omitted; refer to PDF] or [figure omitted; refer to PDF] is satisfied. For example, the usual parameter choice [figure omitted; refer to PDF] satisfies the inequality. On the other hand, we have [figure omitted; refer to PDF] if the inequality [figure omitted; refer to PDF] or [figure omitted; refer to PDF] is satisfied. For example, the parameter choices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy the inequality.

In the case that [figure omitted; refer to PDF] is larger than [figure omitted; refer to PDF] , using (50) in the cosmological constant [figure omitted; refer to PDF] gives [figure omitted; refer to PDF] which is [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . In the case that [figure omitted; refer to PDF] is larger than [figure omitted; refer to PDF] , using (54) in the cosmological constant [figure omitted; refer to PDF] gives [figure omitted; refer to PDF] which is [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

5. Conclusions and Discussion

In the Schrödinger picture, we have considered the theory of a generically coupled free real scalar field in de Sitter space. To investigate the possible effects of trans-Planckian physics on the quantum evolution of the vacuum state of scalar field, we focus on the massive nonminimally coupled scalar field in slow-roll inflation and consider the CJ type dispersion relations with quartic or sextic correction.

We obtain the time evolution of the vacuum state wave functional and calculate the expectation value of the corresponding vacuum energy density. We find that the vacuum energy density is finite and has improved ultraviolet properties. For the usual dispersion parameter choice, the vacuum energy density for quartic correction to the dispersion relation is larger than for sextic correction. For some other parameter choices, the vacuum energy density for quartic correction is smaller than for sextic correction.

We also use the backreaction to constrain the magnitude of parameters in nonlinear dispersion relation and show how the cosmological constant depends on the parameters and the energy scale during the inflation at the grand unification phase transition.

From (50) and (54) we see that the value of the cosmological constant can be reduced significantly through increasing the dispersion parameters in nonlinear dispersion relation and decreasing the cutoff energy scale associated with phase transition. However, the fact that the dispersion relation of a scalar field can not be modified on energy scales smaller than 1 TeV makes the cosmological problem still unsolved.

Acknowledgments

The author thanks M.-J. Wang for stimulating discussions on the cosmological constant and his colleagues at Ming Chi University of Technology for useful suggestions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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