Table of Contents Author Guidelines Submit a Manuscript
Advances in High Energy Physics
Volume 2015, Article ID 569789, 7 pages
http://dx.doi.org/10.1155/2015/569789
Research Article

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

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

Academic Editor: Filipe R. Joaquim

Copyright © 2015 Jung-Jeng Huang. 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.

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.

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 [1214]. 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 [1923] 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 .

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 where is a real scalar field, is the potential, is the mass of the scalar quanta, is the Ricci scalar curvature, is the coupling parameter, and = , . For a spatially flat ()-dimensional Robertson-Walker spacetime with scale factor , we have In the ()-dimensional de Sitter space we have , where is the Hubble parameter which is a constant.

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

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

For , we have , , and the mode index in carries labels which will be suppressed below. Furthermore, from (3)–(8) we get in the continuum limit ()

3. Trans-Planckian Effects on Vacuum Wave Functional

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

For the mass in the tree-level potential, we have GeV [25]. Moreover, the recent BICEP2 experiment suggests that GeV [2628]. Therefore, from , we have and , which will be used below. To study further the effects of trans-Planckian physics, we use the CJ type dispersion relation where is a cutoff scale, is an integer, and is an arbitrary coefficient [1214].

3.1. CJ Type Dispersion Relations with Quartic Correction

First, we use the CJ type dispersion relation (14) with and 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 [1518].

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

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

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

3.2. CJ Type Dispersion Relations with Sextic Correction

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

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 , where . Therefore the relation between the expectation value of the vacuum energy density and the vacuum wave functional in (16) is where we use a field redefinition , denotes the real part of , and the factor in (43) appears through the normalization condition

For and , in region I, we have with . Then, using and (20) in (42), we obtain where and (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 where is defined as with . From (27), (31), and (32), we note that can be approximated by as decreases from to (horizon exit). Then, using and (22) in (42), we obtain From (45) and (48) we have For and , (49) becomes 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, In the slow-roll approximation, using and (50) in (51) gives the constraint on the parameter as . For GeV (the energy scale during inflation implied by the BICEP2 experiment [26, 27]), we have .

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

Comparing (50) with (54), we find that if the inequality or is satisfied. For example, the usual parameter choice satisfies the inequality. On the other hand, we have if the inequality or is satisfied. For example, the parameter choices and satisfy the inequality.

In the case that is larger than , using (50) in the cosmological constant gives which is for and for . In the case that is larger than , using (54) in the cosmological constant gives which is for and for .

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.

Conflict of Interests

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

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.

References

  1. A. D. Linde, Particle Physics and Inflationary Cosmology, Harwood Academic Publishers, Chur, Switzerland, 1990.
  2. R. Fakir and W. G. Unruh, “Improvement on cosmological chaotic inflation through nonminimal coupling,” Physical Review D, vol. 41, no. 6, pp. 1783–1791, 1990. View at Publisher · View at Google Scholar · View at Scopus
  3. D. I. Kaiser, “Primordial spectral indices from generalized Einstein theories,” Physical Review D, vol. 52, no. 8, pp. 4295–4306, 1995. View at Publisher · View at Google Scholar · View at Scopus
  4. E. Komatsu and T. Futamase, “Complete constraints on a nonminimally coupled chaotic inflationary scenario from the cosmic microwave background,” Physical Review D, vol. 59, no. 6, Article ID 064029, 1999. View at Google Scholar
  5. J.-C. Hwang and H. Noh, “COBE constraints on inflation models with a massive nonminimal scalar field,” Physical Review D—Particles, Fields, Gravitation and Cosmology, vol. 60, no. 12, Article ID 123001, 1999. View at Google Scholar · View at Scopus
  6. T. Qiu and K.-C. Yang, “Non-Gaussianities of single field inflation with nonminimal coupling,” Physical Review D, vol. 83, no. 8, Article ID 084022, 2011. View at Publisher · View at Google Scholar
  7. R. Kallosh and A. D. Linde, “New models of chaotic inflation in supergravity,” Journal of Cosmology and Astroparticle Physics, vol. 11, article 011, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. A. D. Linde, M. Noorbala, and A. Westphal, “Observational consequences of chaotic inflation with nonminimal coupling to gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2011, article 013, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. D. C. Edwards and A. R. Liddle, “The observational position of simple non-minimally coupled inflationary scenarios,” http://arxiv.org/abs/1406.5768.
  10. U. H. Danielsson, “Note on inflation and trans-Planckian physics,” Physical Review D, vol. 66, no. 2, Article ID 023511, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. C. Armendariz-Picon and E. A. Lim, “Vacuum choices and the predictions of inflation,” Journal of Cosmology and Astroparticle Physics, vol. 2003, article 006, 2003. View at Publisher · View at Google Scholar
  12. J. Martin and R. Brandenberger, “Trans-planckian problem of inflationary cosmology,” Physical Review D, vol. 63, no. 12, Article ID 123501, 16 pages, 2001. View at Publisher · View at Google Scholar
  13. R. H. Brandenberger and J. Martin, “The robustness of inflation to changes in super-Planck-scale physics,” Modern Physics Letters A, vol. 16, no. 15, pp. 999–1006, 2001. View at Publisher · View at Google Scholar · View at Scopus
  14. J. Martin and R. H. Brandenberger, “Corley-Jacobson dispersion relation and trans-Planckian inflation,” Physical Review D, vol. 65, no. 10, Article ID 103514, 2002. View at Publisher · View at Google Scholar · View at Scopus
  15. P. Hořava, “Quantum gravity at a Lifshitz point,” Physical Review D: Particles, Fields, Gravitation, and Cosmology, vol. 79, no. 8, p. 084008, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  16. P. Hořava, “Spectral dimension of the universe in quantum gravity at a Lifshitz point,” Physical Review Letters, vol. 102, no. 16, Article ID 161301, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  17. G. Amelino-Camelia, M. Arzano, G. Gubitosi, and J. Magueijo, “Dimensional reduction in the sky,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 87, no. 12, Article ID 123532, 2013. View at Publisher · View at Google Scholar · View at Scopus
  18. G. Amelino-Camelia, M. Arzano, G. Gubitosi, and J. Magueijo, “Rainbow gravity and scale-invariant fluctuations,” Physical Review D—Particles, Fields, Gravitation and Cosmology, vol. 88, no. 4, Article ID 041303, 2013. View at Publisher · View at Google Scholar · View at Scopus
  19. J.-J. Huang and M.-J. Wang, “Green’s functions of scalar field in de Sitter space: discrete lattice formalism.-I,” Il Nuovo Cimento A, vol. 100, no. 5, pp. 723–734, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. J.-J. Huang, “Excited states of de Sitter space scalar fields: lattice Schrödinger picture,” Modern Physics Letters A, vol. 21, no. 22, pp. 1717–1725, 2006. View at Publisher · View at Google Scholar · View at Scopus
  21. J.-J. Huang, “Inflation and squeezed states: lattice Schrödinger picture,” Modern Physics Letters A, vol. 24, no. 7, pp. 497–508, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. J.-J. Huang, “Pilot-wave scalar field theory in de Sitter space: lattice Schrödinger picture,” Modern Physics Letters A, vol. 25, no. 1, pp. 1–13, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. J.-J. Huang, “Bohm quantum trajectories of scalar field in trans-Planckian physics,” Advances in High Energy Physics, vol. 2012, Article ID 312841, 19 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. S. Tsujikawa, J. Ohashi, S. Kuroyanagi, and A. De Felice, “Planck constraints on single-field inflation,” Physical Review D: Particles, Fields, Gravitation and Cosmology, vol. 88, no. 2, p. 023529, 2013. View at Publisher · View at Google Scholar · View at Scopus
  25. P. A. R. Ade, N. Aghanim, M. Arnaud et al., “Planck 2015 results. XIII. Cosmological parameters,” http://arxiv.org/abs/1502.01589.
  26. P. A. R. Ade, R. W. Aikin, D. Barkats et al., “Detection of B-mode polarization at degree angular scales by BICEP2,” Physical Review Letters, vol. 112, Article ID 241101, 2014. View at Publisher · View at Google Scholar
  27. C. M. Ho and S. D. H. Hsu, “Does the BICEP2 observation of cosmological tensor modes imply an era of nearly Planckian energy densities?” Journal of High Energy Physics, vol. 2014, no. 7, article 60, 2014. View at Publisher · View at Google Scholar
  28. P. A. R. Ade, N. Aghanim, C. Armitage-Caplan et al., “Planck 2013 results. XXII. Constraints on inflation,” Worldwide Astronomical and Astrophysical Research, vol. 571, article A22, 2013. View at Publisher · View at Google Scholar
  29. J.-J. Huang, “Effect of the generalized uncertainty principle on the primordial power spectrum: lattice Schrödinger picture,” Chinese Journal of Physics, vol. 51, no. 6, pp. 1151–1161, 2013. View at Google Scholar · View at MathSciNet · View at Scopus