Quantum correlators in Friedmann spacetimes: The omnipresent de Sitter spacetime and the invariant vacuum noise

Kinjalk Lochan, Karthik Rajeev, Amit Vikram, and T. Padmanabhan

Phys. Rev. D 98, 105015 – Published 28 November 2018

Abstract

We discuss several aspects of quantum field theory of a scalar field in a Friedmann universe. (i) We begin by showing that it is possible to map the dynamics of a scalar field with a given mass, in a given Friedmann background to another scalar field of a different mass in another Friedmann universe. In particular one can map the dynamics of (1) a massless scalar field in a universe with power-law expansion to (2) a massive scalar field in the de Sitter spacetime. This allows us to understand several features of either system in a simple manner and clarifies several issues related to the massless limit. (ii) We relate the Euclidean Green’s function for the de Sitter spacetime to the solution of a hypothetical electrostatic problem in $D = 5$ and obtain, in a very simple manner, a useful integral representation for Green’s function. This integral representation is helpful in the study of several relevant limits and in recovering some key results which are—though known earlier—not adequately appreciated. One of these results is the fact that, in any Friedmann universe, sourced by a negative pressure fluid, the Wightman function for a massless scalar field is divergent. This shows that the divergence of Wightman function for the massless field in the de Sitter spacetime is just a special limiting case of this general phenomenon. (iii) We provide a generally covariant procedure for defining the power spectrum of vacuum fluctuations in terms of the different Killing vectors present in the spacetime. This allows one to study the interplay of the choice of vacuum state and the nature of the power spectrum in different coordinate systems in the de Sitter universe in a unified manner. (iv) As a specific application of this formalism, we discuss the power spectra of vacuum fluctuations in the static (and Painlevé) vacuum states in the de Sitter spacetime and compare them with the corresponding power spectrum in the Bunch-Davies vacuum. We demonstrate how these power spectra are related to each other in a manner similar to the power spectra detected by the inertial and Rindler observers in flat spacetime. This also gives rise to a notion of an invariant vacuum noise in the corresponding spacetimes which is observer independent. (v) In addition, several conceptual and technical issues regarding quantum fields in general cosmological spacetimes are clarified as a part of this study.

Received 13 June 2018

DOI:https://doi.org/10.1103/PhysRevD.98.105015

Physics Subject Headings (PhySH)

Quantum cosmology

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Kinjalk Lochan^1,*, Karthik Rajeev^2,†, Amit Vikram^3,‡, and T. Padmanabhan^2,§

¹Department of Physical Sciences, IISER Mohali, Sector 81, Manauli 140306, India
²IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India
³Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India

^*kinjalk@iisermohali.ac.in
^†karthik@iucaa.in
^‡amitvikram523@gmail.com
^§paddy@iucaa.in

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Vol. 98, Iss. 10 — 15 November 2018

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Images

Figure 1
Contours of curves corresponding to constant values of Painlevé coordinates [defined in Eq. (5)] are plotted in the comoving $(t, x)$ coordinates. The solid lines correspond to curves of constant Painlevé time (which coincides with the comoving time), and the dashed curves correspond to curves of constant Painlevé radial coordinate which is related to the comoving coordinate $x$ as $r = a (t) x$ .
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Figure 2
Contours of curves corresponding to constant values of static coordinates [as defined in Eq. (8)] are plotted in the comoving $(t, x)$ coordinates. The solid curves correspond to curves of constant static coordinate time $τ$ [as given in Eq. (9)], and the dashed curves correspond to curves of constant radial coordinate which is related to the comoving coordinate $x$ as $r = a (t) x$ . The dark wavy line corresponds to the surface $H r (t, x) = 1$ , the static coordinate patch.
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Figure 3
Contours of curves corresponding to constant values of the geodesic coordinates [as defined in Eq. (12)] are plotted in the comoving $(t, x)$ coordinates. The solid curves correspond to curves of constant timelike coordinate $τ$ , and the dashed curves correspond to curves of constant spacelike coordinate $l$ . The coordinates $(τ, l)$ are related to the comoving coordinates through conformal time $η = - e^{- H t} / H$ and Eq. (13). The shaded regions are not covered by this coordinate system because: (i) the shaded region corresponding to $Z > 1$ are timelike separated from the origin $O$ of the comoving coordinates and (ii) no points in the shaded region marked $Z < - 1$ can be connected to $O$ by a spacelike geodesic segment [see the discussion around Eq. (18)].
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Figure 4
Plots for (i) $p (w) = 2 / (3 + 3 w)$ , (ii) $q (w) = - 2 / (1 + 3 w)$ and (iii) $ν (w) = 3 (w - 1) / 2 (1 + 3 w)$ . The parameters $q$ and $p$ are defined through the expansion factor as $a (t) = (1 + H t / p)^{p} = (- H η / q)^{- q}$ , where $η$ is the conformal time, $t$ is the comoving time and $ν$ is just $q + (1 / 2)$ . We have taken $w \in [- 1, 1]$ , which corresponds to the physical range of the equation of states. The values of $p$ , $ν$ and $q$ for three special cases of the parameter $w$ , viz., (i) $w = 0$ , the dust; (ii) $w = 1 / 3$ , the radiation; and (iii) $w = - 1$ , the de Sitter that are of main interest in cosmology are also marked. The region $- 3 / 2 < ν < 3 / 2$ , where the field theory correlators are finite, which is the region outside the shaded part, maps to the range $0 < w < \infty$ when we exclude the unphysical (“phantom”) range of $w < - 1$ . So the field theory correlators are well-defined only if the equation of state parameter is positive semidefinite.
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Physical Review D

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