Unruh–DeWitt Detectors in Spherically Symmetric Dynamical Space-Times

Abstract

In the present paper, Unruh–DeWitt detectors are used in order to investigate the issue of temperature associated with a spherically symmetric dynamical space-times. Firstly, we review the semi-classical tunneling method, then we introduce the Unruh–DeWitt detector approach. We show that for the generic static black hole case and the FRW de Sitter case, making use of peculiar Kodama trajectories, semiclassical and quantum field theoretic techniques give the same standard and well known thermal interpretation, with an associated temperature, corrected by appropriate Tolman factors. For a FRW space-time interpolating de Sitter space with the Einstein–de Sitter universe (that is a more realistic situation in the frame of ΛCDM cosmologies), we show that the detector response splits into a de Sitter contribution plus a fluctuating term containing no trace of Boltzmann-like factors, but rather describing the way thermal equilibrium is reached in the late time limit. As a consequence, and unlike the case of black holes, the identification of the dynamical surface gravity of a cosmological trapping horizon as an effective temperature parameter seems lost, at least for our co-moving simplified detectors. The possibility remains that a detector performing a proper motion along a Kodama trajectory may register something more, in which case the horizon surface gravity would be associated more likely to vacuum correlations than to particle creation.

Appendix: Response Function for ΛCDM Model

In order to obtain (5.15), we define the variable $x=\exp( -\frac{3}{2}h \varDelta \tau)$ and expand the inverse σ ²(x,s), given by the scale factor (5.13), around x=0 (i.e. Δτ→∞). We obtain a reasonably simple expansion in even powers of x given by

$$\frac{1}{\sigma^2(x,s)} = \frac{1}{\sigma_{dS}^2(s)} -h^2\,\sum _{n=1}^{\infty} \left( x^{2n}\, \sum_{k=1}^{3n-1}\, g(n,k)\, e^{k\, h s} \right). $$

(A.1)

On the right hand side, the first term is the constant term of the expansion and happens to be the pure de Sitter contribution, i.e.

(A.2)

with the effective Hubble constant $h=\sqrt{\varOmega_{\varLambda }}H_{0}$. Numerical hints given by the coefficients of the expansion up to the 10^th order in x, allow us to make a conjecture that the g(n,k)’s in the second term have a mean decreasing behavior and are bounded in the interval (0,1), but the main point is that the series in (A.1) is absolutely convergent with a finite radius of convergence which includes any t>0, namely the entire range of integration.

Hence, integrating term by term the expression (A.1), and making use of (3.14), for finite Δτ one has (5.15).

In the Δτ→∞ limit, we can focus on the leading exponentials contained in the last term of this expression: these are the k=(3n−1) terms, which are all dominated by a common factor exp(−hkΔτ). All the other terms are even more damped, so the convergence to zero is evident.