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Cosmological Horizons

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Cosmological and Black Hole Apparent Horizons

Part of the book series: Lecture Notes in Physics ((LNP,volume 907))

Abstract

FLRW spacetimes are much simpler than black hole spacetimes but still contain horizons. Cosmological horizons have been studied in inflationary scenarios of the early universe. In general, FLRW spaces contain time-dependent apparent horizons expressed by simple equations. This chapter discusses such cosmological apparent horizons and their dynamics.

Simplicity is the ultimate sophistication.

—Leonardo da Vinci

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Notes

  1. 1.

    Also called “r-gauge” (e.g., [30]).

  2. 2.

    As in most situations, the integrating factor is not unique.

  3. 3.

    The coordinate system in which the metric assumes the form (3.21) is sometimes referred to as Nolan gauge in the literature. Nolan [69–71] studied this coordinate system in the case of the McVittie metric [61] representing a central object embedded in a FLRW universe, eventually restricting to k = 0 or − 1. The McVittie metric reduces to that of Eq. (3.21) in the limit in which the mass of the central object vanishes, hence (3.21) is a trivial case of the McVittie line element in the Nolan gauge.

  4. 4.

    The literature contains ambiguous terminology for general FLRW spaces (e.g., [20]), while the de Sitter case does not lend itself to these ambiguities [62, 72].

  5. 5.

    See, e.g., Ref. [36]. The factor 2 in Eqs. (3.52) and (3.53) does not appear in Ref. [5] and in other works because of different normalizations of l a and n a. The expansions of the congruences definitely depend on the choice made for \(l^{c}n_{c}\).

  6. 6.

    More realistically, photons propagate freely in the universe only after the time of the last scattering or recombination, before which the Compton scattering due to free electrons in the cosmic plasma makes it opaque. Therefore, cosmologists introduce the optical horizon with radius \(a(t)\int _{t_{ \text{recombination}}}^{t} \frac{dt'} {a(t')}\) [66]. However, the optical horizon is irrelevant for our purposes and will not be used here.

  7. 7.

    The notation for the proper radius \(R \equiv a(t)r = a(t)f(\chi )\) is consistent with our previous use of this symbol to denote an areal radius because a(t)r is in fact an areal radius, as is obvious from the inspection of the FLRW line element (3.10). If \(k\neq 0\), the proper radius \(a(t)\chi\) and the areal radius \(a(t)f(\chi )\) do not coincide.

  8. 8.

    The Einstein-Friedmann equation (3.6) gives \(\ddot{a} < 0\) in this case.

  9. 9.

    These two examples, together with de Sitter space for k = 0, are presented in Ref. [26]. However, contrary to what is stated in this reference, in both cases the event horizon is not constant: it is the apparent horizon which is constant.

  10. 10.

    Cf. Table 1 of Ref. [39].

  11. 11.

    The coordinates \(\left (t,R,\theta,\varphi \right )\) are called “Painlevé-de Sitter coordinates” [72].

  12. 12.

    In general, the entropy density of a perfect fluid is \(s = \frac{P+\rho } {T}\) [54].

  13. 13.

    Cf. Refs. [15, 85].

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  1. Physics Department, Bishop’s University, Sherbrooke, QC, Canada

    Valerio Faraoni

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Faraoni, V. (2015). Cosmological Horizons. In: Cosmological and Black Hole Apparent Horizons. Lecture Notes in Physics, vol 907. Springer, Cham. https://doi.org/10.1007/978-3-319-19240-6_3

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