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5.3 Cosmological geometries

Analogue model techniques have also been applied to cosmology. In a cosmological framework the key items of interest are the Friedmann-Robertson-Walker (FRW) geometries, more properly called the Friedmann-Lemaître-Robertson-Walker (FLRW) geometries. The simulation of such geometries has been considered in various works such as [17Jump To The Next Citation Point18Jump To The Next Citation Point116Jump To The Next Citation Point115Jump To The Next Citation Point114Jump To The Next Citation Point242Jump To The Next Citation Point58Jump To The Next Citation Point59Jump To The Next Citation Point424Jump To The Next Citation Point] with a specific view to enhancing our understanding of “cosmological particle production” driven by the expansion of the universe.

The acoustic metric can be written as

ds2 = r-[-(c2 - v2)dt2 - 2v .dx dt + dx2]. (256) cs s
Essentially there are two ways to use this metric to reproduce cosmological spacetimes: One is based on physical explosion, the other on rapid variations in the “effective speed of light”.

Following [18Jump To The Next Citation Point115Jump To The Next Citation Point20273203] one can take a homogeneous system r(t),cs(t) and a radial profile for the velocity v = (b/b)r, with b a scale factor depending only on t. Then, defining a new radial coordinate as r = r/b b the metric can be expressed as

2 r-[ 2 2 2 2 2 2 ] ds = cs -c s dt + b (drb + r b d_O_ 2) . (257)
Introducing a Hubble-like parameter,
b(t) Hb(t) = ----, (258) b(t)
the equation of continuity can be written as
-r0-- r + 3Hb(t) r = 0; ==> r(t) = b3(t), (259)
with r0 constant. Finally we arrive at the metric of a flat FLRW geometry
ds2 = - T 2(t) dt2 + a2s(t) (dr2b + r2b d_O_22), (260)
V~ ---- V~ r-- T (t) =_ r cs; as(t) =_ --b. (261) cs
The proper Friedmann time, t, is related to the laboratory time, t, by
integral t = T(t) dt. (262)

The other avenue starts from a fluid at rest v = 0 with respect to the laboratory at all times:

2 2 r 2 ds = - rcs dt + c- dx . (263) s
Now it is not difficult to imagine a situation in which r remains constant, in a sufficiently large region of space, while the speed of sound decreases with time (this can be made in BECs for example by changing with time the value of the scattering length [17Jump To The Next Citation Point18Jump To The Next Citation Point]). This again reproduces an expanding flat FLRW Universe.

Models considered to date focus on variants of the BEC-inspired analogues:

In all of these models the general expectations of the relativity community have been borne out - theory definitely predicts particle production, and the very interesting question is the extent to which the formal predictions are going to be modified when working with real systems experimentally [18]. We expect that these analogue models provide us with new insights as to how their inherent modified dispersion relations affect cosmological processes such as the generation of a primordial spectrum of perturbations (see for example [4241434445464770107158177178207], [229230244252249250251274275296349361362363371] where analogue-like ideas are applied to cosmological inflation).

An interesting side-effect of the original investigation, is that birefringence can now be used to model “variable speed of light” (VSL) geometries [28Jump To The Next Citation Point108Jump To The Next Citation Point]. Since analogue models quite often lead to two or more “excitation cones”, rather than one, it is quite easy to obtain a bimetric or multi-metric model. If one of these metrics is interpreted as the “gravitational” metric and the other as the “photon” metric, then VSL cosmologies can be given a mathematically well-defined and precise meaning [28108].

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