2.5 Cosmological metrics

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 [46Jump To The Next Citation Point, 47Jump To The Next Citation Point, 105Jump To The Next Citation Point, 106Jump To The Next Citation Point, 194Jump To The Next Citation Point, 195Jump To The Next Citation Point, 196Jump To The Next Citation Point, 328Jump To The Next Citation Point, 403Jump To The Next Citation Point, 674Jump To The Next Citation Point, 675Jump To The Next Citation Point, 676Jump To The Next Citation Point, 683Jump To The Next Citation Point, 677Jump To The Next Citation Point] with a specific view to enhancing our understanding of “cosmological particle production” driven by the expansion of the universe.

Essentially there are two ways to use the acoustic metric, written as

ds2 = ρ-[− (c2− v2)dt2 − 2v ⋅ dxdt + dx2] , (95 ) cs s
to reproduce cosmological spacetimes. One is based on physical explosion, the other on rapid variations in the “effective speed of light”.

2.5.1 Explosion

We can either let the explosion take place more or less spherically symmetrically, or through a pancake-like configuration, or through a cigar-like configuration.

Three-dimensional explosion:
Following the cosmological ideas of [46Jump To The Next Citation Point, 47Jump To The Next Citation Point, 105Jump To The Next Citation Point, 106Jump To The Next Citation Point, 195Jump To The Next Citation Point, 674Jump To The Next Citation Point, 675Jump To The Next Citation Point, 676Jump To The Next Citation Point], and the BEC technologies described in [126, 336, 337], one can take a homogeneous system ρ(t), cs(t) and a radial profile for the velocity v = (˙b∕b)r, with b a scale factor depending only on t. (This is actually very similar to the situation in models for Newtonian cosmology, where position is simply related to velocity via “time of flight”.) Then, defining a new radial coordinate as rb = r∕b the metric can be expressed as

ρ [ ] ds2 = -- − c2sdt2 + b2(dr2b + r2bdΩ22) . (96 ) cs
Introducing a Hubble-like parameter,
˙ H (t) = b(t), (97 ) b b(t)
the equation of continuity can be written as
˙ρ + 3Hb (t)ρ = 0; ⇒ ρ(t) = -ρ0-, (98 ) b3(t)
with ρ0 constant. Finally, we arrive at the metric of a spatially-flat FLRW geometry
ds2 = − T 2(t)dt2 + a2s(t)(dr2b + r2bd Ω22), (99 )
with
√ --- ∘ ρ-- T (t) ≡ ρcs; as(t) ≡ --b. (100 ) cs
The proper Friedmann time, τ, is related to the laboratory time, t, by
∫ τ = T (t)dt. (101 )
Then,
2 2 2 2 2 2 ds = − d τ + as(τ )(dr b + rbd Ω2). (102 )
The “physical” Hubble parameter is
H = -1-das. (103 ) as dτ
If one now wishes to specifically mimic de Sitter expansion, then we would make
as(τ) = a0exp (H0τ ). (104 )
Whether or not this can be arranged (in this explosive model with comoving coordinates) depends on the specific equation of state (which is implicitly hidden in cs(t)) and the dynamics of the explosion (encoded in b(t)).

Two-dimensional explosion:
By holding the trap constant in the z direction, and allowing the BEC to expand in a pancake in the x and y directions (now best relabeled as r and ϕ) one can in principle arrange

ρ(x,t) [ ] ds2 = ------- − {c2s(x, t) − v(r,t)2}dt2 − 2v(r,t)drdt + dr2 + r2d ϕ2 + dz2 . (105 ) cs(x,t)

One-dimensional explosion:
An alternative “explosive” route to FLRW cosmology is to take a long thin cigar-shaped BEC and let it expand along its axis, while keeping it trapped in the transverse directions [196Jump To The Next Citation Point, 194Jump To The Next Citation Point]. The relevant acoustic metric is now

ds2 = ρ(x,-t)-[− {c2(x,t) − v (x,t)2}dt2 − 2v (x, t)dxdt + dx2 + {dy2 + dz2} ]. (106 ) cs(x,t) s
The virtue of this situation is that one is keeping the condensate under much better control and has a simpler dimensionally-reduced problem to analyze. (Note that the true physics is 3+1 dimensional, albeit squeezed along two directions, so the conformal factor multiplying the acoustic metric is that appropriate to 3+1 dimensions. See also Section 2.7.2.)

2.5.2 Varying the effective speed of light

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

ds2 = − ρcsdt2 + -ρdx2. (107 ) cs
Now it is not difficult to imagine a situation in which ρ remains spatially and temporally constant, in a sufficiently large region of space, while the speed of sound decreases with time (e.g., we shall see that this can be made in analogue models based on Bose–Einstein condensates by changing with time the value of the scattering length [46Jump To The Next Citation Point, 47Jump To The Next Citation Point, 328Jump To The Next Citation Point, 676Jump To The Next Citation Point, 683Jump To The Next Citation Point, 677Jump To The Next Citation Point]). This again reproduces an expanding spatially-flat FLRW Universe. The proper Friedmann time, τ, is again related to the laboratory time, t, by
∫ √ --- τ = ρcsdt. (108 )
Then, defining ∘ ---- as(t) ≡ ρ∕cs, we have
2 2 2 2 2 2 ds = − d τ + as(τ )(dr b + rbd Ω2). (109 )
If one now specifically wishes to specifically mimic de Sitter expansion then we would wish
a (τ) = a exp (H τ ). (110 ) s 0 0
Whether or not this can be arranged (now in this non-explosive model with tunable speed of sound) depends on the specific manner in which one tunes the speed of sound as a function of laboratory time.
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