### 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 [46, 47, 105, 106, 194, 195, 196, 328, 403, 674, 675, 676, 683, 677] 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

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 [46, 47, 105, 106, 195, 674, 675, 676], and the BEC technologies
described in [126, 336, 337], one can take a homogeneous system , and a radial profile for the
velocity , with a scale factor depending only on . (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 the metric can be expressed as
Introducing a Hubble-like parameter,
the equation of continuity can be written as
with constant. Finally, we arrive at the metric of a spatially-flat FLRW geometry
with
The proper Friedmann time, , is related to the laboratory time, , by
Then,
The “physical” Hubble parameter is
If one now wishes to specifically mimic de Sitter expansion, then we would make
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 ) and the dynamics of the explosion (encoded
in ).

##### Two-dimensional explosion:

By holding the trap constant in the direction, and allowing the BEC to
expand in a pancake in the and directions (now best relabeled as and ) one can in principle
arrange

##### 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 [196, 194]. The relevant acoustic metric is now
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 with respect to the laboratory at all times:

Now it is not difficult to imagine a situation in which 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 [46, 47, 328, 676, 683, 677]). This again reproduces an expanding spatially-flat
FLRW Universe. The proper Friedmann time, , is again related to the laboratory time, , by
Then, defining , we have
If one now specifically wishes to specifically mimic de Sitter expansion then we would wish
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.