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Why does the scale factor $a(t)$ in FLRW metric has only temporal dependence but not directional?

If $a(t)$ had a directional dependence, I think it would violate the cosmological principle since

The cosmological principle is usually stated formally as 'Viewed on a sufficiently large scale, the properties of the universe are the same for all observers.

So it makes sense to state that it has no directional dependence to preserve the homogeneity and isotropicity of the universe considering that FLRW metric describes homogeneous, isotropic and expanding universe. Is there a more fundamental/mathematical reason not to assume a directional dependence in scale factor $a(t)$?

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I can't imagine a more fundamental reason than that the FLRW metric is derived under the assumption of spatial isotropy. It is fairly obvious that the corresponding scale factor cannot have a preferred spatial direction.

You can examine possible spacetimes which are homogeneous but not isotropic - see e.g. Bianchi universes - but those explicitly violate the postulates underlying the FLRW spacetime.

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  • $\begingroup$ I was suspecting the same thing but I thought maybe there's a deeper reason behind this. I will check out Bianchi universes. Thanks for the clarification. $\endgroup$
    – Monopole
    Feb 2 at 18:42
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As you mentioned, the key underlying assumption/definition used in creation of the FLRW metric is the assumption of isotropy and spatial homogeneity. This identically means that the scale factor at each point in space at a given time should have the same scale factor.

However, it is interesting to note that the universe is definitely not spatially homogeneous, as any probe of large scale structure will tell you. While in global averaged sense we can use the FLRW metric (i.e. at large spatial scales), locally it isn't so useful. However, we can generalize the fundemental idea of the FLRW metric in a nice way to these local patches through the "separate universe" formalism. The evolution of an over-dense region region can be describes as a mini "universe" which has a correspondingly larger $\Omega_M$. This concept can be used to approximate the evolution and distribution of large scale structure. In this case the solution is definitely not homogeneous.

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FLRW metric have isotropic expansion and spatially homogeneous spacetime. For the anisotropic expandable universe you can use Kasner metric. Proper distance $ds^2$ in Kasner metric is described as: $$ds^2=dt^2-\sum^3_{i=1}a^2_{i}(t)(dx^i)^2$$ Where $a_i$ is an Kasner exponent, or directed scale factor.

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