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Cosmological Principle

The universe is homogeneous and isotropic on large scales, a fundamental assumption in cosmology and relevant to dark matter and energy distributions.
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The statement of the theorem

The Cosmological Principle assumes that the universe is statistically homogeneous and isotropic on sufficiently large scales. Mathematically, this implies that the spacetime metric gμνg_{\mu\nu} must take the form of the Friedmann-Lemaître-Robertson-Walker (FLRW) metric:\nds2=−c2dt2+a(t)2[dr21−kr2+r2(dθ2+sin⁡2θdϕ2)]ds^2 = -c^2 dt^2 + a(t)^2 \left[ \frac{dr^2}{1 - k r^2} + r^2 (d\theta^2 + \sin^2\theta d\phi^2) \right]\nWhere a(t)a(t) is the scale factor, and kk is the spatial curvature constant (k∈{−1,0,1}k\in\{-1, 0, 1\}). The assumption dictates that the physical laws and statistical properties are independent of the spatial coordinates (r,θ,ϕ)(r, \theta, \phi).