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- Summary
- What will the Friedmann Equations be if we assume an anisotropic universe?

The Friedman Equations is based on the cosmological principle, which states that the universe at sufficiently large scale is homogeneous and isotropic.

But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how would Friedman Equations be modified.

I guess we would have to redefine the Friedmann metric tensor. But how?

The Friedmann metric tensor is:

$$g = -dt \otimes dt + (- \frac {a(t)^2} {kr^2-1}) dr \otimes dr + r^2 a(t)^2 d\theta \otimes d\theta + r^2 a(t)^2 sin(\theta)^2 d\phi \otimes d\phi$$

And the Friedmann Equations are:

The first one:

$$ H^2 \equiv (\frac{\dot a}{a})^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} $$

and the second one:

$$ \frac{\ddot a}{a} = - \frac{4\pi G}{3} (\rho + \frac{3p}{c^2}) + \frac{\Lambda c^2}{3} $$

And the derived Hubble parameter is:

$$ H_0 = \sqrt{(\Omega_c +\Omega_b)a^{-3} + \Omega_{rad} a^{-4} + \Omega_k a^{-2} + \Omega_{DE} a^{-3(1+w)}} $$

So how exactly would those equations need to be modified to account for anisotropy and an axial pole?

But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how would Friedman Equations be modified.

I guess we would have to redefine the Friedmann metric tensor. But how?

The Friedmann metric tensor is:

$$g = -dt \otimes dt + (- \frac {a(t)^2} {kr^2-1}) dr \otimes dr + r^2 a(t)^2 d\theta \otimes d\theta + r^2 a(t)^2 sin(\theta)^2 d\phi \otimes d\phi$$

And the Friedmann Equations are:

The first one:

$$ H^2 \equiv (\frac{\dot a}{a})^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3} $$

and the second one:

$$ \frac{\ddot a}{a} = - \frac{4\pi G}{3} (\rho + \frac{3p}{c^2}) + \frac{\Lambda c^2}{3} $$

And the derived Hubble parameter is:

$$ H_0 = \sqrt{(\Omega_c +\Omega_b)a^{-3} + \Omega_{rad} a^{-4} + \Omega_k a^{-2} + \Omega_{DE} a^{-3(1+w)}} $$

So how exactly would those equations need to be modified to account for anisotropy and an axial pole?