According to cosmological principle , space is homogeneous and isotropic which results in our universe being one of the three types of space(with positive,negative and zero curvature) . If space is homogeneous on large scale, isn't it also homogeneous in a small region?(At least,I can't imagine a 2-sphere which is non homogeneous locally but homogeneous on a large scale) If yes, why is that we don't find the space to be homogeneous around us?


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The universe is definitely not homogeneous on small scales, clearly if you look up and down you will see different things. This principle applies on large scales (cosmological scales), where the clustering of galaxies becomes weak. This can be observed from the two-point correlation function

$$ \xi(r) = \left(\frac{r}{r_0}\right)^{-\gamma} ~~~~\mbox{with}~~~ r_0\sim 10h^{-1}~ {\rm Mpc} $$

On scales much larger than $r_0$ the clustering of galaxies strongly decreases and the universe can be considered uniform.

The question is, what gave rise to this? The answer is given by inflation. Really tiny (Quantum) fluctuations in the density field are exponentially grown during the inflationary epoch, after that, gravity takes over and overdensities become even larger, leading to our standard picture of galaxy formation


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