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The cosmological principle says that on large scales the universe is homogeneous and isotropic. Therefore there should be a way to define and measure inhomogenities and anisotropies.
Regarding the definition I see the following problem:
Usually we would like to define
##M=\int_V dV\,\mu##
##P^i=\int_V dV\,\pi^i##
##L^i=\int_V dV\,\lambda^i##
and
##\bar{\mu} = \frac{M}{V}##
##\tilde{\mu}=\mu-\bar{\mu}##
...
But we know that these integrals cannot be defined mathematically for arbitrary spacetimes (not asymptotically flat, ...) For the mass we have some definitions like Komar mass, for momentum and angular momentum it becomes even more complicated.
So how do we define inhomogenities and anisotropies mathematically?
A related problem is the definition of "large scales ...". What does that mean exactly? How would one classify and distinguish self-similar/ fractal-like / scale-free structures (with voids of every size)?
Having a mathematical definition at hand the problem is to measure it. Obviously we do not have data on a space-like section, but light rays (from galaxies and CMB). Therefore an appropriate definition should be based on light-like data samples.
The next question is whether we have data analysis for inhomogenities (fluctuations in mass or energy density, ...) and for anisotropies (fluctuations in momentum and angular momentum density, polarization, ...)
A remark regarding CMB and Planck data: of course they tell us something about inhomogenities - but only for a very special data set, namely the visible celestial sphere centered at the earth. As indicated by the integrals mentioned above I would like to have a more general definition using e.g. arbitrary volumes. I think this is not possible based on CMB.
Regarding the definition I see the following problem:
Usually we would like to define
##M=\int_V dV\,\mu##
##P^i=\int_V dV\,\pi^i##
##L^i=\int_V dV\,\lambda^i##
and
##\bar{\mu} = \frac{M}{V}##
##\tilde{\mu}=\mu-\bar{\mu}##
...
But we know that these integrals cannot be defined mathematically for arbitrary spacetimes (not asymptotically flat, ...) For the mass we have some definitions like Komar mass, for momentum and angular momentum it becomes even more complicated.
So how do we define inhomogenities and anisotropies mathematically?
A related problem is the definition of "large scales ...". What does that mean exactly? How would one classify and distinguish self-similar/ fractal-like / scale-free structures (with voids of every size)?
Having a mathematical definition at hand the problem is to measure it. Obviously we do not have data on a space-like section, but light rays (from galaxies and CMB). Therefore an appropriate definition should be based on light-like data samples.
The next question is whether we have data analysis for inhomogenities (fluctuations in mass or energy density, ...) and for anisotropies (fluctuations in momentum and angular momentum density, polarization, ...)
A remark regarding CMB and Planck data: of course they tell us something about inhomogenities - but only for a very special data set, namely the visible celestial sphere centered at the earth. As indicated by the integrals mentioned above I would like to have a more general definition using e.g. arbitrary volumes. I think this is not possible based on CMB.
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