Friday, February 3, 2012

Gaussianity

why does structure formation in the linear regime conserve Gaussianity? does it conserve other statistical properties of the initial density field?

2 comments:

  1. Structure formation in the linear regime amounts to a multiplication with the growth factor D_+, which depends on time only. When the random variable X is Gaussian, so is D_+X.

    Isotropy is the rotational invariance of the statistical properties of the field. The rotations commute with D_+, so isotropy is conserved. Similarly for homogeneity. In fact, it would be very strange, if multiplication with a spatial constant singled out points our directions in space.

    This illustrates how closely Gaussianity, Linearity and Homogeneity/Isotropy are related!

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  2. all statistical properties are conserved including Gaussianity. That's because the density field grows homogeneously in the linear regime, i.e. at the same rate D+(a) everywhere. So the amplitudes are just scaled and all n-point functions are equal to their initial values times D+(a)^n, so they're homogeneous functions of order n. For that lucky instance it is possible to investigate inflationary non-Gaussianities in the CMB (and partially in the large-scale structure on large scales) - in the 10^21 decades of redshift since the end of inflation until release of the CMB the density field has conserved exact knowledge of all non-Gaussianities!

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