Friday, February 3, 2012


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


  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!

  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!