Talk:Matter power spectrum

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definition does not make sense

Latest comment: 1 month ago2 comments2 people in discussion

The definition does not make sense. What is the relationship between ${\displaystyle \mathbf {k} ,\mathbf {k} '}$  and ${\displaystyle k}$ ? Silsool (talk) 16:05, 25 March 2025 (UTC)Reply

${\displaystyle k}$  is the magnitude of ${\displaystyle \mathbf {k} }$  (or ${\displaystyle \mathbf {k} ',}$  due to the Dirac delta)

If you are talking about ${\displaystyle \langle {\tilde {\delta }}(\mathbf {k} ){\tilde {\delta }}^{*}(\mathbf {k} ')\rangle =(2\pi )^{3}P(k)\delta ^{3}(\mathbf {k} -\mathbf {k} ')}$  then I would like to say that the units don't make sense. The LHS is dimensionless (I think, as it is just the expected value of a unitless product), but the RHS has ${\displaystyle P(k)}$  with units of volume and ${\displaystyle \delta ^{(3)}(\mathbf {k} -\mathbf {k} ')}$  also has units of volume.

Edit: the Fourier transform of a unitless function is not unitless anymore in general, as the units of its argument interfere, but it depends on the specfic FT convention/definition used, as sometimes it is defined to be unitless for some quantities by introducing a factor of ${\displaystyle 1/V}$  for example. Huterer does not do this though.

A Course in Cosmology by Dragan Huterer has the same equation (9.53), but according to its errata, there should be a factor of ${\displaystyle 1/V}$  on the RHS, and the LHS is not unitless, but also has units of volume. Then there's also equation 9.56 in Huterer: ${\displaystyle P(k){\overset {(!?)}{=}}\langle \delta _{\mathbf {k} }\delta _{\mathbf {k} }^{*}\rangle =\langle \delta _{\mathbf {k} }\delta _{-\mathbf {k} }\rangle }$  which completely contradicts equation 9.53. The second equality there uses the fact that the FT of a real function is Hermitian. Yodo9000 (talk) 19:04, 9 July 2025 (UTC)Reply