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I am trying to understand why, in cosmology, it is said that the presence of fluctuations at scales above the Hubble distance would not be expected in the absence of inflation or something like it.

We treat density fluctuations using the density contrast $\delta = (\rho - \bar{\rho})/\bar{\rho}$ and then we Fourier analyse. Let's write $\delta({\bf k}, t)$ for the amplitude of a Fourier component. We then have a distance scale $2\pi/k$ associated with this Fourier component, and also a distance scale $c/H$ (the Hubble distance) associated with the rate of expansion. Also, the particle horizon would be of a similar order of magnitude to that if it were calculated on the assumption that ordinary GR applies all the way down to $a=0$ (a highly questionable assumption of course). Anyway the main point is that it is asserted that one would not expect fluctuations $\delta({\bf k},t)$ for $2\pi/k$ larger than the separation between points that have had no causal contact. At least that is what I understand to assertion to be.

But I think that if independent random processes gave rise to independent fluctuations in separate spatial regions, then when one Fourier analysed the outcome one can get $\delta({\bf k},t) \ne 0$ for values of $k$ of any size. The independent random processes just happen to do this.

But perhaps the assertion is something more like, "if we treat the fluctuations on all scales as independent random processes, then the result is not compatible with what is observed". But is that true?

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  • $\begingroup$ Could this be a matter of terminology? If there is no causal contact between regions above certain scale then this could be called inhomogeneous cosmology rather than fluctuations since there is no common equilibrium to fluctuate away from. $\endgroup$
    – A.V.S.
    Jun 26, 2020 at 16:25
  • $\begingroup$ It may be true. What this means, though, is not that we need inflation. We need more data about the actual spectrum rather than an unfounded assumption about its randomness. $\endgroup$ Apr 21, 2023 at 16:01

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The answer is as you suggest in the last paragraph.

Density variations in our universe are statistically correlated over superhorizon scales.

You can evaluate the correlation function $$\xi(r)\equiv\langle\delta(\vec x)\delta(\vec x+\vec r)\rangle,$$ where the angle brackets average over positions $\vec x$ and the angles of the separation vector $\vec r$. If $\xi(r)$ is nonzero for some $r$, then we have to explain how density variations became correlated on that scale. This is what inflation can do (although it's not the only way).

The correlation function $\xi(r)$ is indeed nonzero for all scales $r$ that we can measure.

To read this result from the standard cosmological literature, we may note that the primordial curvature power spectrum is $$P(k)\propto k^{n_\mathrm{s}-4},$$ where $n_\mathrm{s}\simeq 0.96$ is the primordial spectral index, which is measured precisely by e.g. the Planck mission. The correlation function is the inverse Fourier transform of this, $$\xi(r) = \int\frac{\mathrm{d}^3\vec k}{(2\pi)^3} \,\mathrm{e}^{\mathrm{i}\vec k\cdot\vec r}P(k)\propto \Gamma(n_\mathrm{s}-2)\sin(n_\mathrm{s}\pi/2)\,r^{1-n_\mathrm{s}}.$$ The prefactor is nonzero for the measured $n_\mathrm{s}$. More generally $\xi(r)$ is only zero (or rather a delta function) if $n_\mathrm{s}=4$.

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You can have field fluctuations on any scale, but in a non-inflationary spacetime they just return to the vacuum. You need inflation to convert the fluctuation to a classical curvature perturbation.

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  • $\begingroup$ Thanks, but any chance of a bit more info? I know, for example, that dark matter density fluctuations can both grow and fade, so some of them grow. So by "field" here do you mean something to do with quantum field theory, or with the metric? But surely as soon as there is any density fluctuation, then it can grow from there? $\endgroup$ Jun 27, 2020 at 12:52
  • $\begingroup$ Right. I’m referring to inflaton field fluctuations, which are the seeds of density perturbations in inflation models. These couple to the metric but you need inflation to amplify them and cause them to decohere into classical perturbations. Once formed, these perturbations evolve according to the type of matter (CDM, baryons, etc) and the expansion rate. $\endgroup$
    – bapowell
    Jun 27, 2020 at 13:16
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    $\begingroup$ To me that's an argument from ignorance. We have absolutely no knowledge about the structure of the vacuum at these energies. Yes, in somebody's pet model of the quantum gravity vacuum that might be true. So what? $\endgroup$ Apr 21, 2023 at 16:00

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