# Why are the primordial fluctuations called scale invariant?

## Main Question or Discussion Point

The slope of the primordial power spectrum (the power spectrum of density fluctuations produced by inflation in the very early Universe before it had been modified by gravitational/hydro dynamics) is often written,
P(k) = A * k
and then in the same line referred to as scale invariant or the Harrison-Zeldovich scale invariant spectrum. How is this scale invariant when it is a function of the wave number k? Anyone know this?

Chalnoth
The slope of the primordial power spectrum (the power spectrum of density fluctuations produced by inflation in the very early Universe before it had been modified by gravitational/hydro dynamics) is often written,
P(k) = A * k
and then in the same line referred to as scale invariant or the Harrison-Zeldovich scale invariant spectrum. How is this scale invariant when it is a function of the wave number k? Anyone know this?
Uh, I think you're mistaken. The way it's normally written:
$$P(k) = A k^{n_s-1}$$

...such that if $$n_s = 1$$ (scale invariant spectrum), then $$P(k) = A$$.

From Eisenstein and Hu 1998, second paragraph of the Appendix.
http://arxiv.org/abs/astro-ph/9709112

"
The power spectrum of the density fluctuations is then proportional to the initial power spectrum times the square of the transfer function. In the most usual case, the initial power spectrum is taken to be a power law, so that P(k) ~ k^n T^2(k), where n=1 is the familiar Harrison-Zeldovich-Peebles scale-invariant case.
"

I was reading Dodelson's Modern Cosmology and I think the scale invariance might refer to the fluctuations of the scalar potential instead of the density but I have to read it a little more carefully.

Chalnoth
Well, whether or not P(k) is a constant depends upon how you define P(k). But regardless, a scale invariant power spectrum is one wherein if we have an inflationary period that is expanding at a constant rate, and the physics that govern the production of perturbations are also constant, then scale-invariant perturbations are, by definition, produced. Exactly how P(k) depends upon k, then, just depends upon how we define the function, and there are a couple of different conventions for doing so.

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I don't know if you are looking for a specific mathematical answer or just an intuitive one. Ned Wright's http://www.astro.ucla.edu/~wright/cosmo_04.htm" [Broken] explains the idea of scale invariance very simply.

Today we see various sized circular static fluctuations in the CMB radiation pattern which are thought to represent greatly expanded artifacts of the original tiny quantum fluctuations. The apparent size today of each circle tells when it occurred during the brief inflation period, i.e. relatively early or late. Scale invariance means that each circle is measured to have the same amount of radiation amplitude (power), regardless of the size of the circle. Which is consistent with the theory that each circle originated as a quantum fluctuation of identical amplitude.

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So I found the answer in Dodelson on pg. 184 today (and also someone on cosmo coffee gave an explanation).

The scale free object is the dimensionless power spectrum of scalar fluctuations to the metric which are equivalent to fluctuations in the gravitational potential in the weak field limit.
$$\Delta_{\phi} \propto k^3 P_{\phi} = Const.$$

then the Poisson equation in k space relating potential to density fluctuations introduces two factors of k
$$\delta \propto k^2 \phi$$

and the power spectrum of density fluctuations squares that,
$$P_{\delta} \propto <|\delta|^2> \propto k^4 <|\phi|^2> \propto k^4 P_{\phi}$$

so that finally you get,
$$P_{\delta} \propto Const. \; k^4 / k^3$$

meaning fluctuations in the gravitational potential which have the same amplitude no matter what scale produce density fluctuations with a power spectrum that has a slope of unity.