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All I know is that the spectral index is related to the CMB, and that it has to be slightly less than 1 to favor inflation.

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The spectral index $n_s$ describes how the clumpiness of stuff varied on various scales just after cosmic inflation. The primordial power spectrum $P(k)$ of the fluctuations (where the wave number is $k=2\pi/\lambda$ with $\lambda$ being the physical scale), is predicted by many inflationary models to be: $$P(k) \propto k^{n_s-1}.$$ If $n_s=1$, the fluctuations are scale invariant.

If $n_s$ is not a constant, but instead changes with $k$, that is if $$\frac{dn_s}{d \ln k} \neq 0,$$ it is called a "running spectral index". And in fact it seems that that $n_s$ does chance with $k$ (see e.g. Cherny et al. 2014).

The term "the running of the spectral index" refers to the quantity $dn_s\,/\,d\ln k$.

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    $\begingroup$ Why are the fluctuations called scale invariant only if $n_s = 1$? In other contexts, power spectra are called scale invariant if the exponent of k takes any constant value ... $\endgroup$
    – Dilaton
    Mar 16, 2015 at 22:20
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    $\begingroup$ Hmm, that's a good question, @Dilaton. I think cosmology is the only place that uses the label "scale invariance" for a flat spectrum, not just any power law. Let me ask some clever people and get back to you. $\endgroup$
    – pela
    Mar 17, 2015 at 10:33
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    $\begingroup$ Googling "why is the primordial power spectrum called scale invariant" led me to this discussion that I don't have the time to read in detail right now, but maybe that helps. $\endgroup$
    – pela
    Mar 17, 2015 at 11:39

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