# What do cosmologists mean when they talk about "the running of the spectral index"?

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.

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 $$dn_s\,/\,d\ln k$$.

• 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 ... Mar 16 '15 at 22:20
• 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.
– pela
Mar 17 '15 at 10:33
• 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.
– pela
Mar 17 '15 at 11:39