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

<p>If <span class="math-container">$n_s$</span> is not a constant, but instead changes with <span class="math-container">$k$</span>, that is if
<span class="math-container">$$\frac{dn_s}{d \ln k} \neq 0,$$</span>
it is called a "running spectral index". And in fact it seems that that <span class="math-container">$n_s$</span> <em>does</em> chance with <span class="math-container">$k$</span> (see e.g. <a href="http://arxiv.org/abs/1403.4589" rel="nofollow noreferrer">Cherny et al. 2014</a>).</p>

<p>The term "<em>the running of the spectral index</em>" refers to the quantity <span class="math-container">$dn_s\,/\,d\ln k$</span>.</p>

CC BY-SA 3.0
2015-03-16T22:20:28.700Z
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 ...
Dilaton
CC BY-SA 3.0
2015-03-17T10:33:52.503Z
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
CC BY-SA 3.0
2015-03-17T11:39:59.610Z
Googling ["why is the primordial power spectrum called scale invariant"](https://www.google.dk/#q=why+is+the+primordial+power+spectrum+called+scale+invariant) led me to [this discussion](https://www.physicsforums.com/threads/why-are-the-primordial-fluctuations-called-scale-invariant.315749/) that I don't have the time to read in detail right now, but maybe that helps.
pela
CC BY-SA 4.0
2015-03-16T20:51:53.797Z
2019-05-13T15:04:51.960Z
2019-05-13T15:04:51.960Z
pela
<p>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.</p>

<cosmology><cosmological-inflation><cosmic-microwave-background>
What do cosmologists mean when they talk about "the running of the spectral index"?
mick
# What do cosmologists mean when they talk about "the running of the spectral index"?

(Asked by: username_0 on 2015-03-16)

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.

**Tags:** cosmology, cosmological-inflation, cosmic-microwave-background

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## 1 answers

### Answer by username_1 on 2015-03-16. Score: 1

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](http://arxiv.org/abs/1403.4589)).

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

> **Comments:**
> **username_2**: 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 ...  
> **username_1**: 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.  
> **username_1**: Googling ["why is the primordial power spectrum called scale invariant"](https://www.google.dk/#q=why+is+the+primordial+power+spectrum+called+scale+invariant) led me to [this discussion](https://www.physicsforums.com/threads/why-are-the-primordial-fluctuations-called-scale-invariant.315749/) that I don't have the time to read in detail right now, but maybe that helps.  

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