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From the cosmic microwave background, one can extract the scalar spectral index $n_s$. It is measured to be smaller than 1 by several standard deviations. Wikipedia says that it is a parameter of the Λ-CDM model. This is puzzling me. I thought that Λ-CDM includes a cosmological constant (i.e. no inflaton field necessary), standard model matter, radiation and WIMP dark matter. How do these parameters determine $n_s$?

Note: I don't want to get into the argument "Oh, but to explain the cosmological constant, we need an inflaton field", I think that's not relevant to this question.

Edit: If the answer is "You're wrong, Λ-CDM is always with an inflaton and a constant Λ has been ruled out", that's a perfectly valid answer to me, if you can point at some resources.

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$n_s$ is a parameter of the standard model of cosmology in the sense that the standard model doesn't predict it. If you want to calculate, for example, matter distribution you start with a spectrum of initial fluctuations and evolve forwards in time. $n_s$ describes the spectrum of those initial fluctuations, so it's an input to the modelling not something that can be predicted.

You ask:

I thought that Λ-CDM includes a cosmological constant (i.e. no inflaton field necessary), standard model matter, radiation and WIMP dark matter. How do these parameters determine $n_s$?

And the answer is simply that they don't. The value of $n_s$ is one of the predictions of inflation and is a result of quantum fluctuations being magnified by inflation. The standard model takes over when inflation ends, at which point the value of $n_s$ has already been set.

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