Gaussian Smoothing of the Matter Power Spectrum

[BOAS]

Homework Statement

Consider the density perturbation smoothed with a Gaussian of scale $\sigma$,

$\Delta_{\sigma}(\vec x') = \int d^3 \vec x \frac{e^{- \frac{(\vec x - \vec x')^2}{2 \sigma^2}}}{(2 \pi \sigma)^{3/2}} \Delta (\vec x)$

Calculate the power spectrum $P_{\Delta_{\sigma}}$ of $\Delta_{\sigma}$ in terms of $P_{\Delta}$ and sketch the form of $P_{\Delta_{\sigma}}$ compared to $P_{\Delta}$ (where this is the linear-theory matter power spectrum)

Homework Equations

The Attempt at a Solution

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I have calculated the smoothed power spectrum using the convolution theorem and found that $P_{\Delta_{\sigma}} = |W(\vec k)|^2 P_{\Delta}$, where $W(\vec k)$ is the Fourier transform of my gaussian.

My question is really that I don't know why you would want to do this, and what effect it has on the power spectrum. My only experience of Gaussian smoothing comes from image processing, where one might use a Gaussian filter to soften an image and reduce noise.

My sketch for the linear matter power spectrum essentially looks like this:

So what exactly is being smoothed out? Is it the small oscillations at large k? (that aren't really visible in that picture, but mine does have them)

 

[NateD]

And what does it mean for the power spectrum to be smoothed? Does it just get a bit more flat at large k?