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I have a difficulty understanding the various explanations of baryon acoustic oscillations that I've been reading. Even on this site, I haven't been able to find answers that directly address the following issues:

My image of these oscillations goes something like the following: I imagine an overdensity in the baryon-photon fluid oscillating in and out kind of like the oscillations of the surface of a drum, these oscillations being due to the gravitational attraction of the overdensity pulling it together, and radiation pressure pulling it apart. Just like the surface of the drum pushes on the surrounding air to create travelling sound waves, I imagine that this overdensity pushes on the surrounding fluid to create travelling sound waves in it, travelling at a little over half the speed of light. With this in mind, here are some issues:

It is explained that the biggest spherical sound waves are those with a radius equal to the sound horizon distance. This is often demonstrated empirically with the BAO bump in the correlation function of the distribution of galaxies. But if these overdensities create continous travelling sound waves, then why dont we get several consecutive bumps, separated by the (current) proper wavelength of the waves? Is it because various overdensities have various frequences of oscillation, thus producing sound waves of different wavelengths such that they deconstructively interfere, except for the first bump, which is common for all oscillations (radius equal to the sound horizon distance)? But if this is true, then why is it said that the various acoustic peaks in the CMB power spectrum corresponds to various states of oscillation (for instance that the second peak corresponds to a compression and then a rarefaction, or a compression, rarefaction and then compression for the third peak etc.)? Shouldn't there be some kind of destructive interference here too (except for the first peak)?

Also, it's been said that the first peak in the CMB spectrum corresponds to overdensities that have just got enough time at recombination to reach maximum compression. But if the radius of the sound waves that this peak corresponds to is the sound horizon distance, then surely these waves must have travelled since the Big Bang, meaning the compression that lead to the emission of these sound waves must have happened right after Big Bang.

Yash Bhusare
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Physics2718
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If you think about the power spectrum of CMB anisotropies, you see various peaks corresponding to different values of l, which represent a particular oscillation mode of the total wave (it is just a Fourier superposition in spherical coordinates). So sure it can be a destructive interference between different modes, but in the approximation of our analysis every mode evolves independently, and in the spectrum we are not representing all the intensities summed up, but only the "intensity" of anysotropies for each mode. It is just an aspect not relevant in our analysis, because we only want to understand the physical phenomenon that "created" different overdensities on different lenght scales. Another thing to notice is that oscillations are possible as long as the mean free path of the photons is much larger compared to the wave lenght of a particular mode, so for modes not corresponding to this condition we observe a damping. This can be explained by the fact that photons interact thruogh scattering in small (comoving) distances and they can thermalize the plasma, removing any kind of overdensity. Lastly, we suppose that the origin of the overdensities lies in the primordial quantum fuctuations that led to inflation. The modes of this fluctuation can evolve only if $k^{-1}<<{(aH)}^{-1}$, where ${(aH)}^{-1}$, is the hubble radius. This means that in this condition modes are causally connected and can generate perturbations. The larger modes, which satisfy $k^{-1}>>{(aH)}^{-1}$, are never causally connected, and remain almost unperturbed until recombination (they correspond to the "flat" spectrum for small l). Here is an interesting article I used when I studied this topic and had the same issues: arXiv:astro-ph/9407093

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