# Understanding Power Spectrum

1. Apr 3, 2012

### ditzycloud

Could anybody explain what "power spectrum" in cosmology means. I am trying to understand the WMAP power spectrum graph, in particular. For example, what do the multiple moments physically represent?

2. Apr 3, 2012

### bapowell

The power spectrum describes how the amplitudes of the primordial perturbations vary with scale. So, for example, a scale-invariant power spectrum is one for which the amplitudes are independent of scale. The primordial perturbations are what gave rise to the large scale structures of today's universe. It's best to start with the correlation function,
$$\xi({\bf r}) = \langle \mathcal{R}({\bf x},\tau)\mathcal{R}^\dagger({\bf x}+{\bf r},\tau)\rangle$$
where x and tau are the space and time coordinates, respectively, and $\mathcal{R}$ is the amplitude of the curvature perturbation. The curvature perturbations are the tiny primordial ripples in spacetime that were the seeds of structure. So the correlation function measures the amount of correlation between an overdensity here with an overdensity there, a distance r away. It is a statistic for describing the clumpiness of the universe as a function of scale. The power spectrum, it turns out, is just the Fourier transform of the correlation function,
$$\xi({\bf r}) =\frac{1}{(2\pi)^3}\int \mathcal{P}_\mathcal{R}(k) e^{-i{\bf k}\cdot{\bf r}}d^3k$$
So, for example, if we had overdensities arranged in a perfectly periodic fashion, the power spectrum would just be a delta function with k equal to the spacing. But, general overdensities are given in terms of a power spectrum across a range of scales. In fact, what I've defined here, while technically a power spectrum, is not what cosmologists mean when they say "power spectrum." Instead, they mean this:
$$P_\mathcal{R}(k) = \frac{d\langle|\mathcal{R}({\bf x},\tau)|^2\rangle}{d{\rm ln}k}$$
which is really the variance of the curvature perturbation. Nice, I know. But don't let that confuse you. Instead just think of the power spectrum as the amplitude of the perturbation as a function of length scale (or k).

Now, in addition to ultimately giving rise to structure, the first thing the primordial perturbations did was imprint themselves on the CMB, observable today as temperature anisotropies. Now, the CMB today comes to us from all directions, appearing to emanate from a giant sphere with Earth at the center. The inside surface of this sphere -- the so-called last scattering surface, is what we measure with our satellites. It is this surface that is cut open and spread out to form the spectacular WMAP images like this one:

The beautiful curve to which you refer, is a way of representing the degree of clumpiness of this map, much like the correlation function measures the clumpiness of 3D structures in today's universe. But, since the CMB sky is really a spherical surface, we use the angle $\theta$ between points on the sky to parameterize the correlations. And we can likewise take the Fourier transform of this temperature correlation function, just as we did for the 3D correlation function, and we obtain the temperature power spectrum, but this time parameterized in terms of multipole number, $\ell$, instead of k. Here, small $\ell$ corresponds to large scales (small k). So, for $\theta=90$ degrees, we are measuring correlations on very large scales, in fact, on scales of order the size of the observable universe. These are the largest scale correlations known, the quadrupole. As we move to smaller angular separations, we observe the octupole, and higher multipoles. Here are the first few multipole moments:

So the CMB sky is really built up out of all of these fluctuations, each multipole moment giving the amplitude of the correlation on a different scale. That's what your curve is showing -- the amplitudes of these fluctuations as a function of angular scale on the CMB sky. It is the Fourier decomposition of the CMB temperature fluctuations on a spherical surface. I've not included the relevant math here for brevity and since I don't want to overly complicate things. In summary, we have a primordial power spectrum, $P(k)$, that describes the scale dependence of the amplitude of the initial perturbations. These give rise to temperature fluctuations in the CMB, which are described by another power spectrum (the $C_{\ell}$'s). Of course the two power spectra are related: the temperature power spectrum is the primordial perturbation spectrum as an adolescent, after the fluctuations have grown and evolved somewhat, projected onto the sphere.

We can also delve into the different features of this curve, and talk about what this curve teaches us about the history and evolution of the universe. It's a fascinating story, and I'd be happy to discuss more if you're interested.

Last edited: Apr 3, 2012
3. Apr 4, 2012

### clamtrox

tl;dr: Power spectrum tells you how the amplitude of fluctuations changes with their physical scale. The variable k corresponds roughly to fluctuation size 1/k²:th of the entire sky. In the CMB power spectrum, the first acoustic peak size is roughly 1 square degree in the sky, and that is the scale where you expect to see largest temperature fluctuations.

4. Apr 4, 2012

### ditzycloud

Thank you bapowell for your very detailed and informative answer. It will take some time to digest these (I am also working my way through Dodelson's book). And I am definitely interested in further discussion. But I want to add another question here. I was reading Wayne Hu's Cosmology Tutorial and I came across the following: "Astronomers divide up the sky into angular degrees, so that 90 degrees is the distance from the horizon to a point directly overhead. COBE measured temperature ripples from the 10 degree to 90 degree scale. This scale is so large that there has not been enough time for structures to evolve." (In this page: http://background.uchicago.edu/~whu/intermediate/intermediate.html). Now why do large scales correspond to short times? Any references you can give for me to understand the ideas here? Thank you.

5. Apr 4, 2012

### clamtrox

Big structures take longer time to form than small ones. This is a nontrivial statement, involving the initial power spectrum, equation of state of dark matter, etc.

You can find a nice visualization of structure formation from the millennium simulation webpage, http://www.mpa-garching.mpg.de/galform/millennium/

6. Apr 4, 2012

### bapowell

The process of structure formation is indeed nontrivial, as clamtrox says, but I think there are some simple concepts that should help elucidate things. They key here is scale. All of the fluctuations seen in the CMB today were once outside our cosmological horizon (they were stretched to superhorizon scales during inflation.) Fluctuations on superhorizon scales do not evolve, because their wavelength's span acausal distances. Once inflation ended and the universe began to evolve according to standard radiation-dominated FRW cosmology, these fluctuations began to fall back inside the horizon. Once inside, they suddenly "come back to life", causally evolving according to all the gory laws of general relativistic fluid dynamics. Initially these fluctuations oscillate, because the photons and baryons are still tightly bound when the smallest scale fluctuations begin to enter the horizon (the largest scales are still way outside). These are the famous acoustic oscillations seen in the CMB from around $\ell \sim 400$ on. Then, the CMB photons decoupled from the baryons -- and we obtain the CMB snapshot that we all know and love. So the WMAP images are pictures of the CMB photons more or less as they looked when they first decoupled from the primordial plasma. At this time, some of the fluctuations were oscillating happily inside the horizon. But some were still outside the horizon. Some were just entering the horizon -- that's the broad central peak at around $\ell \sim 200$. It's a perturbation that has just entered the horizon and had time to oscillate through half a period. So, everying to the left of the central peak are fluctuations existing on superhorizon scales at the time of decoupling; everying to the right, subhorizon and oscillating. That's why we see no oscillations to the left: everythings frozen.

So that brings us to modern day. The largest scales that we can probe in the CMB are those on the scale of our present-day horizon -- the quadrupole (90 degrees). These are fluctuations that have just entered today (the quadrupole is a snapshot of this fluctuation at the time of decoupling, when it was still way superhorizon.) Meanwhile, those small scale oscillations have long since stopped oscillating (they stopped this once the photons decoupled, at which point they began to condense via gravitational instability to form large scale structure, with the help of dark matter.) So the longer something has been inside the horizon, the longer it has had time to grow into bound structure. That's what they mean when they say that the angular scales probed by COBE are too big to have evolved into large scale structure: the longer the length scale of the fluctuation, the shorter time it has been inside the horizon, and hence, the less time it's had to grow into structure.

7. Apr 10, 2012

### ditzycloud

The final post clarified things greatly for me. I just want to mention some issues which are no doubt very basic for many people (and I hope you would indulge me.)

So it seems the concept of horizon plays an essential role in our interpretation of the CMB snapshot for the problem of structure formation, for only when fluctuations are "within" horizon, they can causally interact and become the seeds of structure. Outside the horizon distance, they do not communicate. And I guess the cosmological principle plays a role here in the sense that it is not just "our" horizon distance, but "horizon in general", that is, regardless of the observation point. So the sentence "... the longer something has been inside the horizon, the longer it has had time to grow into bound structure" means to say, the longer the fluctuations are in causal contact, the longer etc. Do I understand this correctly?

And, when we say "The largest scales that we can probe in the CMB are those on the scale of our present-day horizon -- the quadrupole (90 degrees)", do we have in mind the angular-distance relation: Angle = (Width of the observed object) \ (Distance to the object) = W \ L (I think this formula presupposes flat geometry.) In this case, if we fix the angle to 90 degrees, there are still two numbers left to fix. I may look at (in principle) a very distant object whose apparent width is tiny (would this be a large scale because the object is far?) or I can look at a wide object which is less far away (and this would be a large scale in another sense). Provided that this all makes sense, a naive question is: cannot we look at larger angles, say 150 degrees etc.? Maximum L would be our horizon distance for we cannot see beyond this distance, but we would get a larger "width" on CMB?

(I hope what I wrote is not too far off to fix.)

8. Apr 10, 2012

### Tanelorn

I thought that the dipole was as a result of the movement of our galaxy relative to the CMBR?

Is the followong correct? When you subract this dipole you are left with a quadrapole? what is the source of the quadrapole and later the octopole et al?

9. Apr 10, 2012

### bapowell

Yes, that's right. It's not a matter of my horizon versus your horizon, but a matter of causality: fluctuations on scales smaller than $H^{-1}$ evolve.

Good questions...you've definitely got the idea. Yes, the angle that the temperature correlation subtends does indeed relate to physical scale. In fact, we use the location of the broad central peak (by location, I mean where on the $\ell$-axis it lies, which is in turn related to the angle that this correlation subtends) to provide information on the geometry of the universe! If we know the size of the horizon at decoupling, and if we know the distance to the last scattering surface (sloppy jargon for the distance that CMB light has traveled since decoupling), together with the $\ell$ value of the fluctuation on this scale, we can test the very equation you wrote above! If the geometry is flat, then we should recover the Euclidean relationship between these 3 quantities. And so yes, the angular scale of the fluctuation corresponds to physical length scales, and these are very powerful yardsticks for doing cosmology.

Now for the angles. Yes, we can observe correlations on larger angular scales than 90 degrees. The next largest angle would 180 degrees, corresponding to the dipole. While there should indeed be a primordial dipole, it is contaminated by our motion relative to the CMB: via the Doppler effect, those photons in the direction of motion appear bluer than those behind.

10. Apr 10, 2012

### bapowell

You always have the quadrupole and higher multipoles even before you subtract the dipole. The higher multipole moments are the Fourier components of primordial temperature fluctuations -- those imprinted in the CMB before decoupling (if these are inflationary in origin, then the temperature fluctuations arose just after inflation ended, during reheating). The dipole, on the other hand, has a component that is not primordial; it includes a Doppler contribution on account of our motion relative to the CMB.

11. Apr 11, 2012