CMB Power spectrum/CMB anisotropies -> Geometry of Universe

[DanAbnormal]

Here is my understanding of how the anisotropies of the CMB are used to determine the geometry of the universe:

The fluctuations represent fluctuations in temperature just at the moment of last scattering, and therefore are a 'fingerprint' for the fluctuations in density of particles.
We can measure how big these fluctuations should be and therefore compare the angular size of what we observe to what we expect, and the geometry of the universe determines how the photon path will distort the image, ie angular size of fluctuations should be unchanged form our prediction for a flat universe, or appear larger/smaller for a closed/open universe.

Firstly, am I right in thinking the above is true, and accurate to a simple degree?
Also, my main question is how can we calculate the expected size of a fluctuation on the sky? Is it something to do with the intrinsic properties of the material at that time, eg can we determine the temperature etc? if so,how?

Another issue I have is with the cmb power spectrum.
Im not really sure what this graph represents. The x-axis is of increasing spherical harmonic number or something, is it not? Ther more of them there are, the more fluctuations, and the fluctuations get smaller with increasing value of x-axis.
I don't really get what the y-axis is, and therefore have no idea what the physical interpretation of the "first peak" is.
Can someone explain this graph to me, in somewhat simple terms, and how it is used to determine the geometry of the universe??

Thanks
Dan

 

[bapowell]

I don't really get what the y-axis is, and therefore have no idea what the physical interpretation of the "first peak" is.
Can someone explain this graph to me, in somewhat simple terms, and how it is used to determine the geometry of the universe??

The y-axis is the amplitude of the acoustic oscillations. The x-axis essentially corresponds to the wavelengths of the Fourier modes, or equivalently, the angular separation between correlations in the sky (think 3D waves projected onto 2D spherical surface). The first peak corresponds to a wave that has had time to fully compress just once before decoupling. Its wavelength is twice the horizon size at decoupling. The other peaks are waves of smaller wavelength that have gone through several oscillations before decoupling (these entered the horizon earlier than the longer wavelength modes, and hence go through more cycles before decoupling). They are smaller in amplitude because they damp out over time. By measuring the angle subtended by the correlations corresponding to the first peak, we get an idea of the horizon size at decoupling. If we assume a homogeneous universe, then we have a theoretical expression for the horizon size at decoupling in terms of the density of matter and the curvature of the universe. We can therefore use the measurement of the 1st peak (in addition to data on the matter density (also given my the CMB as well as other independent measurements)) to constrain the geometry of the universe. So, yes, your thinking in the first part of your message is correct.

EDIT: In case you're interested, the approximate relation between multipole number of the first peak and the density of the universe is \ell_{\rm peak} \approx 220/\sqrt{\Omega}, where \Omega is the density parameter. \Omega = 1 in a flat universe. See http://arxiv.org/abs/astro-ph/0011147 for more information.

 

[Chalnoth]

Also, my main question is how can we calculate the expected size of a fluctuation on the sky? Is it something to do with the intrinsic properties of the material at that time, eg can we determine the temperature etc? if so,how?

The expected typical length of the fluctuations on the sky is given by what is called the sound horizon: this is the distance that a sound wave could have traveled since the end of inflation. This sound horizon, then, depends critically upon how the universe expanded at early times.

It should be no surprise, then, that CMB data alone don't actually constrain the universe to be flat at all. Instead it's the combination of CMB data with data from the nearby universe that provides the high-accuracy measurement of flatness. For example, simply measuring the current Hubble expansion rate provides tremendous limits upon the flatness of the CMB.

But the measurement that most tightly constrains the flatness is the combination of so-called baryon acoustic oscillation (BAO) data with CMB data. BAO data stem from our understanding of how matter behaved in the early universe, before the CMB was emitted. Dark matter tended to just fall into potential wells, experiencing no pressure. By contrast, normal matter interacts with photons, experienced pressure, and thus tended to bounce (hence the acoustic oscillation part: acoustic oscillations are pressure waves, also known as sound). This different behavior between normal matter and dark matter leads to a characteristic relationship in the average distance between galaxies, setting a length scale for the nearby universe.

Crucially, this length scale in the nearby universe can be directly linked to the length scales we see imprinted on the CMB, giving us two different measures of the same length scale at different distances. The comparison between these two length scales gives us our tightest current bounds on the overall spatial curvature to be flat to within less than a percent.

 

[Simfish]

The x-axis essentially corresponds to the wavelengths of the Fourier modes, or equivalently, the angular separation between correlations in the sky (think 3D waves projected onto 2D spherical surface)

Hm, what exactly is the angular separation between correlations in the sky? Aren't correlation values scalars, making an angular separation undefinable?

 

[Chalnoth]

Hm, what exactly is the angular separation between correlations in the sky? Aren't correlation values scalars, making an angular separation undefinable?

What it is, specifically, is the variance of the spherical harmonic coefficients for a given \ell value. The spherical harmonics for a given \ell are essentially waves across the sky with a wavelength equal to 2\pi/\ell. So basically, the power spectrum of the CMB is the size of the temperature fluctuations as a function of their angular size across the sky.

 

[bapowell]

Hm, what exactly is the angular separation between correlations in the sky? Aren't correlation values scalars, making an angular separation undefinable?

You basically take two antennae and point them in different directions on the sky. They are separated by some angle \theta. You measure the correlation between temperature fluctuations in these two directions. For each angular separation, decompose the correlation into spherical harmonics (indexed by \ell). While many spherical harmonics of different \ell contribute to a given angular correlation, the dominant harmonic is that for which \ell \sim 2\pi/\theta. Now, for a given \theta, there are only 2\ell + 1 statistically independent measurement that one can make on the sky. The angular temperature power spectrum for angle \theta is the variance of these independent 2\ell + 1 measurements.

Note that on large angular scales (small \ell), there are fewer measurements that can be made than at smaller angular scales. This gives rise to the famous cosmic variance that dominates the error in CMB maps at the low multipoles. It places a fundamental limit on the precision with which the spectrum at these lower multipoles can be measured.