How do we learn about the properties of the Universe from the CMB?
In the What causes the CMB pattern? section, we showed a simple way of extracting some information from the CMB. Here we will explore a more sophisticated approach, based on Fourier transforms. Fourier transforms are incredibly useful and have diverse applications from the mobile phone communications to the cosmos! This section builds up the key ideas underpinning how we go from CMB images to cosmological constraints! We utilize the following interactive exercises:
Part 1
What is a Fourier Mode?
One of the most powerful, but counterintuitive, tools scientists use is Fourier analysis. The key idea here is to decompose your original signal into a series of waves. The app below shows what all the possible 2D waves look like. Each wave is characterised by three properties:
- The frequency: how fast is the wave oscillating?
- The amplitude: how large is the wave?
- The phase: when is the wave at its lowest point?
Part 2
Image Analysis 101
In Fourier analysis, we take an image and express it in terms of the amplitudes and phases for every possible wavelength! Try this out for yourself. Upload a picture and see how it looks in Fourier space. The left image shows your uploaded image, the center panel is the amplitude for each different frequency and the final panel shows the phase.
The filter option demonstrates a few operations we can apply to the image in Fourier space and how they impact the original image. These provide some insight into the different roles of Fourier scale and information in the phases vs amplitudes.
Real Space
Amplitude
Phase
Part 3
Image Analysis applied to the CMB
Next we apply these methods to a simulated observation of the CMB. Use the app below to generate a mock CMB. What do you notice about the Fourier space amplitudes and phases?
CMB Map
Amplitude
Phase
The power spectrum
The CMB is random. Each realization looks different. The CMB phases carry no information. However, on average the CMB amplitude looks the same in every direction in Fourier space. This embodies a physical principle that the sky is isotropic! This means we can reduce the 2D Fourier space object to 1D, which we call the power spectrum! This is the average 2D Fourier amplitude in a radial ring. The structure of power spectrum contains the information of the CMB.
Part 4
Realize your own power spectrum
To get a better idea of what the power spectrum contains, try drawing different power spectra. Pressing the generate button will generate a new random draw. Each with the same power spectrum. Similarly different structures in the power spectrum lead to different types of features in the maps.
Power Spectrum P(k)
Draw and generate!
Gaussian Random Field
Part 5
The cosmic test
Judging whether two maps are the same is hard. Try to guess the power spectrum from one realization of the map. Even if you have the exact power spectrum, do the maps look the same?
Draw the Power Spectrum
Press “New Challenge” to begin. This editor uses logarithmic k and P(k) axes.
Target
Your GRF
Part 6
Connecting to the Universe's properties
Taken together these show that judging the information in a CMB map directly is hard! The CMB map is random, but that does not mean it contains no information. Fourier tools allow us to access the coherent structure via the power spectrum. The final question is how cosmic information is contained in the power spectrum? That is explored below!