Power spectrum exercise

Transcription

1 Power spectrum exercise In this exercise, we will consider different power spectra and how they relate to observations. The intention is to give you some intuition so that when you look at a microwave background power spectrum with extra power at particular scales (as in the acoustic peaks) you are able to develop a visual picture of what the measurements look like. We shall construct a number of different power spectra and generate data for each. The reverse process is much more complicated, reconstructing the power spectrum from the data, and we will discuss these methods at the end. This exercise only considers power spectra in one-dimension. The angular 2D reconstruction required for the microwave background has some additional mathematical complications and we will also discuss those. Power spectra A power spectrum refers to the amount of power on different scales. The simplest example is a delta function, where all the power is on a single scale. This means the fluctuations in the data will have a single frequency (inverse wavelength, so has units 1/distance). The fluctuations look like a single sine wave. Sine waves are a convenient way of representing the frequencies since every function can be represented as a sum of different sine waves.

2 The function which turns a delta function into a sine wave is called a Fourier transform. Fourier transforms have the convenient property that if you sum two functions in frequency space, the Fourier transform of the total equals the sum of the Fourier transforms of the two individuals. With a power spectrum where all the power is on two scales (two delta functions), the Fourier transform is the sum of two sine waves. In this exercise, we will use this property of Fourier transforms to compute data for several power spectra.

3 Every worksheet in the Excel file corresponds to a different power spectrum. The worksheets are 1 = 3 delta functions 2 = power proportional to 1/frequency 3 = as 2, but with extra noise 4 = power proportional to the frequency with extra noise In each worksheet column a is the x coordinate column b is random noise, average magnitude 1 in worksheets 1 and 2, and average magnitude 30 in worksheets 3 and 4 in all the rest of the columns except the final column represents one frequency component in the power spectrum. Row 1 is the amplitude. Row 2 is the frequency. The other rows are the sine function for each x. the final column is the sum of all the sine functions, which represents the data. The figures are power spectrum for worksheet 1 data for worksheet 1 power spectrum for worksheet 2 data for worksheet 2 data for worksheet 3 (two figures) power spectrum for worksheet 4 data for worksheet 4

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12 Fast Fourier transform Numerical algorithm to recover power spectrum from discrete data.

13 Spherical harmonics Set of basis functions convenient for solving differential equations on the surface of a sphere.

14 Planck microwave background map

15 Angular power spectrum

16 COBE DMR The COBE satellite was launched in 1989 to look for anisotropies in the microwave background. The Differential Microwave Radiometer made observations at three wavelengths 3.3 mm, 5.7 mm, 9.6 mm and determine the difference in order to subtract out the effects of the Milky Way and of secondary anisotropies. Anisotropies at an angular resolution of 7 were discovered at the level of 1 part in 100,000.

17 Balloon experiments Balloon experiments could give extremely high signal-to-noise ratios on small-scales. The Boomerang experiment was launched in 1998 and 2003 from the South Pole and flew at a height of 40,000 m consisting of 16 horns operating at 145 GHz (2.06 mm), 245 GHz (1.22 mm), and 345 GHz (0.87 mm). Anisotropies were detected on scales of 1. The second peak at a multipole is indicative of a flat Universe.

18 WMAP The Wilkinson Microwave Anisotropy Probe launched in 2001 was the successor to COBE. Measurements were taken at four frequencies between 33 GHz and 94 GHz. The main intention was to see structure in the microwave background on smaller scales than COBE, specifically the higher-order peaks that would allow very precise measurements of the cosmological parameters. WMAP measured the first two acoustic peaks at extremely high precision (the error bars are very small), which determined the cosmological parameters very precisely. The red line represents a lambda-cdm model with 71.4% dark energy, 24.0% CDM, and 4.6% normal baryonic matter.

19 Planck The Planck mission was launched in 2009 to map anisotropies in the microwave background at high signal to noise on small angular scales, improving on the results from WMAP. An additional capability was measuring polarization. Inflation makes predictions about the polarization patterns of the photons which later generate the microwave background. Planck measurements most significantly extended previous WMAP at higher multipoles (smaller scales). The Planck data was good enough to constrain most parameters in lambda-cdm cosmology to 1%.

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21 South Pole Telescope The South Pole Telescope is a 10 m millimetre and microwave detector located at the Amundsen-Scott South Pole Station. The SPT measures the microwave background on small angular scales (high multipoles). it is optimised for studying the Sunyaev-Zeldovich effect and B-mode polarization in the CMB. The SPT-SZ camera mapped the sky at three frequencies between 80 GHz and 220 GHz to construct a 2500 square degree survey of galaxy clusters. The scalar field that operates during inflation generates a signature in the polarization pattern of the microwave background (B-modes). Detection of this signature would be a confirmation of inflation.

22 Acoustic (Doppler) peaks The power spectrum of the microwave background anisotropies is determined by acoustic oscillations and diffusion damping. The acoustic oscillations arise from the competition between (1) gravitational attraction of the baryons, and (2) the photons-baryon plasma pressure smoothing out the fluctuations. The oscillations arise as these two effects periodically win or lose relative to each other. Satellite experiments measure the fundamental and first harmonic very accurately. Balloon and ground-based experiments measure the second and higher order harmonics less accurately.

23 Baryonic Acoustic Oscillations The characteristic spacing between overdensities in the early Universe correspondence to the mean free path of photons between collisions with electrons. This mean free path becomes larger than the size of the Universe following recombination when most of the electrons were bound into atoms. Periodicity in the galaxy distribution is seen in redshift surveys on this scale. The Sloan Digital Sky Survey measured extra power in the galaxy correlation function on scales about 100 Mpc. In comoving space this is comparable to the mean free path of photons in the early Universe.

24 Cosmological parameters from the microwave background The different regions of the power spectrum depend on the cosmological parameters in different ways. The tight constraints from Planck follow from the increased precision of the higher order acoustic peaks.

25 B-mode polarization The polarization pattern in the microwave background can be decomposed into two components, E and B modes. Density perturbations only contribute to the E modes. B modes are generated by gravitational waves created during inflation. Measurements are complicated by contamination from secondary gravitational lensing on small scales and from Galactic emission.