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Power Spectrum


Contents

The CMBR Power Spectrum
Generation of the CMBR Power Spectrum
Observational Data
Photon Fluid Approximation
Theoretical Models

The CMBR Power Spectrum

power spectrum Theoretical physicists use the power spectrum from the observational data to determine the cosmic parameters. Essentially, the power spectrum is obtained by taking the temperature (or density) fluctuations k between two points in the sky with angular separation (,) = (2,2) - (1,1), e.g., 2 = [(T2 - T0)/T0] and 1 = [(T1 - T0)/T0], with F(,) = 21 (where T0 = 2.726oK is the average temperature). The power spectrum at that particular angular separation (,) is calculated by averaging F(,) over the whole sky. Figure 01a shows the fluctuation of temperature at different angular scales ~ 2/. It is a theoretical model based on several parameters such as the total cosmic density, the baryon density (luminous matter) and the Hubble's constant as explained in more details below. There are literally millions of such models. The task is to obtain one that is best fit to the observational data. The shape of the power spectrum in Figure 01a can be separated into sections corresponding to different underlying physical processes as summarized below:

Figure 01a Power Spectrum [view large image]

  1. ISW (Integrated Sachs-Wolfe Effect) Rise - This effect arose from the time-dependent perturbations of the gravitational field. The effect is the sum from contributions along the path of the photons. It has been confirmed through correlations between the large-angle anisotropies and large-scale structure.
  2. Sachs-Wolfe Plateau - Perturbation of the gravitation field at large scale is responsible for this near constant appearance at lower s. Anisotropies at this scale have not evolved significantly, and hence directly reflect the "initial conditions".
  3. Acoustic (Doppler) peaks - The rich structure in this region is the consequence of the acoustic oscillation driven by repulsive radiation pressure and attractive gravity (as explained in more details later). The main peak is the oscillatory mode that went through 1/4 of a period (reaching maximal compression) at the time of recombination (between electrons and protons to form neutral atoms). The lower peaks correspond to the harmonic series of the main peak frequency. An additional effect comes from geometrical projection such that the angular position of the peaks is sensitive to the spatial curvature of the universe.
  4. Damping Tail (Doppler Foothills) - The recombination process is not instantaneous, giving a thickness to the last scattering surface. This leads to a damping of the anisotropies at the high s, corresponding to scales smaller than that subtended by this thickness. The damping cuts off the anisotropies at multipoles above ~ 2000.
Density Power Spectrum Sound Wave Spectrum Figure 01b shows the power spectrum for density fluctuations of the CMBR and other astronomical objects of various size as explained in the topic of "Superclusters". The corresponding sound wave spectrum is depicted in Figure 01c. Actually, this is not the kind of sound wave we hear on Earth. Its wavelength is very long in the order of 1 - 1000 Mpc, and its medium is not the air but hot plasma with a mixture of photons and other elementary particles.

Figure 01b Density Power Spectrum [view large image]

Figure 01c Sound Wave Spectrum

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Generation of the CMBR Power Spectrum