Power Spectrum

Contents

The CMBR Power Spectrum

Theoretical physicists use the power spectrum from the observational data to determine the cosmic parameters. Essentially, the power spectrum is a plot of the amount of fluctuation against the angular (or linear) size. The fluctuation is the difference in the two measurements at the corresponding points. It can be the fluctuation of temperature or density or any other kind of measurable quantity. Figure 01a shows the fluctuation of temperature at different angular scales (inverted). 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

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 ls. 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 ls, corresponding to scales smaller than that subtended by this thickness. The damping cuts off the anisotropies at multipoles above ~ 2000.

		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

Generation of the CMBR Power Spectrum

The structure in the universe is seeded by random quantum fluctuations in the very beginning of the Big Bang. A period of rapid expansion, called inflation, caused these quantum fluctuations to be stretched into cosmic scales. The gravitational attraction in the density enhanced regions and radiation repulsion acted together to produce the incoherent acoustic oscillations (noise). Compressing a gas heats it up; letting it expand cools it down - this is the origin of the temperature variation. As shown in Figure 02, if the gravity and sonic motion (the alternate compression and rarefication) work together then the photons and baryons are compressed in the trough producing the first peak with large temperature fluctuation. However, if they counteract each others, a smaller second peak will be created.

Figure 02 Acoustic Oscillations [view large image]

The size of these oscillations occurred on all scales (wavelength). Mathematically, the size of the variation at location x, i.e., F(x) = T/T can be expressed by the Fourier series:
F(x) = _k{G_k cos(kx)} ---------- (1)
where the sum is over all values of k = 2/L, L is the wavelength. The coefficient G_k can be calculated from the inverse relation: G_k = {F(x) cos(kx)}, where the sum is over all x. For a given value of k, its harmonics are 2k, 3k, ...; k is called the fundamental mode.

Theory of Inflation predicts that there should be as many hot spots as cold spots, i.e., its distribution curve is Gaussian (Figure 03a). Recent (2008) analysis of the WMAP data suggests that it may not be the case. The skewing, known as non-Gaussianity, shows up as a tiny effect with distortion in temperature distribution of the order 1 in 100000. More observations are needed to confirm such finding, which would falsify the theory of inflation. However, other researches indicated that the non-Gaussianity is caused by a large cold spot. The distribution remains Gaussian after removing this abnormal data.

Figure 03a Gaussian Dis-tribution [view large image]

As the universe expands and cools, the average energy of a photon falls until eventually hydrogen atoms are able to form. This is the epoch of recombination (Figure 03b) when the photons are released and stream off unimpeded as CMBR today. The acoustic oscillations stop at recombination (no more radiation pressure to produce the expansion). There is a special mode k1 for which the fluid just had enough time to compress once before frozen in at recombination (thus producing the maximum variation). The corresponding distance is called the sound horizon: k1 = /(2 sound horizon). Modes caught at oscillations with such

Figure 03b Recombination [view large image]

wavelength become the peaks in the CMBR power spectrum and form a harmonic series based on k1.

Observational Data

Any map drawn on a sphere, whether it be the CMBR's temperature or the topography of the earth, can be broken down into multipoles. The lowest multipoles are the largest-area, continent- and ocean-size undulations on the temperature map. Higher multipoles are like successively smaller-area plateaus, mountains and hills (and trenches and valleys) inserted on top of the larger features. The entire complicated topography is the sum of the individual multipoles. The lowest mode (l=0) is the monopole - the entire sphere pulses as one. This is the average temperature (2.726oK) of the CMBR. The next lowest mode (l=1) is the dipole, in which the temperature goes up in one hemisphere and down in the other. In the CMBR mapping, the dipole is dominated by the Doppler shift of the solar system's motion relative to the CMBR; the sky appears slightly hotter in the direction the sun is traveling (see Figure 02-05 in Topic 02, Observable Universe). The CMBR power spectrum begins at Cl=2 because the real information about cosmic fluctuations begins with the quadrupole (l = 2). Note that the peak variation occurs at about l = 200 corresponding to an angular size of about 1 degree. Figure 04 shows the multipoles with l = 0, 1, 2. The red color represents variation above the average (green); while the blue color denots less.

Figure 04 Multipoles [view large image]

Photon Fluid Approximation

		In the photon fluid approximation, the medium for sound propagation is a fluid of pure photons without taking into account the matter and expansion effects. Figure 05 is a plot of the displacement (red) and its square (blue) of the sound wave at the moment of recombination as a function of k, i.e., it is a much simplified version of the power spectrum. It shows many differences when compares to the observed power spectrum in Figure 06.
Figure 05 Simplified Power Spectrum [view large image]	Figure 06 Observed Power Spectrum [view large image]

The relationship between the sound wave and the Hubble horizon is crucial to understand the differences between the simplified and observed power spectrum. Figure 07 plots the inverse of the Hubble horizon (in a comoving frame) against the conformal time in the inflationary era (blue), the radiation era (orange), and the matter era (red). For those values of k under the colored curves, the corresponding wavelength is greater than the Hubble horizon. These kinds of sound wave are frozen and cannot oscillate. As time progresses beyond the inflationary era, sound wave with longer and longer wave-length can re-appear first into the radiation era then to the matter era. The curious state

Figure 07 Hubble Horizon and Sound Wave [view large image]

of shrinking horizon in the inflationary era is related to the acceleration of the cosmic expansion in this phase. It can be illustrated mathematically by the Hubble's formula: d = v/H, where H = (dR/dt)/R and R is a time dependent function called the scale

The shape of the observed power spectrum is determined by a number of factors:
1. The minima in Figure 05 between the peaks always reach down to zero. They are lifted upward by Doppler shift, which makes an out-of-phase contribution filling in the zeros (see Figure 06).
2. The varying height of the peaks in Figure 06 is due to the presence of attractive gravity, which causes more compression and less stretching, hence the odd peaks (#1, 3, ...) are higher (more compression) while the even peaks (#0, 2, ...) are lower (less stretching).
3. The first non-zero peak in Figure 06 corresponds to a flat space geometry. For a universe with positive curvature, it would shift to smaller k (to the left of the diagram); while the shift is to larger k for negative curvature.
4. For those waves (with higher k or shorter wavelength) emerging into the radiation era, they encounter a world of diluting density and gravity due to the cosmic expansion. The net effect is to cause the peaks to decrease with increasing k. The power spectrum eventually trails off at very high value of k.
5. For those waves (with lower k or longer wavelength) re-entering into the matter era when the increase in density is almost exactly balanced by the cosmic expansion. As a result, density, sound amplitude, and gravitational potential remain fairly constant through the matter era for small values of k.
6. Absence of the 0^th peak at k = 0 is related to the fact that the sound wave re-enters into the matter era after recombination with no more radiation pressure, so there is no oscillation.

Theoretical Models

As the amplitude and position of the primary and secondary peaks are intrinsically determined by the number of electron scatterers (density) and by the geometry of the Universe, they can be used to calculate the density of baryons and dark matter, as well as other cosmological constants. Specifically, the first and second peaks yield information about the total density, baryon density and the Hubble's constant. Figure 08 shows the different theoretical models - low Hubble's constant H0, dominant cosmologic repulsion, neutrino with mass (Hot Dark Matter), high baryon density, open universe, and early universe with textures (which is a theory different from the inflationary model and based on topological defects1).

Figure 08 Power Spectrum Models [view large image]

Theoretical power spectrum has become the modern computational tool for cosmology. There are essentially four components in its framework:


1. Friedmann-Robertson-Walker (FRW) Universe - It is used as the base for cosmic expansion. The effect of open, closed, or flat space are taken into consideration via the corresponding solutions in FRW.
2. Linearized equations of general relativity - The small perturbations of the metric tensor g produce the temperature fluctuations on the photons sitting in the photon-baryon fluid. This is called the SW (Sachs-Wolfe) effect.
3. Fluid equations - It is believed that the structure of the present universe has evolved from very small initial perturbations, which have grown due to gravity. The universe consists of several different particle species (e.g. photons, neutrinos, baryons and cold dark matter), which interact with each other and have different equations of state. Hence it is necessary to consider the coupled evolution of individual particle species as multicomponent fluid.
4. Boltzmann equation - The equation that governs the temperature fluctuations is derived from the Boltzmann equation. The collision term describes the interaction of the photon with the electrons. The initial power spectrum is usually assumed to be in the form: k^n-1, where k denotes the momentum of the photon, and n = 1 for flat space (n is called the tilt). There are altogether three terms in determining the shape of the power spectrum. The SW effect is the major contribution with the temperature fluctuation T/T ~ -, where is gravitational potential. Then there is the ISW (Integrated Sachs-Wolfe) effect, which modifies the energy of the photons as they climb in and out of the potential well associated with large scale structures. The ISW effect is seen mainly in the lowest multipoles in the power spectrum. The last one is the Doppler effect. It is caused by electron movements in the plasma, because some of the electrons are moving towards the observer and some move away when they last scatter radiation. The temperature fluctuation is given by the formula: T/T ~ v/c with an angular size around 1^o - 2^o

There are altogether 10 parameters in these equations, including the densities of CDM, baryons, neutrinos, vacuum energy and curvature, the reionization optical depth, and the normalization and tilt for both scalar (unpolarized) and tensor (polarized) fluctuations, etc. Usually, numerical computation is used to construct models with various values of the parameters. There is an "industrial standard" computer program called CMBFAST, which can be downloaded free of charge from: http://www.cmbfast.org/. It will crank out power spectrum with input parameters. Figure 09 is one of the animated graphs from: http://space.mit.edu/home/tegmark/cmb/movies.html. It shows the effects of varying the parameters on the theoretical curves. The graph on the top is the CMBR power spectrum, while the one below shows the power spectrum of the large scale structures. Click the STOP button on the toolbar (the ) to view a stationary graph.

Figure 09 Power Spectrum Animation [view large animation]

Some comments on the parameters in the animation:


• - This is the optical depth. It is used to measure the average distance a photon travelled before its original path is altered by hitting something, e.g., an electron. The CMBR would experiences a certain optical depth, if the Universe was reioninized long ago by quasars or early stars. This effect smears out small-scale features in the CMBR power spectrum, suppressing all accoustic peaks by a constant factor exp(-), while leaving the power spectrum on the large scale structures unaffected.
• _k - The space curvature is measured by this parameter from a negative number (open universe) to zero (flat universe), and a positive number (closed universe).
• - This is the cosmological constant to simulate the effect of cosmic repulsion discovered lately. It is sometimes referred to as vacuum energy density.
• _cd - This is the undetected cold matter density in term of the critical density _c, i.e., _cd = _cd/_c
• _b - This is the observable luminous matter (baryons) density in term of the critical density _c, i.e., _b = _b/_c
• f - This is the undetected hot matter density such as neutrinos.
• n_s - It describes the initial clumping of matter in space on different scales. This clumping in turn produced the galaxies and clusters of galaxies we see today. If the amplitude (A_s) of these tiny, "primordial" perturbations was the same on all physical scales, leading to the formation of similar structures on all scales, their spectrum is said to have no tilt (n_s = 1). The subscript "s" stands for scalar, which means the CMBR is un-polarized.
• n_t - This is the tilt for polarized CMBR.
• h - This is the Hubble's constant in unit of 100 km/sec-Mpc.

Table 01 Cosmological Parameters from the WMAP data.