Understanding the Cosmic Microwave Background Temperature Power Spectrum

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2 However, the vast majority of information lies not in the frequency spectrum of the CMB, but in its temperature field. Although the observed average temperature is amazingly uniform across the sky, a good signal-to-noise experiment will reveal small fluctuations around this average. These fluctuations are small (1 part in 10,000!), and in 2003 the satellite experiment WMAP provided the first high resolution, high signal-to-noise, full-sky map of these fluctuations. Since we are interested in the deviation from the average temperature, we generally define a dimensionless quantity Θ(ˆn) = T(ˆn) T T, where ˆn is a direction in the sky, ˆn (θ, φ) We see these temperature fluctuations projected in a 2D spherical surface sky, and so it has become common in the literature to expand the temperature field using spherical harmonics. The spherical harmonics form a complete orthonormal set on the unit sphere and are defined as Y lm = 2l + 1 (l m)! 4π (l + m)! P l m (cosθ)eimφ (2) where the indices l = 0,..., and l m l and Pl m are the Legendre polynomials. l is called the multipole and represents a given angular scale in the sky α, given approximately by α = π/l (in degrees). We can expand our temperature fluctuations field using these functions where Θ(ˆn) = l= l l=0 m= l a lm Y lm (ˆn) (3) a lm = π θ= π 2π φ=0 Θ(ˆn)Y lm(ˆn)dω (4) and, analogously to what we do in Fourier space, we can define a power spectrum of these fluctuations, C l, as the variance of the harmonic coefficients a lm a l m = δ ll δ mm C l (5) where the above average is taken over many ensembles and the delta functions arise from isotropy (we can sample from our own Universe). We only have one Universe, so we are intrinsically limited on the number of independent m-modes we can measure - there are only (2l + 1) of these for each multipole. We can write the following expression for the power spectrum: C l = 1 2l + 1 l a lm 2 (6) m= l This leads to an unavoidable error in our estimation of any given C l of C l = 2/(2l + 1): how well we can estimate an average value from a sample depends on how many points we have on the sample. This is normally called the cosmic variance. In real space, the power spectrum is related to the expectation value of the correlation of the temperature between two points in the sky: ξ ΘΘ (θ) = Θ(ˆn)Θ(ˆn ) = 1 4π (2l + 1)C l P l cosθ, ˆn.ˆn = cosθ (7) l=0 Cosmological models normally predict what the variance of the a lm coefficients is over an ensemble, so they predict the power spectrum. Each Universe is then only one realisation of a given model. 2

3 Figure 1: The CMB power spectrum as a function of angular scale. Red line is our best fit to the model, and the grey band represents the cosmic variance (see text). Under the Inflation paradigm, the temperature fluctuations are Gaussian, which means that the harmonic coeffcients have Gaussian distributions with mean zero and variance given by C l. In this case, all we need to characterise the statistics of our temperature fluctuations field is the power-spectrum - all higher-point statistics will be zero and contain no extra information. The Gaussian hypothesis is now being questioned by detections of non-gaussianity and deviations from isotropy in the WMAP first-year data. However, none of the detections are yet confirmed to be of cosmological origin and we continue to assume Gaussianity throughout this lecture. The sum in equation (3) will generally start at l = 2 and go on to a given l max which is dictated by the resolution of the data. The reasons why we exclude the first two terms are as follows. The monopole (l = 0) term is simply the average temperature over the whole sky (Y 00 = 1/2 π which makes Θ(ˆn) l=0 = 1/4π Θ(ˆn)dφdcosθ Θ(ˆn), where the integrals are done over the entire surface), and so from our definition of Θ(ˆn) it should average to zero. The monopole temperature term would be a valuable source of cosmological information in its own right, but its value can never be determined accurately because of cosmic variance - essentially we have no way of telling if the average temperature we measure locally is different from the average temperature of the Universe. The dipole term (l = 1, α 180 ) is affected by our own motion across space - CMB photons coming towards us will appear blueshifted and those going away from us will appear redshifted. This creates an anisotropy at this scale which dominates over the intrinsic cosmological dipole signal and therefore we normally subtract the monopole/dipole from a CMB map or discard the first two values of the power spectrum prior to any analysis. Our best estimate at what the power spectrum of the observed CMB fluctuations looks like can be seen in Figure 1. It is usually plotted as l(l+1)c l /2π. This is related to the contribution towards the variance of the temperature fluctuations in a patch of sky of size 1/l: Θ 2 = ξ ΘΘ (0) = 1 4π l (2l+1)C l (since P l (1) = 1). The contribution over a range of values of l is given approximately given by 2l Cl dl = 2l 2 C l l l dl l (for l 1) and so 2l 2 C l is proportional to the contribution to the variance per unit lnl. This gives a flat plateau at large angular scales, and brings out a lot of the structure at smaller scales (see later). Relating angular sizes with linear scales Having had a look at what how we observe the CMB radiation today and having looked at some of the formalism we need to analyse it, we would like to start relating what we observe to what was happening at the time of last scattering. One of the first things we can do is to ask how we can relate angular scales in the sky to linear sizes at the time of last scattering. We take the LSS as being a spherical surface at a distance z LS from us. We will take the comoving distance to this surface as being r LS. We want to relate a small angle in the sky θ to the linear comoving distance x 3

4 at last scattering, such that θ x/r (for θ 1 and in flat space). The comoving distance-redshift relation is given by R 0 r = c H 0 dz H(z) (8) where H(z) is the Hubble factor and is given by H(z) = [Ω(1 + z) 2 + Ω v + Ω m (1 + z) 3 + Ω r (1 + z) 4 ] 1/2. This integration can only be done numerically for most cases, but for the case of a matter dominated, flat Universe then the Hubble factor simplifies to H(z) = (1 + z) 3/2 and we get r LS = c R 0 H 0 zls 0 (1 + z) 3/2 dz = 2c (1 + z) 1/2 H 0 R 0 z=zls z=0 = 2c H 0 R 0 (1 (1 + z LS ) 1/2 ) (9) 2c For z LS 1 then r LS is given simply by H 0R 0. Formally, by taking z LS to infinity we are effectively calculating the present-day particle horizon length which is the maximum comoving distance light could have travelled since the Big Bang, d H. So a small angle in the sky θ corresponds to a linear comoving distance at last scattering given by (in radians) θ = x R 0H 0 2c (10) One particular comoving distance at the time of last scattering which we might be interested in is the particle horizon length, which is given by d H (z = z LS ) = z LS which, from (10) and for z LS 1100, means that dz H(z) = 2c (1 + z LS ) 1/2 (11) H 0 R 0 θ LS H = (1 + z LS ) 1/2 1.7 (12) This tells us that scales larger than 1.7 in the sky were not in causal contact at the time of last scattering. However, the fact that we measure the same mean temperature across the entire sky suggests that all scales were once in causal contact - this led to the idea of Inflation, which suggests that the Universe went through a period of very fast expansion, which would have stretched a small, causally connected patch of the Universe into a region of size comparable to the size of the observable Universe today. Inflation theories usually make use of one or more repulsive potential fields to drive the inflationary period. Quantum fluctuations in these fields can then be stretched and propagated into gravitational potential perturbations, which will in turn leave their mark in the temperature and density fields (these are the seeds which will evolve under gravity to give rise to the large-scale structure of the Universe we see today). The whole subject of Inflation is extremely vast and complex and is beyond the point of this lecture. As far as we re concerned, Inflation is a mechanism which provides us with primordial spatial inhomogeneities in the gravitational field, δφ, and with uniformity across the whole sky. Physical Mechanisms: the origin of the anisotropies CMB anisotropies can be classified into primary or secondary anisotropies, according to whether they were created at last scattering or during the photon s path along the line of sight. Photons can be affected by a range of things after last-scattering e.g. re-ionization, passing through hot clusters gas, evolving potential wells, gravitational lensing, etc. While secondary anisotropies hold a good deal of information about the more recent Universe, they are not the subject of this lecture. Mostly, their effect on the temperature power spectrum lies at very small scales (very large l) just beyond our current technical abilities. The exception 4

6 where the derivative is with respect to conformal time η cdt/r(t), which scales out the expansion, and v γ is the photon fluid velocity. Taking into account the Universe s expansion, what is actually conserved is n γ /R 3, and so ṅ γ + 3n γ Ṙ R +.(n γv γ ) = 0 For linear perturbations δn γ = n γ n γ this reduces to ( δnγ ) =.v γ (19) n γ We can relate this to temperature fluctuations by n γ T 3 to give 3Θ = δn γ /n γ. This reduces our continuity equation to Θ = (1/3).v γ or, in Fourier space, to Θ = 1 3 ik.v γ (20) We now consider the momentum of the radiation. Momentum is given by q = (ρ γ + p γ )v γ where (ρ γ + p γ ) is the effective mass and, for radiation, p γ = (1/3)ρ γ. Ignoring gravitational effects and viscosities, the only force is given by the pressure gradient p γ = (1/3) ρ γ. We can then write q = F as and so v γ = Θ or, in Fourier space, 4 3 ρ γ v γ = 1 3 ρ γ (21) v γ = ikθ (22) We now consider only the velocity component along the direction ˆk, as this is the only one with a gravitational source and we write our final continuity and Euler equations as Θ = 1 3 kv γ (Continuity) (23) v γ = kθ (Euler) (24) These can quickly be combined to give Θ k2 Θ = 0 (25) which is a simple harmonic oscillator equation. The 1/3 factor is generally the adiabatic sound speed which is defined as c 2 s ṗ γ/ ρ γ which in this case is equal to 1/3. The general solution for equation (25) is given by Θ(η) = Θ(0)cos(kc s η) + Θ(0) kc s sin (kc s η) (26) By assuming negligible initial velocities and by defining a sound horizon as s c s dη, we simplify our solution to Θ(η) = Θ(0) cos(ks). Let us take some time to revise what we have done. We are trying to analyse the dynamical behaviour of a photon-baryon fluid, and study how temperature fluctuations behave in this system. We took some very constraining assumptions (such as ignoring gravity and the baryons!) and worked on a system whose only force was given by radiation pressure gradients. What we found is that this pressure acts as a restoring force to initial perturbations and we are left with oscillations which propagate at the speed of sound. This is an important results, which holds even when we take into account other effects to make our system a realistic one. This behaviour continues until we hit the temperature of recombination, at which time matter and radiation de-couple and any temperature fluctuations are essentially frozen into the photons temperature, which we measure (nearly unchanged!) today. 6

7 I emphasise that these oscillations are happening at all scales, and we are interested in those which at the time of recombination happen to be at their maxima or minima (remember, we re interested in the temperature squared). If this happens at a conformal time η rec (corresponding to a sound horizon s rec ), then modes will be frozen with an amplitude given by Θ(η rec ) = Θ(0)cos(ks rec ) (27) and those caught at their extrema will have k n s rec = nπ. We can therefore find a fundamental scale of oscillation by taking n = 1 k F = k 1 = π s rec (28) This is our largest oscillating mode, and of course, all of the corresponding overtones will be caught at their extrema too. These will correspond to higher values of k n and are simply oscillations which have had time to go another complete half-oscillation: k 1 corresponds to the oscillation which has had time to compress fully once, k 2 = 2k 1 to the oscillation which has had time to compress and then decompress fully, and so on. We see that the maximum scale at which these fluctuations will happen (related to 1 k F ) is related to the sound horizon at the time of recombination, which was close to the particle horizon. This means that scales larger than this won t be affected by acoustic oscillations, and we wouldn t expect otherwise given that acoustic oscillations can only happen in regions which are causally connected. However, there is a caveat to this toy model. These oscillations also set up velocities in the fluid, which will in turn produce Doppler shifts in the frequencies of the photons. Velocity oscillations are precisely π/2 out of phase with acoustic oscillations (fluid stationary at maximum compression/extension), which in this case cancels the temperature oscillations in the radiation completely and gives a flat Θ. A full treatment should take into account gravity, mass and inertia of the baryons, the evolution of the photon/baryon ratio, viscosity, diffusion and so on. A full solution looks more like (Hu, 1995): < T T 2 [ ] 2 [ ] (η) >= 3 (1 + 3R)Φ cos(kc sη) RΦ + 3 (1 + R)(1 + R) 1/2 Φ sin(kc s η) (29) where R is fluid momentum density and R 3ρ b 4ρ γ z Ω b h The first term represents the acoustic oscillations which are equivalent to what we derived with our toy model and the second term represents the Doppler oscillations, which in this case are much smaller than the acoustic oscillations, but still π/2 out of phase (notice the sine instead of the cosine). However, we don t need a full treatment to understand what the effect of gravity and the introduction of baryons does to this system, at least qualitatively. Gravity s main effect is to introduce another force into the system, F grav = m Φ, which adds a term to our Euler equations. This in effect creates a potential well and changes the range of the temperature oscillations, effectively their amplitude. If we now introduce baryons to the system, then we are introducing mass into the system. As in a classical system consisting of a spring (restoring force) and a mass attached to the end of ths spring (the photon-baryon fluid), increasing the mass will cause the spring to fall further, but it won t change the maximum rebound height. Recall that our peaks in the temperature fluctuations alternate between compression and expansion of the plasma, so introducing matter into the system changes only every other peak - those corresponding to compression of the fluid (matter is falling further into the potential well). Hence we expect even numbered peaks to be suppressed in relation to odd numbered peaks. We also expect curvature to affect the observed angular temperature anisotropies, as it affects the path the CMB photons will have taken to get to us. It is essentially a problem of geometry: in a closed Universe, a given angle subtended in the sky will correspond to a smaller linear distance at last scattering than in a flat Universe, and so curvature shifts the peaks along the multipole axis. The detection of the first acoustic peak at an l 200 provided a good constraint on the flatness of our Universe. 7

8 At smaller scales, however, we see that these oscillations are clearly damped. This comes from the fact that the last scattering surface has a finite thickness, and therefore recombination and last scattering don t happen at the same time. This damps out small scale fluctuations (where the scale is related to the thickness of the scattering surface), as photons still have to random walk out of this shell before they are essentially free, smoothing out the fluctuations. Hopefully, it is clear now that the temperature fluctuations we see in the sky hold a vast amount of information about our Universe s content and evolution. And in fact, if the CMB is Gaussian, all of this information will be imprinted in the angular power spectrum C l. The extraction of information about cosmological parameters from the power spectrum is a complicated process, and it is worth mentioning that there are a number of degeneracies between these parameters which can only be lifted by combining CMB data with other data sets (such as supernova experiments or galaxy surveys). Nevertheless, CMB analysis is now precision Cosmology, which promises to continue to help us to construct a better image of our Universe. 8

9 Bibliography P. Castro. The Statistics of the Cosmic Microwave Background. Ph.D. Thesis, W. T. Hu. CMB Temperature and Polarization Anisotroopy Fundamentals. W. T. Hu. Wandering in the Background: a Cosmic Microwave Background Explorer. Ph.D. Thesis, Jan W. T. Hu and S. Dodelson. Cosmic Microwave Background Anisotropies. J. A. Peacock. Cosmological Physics. Cosmological Physics, by John A. Peacock, pp ISBN X. Cambridge, UK: Cambridge University Press, January 1999., Jan G. F. Scott and Smoot. Cosmic Microwave Background Mini-Review. Physics Letters B,

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