### 3. ACOUSTIC PEAKS

When the temperature of the Universe was ~ 3000K at a redshift z* 103, electrons and protons combined to form neutral hydrogen, an event usually known as recombination ([Peebles, 1968, Zel'dovich et al, 1969]; see [Seager et al, 2000] for recent refinements). Before this epoch, free electrons acted as glue between the photons and the baryons through Thomson and Coulomb scattering, so the cosmological plasma was a tightly coupled photon-baryon fluid [Peebles & Yu, 1970]. The spectrum depicted in Plate 1 can be explained almost completely by analyzing the behavior of this pre-recombination fluid.

In Section 3.1, we start from the two basic equations of fluid mechanics and derive the salient characteristics of the anisotropy spectrum: the existence of peaks and troughs; the spacing between adjacent peaks; and the location of the first peak. These properties depend in decreasing order of importance on the initial conditions, the energy contents of the Universe before recombination and those after recombination. Ironically, the observational milestones have been reached in almost the opposite order. Throughout the 1990's constraints on the location of the first peak steadily improved culminating with precise determinations from the TOCO [Miller et al, 1999], Boomerang, [de Bernardis et al, 2000] and Maxima-1 [Hanany et al, 2000] experiments (see Plate 1 top). In the working cosmological model it shows up right where it should be if the present energy density of the Universe is equal to the critical density, i.e. if the Universe is flat. The skeptic should note that the working cosmological model assumes a particular form for the initial conditions and energy contents of the Universe before recombination which we shall see have only recently been tested directly (with an as yet much lower level of statistical confidence) with the higher peaks.

In Section 3.2 we introduce the initial conditions that apparently are the source of all clumpiness in the Universe. In the context of ab initio models, the term "initial conditions" refers to the physical mechanism that generates the primordial small perturbations. In the working cosmological model, this mechanism is inflation and it sets the initial phase of the oscillations to be the same across all Fourier modes. Remarkably, from this one fact alone comes the prediction that there will be peaks and troughs in the amplitude of the oscillations as a function of wavenumber. Additionally the inflationary prediction of an approximately scale-invariant amplitude of the initial perturbations implies roughly scale-invariant oscillations in the power spectrum. And inflation generically predicts a flat Universe. These are all falsifiable predictions of the simplest inflationary models and they have withstood the test against observations to date.

The energy contents of the Universe before recombination all leave their distinct signatures on the oscillations as discussed in Section 3.3-Section 3.5. In particular, the cold dark matter and baryon signatures have now been seen in the data [Halverson et al, 2001, Netterfield et al, 2001, Lee et al, 2001]. The coupling between electrons and photons is not perfect, especially as one approaches the epoch of recombination. As discussed in Section 3.6, this imperfect coupling leads to damping in the anisotropy spectrum: very small scale inhomogeneities are smoothed out. The damping phenomenon has now been observed by the CBI experiment [Padin et al, 2001]. Importantly, fluid imperfections also generate linear polarization as covered in Section 3.7. Because the imperfection is minimal and appears only at small scales, the polarization generated is small and has not been detected to date.

After recombination the photons basically travel freely to us today, so the problem of translating the acoustic inhomogeneities in the photon distribution at recombination to the anisotropy spectrum today is simply one of projection. This projection depends almost completely on one number, the angular diameter distance between us and the surface of last scattering. That number depends on the energy contents of the Universe after recombination through the expansion rate. The hand waving projection argument of Section 3.1 is formalized in Section 3.8, in the process introducing the popular code used to compute anisotropies, CMBFAST. Finally, we discuss the sensitivity of the acoustic peaks to cosmological parameters in Section 3.9.

For pedagogical purposes, let us begin with an idealization of a perfect photon-baryon fluid and neglect the dynamical effects of gravity and the baryons. Perturbations in this perfect fluid can be described by a simple continuity and an Euler equation that encapsulate the basic properties of acoustic oscillations.

The discussion of acoustic oscillations will take place exclusively in Fourier space. For example, we decompose the monopole of the temperature field into

 (6)

and omit the subscript 00 on the Fourier amplitude. Since perturbations are very small, the evolution equations are linear, and different Fourier modes evolve independently. Therefore, instead of partial differential equations for a field (x), we have ordinary differential equations for (k). In fact, due to rotational symmetry, all (k) for a given k obey the same equations. Here and in the following sections, we omit the wavenumber argument k where no confusion with physical space quantities will arise.

Temperature perturbations in Fourier space obey

 (7)

This equation for the photon temperature , which does indeed look like the familiar continuity equation in Fourier space (derivatives become wavenumbers k), has a number of subtleties hidden in it, due to the cosmological setting. First, the "time" derivative here is actually with respect to conformal time dt / a(t). Since we are working in units in which the speed of light c = 1, is also the maximum comoving distance a particle could have traveled since t = 0. It is often called the comoving horizon or more specifically the comoving particle horizon. The physical horizon is a times the comoving horizon.

Second, the photon fluid velocity here v has been written as a scalar instead of a vector. In the early universe, only the velocity component parallel to the wavevector k is expected to be important, since they alone have a source in gravity. Specifically, v = - iv . In terms of the moments introduced in Section 2, v represents a dipole moment directed along k. The factor of 1/3 comes about since continuity conserves photon number not temperature and the number density n T3. Finally, we emphasize that, for the time being, we are neglecting the effects of gravity.

The Euler equation for a fluid is an expression of momentum conservation. The momentum density of the photons is ( + p) v, where the photon pressure p = / 3. In the absence of gravity and viscous fluid imperfections, pressure gradients p = / 3 supply the only force. Since T4, this becomes 4k / 3 in Fourier space. The Euler equation then becomes

 (8)

Differentiating the continuity equation and inserting the Euler equation yields the most basic form of the oscillator equation

 (9)

where cs ( / )1/2 = 1 / 31/2 is the sound speed in the (dynamically baryon-free) fluid. What this equation says is that pressure gradients act as a restoring force to any initial perturbation in the system which thereafter oscillate at the speed of sound. Physically these temperature oscillations represent the heating and cooling of a fluid that is compressed and rarefied by a standing sound or acoustic wave. This behavior continues until recombination. Assuming negligible initial velocity perturbations, we have a temperature distribution at recombination of

 (10)

where s = cs d / 31/2 is the distance sound can travel by , usually called the sound horizon. Asterisks denote evaluation at recombination z*.

In the limit of scales large compared with the sound horizon ks << 1, the perturbation is frozen into its initial conditions. This is the gist of the statement that the large-scale anisotropies measured by COBE directly measure the initial conditions. On small scales, the amplitude of the Fourier modes will exhibit temporal oscillations, as shown in Figure 1 [with = 0, i = 3(0) for this idealization]. Modes that are caught at maxima or minima of their oscillation at recombination correspond to peaks in the power, i.e. the variance of (k, *). Because sound takes half as long to travel half as far, modes corresponding to peaks follow a harmonic relationship kn = n / s*, where n is an integer (see Figure 1a).

 Figure 1. Idealized acoustic oscillations. (a) Peak scales: the wavemode that completes half an oscillation by recombination sets the physical scale of the first peak. Both minima and maxima correspond to peaks in power (dashed lines, absolute value) and so higher peaks are integral multiples of this scale with equal height. Plotted here is the idealization of Equation (15) (constant potentials, no baryon loading). (b) Baryon loading. Baryon loading boosts the amplitudes of every other oscillation. Plotted here is the idealization of Equation (16) (constant potentials and baryon loading R = 1/6) for the third peak.

How does this spectrum of inhomogeneities at recombination appear to us today? Roughly speaking, a spatial inhomogeneity in the CMB temperature of wavelength appears as an angular anisotropy of scale / D where D(z) is the comoving angular diameter distance from the observer to redshift z. We will address this issue more formally in Section 3.8. In a flat universe, D* = 0 - * 0, where 0 (z = 0). In harmonic space, the relationship implies a coherent series of acoustic peaks in the anisotropy spectrum, located at

 (11)

To get a feel for where these features should appear, note that in a flat matter dominated universe (1 + z)-1/2 so that * / 0 1/30 2°. Equivalently 1 200. Notice that since we are measuring ratios of distances the absolute distance scale drops out; we shall see in Section 3.5 that the Hubble constant sneaks back into the problem because the Universe is not fully matter-dominated at recombination.

 Figure 2. Angular diameter distance. In a closed universe, objects are further than they appear to be from Euclidean (flat) expectations corresponding to the difference between coordinate distance d and angular diameter distance D. Consequently, at a fixed coordinate distance, a given angle corresponds to a smaller spatial scale in a closed universe. Acoustic peaks therefore appear at larger angles or lower in a closed universe. The converse is true for an open universe.

In a spatially curved universe, the angular diameter distance no longer equals the coordinate distance making the peak locations sensitive to the spatial curvature of the Universe [Doroshkevich et al, 1978, Kamionkowski et al, 1994]. Consider first a closed universe with radius of curvature R = H0-1 |tot - 1|-1/2. Suppressing one spatial coordinate yields a 2-sphere geometry with the observer situated at the pole (see Figure 2). Light travels on lines of longitude. A physical scale at fixed latitude given by the polar angle subtends an angle = / R sin . For << 1, a Euclidean analysis would infer a distance D = R sin , even though the coordinate distance along the arc is d = R; thus

 (12)

For open universes, simply replace sin with sinh. The result is that objects in an open (closed) universe are closer (further) than they appear, as if seen through a lens. In fact one way of viewing this effect is as the gravitational lensing due to the background density (c.f. Section 4.2.4). A given comoving scale at a fixed distance subtends a larger (smaller) angle in a closed (open) universe than a flat universe. This strong scaling with spatial curvature indicates that the observed first peak at 1 200 constrains the geometry to be nearly spatially flat. We will implicitly assume spatial flatness in the following sections unless otherwise stated.

Finally in a flat dark energy dominated universe, the conformal age of the Universe decreases approximately as 0 0(1 + lnm0.085). For reasonable m, this causes only a small shift of 1 to lower multipoles (see Plate 4) relative to the effect of curvature. Combined with the effect of the radiation near recombination, the peak locations provides a means to measure the physical age t0 of a flat universe [Hu et al, 2001].

As suggested above, observations of the location of the first peak strongly point to a flat universe. This is encouraging news for adherents of inflation, a theory which initially predicted tot = 1 at a time when few astronomers would sign on to such a high value (see [Liddle & Lyth, 1993] for a review). However, the argument for inflation goes beyond the confirmation of flatness. In particular, the discussion of the last subsection begs the question: whence (0), the initial conditions of the temperature fluctuations? The answer requires the inclusion of gravity and considerations of causality which point to inflation as the origin of structure in the Universe.

The calculations of the typical angular scale of the acoustic oscillations in the last section are familiar in another context: the horizon problem. Because the sound speed is near the speed of light, the degree scale also marks the extent of a causally connected region or particle horizon at recombination. For the picture in the last section to hold, the perturbations must have been laid down while the scales in question were still far outside the particle horizon 2. The recent observational verification of this basic peak structure presents a problem potentially more serious than the original horizon problem of approximate isotropy: the mechanism which smooths fluctuations in the Universe must also regenerate them with superhorizon sized correlations at the 10-5 level. Inflation is an idea that solves both problems simultaneously.

The inflationary paradigm postulates that an early phase of near exponential expansion of the Universe was driven by a form of energy with negative pressure. In most models, this energy is usually provided by the potential energy of a scalar field. The inflationary era brings the observable universe to a nearly smooth and spatially flat state. Nonetheless, quantum fluctuations in the scalar field are unavoidable and also carried to large physical scales by the expansion. Because an exponential expansion is self-similar in time, the fluctuations are scale-invariant, i.e. in each logarithmic interval in scale the contribution to the variance of the fluctuations is equal. Since the scalar field carries the energy density of the Universe during inflation, its fluctuations induce variations in the spatial curvature [Guth & Pi, 1985, Hawking, 1982, Bardeen et al, 1983]. Instead of perfect flatness, inflation predicts that each scale will resemble a very slightly open or closed universe. This fluctuation in the geometry of the Universe is essentially frozen in while the perturbation is outside the horizon [Bardeen, 1980].

Formally, curvature fluctuations are perturbations to the space-space piece of the metric. In a Newtonian coordinate system, or gauge, where the metric is diagonal, the spatial curvature fluctuation is called gij = 2a2 ij (see e.g. [Ma & Bertschinger, 1995]). The more familiar Newtonian potential is the time-time fluctuation gtt = 2 and is approximately - . Approximate scale invariance then says that 2 k3 P(k) / 22 kn-1 where P(k) is the power spectrum of and the tilt n 1.

Now let us relate the inflationary prediction of scale-invariant curvature fluctuations to the initial temperature fluctuations. Newtonian intuition based on the Poisson equation k2 = 4 Ga2 tells us that on large scales (small k) density and hence temperature fluctuations should be negligible compared with Newtonian potential. General relativity says otherwise because the Newtonian potential is also a time-time fluctuation in the metric. It corresponds to a temporal shift of t / t = . The CMB temperature varies as the inverse of the scale factor, which in turn depends on time as a t2/[3(1+p/)]. Therefore, the fractional change in the CMB temperature

 (13)

Thus, a temporal shift produces a temperature perturbation of - / 2 in the radiation dominated era (when p = / 3) and -2 / 3 in the matter dominated epoch (p = 0) ([Peacock, 1991]; [White & Hu, 1997]). The initial temperature perturbation is therefore inextricably linked with the initial gravitational potential perturbation. Inflation predicts scale-invariant initial fluctuations in both the CMB temperature and the spatial curvature in the Newtonian gauge.

Alternate models which seek to obey the causality can generate curvature fluctuations only inside the particle horizon. Because the perturbations are then not generated at the same epoch independent of scale, there is no longer a unique relationship between the phase of the oscillators. That is, the argument of the cosine in Equation (10) becomes ks* + (k), where is a phase which can in principle be different for different wavevectors, even those with the same magnitude k. This can lead to temporal incoherence in the oscillations and hence a washing out of the acoustic peaks [Albrecht et al, 1996], most notably in cosmological defect models [Allen et al, 1997, Seljak et al, 1997]. Complete incoherence is not a strict requirement of causality since there are other ways to synch up the oscillations. For example, many isocurvature models, where the initial spatial curvature is unperturbed, are coherent since their oscillations begin with the generation of curvature fluctuations at horizon crossing [Hu & White, 1996]. Still they typically have 0 (c.f. [Turok, 1996]). Independent of the angular diameter distance D*, the ratio of the peak locations gives the phase: 1 : 2 : 3 ~ 1 : 2 : 3 for = 0. Likewise independent of a constant phase, the spacing of the peaks n - n-1 = A gives a measure of the angular diameter distance [Hu & White, 1996]. The observations, which indicate coherent oscillations with = 0, therefore have provided a non-trivial test of the inflationary paradigm and supplied a substantially more stringent version of the horizon problem for contenders to solve.

We saw above that fluctuations in a scalar field during inflation get turned into temperature fluctuations via the intermediary of gravity. Gravity affects in more ways than this. The Newtonian potential and spatial curvature alter the acoustic oscillations by providing a gravitational force on the oscillator. The Euler equation (8) gains a term on the rhs due to the gradient of the potential k. The main effect of gravity then is to make the oscillations a competition between pressure gradients k and potential gradients k with an equilibrium when + = 0.

Gravity also changes the continuity equation. Since the Newtonian curvature is essentially a perturbation to the scale factor, changes in its value also generate temperature perturbations by analogy to the cosmological redshift = - and so the continuity equation (7) gains a contribution of - on the rhs.

These two effects bring the oscillator equation (9) to

 (14)

In a flat universe and in the absence of pressure, and are constant. Also, in the absence of baryons, cs2 = 1/3 so the new oscillator equation is identical to Equation (9) with replaced by + . The solution in the matter dominated epoch is then

 (15)

where md represents the start of the matter dominated epoch (see Figure 1a). We have used the matter dominated "initial conditions" for given in the previous section assuming large scales, ksmd << 1.

The results from the idealization of Section 3.1 carry through with a few exceptions. Even without an initial temperature fluctuation to displace the oscillator, acoustic oscillations would arise by the infall and compression of the fluid into gravitational potential wells. Since it is the effective temperature + that oscillates, they occur even if (0) = 0. The quantity + can be thought of as an effective temperature in another way: after recombination, photons must climb out of the potential well to the observer and thus suffer a gravitational redshift of T / T = . The effective temperature fluctuation is therefore also the observed temperature fluctuation. We now see that the large scale limit of Equation (15) recovers the famous Sachs-Wolfe result that the observed temperature perturbation is / 3 and overdense regions correspond to cold spots on the sky [Sachs & Wolfe, 1967]. When < 0, although is positive, the effective temperature + is negative. The plasma begins effectively rarefied in gravitational potential wells. As gravity compresses the fluid and pressure resists, rarefaction becomes compression and rarefaction again. The first peak corresponds to the mode that is caught in its first compression by recombination. The second peak at roughly half the wavelength corresponds to the mode that went through a full cycle of compression and rarefaction by recombination. We will use this language of the compression and rarefaction phase inside initially overdense regions but one should bear in mind that there are an equal number of initially underdense regions with the opposite phase.

So far we have been neglecting the baryons in the dynamics of the acoustic oscillations. To see whether this is a reasonable approximation consider the photon-baryon momentum density ratio R = (pb + b / (p + ) 30b h2(z / 103)-1. For typical values of the baryon density this number is of order unity at recombination and so we expect baryonic effects to begin appearing in the oscillations just as they are frozen in.

Baryons are conceptually easy to include in the evolution equations since their momentum density provides extra inertia in the joint Euler equation for pressure and potential gradients to overcome. Since inertial and gravitational mass are equal, all terms in the Euler equation save the pressure gradient are multiplied by 1 + R leading to the revised oscillator equation [Hu & Sugiyama, 1995]

 (16)

where we have used the fact that the sound speed is reduced by the baryons to cs = 1 / [3(1 + R)]1/2.

To get a feel for the implications of the baryons take the limit of constant R, and . Then d2(R ) / d2(= 0) may be added to the left hand side to again put the oscillator equation in the form of Equation (9) with + (1 + R). The solution then becomes

 (17)

Aside from the lowering of the sound speed which decreases the sound horizon, baryons have two distinguishing effects: they enhance the amplitude of the oscillations and shift the equilibrium point to = - (1 + R) (see Figure 1b). These two effects are intimately related and are easy to understand since the equations are exactly those of a mass m = 1 + R on a spring in a constant gravitational field. For the same initial conditions, increasing the mass causes the oscillator to fall further in the gravitational field leading to larger oscillations and a shifted zero point.

The shifting of the zero point of the oscillator has significant phenomenological consequences. Since it is still the effective temperature + that is the observed temperature, the zero point shift breaks the symmetry of the oscillations. The baryons enhance only the compressional phase, i.e. every other peak. For the working cosmological model these are the first, third, fifth... Physically, the extra gravity provided by the baryons enhance compression into potential wells.

These qualitative results remain true in the presence of a time-variable R. An additional effect arises due to the adiabatic damping of an oscillator with a time-variable mass. Since the energy/frequency of an oscillator is an adiabatic invariant, the amplitude must decay as (1 + R)-1/4. This can also be understood by expanding the time derivatives in Equation (16) and identifying the term as the remnant of the familiar expansion drag on baryon velocities.

We have hitherto also been neglecting the energy density of the radiation in comparison to the matter. The matter-to-radiation ratio scales as m / r 24m h2(z / 103)-1 and so is also of order unity at recombination for reasonable parameters. Moreover fluctuations corresponding to the higher peaks entered the sound horizon at an earlier time, during radiation domination.

Including the radiation changes the expansion rate of the Universe and hence the physical scale of the sound horizon at recombination. It introduces yet another potential ambiguity in the interpretation of the location of the peaks. Fortunately, the matter-radiation ratio has another effect in the power spectrum by which it can be distinguished. Radiation drives the acoustic oscillations by making the gravitational force evolve with time [Hu & Sugiyama, 1995]. Matter does not.

The exact evolution of the potentials is determined by the relativistic Poisson equation. But qualitatively, we know that the background density is decreasing with time, so unless the density fluctuations in the dominant component grow unimpeded by pressure, potentials will decay. In particular, in the radiation dominated era once pressure begins to fight gravity at the first compressional maxima of the wave, the Newtonian gravitational potential and spatial curvature must decay (see Figure 3).

 Figure 3. Radiation driving and diffusion damping. The decay of the potential drives the oscillator in the radiation dominated epoch. Diffusion generates viscosity , i.e. a quadrupole moment in the temperature, which damps oscillations and generates polarization. Plotted here is the numerical solution to Equation (18) and Equation (19) for a mode with wavelength much smaller than the sound horizon at decoupling, ks* >> 1.

This decay actually drives the oscillations: it is timed to leave the fluid maximally compressed with no gravitational potential to fight as it turns around. The net effect is doubled since the redshifting from the spatial metric fluctuation also goes away at the same time. When the Universe becomes matter dominated the gravitational potential is no longer determined by photon-baryon density perturbations but by the pressureless cold dark matter. Therefore, the amplitudes of the acoustic peaks increase as the cold dark matter-to-radiation ratio decreases [Seljak, 1994, Hu & Sugiyama, 1995]. Density perturbations in any form of radiation will stop growing around horizon crossing and lead to this effect. The net result is that across the horizon scale at matter radiation equality (keq (4 - 22) / eq) the acoustic amplitude increases by a factor of 4-5 [Hu & Sugiyama, 1996]. By eliminating gravitational potentials, photon-baryon acoustic oscillations eliminate the alternating peak heights from baryon loading. The observed high third peak [Halverson et al, 2001] is a good indication that cold dark matter both exists and dominates the energy density at recombination.

The photon-baryon fluid has slight imperfections corresponding to shear viscosity and heat conduction in the fluid [Weinberg, 1971]. These imperfections damp acoustic oscillations. To consider these effects, we now present the equations of motion of the system in their full form, including separate continuity and Euler equations for the baryons. Formally the continuity and Euler equations follow from the covariant conservation of the joint stress-energy tensor of the photon-baryon fluid. Because photon and baryon numbers are separately conserved, the continuity equations are unchanged,

 (18)

where b and vb are the density perturbation and fluid velocity of the baryons. The Euler equations contain qualitatively new terms

 (19)

For the baryons the first term on the right accounts for cosmological expansion, which makes momenta decay as a-1. The third term on the right accounts for momentum exchange in the Thomson scattering between photons and electrons (protons are very tightly coupled to electrons via Coulomb scattering), with ne T a the differential Thomson optical depth, and is compensated by its opposite in the photon Euler equation. These terms are the origin of heat conduction imperfections. If the medium is optically thick across a wavelength, / k >> 1 and the photons and baryons cannot slip past each other. As it becomes optically thin, slippage dissipates the fluctuations.

In the photon Euler equation there is an extra force on the rhs due to anisotropic stress gradients or radiation viscosity in the fluid, . The anisotropic stress is directly proportional to the quadrupole moment of the photon temperature distribution. A quadrupole moment is established by gradients in v as photons from say neighboring temperature crests meet at a trough (see Plate 3, inset). However it is destroyed by scattering. Thus = 2(kv / ) Av, where the order unity constant can be derived from the Boltzmann equation Av = 16/15 [Kaiser, 1983]. Its evolution is shown in Figure 3. With the continuity Equation (7), kv -3 and so viscosity takes the form of a damping term. The heat conduction term can be shown to have a similar effect by expanding the Euler equations in k / . The final oscillator equation including both terms becomes

 (20)

where the heat conduction coefficient Ah = R2 / (1 + R). Thus we expect the inhomogeneities to be damped by a exponential factor of order e-k2 / (see Figure 3). The damping scale kd is thus of order ( / )1/2, corresponding to the geometric mean of the horizon and the mean free path. Damping can be thought of as the result of the random walk in the baryons that takes photons from hot regions into cold and vice-versa [Silk, 1968]. Detailed numerical integration of the equations of motion are required to track the rapid growth of the mean free path and damping length through recombination itself. These calculations show that the damping scale is of order kds* 10 leading to a substantial suppression of the oscillations beyond the third peak.

How does this suppression depend on the cosmological parameters? As the matter density m h2 increases, the horizon * decreases since the expansion rate goes up. Since the diffusion length is proportional to (*)1/2, it too decreases as the matter density goes up but not as much as the angular diameter distance D* which is also inversely proportional to the expansion rate. Thus, more matter translates into more damping at a fixed multipole moment; conversely, it corresponds to slightly less damping at a fixed peak number. The dependence on baryons is controlled by the mean free path which is in turn controlled by the free electron density: the increase in electron density due to an increase in the baryons is partially offset by a decrease in the ionization fraction due to recombination. The net result under the Saha approximation is that the damping length scales approximately as (b h2)-1/4. Accurate fitting formulae for this scale in terms of cosmological parameters can be found in [Hu & White, 1997c].

The dissipation of the acoustic oscillations leaves a signature in the polarization of CMB in its wake (see e.g. [Hu & White, 1997a] and references therein for a more complete treatment). Much like reflection off of a surface, Thomson scattering induces a linear polarization in the scattered radiation. Consider incoming radiation in the - x direction scattered at right angles into the z direction (see Plate 2, left panel). Heuristically, incoming radiation shakes an electron in the direction of its electric field vector or polarization causing it to radiate with an outgoing polarization parallel to that direction. However since the outgoing polarization must be orthogonal to the outgoing direction, incoming radiation that is polarized parallel to the outgoing direction cannot scatter leaving only one polarization state. More generally, the Thomson differential cross section dT / d | . |2.

Unlike the reflection of sunlight off of a surface, the incoming radiation comes from all angles. If it were completely isotropic in intensity, radiation coming along the would provide the polarization state that is missing from that coming along leaving the net outgoing radiation unpolarized. Only a quadrupole temperature anisotropy in the radiation generates a net linear polarization from Thomson scattering. As we have seen, a quadrupole can only be generated causally by the motion of photons and then only if the Universe is optically thin to Thomson scattering across this scale (i.e. it is inversely proportional to ). Polarization generation suffers from a Catch-22: the scattering which generates polarization also suppresses its quadrupole source.

The fact that the polarization strength is of order the quadrupole explains the shape and height of the polarization spectra in Plate 1b. The monopole and dipole and v are of the same order of magnitude at recombination, but their oscillations are / 2 out of phase as follows from Equation (9) and Equation (10). Since the quadrupole is of order kv / (see Figure 3), the polarization spectrum should be smaller than the temperature spectrum by a factor of order k / at recombination. As in the case of the damping, the precise value requires numerical work [Bond & Efstathiou, 1987] since changes so rapidly near recombination. Calculations show a steady rise in the polarized fraction with increasing l or k to a maximum of about ten percent before damping destroys the oscillations and hence the dipole source. Since v is out of phase with the monopole, the polarization peaks should also be out of phase with the temperature peaks. Indeed, Plate 1b shows that this is the case. Furthermore, the phase relation also tells us that the polarization is correlated with the temperature perturbations. The correlation power CE being the product of the two, exhibits oscillations at twice the acoustic frequency.

Until now, we have focused on the polarization strength without regard to its orientation. The orientation, like a 2 dimensional vector, is described by two components E and B. The E and B decomposition is simplest to visualize in the small scale limit, where spherical harmonic analysis coincides with Fourier analysis [Seljak, 1997]. Then the wavevector k picks out a preferred direction against which the polarization direction is measured (see Plate 2, right panel). Since the linear polarization is a "headless vector" that remains unchanged upon a 180° rotation, the two numbers E and B that define it represent polarization aligned or orthogonal with the wavevector (positive and negative E) and crossed at ± 45° (positive and negative B).

 Plate 2. Polarization generation and classification. Left: Thomson scattering of quadrupole temperature anisotropies (depicted here in the - plane) generates linear polarization. Right: Polarization in the - plane along the outgoing axis. The component of the polarization that is parallel or perpendicular to the wavevector k is called the E-mode and the one at 45° angles is called the B-mode.

In linear theory, scalar perturbations like the gravitational potential and temperature perturbations have only one intrinsic direction associated with them, that provided by k, and the orientation of the polarization inevitably takes it cue from that one direction, thereby producing an E -mode. The generalization to an all-sky characterization of the polarization changes none of these qualitative features. The E -mode and the B -mode are formally distinguished by the orientation of the Hessian of the Stokes parameters which define the direction of the polarization itself. This geometric distinction is preserved under summation of all Fourier modes as well as the generalization of Fourier analysis to spherical harmonic analysis.

The acoustic peaks in the polarization appear exclusively in the EE power spectrum of Equation (5). This distinction is very useful as it allows a clean separation of this effect from those occuring beyond the scope of the linear perturbation theory of scalar fluctuations: in particular, gravitational waves (see Section 4.2.3) and gravitational lensing (see Section 4.2.4). Moreover, in the working cosmological model, the polarization peaks and correlation are precise predictions of the temperature peaks as they depend on the same physics. As such their detection would represent a sharp test on the implicit assumptions of the working model, especially its initial conditions and ionization history.

The discussion in the previous sections suffices for a qualitative understanding of the acoustic peaks in the power spectra of the temperature and polarization anisotropies. To refine this treatment we must consider more carefully the sources of anisotropies and their projection into multipole moments.

Because the description of the acoustic oscillations takes place in Fourier space, the projection of inhomogeneities at recombination onto anisotropies today has an added level of complexity. An observer today sees the acoustic oscillations in effective temperature as they appeared on a spherical shell at x = D* at recombination, where is the direction vector, and D* = 0 - * is the distance light can travel between recombination and the present (see Plate 3). Having solved for the Fourier amplitude [ + ](k, *), we can expand the exponential in Equation (6) in terms of spherical harmonics, so the observed anisotropy today is

 (21)

where the projected source a(k) = [ + ](k, *) j(kD*). Because the spherical harmonics are orthogonal, Equation (1) implies that m today is given by the integral in square brackets today. A given plane wave actually produces a range of anisotropies in angular scale as is obvious from Plate 3. The one-to-one mapping between wavenumber and multipole moment described in Section 3.1 is only approximately true and comes from the fact that the spherical Bessel function j(kD*) is strongly peaked at kD* . Notice that this peak corresponds to contributions in the direction orthogonal to the wavevector where the correspondence between and k is one-to-one (see Plate 3).

 Plate 3. Integral approach. CMB anisotropies can be thought of as the line-of-sight projection of various sources of plane wave temperature and polarization fluctuations: the acoustic effective temperature and velocity or Doppler effect (see Section 3.8), the quadrupole sources of polarization (see Section 3.7) and secondary sources (see Section 4.2, Section 4.3). Secondary contributions differ in that the region over which they contribute is thick compared with the last scattering surface at recombination and the typical wavelength of a perturbation.

There is one more subtlety involved when passing from acoustic oscillations to anisotropies. Recall from Section 3.5 that radiation leads to decay of the gravitational potentials. Residual radiation after decoupling therefore implies that the effective temperature is not precisely [ + ](*). The photons actually have slightly shallower potentials to climb out of and lose the perturbative analogue of the cosmological redshift, so the [ + ](*) overestimates the difference between the true photon temperature and the observed temperature. This effect of course is already in the continuity equation for the monopole Equation (18) and so the source in Equation (21) gets generalized to

 (22)

The last term vanishes for constant gravitational potentials, but is non-zero if residual radiation driving exists, as it will in low m h2 models. Note that residual radiation driving is particularly important because it adds in phase with the monopole: the potentials vary in time only near recombination, so the Bessel function can be set to jl(kD*) and removed from the integral. This complication has the effect of decreasing the multipole value of the first peak 1 as the matter-radiation ratio at recombination decreases [Hu & Sugiyama, 1995]. Finally, we mention that time varying potentials can also play a role at very late times due to non-linearities or the importance of a cosmological constant for example. Those contributions, to be discussed more in Section 4.2.1, are sometimes referred to as late Integrated Sachs-Wolfe effects, and do not add coherently with [ + ](*).

Putting these expressions together and squaring, we obtain the power spectrum of the acoustic oscillations

 (23)

This formulation of the anisotropies in terms of projections of sources with specific local angular structure can be completed to include all types of sources of temperature and polarization anisotropies at any given epoch in time linear or non-linear: the monopole, dipole and quadrupole sources arising from density perturbations, vorticity and gravitational waves [Hu & White, 1997b]. In a curved geometry one replaces the spherical Bessel functions with ultraspherical Bessel functions [Abbott & Schaefer, 1986, Hu et al, 1998]. Precision in the predictions of the observables is then limited only by the precision in the prediction of the sources. This formulation is ideal for cases where the sources are governed by non-linear physics even though the CMB responds linearly as we shall see in Section 4.

Perhaps more importantly, the widely-used CMBFAST code [Seljak & Zaldarriaga, 1996] exploits these properties to calculate the anisotropies in linear perturbation efficiently. It numerically solves for the smoothly-varying sources on a sparse grid in wavenumber, interpolating in the integrals for a handful of 's in the smoothly varying C. It has largely replaced the original ground breaking codes [Wilson & Silk, 1981, Bond & Efstathiou, 1984, Vittorio & Silk, 1984] based on tracking the rapid temporal oscillations of the multipole moments that simply reflect structure in the spherical Bessel functions themselves.

 Plate 4. Sensitivity of the acoustic temperature spectrum to four fundamental cosmological parameters (a) the curvature as quantified by tot (b) the dark energy as quantified by the cosmological constant (w = - 1) (c) the physical baryon density b h2 (d) the physical matter density m h2, all varied around a fiducial model of tot = 1, = 0.65, b h2 = 0.02, m h2 = 0.147, n = 1, zri = 0, Ei = 0.

The phenomenology of the acoustic peaks in the temperature and polarization is essentially described by 4 observables and the initial conditions [Hu et al, 1997]. These are the angular extents of the sound horizon a D* / s*, the particle horizon at matter radiation equality eq keq D* and the damping scale d kd D* as well as the value of the baryon-photon momentum density ratio R*. a sets the spacing between of the peaks; eq and d compete to determine their amplitude through radiation driving and diffusion damping. R* sets the baryon loading and, along with the potential well depths set by eq, fixes the modulation of the even and odd peak heights. The initial conditions set the phase, or equivalently the location of the first peak in units of a, and an overall tilt n in the power spectrum.

In the model of Plate 1, these numbers are a = 301 (1 = 0.73a), eq = 149, d = 1332, R* = 0.57 and n = 1 and in this family of models the parameter sensitivity is approximately [Hu et al, 2001]

 (24)

and R* / R* 1.0 b h2 / b h2. Current observations indicate that a = 304 ± 4, eq = 168 ± 15, d = 1392 ± 18, R* = 0.60 ± 0.06, and n = 0.96 ± 0.04 ([Knox et al, 2001]; see also [Wang et al, 2001, Pryke et al, 2001, de Bernardis et al, 2001]), if gravitational waves contributions are subdominant and the reionization redshift is low as assumed in the working cosmological model (see Section 2.1).

The acoustic peaks therefore contain three rulers for the angular diameter distance test for curvature, i.e. deviations from tot = 1. However contrary to popular belief, any one of these alone is not a standard ruler whose absolute scale is known even in the working cosmological model. This is reflected in the sensitivity of these scales to other cosmological parameters. For example, the dependence of a on m h2 and hence the Hubble constant is quite strong. But in combination with a measurement of the matter-radiation ratio from eq, this degeneracy is broken.

The weaker degeneracy of a on the baryons can likewise be broken from a measurement of the baryon-photon ratio R*. The damping scale d provides an additional consistency check on the implicit assumptions in the working model, e.g. recombination and the energy contents of the Universe during this epoch. What makes the peaks so valuable for this test is that the rulers are standardizeable and contain a built-in consistency check.

There remains a weak but perfect degeneracy between tot and because they both appear only in D*. This is called the angular diameter distance degeneracy in the literature and can readily be generalized to dark energy components beyond the cosmological constant assumed here. Since the effect of is intrinsically so small, it only creates a correspondingly small ambiguity in tot for reasonable values of . The down side is that dark energy can never be isolated through the peaks alone since it only takes a small amount of curvature to mimic its effects. The evidence for dark energy through the CMB comes about by allowing for external information. The most important is the nearly overwhelming direct evidence for m < 1 from local structures in the Universe. The second is the measurements of a relatively high Hubble constant h 0.7; combined with a relatively low m h2 that is preferred in the CMB data, it implies m < 1 but at low significance currently.

The upshot is that precise measurements of the acoustic peaks yield precise determinations of four fundamental parameters of the working cosmological model: b h2, m h2, D*, and n. More generally, the first three can be replaced by a, eq, d and R* to extend these results to models where the underlying assumptions of the working model are violated.

2 Recall that the comoving scale k does not vary with time. At very early times, then, the wavelengh k-1 is much larger than the horizon . Back.