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I have read up quite a bit on the CMB power spectrum, but one feature evades my understanding. The first peak is often referred to as the "first compression peak", meaning that the plasma had time to compress exactly once. The way I imagine this is that we have a central overdensity, and then a ring of plasma around it which moves closer, i.e. the ring is being compressed. In the same vein I think of the second peak as corresponding to the ring being compressed and then rarefacted. But I don't see how this fits in with the power spectrum: the first peak corresponds to a *larger* separation between the center and the ring, while the second peak corresponds to a *smaller* separation. With the way I have described it, I believe it should be the other way around: compression should correspond to smaller scales, while rarefaction should correspond to larger scales.

What am I misunderstanding about this?

My attempts at a short answer failed, so here is my best shot at an explanation.

When you look at what the photon-baryon plasma does in the dark matter background, you will bump into an equation that is *almost* the equation for a harmonic oscillator, but it contains a time dependent quantity $chi = frac{3
ho_B}{4
ho_gamma}$ that spoils the fun. It is, however, possible to approximate the solution: $$Theta_k( au) + Phi_k approx A_k cosBig(kint_0^ au c_s( au')d au'Big) - chiPhi_k$$ where $Theta$ is the temperature fluctuation $delta T/T$ (in Fourier space), $ au$ is the conformal time, $Phi$ is the potential caused by dark matter and $c_s$ is the sound speed in the plasma. The observable quantity is $Theta_{0k} = Theta_k + Phi_k$. These are the intrinsic temperature fluctuations plus the background potential. The prefactor $A_k$ is determined by evaluating this expression at $ au = au_{LS}$, the moment of last scattering. This then yields: $$Theta_{0k} = frac{1}{3}(1 + 3chi)Phi_kcos(klambda_s) - chiPhi_k $$ with $lambda_s = int_0^{ au_{LS}} c_s d au$ the sound horizon (i.e. the distance acoustic waves can travel in a certain amount of time). The quantity $Theta_{0k}$ is called the monopole contribution to the CMB. The dipole contribution, that is due to the Doppler effect, is much smaller and I will not consider it here.

Let's take a look at an overdensity ($Phi_k < 0$) and place our coordinate axes right in the middle of it. At the "time of the Big Bang", $ au = 0$ the monopole is $Theta_{0k} = frac{1}{3}Phi_k <0$. The argument of the cosine is 0, so the cosine itself reaches its maximum value. Half a period of the cosine later: $Theta_{0k} = -frac{1}{3}(1 + 6chi)Phi_k$, which is a lot larger! In the standard cosmological model, $Phi_k$ remains constant, so the only thing that could have gotten bigger is the intrinsic temperature fluctuation $Theta$. This temperature fluctuation is directly linked to the density fluctuations of the photons and baryons.

What happens intuitively is the following: the baryons love falling into dark matter potentials. Baryons are approximately pressureless, so nothing would impede them to do so if it weren't for the photons to which they are coupled before $ au_{LS}$. The photons are dragged down into the potential well by the baryons, which makes them hotter (photons do have pressure), so thermodynamics tells them to cool again by escaping the well. It's much harder to get out of a potential well than it is to fall into one, so the amplitudes of compression ($B$ and $gamma$ falling into a well) and expansion ($gamma$ pushing $B$ out of the well) are not equal.

Look again at the cosine at $ au_{LS}$. It will reach maximal compression when $klambda_s = (2n + 1)pi$ for $n = 0, 1, 2… $, which defines the wavenumbers of the fluctuations that reach maximal compression. It will reach maximal expansion when $klambda_s = 2npi$ for $n = 1, 2, 3… $.

The first peak reaches maximal compression at last scattering. It is huge because it is least affected by Silk diffusion. It corresponds to a large angular scale (compared to the other peaks) because its wavelength is $lambda = 2lambda_s$ and is therefore the largest fluctuation.

And finally, I think you already found Wayne Hu's website, but here is the link to a gif that shows you the oscillation of the baryon-photon plasma in a dark matter potential well.

## Angular Power Spectrum of CMB

I was wondering, if anyone could guide me through the different peaks of this spectrum (See below)? I've been reading and reading numerous pages about this, but I can't seem to get my head around this.

I know that the high peak tells us the curvature of the Universe, but I'm not sure why and how, and where it comes from. My somewhat conclusion to where it comes from is, that it comes from "sound waves" from the early Universe, before photons decoupled. So basically, due to fluctuations in density (From even earlier Universe), these more dense areas contract and raises the temperature of the photons in the same area. This heats up the area, giving the photons more energy, and from that more radiation pressure. This low-to-high-pressure region makes some kind of sound wave (If I'm not mistaken), which, somehow, translates into the high peak - but again, I'm not quite sure why and how.

The other peaks, I'm really just confused about, and I'm really not sure what they tell me, and, physically, where they come from.

I know it might be a long answer, but I had to try, since I haven't been able to figure it out myself.

Thanks in advance.

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Research output : Contribution to journal › Article › peer-review

T1 - First-year Wilkinson microwave anisotropy probe (WMAP) observations

T2 - Interpretation of the TT and TE angular power spectrum peaks

N1 - Copyright: Copyright 2008 Elsevier B.V., All rights reserved.

N2 - The CMB has distinct peaks in both its temperature angular power spectrum (TT) and temperature-polarization cross-power spectrum (TE). From the WMAP data we find the first peak in the temperature spectrum at ℓ = 220.1 ± 0. 8 with an amplitude of 74.7 ± 0.5 μK the first trough at ℓ = 411.7 ± 3.5 with an amplitude of 41.0 ± 0.5 μK and the second peak at ℓ = 546 ± 10 with an amplitude of 48.8 ± 0.9 μK. The TE spectrum has an antipeak at ℓ = 137 ± 9 with a cross-power of -35 ± 9 μK2, and a peak at ℓ = 329 ± 19 with cross-power 105 ± 18 μK2. All uncertainties are 1 σ and include calibration and beam errors. An intuition for how the data determine the cosmological parameters may be gained by limiting one's attention to a subset of parameters and their effects on the peak characteristics. We interpret the peaks in the context of a flat adiabatic ΛCDM model with the goal of showing how the cosmic baryon density, Ωbh 2, matter density, Ωmh2, scalar index, ns, and age of the universe are encoded in their positions and amplitudes. To this end, we introduce a new scaling relation for the TE antipeak-to-peak amplitude ratio and recompute known related scaling relations for the TT spectrum in light of the WMAP data. From the scaling relations, we show that WMAP's tight bound on Ωbh2 is intimately linked to its robust detection of the first and second peaks of the TT spectrum.

AB - The CMB has distinct peaks in both its temperature angular power spectrum (TT) and temperature-polarization cross-power spectrum (TE). From the WMAP data we find the first peak in the temperature spectrum at ℓ = 220.1 ± 0. 8 with an amplitude of 74.7 ± 0.5 μK the first trough at ℓ = 411.7 ± 3.5 with an amplitude of 41.0 ± 0.5 μK and the second peak at ℓ = 546 ± 10 with an amplitude of 48.8 ± 0.9 μK. The TE spectrum has an antipeak at ℓ = 137 ± 9 with a cross-power of -35 ± 9 μK2, and a peak at ℓ = 329 ± 19 with cross-power 105 ± 18 μK2. All uncertainties are 1 σ and include calibration and beam errors. An intuition for how the data determine the cosmological parameters may be gained by limiting one's attention to a subset of parameters and their effects on the peak characteristics. We interpret the peaks in the context of a flat adiabatic ΛCDM model with the goal of showing how the cosmic baryon density, Ωbh 2, matter density, Ωmh2, scalar index, ns, and age of the universe are encoded in their positions and amplitudes. To this end, we introduce a new scaling relation for the TE antipeak-to-peak amplitude ratio and recompute known related scaling relations for the TT spectrum in light of the WMAP data. From the scaling relations, we show that WMAP's tight bound on Ωbh2 is intimately linked to its robust detection of the first and second peaks of the TT spectrum.

## 3 Answers 3

OK, here's a brief outline that is very math light (and also somewhat oversimplified): The big idea is to compare the size of the lumpiness now to the size of the lumpiness back then when the CMB was generated, also called at the decoupling time. We measure the size of the lumpiness now by looking a galaxy superclusters and voids. We can then figure out the expansion rate of the universe, if we also measure the time and the distance back to the big bang (decoupling). From Einsteins GR we can convert the expansion rate, and also the sizes, distances and times into a curvature. (We may also assume and use some other things such as approximate uniformity, and spherical symmetry.) If you can truly explain all this, including the GR part, with a math-lite explanation, you are a much better man than I am, Charlie Brown.

Figuring out how big the lumps are is the first part, then measuring how big the lumps look to us is the second part. If they look bigger or smaller than they should, you have curved space.

The answer is in the same Wikipedia article but I feel the need to mention anisotropies:

Anisotropy/ˌænaɪˈsɒtrəpi/ is the property of beingdirectionally dependent, as opposed to isotropy. An example of anisotropy is the light coming through a polarizer. An example of an anisotropic material iswood, which is easier to split along its grain than across its grain.

In general, anisotropies give us an idea about density fluctuations in the early universe which form out the basis(or seeds) of dense matter clusters(galaxies etc) in the universe. How big are these anisotropies(fluctuations)? That measurement is related to parameters of the universe. The answer is in the paragraph above the paragraph you mentioned. This :

The structure of the

cosmic microwave background anisotropiesis principally determined by two effects:acoustic oscillations and diffusion damping(also called collisionless damping or Silk damping). The acoustic oscillations arise because of a conflict in the photon–baryon plasma in the early universe. The pressure of the photons tends to erase anisotropies, whereas the gravitational attraction of the baryons—moving at speeds much slower than light—makes them tend to collapse to form dense haloes.These two effects compete to create acoustic oscillations which give the microwave background its characteristic peak structure.The peaks correspond, roughly, to resonances in which the photons decouple when a particular mode is at its peak amplitude.

Here are necessary details. Some information about the Primordial Density Fluctuations from Wikipedia:

The statistical properties of the primordial fluctuations can be inferred from

observations of anisotropies in the cosmic microwave background and from measurements of the distribution of matter, e.g., galaxy redshift surveys. Since the fluctuations are believed to arise from inflation, such measurements can alsoset constraints on parameterswithin inflationary theory.

Note that it says, these measurements set constraints on the parameters, i.e. they must be less than or greater than some value. An exact opinion is and was given using different techniques and looking for the common areas in results.

If there's something that bothers you or I was unable to explain well, please mention. :)

## QUIET, please…we’re observing the CMB

**• Title:** Second Season QUIET Observations: Measurements of the CMB Polarization Power Spectrum at 95 GHz **• Authors:** The QUIET Collaboration **• First Author’s Institution:** Kavli Institute for Cosmological Physics, The University of Chicago

Since its discovery in 1965, the Cosmic Microwave Background (CMB) has been a treasure trove of information about the universe. The CMB is snapshot of the universe at 380,000 years of age. Its measurement essentially confirmed the Big Bang theory, and since then has told us the values of many parameters that describe our universe, including its curvature. This paper describes an experiment designed to confirm or refute the leading theory about what happened in the instant of time after the Big Bang.

**Inflation and the CMB**

The theory of inflation is absolutely critical to our understanding of modern cosmology. The idea is that in the very beginning of the universe, around 10^-36 seconds after the Big Bang, the universe experienced a violent exponential expansion in which its volume increased by a factor of about 10^78. While it sounds farfetched, inflation solves many cosmological problems: it explains why we don’t see magnetic monopoles, why the CMB is so uniform, and why the universe has a flat geometry. Since it was proposed over 30 years ago, almost all of inflation’s predictions have been detected. The last, unobserved prediction is that quantum fluctuations generated a gravitational wave background in the early universe. If detected, it would be a “smoking gun” for inflation, and could even tell us how powerful the expansion was. But how could we detect these inflationary gravitational waves?

It turns out that the CMB is polarized due to Thomson scattering, most of which is due to density fluctuations in the early universe. A much smaller amount of polarization should be caused by gravitational waves — and luckily, these two sources give rise to different polarization patterns. In analogy to electromagnetism, a polarization map can be decomposed into two orthogonal bases, or modes: an “E-mode” which looks like a gradient, and a “B-mode” which looks like a curl (see Figure 1). E-modes, which were detected 10 years ago, can be generated by both density fluctuations and gravitational waves. B-modes, which remain unseen, can *only* be generated by gravitational waves. So if we make a map of the CMB polarization, decompose it into E and B, and see nonzero B-modes, we have detected the gravitational wave background and have thus confirmed the existence of inflation!

Figure 1: A polarization field decomposed into "pure E" (left) and "pure B" (right).

**The QUIET Instrument**

The Q/U Imaging ExperimenT (QUIET) was built with the purpose of detecting B-modes. Located on the Chajnantor plateau in Chile’s Atacama desert, a site known for excellent radio and microwave seeing, it observed from 2008 to 2010 at 43 GHz and 95 GHz. This paper covers the second season of observations with the 95 GHz data (the 43 GHz data were released in a previous paper). The telescope is a 1.4 meter Dragonian reflector (Figure 2) which illuminates a focal plane populated by polarization-sensitive radiometers.

Figure 2: The QUIET reflector and cryostat featuring a side-fed Dragonian design. The focal plane is inside the cryostat.

Unlike many other CMB experiments, QUIET uses coherent detectors which preserve the wave nature of light (most experiments employ bolometers, devices which which are only sensitive to intensity and not phase). This allows the detectors to simultaneously measure both components of the linear polarization of light, unlike polarization sensitive bolometers. The QUIET detector modules are Monolithic Microwave Integrated Circuits (MMICs), which are essentially “detectors on a chip.” This technology allows a large number of radiometers, each the size of a postage stamp, to be packaged into a focal plane at low cost (see Figure 3). Since more detectors equals more sensitivity, this is a great innovation!

Figure 3: QUIET MMICs. Left: The first 90 GHz module. Top right: The exterior of a 90 GHz module. Bottom right: The interior of a 40 GHz module. This new "detector on a chip" technology makes it cost-effective to deploy large numbers of radiometers in detector arrays.

**Power Spectra and Limits on Inflation**

After observing for the CMB for 5337 hours at 95 GHz, QUIET made polarization maps of four regions on the sky. These maps were then decomposed into E and B maps, and then into angular power spectra: plots showing how much the sky signal varies as a function of angular distance. The top panel of Figure 4 shows the E spectrum. As expected, nonzero E-modes are clearly seen (and the first three acoustic peaks match well with standard cosmological theory).

Figure 4: QUIET E-modes and B upper limits. The QUIET data points are in black. The x-axis denotes angular scale low values of l correspond to large angles on the sky. In the B mode plot, the dashed line shows the theoretical power spectrum for B modes at r = 0.1 (it could be much lower). The dotted line shows the secondary effect of gravitational lensing, which turns E-modes into B-modes at small angular scales. The solid line is the sum of the two.

B-modes were not seen in the data. The standard measure of inflation is the tensor-to-scalar ratio, *r*, the ratio of the amplitudes of the gravitational waves to density perturbations in the early universe. A large value of *r* means that gravitational waves in the early universe was large, and thus that inflation was strong. Since the B spectrum is consistent with zero, it is not possible to give a confidence interval for *r* instead, CMB experiments quote upper limits on the strength of inflation. QUIET’s data reduction returns two upper limits: *r* < 2.7 or *r* < 2.8, depending on the pipeline used. As seen on the bottom panel of Figure 4, many experiments are now pushing down the limits on B-modes. If *r* is in the range of 0.01, as is predicted by some inflationary theories, the next generation of experiments which feature many more detectors should be able to find it!

## Understanding the angular peaks in the CMB power spectrum - Astronomy

When the temperature of the Universe was

3000K at a redshift *z*_{*} 10 3 , 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

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

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* *T* 3 . 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 *T* 4 , this becomes 4*k* / 3 in Fourier space. The Euler equation then becomes

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

where *c*_{s} ( / ) 1/2 = 1 / 3 1/2 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

where *s* = *c*_{s} *d* / 3 1/2 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 *k*_{n} = *n* / *s*_{*}, where *n* is an integer (see Figure 1a).

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

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.

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* = *H*_{0} -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

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 + ln_{m} 0.085 ). 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 *t*_{0} 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 *g*_{ij} = 2*a* 2 _{ij} (see e.g. [Ma & Bertschinger, 1995]). The more familiar Newtonian potential is the time-time fluctuation *g*_{tt} = 2 and is approximately - . Approximate scale invariance then says that 2 *k* 3 *P* (*k*) / 2 2 *k* n-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 *k* 2 = 4 *Ga* 2 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* *t* 2/[3(1+p/)] . Therefore, the fractional change in the CMB temperature

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

In a flat universe and in the absence of pressure, and are constant. Also, in the absence of baryons, *c*_{s} 2 = 1/3 so the new oscillator equation is identical to Equation (9) with replaced by + . The solution in the matter dominated epoch is then

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, *ks*_{md} << 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* = (*p*_{b} + _{b} / (*p* + ) 30_{b} *h* 2 (*z* / 10 3 ) -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]

where we have used the fact that the sound speed is reduced by the baryons to *c*_{s} = 1 / [3(1 + *R*)] 1/2 .

To get a feel for the implications of the baryons take the limit of constant *R*, and . Then *d* 2 (*R* ) / *d* 2 (= 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

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} 24_{m} *h* 2 (*z* / 10 3 ) -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).

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 (*k*_{eq} (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,

where _{b} and *v*_{b} are the density perturbation and fluid velocity of the baryons. The Euler equations contain qualitatively new terms

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 *n*_{e} _{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* / ) *A*_{v}, where the order unity constant can be derived from the Boltzmann equation *A*_{v} = 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

where the heat conduction coefficient *A*_{h} = *R* 2 / (1 + *R*). Thus we expect the inhomogeneities to be damped by a exponential factor of order *e* -k 2 / (see Figure 3). The damping scale *k*_{d} 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 *k*_{d}*s*_{*} 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} *h* 2 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} *h* 2 ) -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 *d*_{T} / *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 *C* E 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*).

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

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).

Projection is less straightforward for other sources of anisotropy. We have hitherto neglected the fact that the acoustic motion of the photon-baryon fluid also produces a Doppler shift in the radiation that appears to the observer as a temperature anisotropy as well. In fact, we argued above that *v*_{b} *v* is of comparable magnitude but out of phase with the effective temperature. If the Doppler effect projected in the same way as the effective temperature, it would wash out the acoustic peaks. However, the Doppler effect has a directional dependence as well since it is only the line-of-sight velocity that produces the effect. Formally, it is a dipole source of temperature anisotropies and hence has an = 1 structure. The coupling of the dipole and plane wave angular momenta imply that in the projection of the Doppler effect involves a combination of *j*_{۫} that may be rewritten as *j* '(*x*) *dj* (*x*) / *dx*. The structure of *j* ' lacks a strong peak at *x* = . Physically this corresponds to the fact that the velocity is irrotational and hence has no component in the direction orthogonal to the wavevector (see Plate 3). Correspondingly, the Doppler effect cannot produce strong peak structures [Hu & Sugiyama, 1995]. The observed peaks must be acoustic peaks in the effective temperature not "Doppler peaks".

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

The last term vanishes for constant gravitational potentials, but is non-zero if residual radiation driving exists, as it will in low _{m} *h* 2 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 *j*_{l}(*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

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.

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} *k*_{eq} *D*_{*} and the damping scale _{d} *k*_{d} *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} = 0.73_{a}), _{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]

and *R*_{*} / *R*_{*} 1.0 _{b} *h* 2 / _{b} *h* 2 . 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} *h* 2 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} *h* 2 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} *h* 2 , _{m} *h* 2 , *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. *****

## Title: Constraints on Cosmological Parameters from the Angular Power Spectrum of a Combined 2500 deg 2 SPT-SZ and Planck Gravitational Lensing Map

Here, we report constraints on cosmological parameters from the angular power spectrum of a cosmic microwave background (CMB) gravitational lensing potential map created using temperature data from 2500 deg$^2$ of South Pole Telescope (SPT) data supplemented with data from *Planck* in the same sky region, with the statistical power in the combined map primarily from the SPT data. We fit the corresponding lensing angular power spectrum to a model including cold dark matter and a cosmological constant ($Lambda$CDM), and to models with single-parameter extensions to $Lambda$CDM. We find constraints that are comparable to and consistent with constraints found using the full-sky *Planck* CMB lensing data. Specifically, we find $sigma_8 Omega_<
m m>^<0.25>=0.598 pm 0.024$ from the lensing data alone with relatively weak priors placed on the other $Lambda$CDM parameters. In combination with primary CMB data from Planck, we explore single-parameter extensions to the $Lambda$CDM model. We find $Omega_k = -0.012^<+0.021>_<-0.023>$ or $M_<
u>< 0.70$eV both at 95% confidence, all in good agreement with results that include the lensing potential as measured by *Planck* over the full sky. We include two independent free parameters that scale the effect of lensing on the CMB: $A_*Planck* temperature data, indicating that the amount of lensing is in excellent agreement with what is expected from the observed CMB angular power spectrum when not including the information from smearing of the acoustic peaks.

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Note. – All values come from fitting a Gaussian shape to just the first peak of the data set specified, and include calibration error. Each data set is considered on its own, without the COBE /DMR data, and so a direct comparison between experiments may be made. The best fit position depends somewhat on the fitting function so the values from different analyses yield different results (e.g., Knox & Page, 2000 Durrer et al., 2003 Ödman et al., 2002 Grainge et al., 2003) . The TOCO, VSA, and BOOMERANG-NA experiments were calibrated with Jupiter. The TOCO and VSA experiments are most affected because of they operate at 30-150 GHz and 35 GHz respectively. With the new calibration of Jupiter (Page et al., 2003) , the peak values above will be reduced ≈ 5 % . The weighted peak amplitude is 71.7 ± 2.4 μ K and the weighted peak position is 218.8 ± 3.5 in good agreement with WMAP . In a separate analysis based on different assumptions, Bond reports ℓ T T 1 = 222 ± 3 (private communication). This was also the value preferred by a concordance model (Wang et al., 2000) that predated all the experiments of the new millennium. Note that WMAP ’s values for the position and amplitude are both more than four times more precise than all the listed measurements combined.

Table 1: Previous Measurements of the First Peak

Quantity | Symbol | ℓ | Δ T ℓ [ μ K ] | Δ T 2 ℓ [ μ K 2 ] | FULL ℓ | FULL Δ T 2 ℓ [ μ K 2 ] |
---|---|---|---|---|---|---|

First TT Peak | ℓ T T 1 | 220.1 ± 0.8 | 74.7 ± 0.5 | 5583 ± 73 | 219.8 ± 0.9 | 5617 ± 72 |

First TT Trough | ℓ T T 1.5 | 411.7 ± 3.5 | 41.0 ± 0.5 | 1679 ± 43 | 410.0 ± 1.6 | 1647 ± 33 |

Second TT Peak | ℓ T T 2 | 546 ± 10 | 48.8 ± 0.9 | 2381 ± 83 | 535 ± 2 | 2523 ± 49 |

First TE Antipeak | ℓ T E 1 | 137 ± 9 | − 35 ± 9 | 151.2 ± 1.4 | − 45 ± 2 | |

Second TE Peak | ℓ T E 2 | 329 ± 19 | 105 ± 18 | 308.5 ± 1.3 | 117 ± 2 |

Note. – The values and uncertainties are the maximum and width of the a posteriori distribution of the likelihood assuming a uniform prior. The uncertainties include calibration uncertainty and cosmic variance. The FULL values are derived from the full CMBFAST-based likelihood analysis using just the WMAP data (Spergel et al., 2003) . The FULL method yields consistent results. Recall that the FULL chains are sensitive to the combined TT and TE spectra and not just the individual peak regions. Numerical errors in CMBFAST will increase the uncertainties, but should not bias the results.

Table 2: WMAP Peak and Trough Amplitudes and Positions

Quantity | Symbol | Value | |
---|---|---|---|

Physical baryon density | ω b | 0.024 ± 0.001 | |

Physical mass density | ω m | 0.14 ± 0.02 | INPUT |

Scalar index | n s | 0.99 ± 0.04 | |

First TT peak phase shift | ϕ 1 | 0.265 ± 0.006 | |

First TT trough phase shift | ϕ 1.5 | 0.133 ± 0.007 | |

Second TT peak phase shift | ϕ 2 | 0.219 ± 0.008 | DERIVED |

Third TT peak phase shift | ϕ 3 | 0.299 ± 0.005 | FROM |

Redshift at decoupling | z d e c | 1088 + 1 − 2 | INPUT |

Redshift at matter radiation equality | z e q | 3213 + 339 − 328 | |

Comoving acoustic horizon size at decoupling (Mpc) | r s | 143 ± 4 | |

Acoustic scale | l A | 300 ± 3 | DERIVED FROM |

Comoving angular size distance to decoupling (Gpc) | d A | 13.7 ± 0.4 | INPUT + PEAKS |

Note. – The cosmological parameters in the top section are derived from just the WMAP data assuming a flat Λ CDM model (Spergel et al., 2003) . The quantities in the middle section are derived from the cosmological parameters in the top section. The quantities in bottom section are calculated using the middle quantities and the measured position of the first peak. The quantity z d e c which we use corresponds to the location of the maximum of the visibility function in CMBFAST. The quantity computed using Hu & Sugiyama (1996) corresponds to τ ( z d e c ) = 1 and is 1090 ± 2 .

Table 3: WMAP Cosmological Parameters for the Peaks Analysis Figure 1: The binned WMAP data is shown in blue, the maximum likelihood peak model from the peak fitting functions in red, and the uncertainty contours in black. The top panel shows the TT angular power spectrum. The bottom panel shows the TE angular cross-power spectrum. For each peak or trough, the contours from the MCMC chains are multiplied by a uniform prior and so they are equal to contours of the a posteriori likelihood of the data given the model. The contours are drawn at Δ χ 2 = 2.3 and 6.18 corresponding to 1 σ and 2 σ . Figure 2: The WMAP data in the Ω m − h plane. The thick solid contours in black are at Δ χ 2 = − 2.3 , − 6.18 ( 1 σ , 2 σ ) of the marginalized likelihood from the full analysis (Spergel et al., 2003) . The filled region is the constraint from the position of the first peak, with ω b = 0.023 fixed. In effect, it shows how Ω m and h must be related to match the observed position of the first peak in a flat geometry, or equivalently to match the measured values of θ A . Blue shows the 1 σ region and green shows the 2 σ region. The dotted lines are isochrons separated by 1 Gyr. It is clear that the WMAP data pick out 13.6 Gyr for the age of the universe in the flat, w = − 1 case. The dashed lines show the 1 σ limits on ω m . The dashed yellow line shows Ω m h 3.4 = c o n s t .

Figure 3: Left: Parameter restrictions from H T T 2 in the ω b − n s plane. The orange swath is the 1 σ band corresponding to H T T 2 = 0.426 ± 0.015 with ω m = 0.14 . The swath is broadened if one includes the uncertainty in ω m . The light orange swath is 2 σ . The solid line in the middle of the swath is for Δ H T T 2 = Δ ω m = 0 . The green contours are from the full analysis of just the WMAP data and are thus more restrictive. Right: The constraints in the ω b − ω c plane from the peak ratios in a flat geometry with n s = 0.99 . The dark shaded regions in each swath are the 1 σ allowed range the light shaded regions show the 2 σ range. Orange is for H T T 2 = 0.426 ± 0.015 , blue is for H T T 3 = 0.42 ± 0.08 , and red is for H T E 2 = 0.33 ± 0.10 . The uncertainty band for H T E 2 is not shown as it is broader than the H T T 3 swath. The heavier central lines correspond to Δ H T T 2 = 0 , Δ H T T 3 = 0 , and Δ H T E 2 = 0 , each with Δ n s = 0 . As the mission progresses, all uncertainties will shrink.

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## Understanding the angular peaks in the CMB power spectrum - Astronomy

**2.4. The Cosmic Microwave Background**

Today, CMB photons, while very numerous (there are about 2 billion photons for every hydrogen atom) account for a negligible fraction of the mass-energy budget (about 0.01%). Still, they play a central role in cosmology. First, at early times, the CMB was the dominant part of the mass-energy budget, from which we ascertain that the infant Universe was a hot thermal bath of elementary particles. Second, photons from the CMB interacted closely with matter until the temperature of the Universe had cooled enough for the ionized plasma to combine and form neutral atoms, allowing the photons to stream past. At this "last-scattering surface" of the CMB, the Universe was about 400,000years old, and about 1100 times smaller than it is today. The CMB is a "snapshot" of the Universe at a much simpler time.

The CMB measurements are a striking example of a new level of precision now being made in cosmology. NASA's COBE satellite, a four-year mission launched in 1989, measured the temperature of the background radiation to better than one part in a thousand, *T*_{0} = 2.725 ± 0.001 K (Mather *et al.*, 1999), and discovered tiny (tens of microKelvin) variations in the temperature of the CMB across the sky. These tiny fluctuations arise from primeval lumpiness in the distribution of matter. In the early Universe, outward pressure from the CMB photons, acting counter to the inward force of gravity due to matter, set up oscillations whose frequencies are now seen imprinted in the CMB fluctuations. Evidence of these "acoustic oscillations" can be seen when the fluctuations are described by their spherical-harmonic power spectrum (see Figures 7-9). In late 2002, the DASI Colloboration detected the last feature predicted for the CMB: polarization (Kovac *et al.*, 2002). Because the CMB radiation is not isotropic (as evidenced by the anisotropy seen across the microwave sky) and Thomson scattering off electrons is not isotropic, CMB anisotropy should develop about a 5% polarization.

**Figure 8.** Anisotropy of the Cosmic Microwave Background: Angular power spectrum of the CMB, incorporating all the pre-WMAP data (COBE, BOOMERanG, MAXIMA, DASI, CBI, ACBAR, FIRS, VSA, and other experiments). Variance of the multipole amplitude is plotted against multiple number as indicated by the top scale, multipole measures the fluctuations on angular scale

**Figure 9.** Anisotropy of the Cosmic Microwave Background: The WMAP angular power spectrum (also includes data from CBI and ACBAR). The curve is the consensus cosmology model the grey band includes cosmic variance. The WMAP measurements up to

350 are cosmic variance limited. The lower panel shows the anisotropy cross polarization power spectrum the high point marked re-ionization is the evidence for re-ionization of the Universe at *z*

The precise shape of the angular power spectrum of anistropy and polarization depends in varying degrees upon all the cosmological parameters in Table I, and so CMB anisotropy encodes a wealth of information about the Universe. With a host of ground-based and balloon-borne CMB experiments following COBE, a NASA space mission (the Microwave Anisotropy Probe, MAP) now taking new data, and with an European Space Agency (ESA) mission planned for launch in 2007, we are in the midst of realizing the potential of the CMB as a probe of cosmological parameters. A summary of the progress includes determination of the curvature, _{0} = 1.03 ± 0.03, the power law index of density perturbations, *n* = 1.05 ± 0.09, the baryon density _{B} = 4.0 ± 0.6 × 10 -31 g cm -3 , and the matter density _{M} = 2.7 ± 0.4 × 10 -30 g cm -3 . The uncertainties in all of these quantities are expected to diminish by at least a factor of ten.

As mentioned above, the CMB value for the baryon density is consistent with that determined from BBN. This not only provides confidence that ordinary matter accounts for a small fraction of the total amount of matter, but also is a remarkable consistency test of the entire framework. The CMB provides independent, corroborating evidence for a significant component of dark energy through the discrepancy between the total amount of matter and energy (critical density) and that in matter (1/3 of the critical density). Finally, the measurements of the CMB multipole spectrum are consistent with the emerging new cosmology: a flat Universe with dark matter and dark energy.

Establishing a reliable accounting of the matter and energy in the Universe (see Figure 10) is a major achievement but, we still have much more to learn about each component and almost everything to understand about the "strange recipe." Moreover, because the energy density of matter, photons and dark energy each change in distinctive ways as the universe expands, the mix we see today must have been different in the past and will be different in the future.

The energy per photon (or per relativistic particle) is redshifted by the expansion (decreasing as *a* -1 ) and the number density of photons is diluted by the increase in volume (as *a* -3 ), resulting in a total decrease in the energy density proportional to *a* -4 . The energy density in matter is diluted by the volume increase of the universe, so that it decreases as *a* -3 . The energy density in dark energy changes little (or not at all) as the universe evolves. This means that the Universe began with photons (and other forms of radiation) dominating the energy density at early times (*t* < 10 4 yrs), followed by an era where matter dominated the energy density, culminating in the present accelerating epoch characterized by a transition to a universe dominated by dark energy. *****

## Understanding the angular peaks in the CMB power spectrum - Astronomy

Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic

Received July 30, 2011 revised September 8, 2011 accepted September 20, 2011

**Keywords:** CMB Radiation, Analysis of CMB Spectrum, Radial Distribution Function of Objects, Early Universe Cluster Structure, Density of Ordinary Matter

A formalism of solid state physics has been applied to provide an additional tool for the research of cosmological problems. It is demonstrated how this new approach could be useful in the analysis of the Cosmic Microwave Background (CMB) data. After a transformation of the anisotropy spectrum of relict radiation into a special two-fold reciprocal space it was possible to propose a simple and general description of the interaction of relict photons with the matter by a “relict radiation factor”. This factor enabled us to process the transformed CMB anisotropy spectrum by a Fourier transform and thus arrive to a radial electron density distribution function (RDF) in a reciprocal space. As a consequence it was possible to estimate distances between Objects of the order of

10 2 [m] and the density of the ordinary matter

10 –22 [kg∙m –3 ]. Another analysis based on a direct calculation of the CMB radiation spectrum after its transformation into a simple reciprocal space and combined with appropriate structure modelling confirmed the cluster structure. The internal structure of Objects may be formed by Clusters distant

10 [cm], whereas the internal structure of a Cluster consisted of particles distant

0.3 [nm]. The work points in favour of clustering processes and to a cluster-like structure of the matter and thus contributes to the understanding of the structure of density fluctuations. As a consequence it may shed more light on the structure of the universe in the moment when the universe became transparent for photons. On the basis of our quantitative considerations it was possible to derive the number of particles (protons, helium nuclei, electrons and other particles) in Objects and Clusters and the number of Clusters in an Object.

The angular power spectrum (anisotropy spectrum) of the Cosmic Microwave Background (CMB) radiation ([1,2]) shows incredible similarity with X-ray or neutron scattering measured on non-crystalline materials ([3], [4]), see Figures 1 and 2.

Astronomers ascribe to various peaks of the anisotropy spectrum of the CMB radiation different processes [5]. It is the Sachs-Wolf effect, Doppler effect, Silk damping, Rees-Sciama effect, Sunyaev-Zeldovich effect, etc. In this connection it should be stated that all theoretical predictions of the standard cosmological model are in very good agreement with the course of the anisotropy spectrum of CMB radiation.

However, the formal similarity in the form of both figures initiates the tempting idea if an analysis of the anisotropy spectrum of relict radiation using an analogous approach as is common in solid state physics, i.e. in the structural analysis of disordered materials, would bring more information on the structure of the early universe.

The inspiration for this approach we found further in the nowadays situation: Although the individual disciplines in physics are highly specialized, nevertheless their methods and results are shared in areas that at the first sight may seem to be far apart. An example of this is the already established use of elementary particle physics

Figure 1 . Anisotropy spectrum of the CMB radiation [1]. The figure describes the dependence of the magnitude of the intensity of microwave background on the multipole moment L = 180˚/α, where α is the angle between two points in which the temperature fluctuations are compared to an overall medium temperature. The description of the Y-axis is for simplicity described in [Arbitrary units]. The original description was given as L(L + 1)C_{L}/2π in [μK 2 ] units, where L is the multipole moment, C_{L} is a function reflecting the width of the window measuring the temperature fluctuations.

Figure 2 . X-ray scattering diagram taken on a sample of a chalcogenide glass of a composition (Ge_{0.19}Ag_{0.25}Se_{0.50}) using the MoK_{α} radiation, see [4] for detail. The reciprocal space scattering vector s is defined in Equation (A5).

Similarly, we hope that it may be time now to apply the formalism of solid state physics to some special cosmological problems and in this way to provide an additional tool for their research.

First of all our new approach may be useful in the analysis of the CMB data. We will show how after the transformation of the anisotropy spectrum of relict radiation into a special two-fold reciprocal space we will be able to process the transformed CMB anisotropy spectrum by a Fourier transform and thus calculate a radial distribution function (RDF) of the matter in a reciprocal space. Because the CMB radiation reflects the fluctuations in the density of the matter, we hope that in this way our study will contribute to the understanding of the structure of these density fluctuations. Simultaneously, as a consequence, it may shed more light on the structure of the universe in the moment when the universe became transparent for photons (see Subsection 5.1.).

Moreover, in contrast to solid state physics where the atomic (coherent) and Compton (incoherent) scattering factors are describing theoretically the interaction of X-rays (or neutrons) with all kinds of atoms, this new formalism will present a general description of the interaction of relict radiation with the matter by a single “relict radiation factor”, which should unify all processes realized during the interaction of relict radiation with various kinds of particles forming the primordial matter, see Subsections 2.3.3. and 5.2.

2. Construction of the Classic and Relict Reciprocal Space

In solid state physics the principal mathematical method during the structure analysis of the matter is the Fourier transform of the intensity of X-rays (or neutrons) scattered by atoms building the material. The experimental data are collected in the reciprocal space and their Fourier transform brings the required information on the distribution of atoms in the real space. In this contribution we will try to apply this approach to the CMB spectrum (see Figure 1 ) and simultaneously point out the complications we have to overcome in this direction.

The necessary basic mathematical apparatus is summarized in the Appendix A, the most important basic equations for the analysis of “scattered” radiation and leading to the radial density distribution function (RDF) are Equations (A1) and (A2). The essential difference in the use of terms “scattering” and “interaction” of photons will be elucidated in the next Subsection 2.1.

2.1. The Relict Radiation Factor

During a conventional structure analysis with X-rays or neutrons, the X-ray or neutron atomic scattering factors are a precise picture of the interaction of radiation with the matter and are known precisely [6]. They enter into the calculation of the RDF in correspondence with the composition of the studied material see Equations (A6), (A7) and (A10). Generally, for coherent scattering, the atomic scattering factor f is the ratio of the amplitude of X-rays scattered by a given atom E_{a} and that scattered according to the classical theory by one single electron E_{e}, i.e. (), where Z is the number of electrons in the atom.

Moreover, there are scattering factors not only for the coherent but also for the incoherent (Compton) type of scattering, see e.g. later on Figure 6 .

In our study, however, the basic obstacle is that with CMB photons we have not a classic scattering process of photons on atoms i.e. a process described in equations of the Appendix A. There are not atoms, there are particles only (e.g. baryons, electrons, etc.), which participate in the formation of the structure of density fluctuations. Therefore we will speak throughout this article about an “interaction” instead of “scattering” in all cases when instead of the classic “atomic scattering factor” the new “relict radiation factor” will be used.

It is true that a part of the interaction of photons with electrons before the recombination may be realised as Thomson scattering (elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism) 1 , but the complex picture of physical processes describing the interaction of relict photons with the non uniform matter composed of various particles (electrons, ions, etc.) is not known to such an extent in order to enable a theoretical calculation of this interaction (on the basis of scattering factors).

It is therefore evident that it will not be possible to use the conventional atomic scattering factors and that a new special factor reflecting the complexity of interaction processes of photons with the primordial matter has to be constructed. We only point out that the description of these interactions is possible only in a special two-fold reciprocal space into which the CMB spectrum is transformed. This new factor will be called the relict radiation factor and substitutes all complicated processes which participate in the formation of the angular power spectrum of CMB radiation.

The construction of the relict radiation factor is presented in Subsection 2.3.3.

2.2. The Wavelength of Radiation

The wavelength of radiation is a quantity of highest importance, too. It follows from Equation (A5), that the greater the wavelength the smaller is the maximal possible value s max of the reciprocal space vector. At the same time the upper limit of the integral in Equation (A2) strongly influences the quality of the Fourier transform.

Although there is a broad distribution of wavelengths of photons (see later on the discussion in Subsection 5.3.) the calculation will be undertaken for the wavelength corresponding to the maximum of the wavelength distribution which corresponds to the temperature 2.725 K of the Universe today (see later on Figure 1 8 ), i.e. for the wavelength λ = 1.9 [mm].

That this wavelength is rational is based on three arguments. First of all photons with this wavelength bring us the information on their last several interactions with particles today, in the second place the CMB radiation spectrum is the same for all wavelengths and in the third place the wavelength corresponding to the maximum of the wavelength distribution secures the highest probability of the interaction process of photons with the matter.

2.3. Preparatory Calculations

2.3.1. The Classic Reciprocal Space

During a classic scattering experiment we measure the intensity of the scattered radiation (e.g. X-rays) as a function of the scattering angle θ_{Classic}. This scattering angle describes in real space the angle between the incident and scattered radiation. Its relation with the scattering vector in reciprocal space was described in Equation (A5).

On the other hand the angle α in the anisotropy spectrum of relict radiation (see already Figure 1 ) is not a scattering angle. It is an angle characterizing the distance between an arbitrary point to another—in those different points the temperature fluctuation is measured and compared with the overall medium one.

In order to overcome the incomparableness between the angles α and θ, we will construct an angle dependent reciprocal space to the angle α. The basic quantity determining this reciprocal space will be the scattering angle θ_{Classic}.

We will suppose that the maximum possible value of the classic scattering angle 90˚, corresponds to the maximum value of the multipole moment L max = 3000.

As a consequence we receive a transformation coefficient Q

, (1)

(its value in this case is Q = 0.03).

We are then able to calculate the whole set of θ_{Classic} angles

(2)

(3)

(4)

(5)

is a coefficient enabling the transition between the space α and the space θ_{Classic} and where the angular space θ_{Classic} is reciprocal to the angular space α.

According Equation (A5) we are now able to construct the whole set of scattering vectors s_{Classic}

, (6)

where λ is the wavelength of the relict radiation. It should be noted that the quantities s_{Classic} and α are in an indirect relation. The space of the vector s_{Classic} will be further on called a “Classic reciprocal space”.

It should be pointed out that in this construction (see Equation (6)) the scattering vector s_{Classic} is defined in the reciprocal space (1/λ) and that this space is now dipped into the reciprocal space (1/α), see Equations (2), (4) and (6). For this “dipping” we will use further on the expression that the space s_{Classic} is a 2-fold reciprocal space to the space α.

The recalculation of the original data presented in Figure 1 using Equations (4) and (6) is shown in Figure 3 . This new intensity dependence is labelled I_{Classic} ().

2.3.2. The Relict Reciprocal Space

There is a possibility to construct another reciprocal space which will be based directly on the angle α. For a better comparison and lucidity we will use now for the angle α the labelling θ_{Relict}. i.e.

, (7)

In close analogy with Equation (A5) we now transform the anisotropy spectrum of CMB (relict) radiation into a reciprocal space (1/λ) described by the parameter S_{Relict}

, (9)

where λ is the wavelength of the relict radiation. The space of the vector S_{Relict} will be further on called the “Relict reciprocal space”.

It should be noted that quantities S_{Relict} and are in a direct relation. The anisotropy spectrum of the CMB radiation rescaled on the basis of Equation (9) is here labelled I_{Relict} (S_{Relict}) and is shown in Figure 4 .

2.3.3. Construction of the Relict Radiation Factor

Generally, a correct scattering factor has to fulfil three criterions:

Figure 3 . Anisotropy spectrum of the relict radiation shown in Figure 1 is recalculated as a function of s_{Classic}, i.e. after a rescaling of the angular moment L and is labelled I_{Classic} (s_{Classic}). The rescaling of the angular moment L is realized on the basis of Equations (2), (4) and (6) and using the MoK_{α} radiation wavelength λ = 0.071609 [nm].

Figure 4 . Anisotropy spectrum of the relict radiation shown in Figure 1 is after a rescaling of the angular moment L, recalculated as a function of the relict reciprocal space vector S_{Relict} and labelled I_{Relict} (S_{Relict}). The rescaling of the angular moment L is realized on the basis of Equations (8), (9) and (11) using the MoK_{α} radiation wavelength λ = 0.071609 [nm]. The dashed line represents a smoothed curve.

1) the I_{norm} (s) curve should oscillate along the I_{gas} (s) curve and as a consequence according Equation (A9)

2) the curve I_{distr} (s) should oscillate along the zero value of the intensity axis

3) the resulting RDF must not be contaminated by parasitic fluctuations due to bad scaling (see Section A2.) as a consequence of a bad course of the scattering factor.

The mutual relation between quantities I_{norm} (s), I_{gas} (s) and I_{distr} (s) is explained in the Appendix A, see Equations (A9), (A10) and (B1) with (B2).

In Figure 5 the calculation of the crucial curve I_{gas} is undertaken for the relict radiation factor f_{Relict}. The form of this factor was determined by the trial and error method and is shown in Figure 6 . In this figure is the factor f_{Relict} compared with the coherent () and incoherent () atomic scattering factor for X-rays corresponding to the Hydrogen atom (according the International Tables for Crystallography [6]).

Similarly as for X-rays we have set the relict radiation factor f_{Relict}

(10)

and further, we have set in Equation (A7) Z = 1 and m = 1, hence in Equation (A6) is K_{m} = 1. From this point of view our construction of the relict radiation factor f_{Relict} should formally correspond to a “hydrogen-like” particle.

Further we have to point out that in connection with the presentation of the quantity I_{gas}(s) in Equation (A10) its course in Figure 5 is given now by the relation

. (11)

In Figure 5 we see that the function I_{norm}(s) is properly oscillating along the function I_{gas}(s) and therefore the function I_{distr}(s) is properly oscillating along the zero line. The consequence is that we will obtain a “proper” radial distribution function, i.e. without any parasitic maxima, see the Subsection 3.1.

2.3.4. Relation between the Classic and Relict Distribution of Distances

We rewrite now the basic Equation (A2) using the scat-

Figure 5 . Calculation of quantities I_{norm} (s)—full line, I_{gas} (s) —dashed line (see Equation (11)) and of I_{distr} (s)—dashed dotted line, according Equations (A9), (A10) and (B1), (B2) using the “artificial” relict radiation factor f_{Relict} for the wavelength λ = 0.071069 [nm]. Oscillations of the curve I_{distr} (s) are along the x-axis hence the criterions set at the beginning of this section are fulfilled. See text for details.

Figure 6 . Behaviour of the relict radiation factor f_{Relict} is shown. For comparison the courses of the classic coherent and incoherent X-ray atomic scattering factors and for Hydrogen are included. The parameter s_{Classic} is defined in Equation (6), the parameter is described in Equation (A5). Data for and are taken from [6]. The calculation is demonstrated for the wavelength λ = 0.071069 [nm].

tering vector in the classic reciprocal space s_{Classic}, see Equation (6).

, (12)

where is the member which is not Fourierdependent and describes the structure-less total disorder depending on the density of the matter.

The parameter r is measured in [nm*] in order to emphasize that the calculation of the RDF ρ(r) is realized on the basis of the parameter s_{Classic}, which is dipped in a 2-fold reciprocal space (see Subsection 2.3.1.). In other words: the calculation of the RDF ρ(r) is realized in the reciprocal space of classic distances, which have the dimension [nm*]. Here we again point out the fact, that classic distances are distances between Objects calculated on the basis of the function I_{Classic} (s_{Classic}), see Figure 3, which we analyze using Equation (A2) or (12).

In order to receive now the information in the real space of classic distances (characterized by the parameter R) we must calculate the reciprocal value of the parameter r, hence the relation between r and R is

. (13)

It would be now possible to rewrite quite formally Equation (A2) using the scattering vector in the relict reciprocal space S_{Relict}, see Equation (9). Similarly as for Equation (12) we would receive

. (14)

Quite hypothetically the RDF ρ(R) would then bring us information on the real space of relict distances, which have the dimension [nm]. Actually, however, a RDF will not be calculated in this case, because the distribution I(S_{Relict}), see Figure 4 , is not convenient for a Fourier transform. The calculation of relict distances in the real space, i.e. of distances between complex Objects (big clusters) will be done on the basis of a theoretical calculation of the function I(S_{Relict})) using the Debye formula (18) calculated for appropriate models, see later on Section 4.

3. Calculations in the Classic Reciprocal Space s_{Classic}

In our first example we calculate in Figure 7 the RDF of Objects corresponding to the Fourier transform of intensities for the wavelength λ = 0.071069 [nm] which is a frequently used wavelength (λ_{MoKα}) in e.g. structure analysis, see Equation (A2) andor (12). The scaling of intensities has been already demonstrated in Figure 5 on the basis of the relict radiation factor f_{Relict} constructed in Figure 6 .

In the same way we calculated RDFs for four more typical wavelengths, i.e. 0.110674 (λ_{SeKα}), 0.154178 (λ_{CuKα}), 0.250466 (λ_{VKα}) and 0.537334 [nm] (λ_{SKα}). From these calculations it follows that, as expected, the dependence of the magnitude of corresponding coordination spheres on the wavelength λ is linear, see Figure 8 , moreover, all RDFs had the same appearance.

In this connection we have to point out, that the distances are measured in reciprocal space distances [nm*] and that, with respect to Equation (13), these distances have to be recalculated to “real space” distances, e.g. in [km]. This recalculation is realized in Table 1 only for the most important distance min r = 0.348 [nm*]. Simultaneously we review this parameter for all wavelengths ( Figure 8 ) and simultaneously extrapolate this distance to the wavelength of relict radiation photons λ = 1.9

Figure 7 . Calculation of the radial distribution function (RDF) according Equations (A2) and-or (12) for the wavelength λ = 0.071069 [nm]. The dashed-dotted line corresponds to the second member in Equation (12), the dashed line is the first member in this equation (dependent on density) and full line is the sum of both components, see text for details. Value of the density D necessary to shift the minimum at 0.348 [nm*] to positive values of the RDF is indicated in the upper right corner.

Figure 8 . Dependences of most important distances, i.e. of coordination spheres 1 r (squares), 2 r (circles) separated by the minimum min r (down triangles) on the wavelength λ in the reciprocal space [nm*] according Figure 7 and from analogical calculations for wavelengths 0.110674, 0.154178, 0.250466 and 0.537334 [nm]. For an easier survey error bars are inserted only for the sphere 1 r.

Real space distances min R calculated in Table 1 are visualized in Figure 9 . The extrapolation to the wavelength of relict photons 1.9 [mm] indicates that for this wavelength the shortest min R distances are of the order 10 2 meters. Later on (see Subsections 4. and 5.1.) the distance min R will be ascribed to the distance between “Objects”.

Figure 9 . Dependence of the real space distances min R (full circles) on the wavelength λ (see Table 1 for details). Simultaneously an extrapolation to a distance corresponding to the wavelength of CMB (relict) photons 1.9 [mm] is visualized (empty circles). Later on (Sections 4. and 5.1.) the quantity min R will describe the distance between “Objects”.

Table 1 . Review of the most important distance min r characterizing the separation of the ordered region from the structure-less one on the wavelength λ (see Figure 8 ). Recalculation to the real space distances min R [km] is included. Extrapolation of this distance to the wavelength of relict radiation photons 1.9 [mm] is computed together with an estimate of final errors.

3.2. Calculation of the Density

The calculation presented in Figure 7 and repeated for four additional wavelengths enabled us to estimate the density of the matter, i.e. the important parameter effecting the first member ρ_{0} Medium (r) in Equation (12). We simply supposed that the fluctuations of the RDF should not be negative. In order to shift in Figure 7 the minimum at min r = 0.348 [nm*] to positive values we had to set the density to a value D = 108.60 [kg∙m –3 ]. In the same way we have determined densities for the remaining four wavelengths.

The results are summarized in Figure 1 0 and Table 2 . In the log-scale is the dependence of density on the wavelength nearly linear and therefore enables again an extrapolation to higher wavelengths. This extrapolation is presented in Table 2 and visualized in Figure 1 1 .

It follows from Table 2 and Figure 1 1 that the most probable medium density of density fluctuations of the matter with which CMB (relict) photons realized their last interaction is

9 × 10 –23 [kg∙m –3 ]. Taking in account the limits of our calculation then the density can be formally written as [kg∙m –3 ], see also Figure 11 and Table 2 .

4. Modelling in the Relict Reciprocal Space S_{Relict}

In the case when Figure 4 should be an X-ray scattering picture of a disordered material (e.g. of a glass) then such record would represent a picture typical for a material with well developed clusters. Their mutual distance should then characterize the position of the “first” massive peak. It follows from theory and experience that it is

Figure 1 0 . Dependence of macroscopic densities on short wavelengths. In the log-scale this dependence is nearly linear. Numerical values are given in Table 2 . Numbers indicate wavelengths, for which the corresponding RDF was calculated.

Table 2 . Review of numerical values of densities according Figure 1 0 is presented. Extrapolation of the sequence of densities to higher wavelengths, especially to the wavelength of relict radiation photons 1.9 [mm] is shown. First five densities D were calculated following the description in Subsection 3.2. Possible final error is estimated and the values of the critical density according [7] and [8] are given.

Figure 1 1 . Extrapolation of the dependence of densities on wavelengths λ to the wavelength of relict (CMB) photons λ = 1.9 [mm]. Empty circles represent values shown already in Figure 1 0 . Full circles are extrapolated values. Dashed lines show the limits of possible extrapolations.

not possible to get from this peak information on the internal structure of Clusters, only on their magnitude and mutual distance.

The method which has to be used for an analysis of this type of scattering is a direct calculation of scattered radiation on the basis of the Debye formula

. (18)

Here f_{i} and f_{j} are the scattering factors of n input particles and d_{ij} are the distances in real space between all available particles in the model and S_{Relict} is the scattering vector in the relict reciprocal space defined in Equation (9). It should be pointed out that as scattering factors f_{i} and f_{j} we have used now the relict radiation factor f_{Relict} calculated in Subsection 2.3.3. The summation is over all n particles in the model. This formula gives the average scattered intensity for an array of particles (or atoms in solid state physics) with a completely random orientation in space to the incident radiation.

Our model was quite simple: For the wavelength λ = 0.071069 [nm] the Cluster was a tetrahedron (5 particles) with an inter-particle distance 0.263 [nm] i.e. located in a cube with an edge 0.607 [nm]. In order to find the best fit with the scattering curve according Equation (18), the distance between Clusters (tetrahedrons) had to be d = 3 [nm], i.e. the tetrahedrons were located in positions of the basic skeleton, see Figure 1 2 , characterized now by a side a = 6.93 [nm]. This model had 22 × 5 particles, i.e. a total of 110 particles. This calculation is shown in Figure 1 3 .

For all other wavelengths (λ ≥ 0.110674 [nm]) we had to increase the dimensions of the Cluster. The Cluster had then the form of the skeleton shown in Figure 1 2 with an edge 0.607 [nm] and consisted of 22 particles (again with an inter-particle distance 0.263 [nm]) embedded in 8 edge-bound tetrahedrons. Only this Cluster occupied then the “positions” of the cubic basic skeleton shown in Figure 1 2 forming now an Object. (A more

Figure 1 2 . The basic skeleton (and-or a part of a Cluster structure) consists of 22 “positions” formed by 8 edgebound tetrahedrons. All positions are identical, for a better graphic representation are the centres of tetrahedrons drawn white. The picture has been constructed using programs [9] and [10].

instructive schematic presentation of an Object is shown in Figure 1 7 where Clusters are presented as small darker circles filled with “particles”.) When changing the dimension of this basic skeleton, we simultaneously changed again the distance d between Clusters. In order to reach for λ = 0.110674 [nm] the correct position of the massive peak at 1.6 [nm –1 ] an inter-Cluster distance d = 4.65 [nm] had to be used, i.e. the dimension of the skeleton was characterized by the side a = 10.74 [nm]. This model had then 22 × 22 particles, i.e. a total of 484 particles and simulated a part of the Object structure. The

Figure 1 3 . Calculation of the profile of the recalculated anisotropy spectrum for λ = 0.071069 [nm] based on a set of 22 Clusters with a mutual distance d = 3 [nm]. The Cluster was formed by a tetrahedron (5 particles) hence there were 110 particles in a model, see text for details. Full line - experiment, empty circles—calculated scaled and smoothed curve, dashed line—calculated scaled scattering.

Figure 1 4 . Calculation of the profile of the recalculated anisotropy spectrum for λ = 0.110674 [nm] and for a set of 22 Clusters with a mutual distance d = 4.65 [nm]. The Cluster consisted of 22 particles, hence there were 484 particles in the model, see text for details. Full line—experiment, empty circles—calculated scaled and smoothed curve, dashed line—calculated scaled scattering.

calculation is shown in Figure 1 4 .

Calculations of Cluster distances for additional wavelengths (0.154178, 0.250466 and 0.537334 [nm]) have shown (see Figure 1 5 ) that the dependence of Cluster distances on the corresponding wavelength is linear. This fact enabled an extrapolation of the Cluster distance d to the wavelength of relict photons λ = 1.9 [nm], see Table 3 . This extrapolated distance is [cm]. The extrapolation is visualized in Figure 1 6 .

It should be noted that the recalculated anisotropy spectrum depends in this case directly on the angle θ_{Relict} which is equal to the angle α (see Equation (8)) and therefore a recalculation of the inter-Cluster distance d into real space distances is not necessary because the Debye formula analyzes the relict reciprocal space represented by the vector S_{Relict} directly in real space distances, see the quantity d_{ij} in Equation (18).

In the following discussion we will concentrate on several important ideas which may arise when reading this paper.

In particular this contribution should demonstrate how the formalism imported from solid state physics could be useful in solving specific cosmological problems: It may shed some new light on the physical processes taking place in the primordial plasma.

First of all, according our opinion, this work points in favour of clustering processes and consequently to a cluster-like structure of the matter in the moment when

Figure 1 5 . Dependence of distances d between Clusters on the wavelength λ (numerical values are shown). With exception of the “0.071 case” models consisted of 22 Clusters with 22 particles in each Cluster, i.e. a model included 484 particles. Inter-Cluster distances characterize the position of the massive peak, see Figures 13 and 14 and the text for details.

Table 3 . Extrapolation of distances d between Clusters to the wavelength of relict photons λ = 1.9 [nm]. These distances influence the position of the massive peak, see Figures 14, 15 and 16. The estimate of the final error is based on errors given in Figure 1 5 .

Figure 1 6 . Extrapolation of distances d between Clusters to the wavelength λ = 1.9 [mm]—full squares calculated values—empty squares (see Figure 1 5 and Table 3 ).

the universe became transparent for photons (see Subsection 5.1.).

In the second place the new formalism enabled us a simple and general description of the interaction of relict radiation with the matter and may help in an improvement of the theoretical predictions of the CMB pattern (see Subsection 5.2.).

Finally this new approach may be useful in the analysis of the CMB data. We have shown that the transformation of the anisotropy spectrum of relict radiation into a special two-fold reciprocal space and into a simple reciprocal space was able to bring quantitative data in real space. Problems with the transformation into reciprocal spaces, mainly with the use of the proper wavelength of relict photons will be discussed in Subsection 5.3.

5.1. The Cluster-Like Structure of the Primordial Matter

The most important consequence of our quantitative results obtained in Sections 3 and 4 is, according our opinion, the idea of clustering processes taking part in the formation of the primordial matter. There we have arrived to three distances, which we interpret in a following way: The first distance

10 2 [m], ( Table 1 ) should indicate the distance between Objects (big clusters), the second one

10 –1 [m], ( Table 3 ) should indicate the distance between smaller Clusters, while the internal struc-

Figure 1 7 . A schematic arrangement of Clusters (darker regions) with particles (small white points) in an Object (white region). In our model the distance between Objects is

10 2 [m], see Table 1 . A detailed structure of a Cluster and of an Object in this model is presented in Figure 1 2 . The most probable model distance between Clusters is

10 –1 [m], see Table 3 . The distance between particles in the model is 0.26 [nm]. We estimate that there are

10 11 particles in one Object and

10 2 particles in one Cluster (see Appendix C1 and C2) and

10 9 Clusters in one Object, see Appendix C3.

ture of a single Cluster was formed in the model by 22 particles with a medium particle distance 0.26 [nm], see Section 4.

In Figure 1 7 we show a schematic picture of this cluster model. The big circle represents an Object. An Object is a clump of Clusters, where only a part of this clump was simulated in the model (the model of an Object had 22 Clusters each consisting of 22 particles, i.e. it consisted of a total of 484 particles).

Although this model gave a sufficiently well agreement with the width of the massive peak, as demonstrated in Figure 1 4 , our estimates (see Appendix C) show that the number of Clusters as well as the number of particles in one Cluster is greater, i.e. that there may be as far as 10 11 particles in one Object and 10 2 particles in one Cluster. That the density plays an important role in these calculations will be discussed in Subsection 5.4.

Even when the cluster model gave a good profile of the massive peak at e.g. 1.62 [nm], than such a model of internal structure of Clusters cannot be a unique one, because the calculation of the profile is not sensitive to the internal cluster structure, nevertheless the cluster-like character of the modelling process has to be maintained.

5.2. The Relict Radiation Factor

We have already pointed out in Subsection 2.1. why during the analysis of the CMB spectrum it has not been possible to apply conventional atomic scattering factors used in solid state physics and why a new special factor reflecting the complexity of interaction processes of photons with the primordial matter has to be constructed. It is important to have on mind that the description of these interactions is possible only in a special two-fold reciprocal space into which the CMB spectrum was transformed. We have called this new factor the relict radiation factor and it had to substitute all complicated processes which participate in the formation of the angular power spectrum of CMB radiation.

Because relict photons realize their interaction with various kinds of particles and we have generated only one radiation factor, this factor represents, as a matter of fact, a medium from all possible individual relict radiation factors. In this way this new formalism offers a general description of the interaction of relict radiation with the matter and simultaneously reflects the complexity of processes which influence the anisotropy spectrum of CMB radiation from the cosmological point of view [5].

During our study we have concentrated on three important facts which may justify the attempt to interpret the anisotropy spectrum of CMB radiation as a conesquence of the interaction of photons with density fluctuations which characterize the distribution of particles before the recombination process.

The first fact is that temperature fluctuations in the CMB spectrum are related to fluctuations in the density of matter in the early universe and thus carry information about the initial conditions for the formation of cosmic structures such as galaxies, clusters or voids [11].

Secondly, it is the fact that the information on these density fluctuations in the distribution of particles (electrons, ions, etc.) has been brought by photons. Photons which we observe from the microwave background have travelled freely since the matter was highly ionized and they realized their last Thomson scattering (see already Subsection 2.1.). If there has been no significant early heat input from galaxy formation then this happened when the Universe became cool enough for the protons to capture electrons, i.e. when the recombination process started [12].

The third fact is that the anisotropy spectrum is angular dependent, see Figure 1 .

Although we know that the anisotropy spectrum of CMB radiation, as presented in Figure 1 , has no direct connection with a scattering process of photons, it was the transformation of the CMB spectrum into a two-fold reciprocal space, which enabled us to interpret the anisotropy spectrum of CMB radiation as a result of an interaction process of photons with density fluctuations of the matter represented by electrons, ions or other particles. This approach enabled us to reach an advantageous approximation of this process.

The process consisted of two steps: First of all we have constructed in Subsection 2.3.1. an angular reciprocal space characterized by the “scattering” angle θ_{Classic}, see Equations (2) and (4). This space is reciprocal to the space characterized by the angle α (α is the angle between two points in which the temperature fluctuations of CMB radiation are compared to an overall medium temperature).

Then, we have constructed an additional “classic” reciprocal space (1/λ) into which the first one (the θ_{Classic} space) was dipped, by defining in this new “two-fold” reciprocal space the classic scattering vector s_{Classic}, see Equation (6). Only after these transformations we treated in this new classic reciprocal space the transformed anisotropy CMB spectrum as a scattering picture of relict photons.

It was only this space in which we simulated (in Subsection 2.3.3.) the interaction of CMB (relict) photons with density fluctuations by the relict radiation factor f_{Relict}.

The criterion for the trial and error construction of the relict radiation factor f_{Relict} has been that this factor had to fulfil the three requirements set at the beginning of Subsection 2.3.3. Only then it was secured that after the Fourier transform, according Equations (A2) and-or (12), there will not be any (or at least small) parasitic fluctuations on the curve ρ(r) and-or ρ Fourier . That we have achieved these demands is documented in Figure 7 where we do not see any parasitic fluctuations on the curve ρ Fourier and as a consequence on the curve ρ(r).

To summarize: It is true that in our formal analogy between scattering of e.g. short-wave radiation on disordered matter ( Figure 2 ) and “scattering” of CMB photons on electrons, ions and other particles ( Figure 1 ) is an essential difference, because the physical processes are completely different, e.g. the scattering process itself, length scales involved, etc., however, the difference between physical processes is reflected and simultaneously eliminated by the special relict radiation factor f_{Relict} (Subsection 2.3.3.), which we have included into all calculations based on the classic two-fold reciprocal space (see Subsection 2.1.). Moreover, additional calculations in the relict reciprocal space (see Subsection 4.) based on the relict radiation factor were done directly for the transformed angular power spectrum of relict radiation (see I_{Relict} (S_{Relict}) in Figure 4 ) and thus present an information on distance relations between Clusters (formed by particles) in real space.

5.3. The Wavelength Problem

The problem is to which wavelength of relict photons we have to relate our calculations. One possibility may be to refer this wavelength to that time when 379.000 years after the Big Bang the Universe cooled down to 3000 K and the ionization of atoms decreased already only to 1%. Then according Wien’s law

(16)

where λ_{max} is the peak wavelength, T is the absolute temperature of the blackbody, and b is a constant of proportionality called Wien’s displacement constant, [mK], we obtain for the temperature 3000 K a wavelength value [nm], see [13].

However, simultaneously we must be aware of the fact that we are analyzing CMB photons now when the temperature of the universe, due to its expansion, is 2.725 K. Then the wavelength of photons according the Wien’s law should be

On the other hand the COsmic Background Explorer (COBE) measured with the Far Infrared Absolute Spectrophotometer (FIRAS) the frequency spectrum of the CMB, which is very close to a blackbody with a temperature 2.725 K [11,14]. The results are shown in Figure 18 in units of intensity (see the text to Figure 1 8 ). It follows that the wavelength corresponding to the maximum is 1.9 [mm].

Figure 1 8 . Dependence of the intensity of the CMB radiation on frequency as measured by the COBE Far InfraRed Absolute Spectrophotometer (FIRAS) ([11,14]). The thick curve is the experimental result the points are theoretically calculated for an absolute black body with a temperature of 2.725 [K]. The x axis variable is the frequency in [cm –1 ]. The y-axis variable is the power per unit area per unit frequency per unit solid angle in MegaJanskies per steradian [sr], (1 [Jansky] is a unit of measurement of flux density used in radioastronomy, abbreviated “Jy” (1 [Jansky] is 10 –26 [W∙m –2 ∙Hz –1 ]).

After all we have decided to relate our results to the wavelength of CMB photons λ = 1.9 [mm] which corresponds to the maximum of the intensity distribution. Because the distribution of the spectrum covers a relatively broad interval of wavelengths, see Figure 1 8 , calculations based on the wavelength 1.9 [mm] should then represent the most probable estimate. Moreover, this consideration is supported by the fact that the angular distribution of CMB radiation is the same for all wavelengths.

However, on the basis of graphs in Figures 10, 12 and 16 an easy recalculation of distances and-or of the density would be possible when another CMB photons wavelength would be considered as more appropriate.

5.4. The Density of the Mass and Distances between Objects, Clusters and Particles

The way how we arrived to numbers characterizing the density of the matter was described in Subsection 3.2. In a conventional X-ray analysis the density is the macroscopic density of the material under study. Therefore we suppose that also in this case the density which influences the parabolic shape of the curve of total disorder (see the first member on the right side of Equation (A2) and-or (12) and Figure 8 ) should be understood as a real medium density of density fluctuations.

The dependence of the density on the wavelength as demonstrated in Figures 11 and 12 is not perfectly linear therefore we have marked in Figure 1 1 the extent of possible linear dependences. This result can be formally written as

[kg∙m –3 ]. (17)

This medium value is about 10 5 times higher than the “critical density” D_{critical}

Further, we should have in mind that the local density in a Cluster or in an Object has to be greater. We are able to document this fact on the basis of our Cluster model. Based on particle distances d_{particles} = 0.263 [nm], we have simulated a part of the Cluster structure by a cube with an edge a_{Cluster} = 0.607 [nm]. There were 22 particles in this cube which can be closed in a sphere with a radius R_{Cluster} = 2d_{particles} = 2 × 0.263 [nm] = 0.53 [nm]. The volume of this sphere is V_{Cluster} = 0.62 [nm 3 ] = 0.62 × 10 –27 [m 3 ]. Supposing that particles are represented according expression (C6) by their medium mass [kg], we obtain for the density in the Cluster the value

[kg∙m –3 ], (18)

i.e. a value approaching density values known from solid state physics (i.e. values lying between the densities of gases and liquids).

At the same time we have to take in account that the estimates concerning the density of matter are really complicated. The microwave light seen by the Wilkinson Microwave Anisotropy Probe (WMAP), suggests that fully 72% of the matter density in the universe appears to be in the form of dark energy [15] and 23% is dark matter. Only 4.6% is ordinary matter. So less than 1 part in 20 is made out of matter we have observed experimenttally or described in the standard model of particle physics. Of the other 96%, apart from the properties just mentioned, we know “absolutely nothing” [16]. In this connection we consider the density value we have received (9 × 10 – 23 [kg∙m –3 ]) as the density of the ordinary matter.

Last remark should be given to the probability of Object interactions in the case of their apparently large mutual distances (

10 2 [m]). It follows from the Maxwell speed distribution that the root mean square particle velocity v corresponding to the temperature T = 3000 [K], is

, (19)

where k is the Boltzmann constant (k = 1.38 × 10 –23 [Joule∙K –1 ]) and m is the mass of the particle, which may be here for example the mass of the proton m = 1.67 × 10 –27 [kg]. Then we obtain

[m∙s – 1 ]

This is already a velocity, which should make possible an intensive interaction of Objects formed by Clusters consisting of particles.

A formalism of solid state physics has been applied to provide an additional tool for the research of cosmological problems. It was demonstrated how this new approach could be useful in the analysis of the CMB data. After a transformation of the anisotropy spectrum of relict radiation into a special two-fold reciprocal space it was possible to propose a simple and general description of the interaction of relict photons with the matter 380.000 years after the Big-Bang by a “relict radiation factor”. This factor, which may help in an improvement of the theoretical predictions of the CMB pattern, enabled us to process the transformed CMB anisotropy spectrum by a Fourier transform and thus arrive to a radial electron density distribution function (RDF) in a reciprocal space.

As a consequence it was possible to estimate distances between Objects of the order of

10 2 [m] and the density of the ordinary matter

10 –22 [kg∙m –3 ]. Another analysis based on a direct calculation of the CMB radiation spectrum after its transformation into a simple reciprocal space and combined with appropriate structure modelling confirmed the cluster structure. It indicated that the internal structure of Objects may be formed by Clusters distant

10 –1 [m], whereas the internal structure of a Cluster consisted of particles distant

In this way the work points in favour of clustering processes and to a cluster-like structure of the matter and thus may contribute to the understanding of the structure of density fluctuations and hence to a refinement of parameters describing the Standard Model of Cosmology [17]. Simultaneously, the work sheds more light on the structure of the universe in the moment when the universe became transparent for photons. On the basis of quantitative considerations it was possible to estimate the number of particles (protons, helium nuclei, electrons and other particles) in Objects and Clusters and the number of Clusters in an Object.

My thanks are due to Prof. Richard Gerber (University of Salford, Manchester) for discussion and proposals directed to the final presentation of this paper, to Mgr. Radomír Šmída, PhD (Institute of Physics, Acad. Sci. of the Czech Republic) for comments, proposals and discussion concerning this article, to Prof. Karel Segeth (Institute of Mathematics, Acad. Sci. of the Czech Republic) for discussions and help in clarifying some aspects of the Fourier transform, to Prof. Jan Kratochvíl (Department of Physics, Faculty of Civil Engineering, Czech Technical University in Prague) for discussions pointing out several inconsistencies in the original conception of the article. Last acknowledgement is due to Dr. Jiří Hybler (Institute of Physics, Acad. Sci. of the Czech Republic) for help in the preparation of Figure 1 2 presenting a part of a Cluster skeleton.

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Appendix A. Basic Equations

Generally the intensity of radiation scattered 2 on a matter (solid, liquid) offers us information on the structure of a material of any kind in the reciprocal space. The relation between the reciprocal and real space is mediated by the Fourier transform of the radiation intensity scattered by a disordered material.

. (A1)

In a more detailed description the quantity ρ(r) is then expressed as

(A2)

and describes the radial electron density distribution function (RDF) in real space in the case when the atomic scattering factor f_{m} (see Equation (A6)) is given in electrons [e]. The parameter r is the distance of an arbitrary atom from the origin in real space units.

In Equation (A2) a_{m} are the concentrations of elements composing the matter, are the elemental contributions of electron density to the overall electron density, i.e. it is the electron density around an atom of kind m, the factor exp(τs 2 ) is an artificial temperature factor in which usually τ = –0.010, is the mean electron density in a totally disordered material, which can be deduced from the macroscopic density via the Avogadro number L

, (A3)

where Z_{m} is the atomic number of kind m, D is the macroscopic density in [g∙cm –3 ] and M is the molecular weight

W_{m} are corresponding atomic weights. The factor 10 –21 in Equation (A3) is a consequence of the fact that the parameter r is in [nm].

The parameter s is in Equation (2) related with the wavelength λ of scattered radiation by the formula

. (A5)

Here is s = s – s_{0}, where s_{0} is the vector of the incident and s the vector of the scattered radiation in the reciprocal space.

Further, θ is the angle between the incident and scattered radiation (X-rays or neutrons) and λ is the wavelength of this radiation and K_{m} is the effective number of electrons in an atom of kind m

, (A6)

where f_{m} is the atomic scattering factor for X-rays for an atom of the kind m (see already Subsection 2.1.) and f_{e} is the atomic scattering power of an electron for X-rays

(A7)

During a conventional experiment (e.g. see Figure 2 ), i.e. using Mo_{K}_{a} radiation, we have = 0.071069 [nm] and the maximum possible value of corresponding to θ = 90˚ is then according Equation (A5)

. (A8)

Here we are starting to use the subscript “Classic”, which should point out that the scattering vector in the reciprocal space s_{Classic} will be considered in the same way as in the “classic” conventional non-crystalline case.

In Equation (A2) is i(s) the experimentally obtained scattered intensity of radiation, I_{corr} is this intensity corrected for various factors 3 and properly scaled for the absolute value of scattering, hence

, (A9)

the parameter is acting here as a sharpening function.

The general formula for the scattering on gas I_{gas} (s) is

, (A10)

where are the scattering factors for the incoherent (Compton) scattering, see Figure 6 .

The labelling I_{distr} for i(s) is used in Appendix B, where the scaling methods, important for a correct Fourier transform, are reviewed.

Appendix B. The Scaling Problem

In Equation (A9) we have already introduced the quantity I_{corr} (s), i.e. the corrected experimental scattered intensity. However, in order to arrive to a correct RDF, I_{corr} (s) must be scaled to the I_{gas} (s) function in the absolute scale of atomic scattering, see Equation (A10).

In the simplest scaling method we suppose that for high s-values there are not any scattering effects on the corrected experimental curve I_{corr} (s) and therefore the I_{corr} (s) and the I_{gas} (s) curves should be equal. Then the scaling parameter a_{HSV} is for easily calculated as

. (B1)

As a consequence we obtain in the whole interval of s-values a scaled scattered intensity represented by the equation

. (B2)

The function oscillates around the I_{gas} (s) curve. Following Equation (A9), we subtract the scattering on gas and obtain the most important function I_{distr}, see Figure 5 .

There are several other scaling methods. An integral method [19] is characterized by a scaling factor and supposes that the areas under the experimental scattering curve I_{corr} (s) and the structure-less I_{gas} (s) curve should be equal. Similarly there is a quadratic integral method [20] characterized by a scaling parameter.

Our long experience in the research of disordered materials documents that the better was the experiment and the better has been the application of scattering factors, the smaller was the difference (only several percent) between the scaling coefficients, and and simultaneously the smaller were the parasitic fluctuations on the RDF. In the present work we have used all three scaling methods and have kept the difference between scaling factors in the limit of 4 percent.

Appendix C. Quantitative Relations between Objects, Clusters and Particles Estimated on the Basis of the Cluster Model

C1. Estimates from Object Distances

In Subsection 3.1, we have defined that the nearest distance between Objects (big clusters) is

108 [m], see Table 1 . In this moment we suppose a relatively simple organization of Objects, i.e. a “cubic body-centred” arrangement, in which an Object in the centre has 8 nearest neighbour Objects distant b_{O} = 108 [m], where b_{O} is the half of the body diagonal in a cube with a side

. (C1)

The volume V_{O} of this cube is therefore

. (C2)

Using now our result on the density of the matter, see Table 2 ,

(C3)

we are able to calculate in this model the mass m_{2O} of Objects embedded in a cube with the volume V_{O}.

(C4)

At the same time, however, we have to take in account that, as a matter of fact, there are two Objects in the space of the cube (in each cube corner there is only 1/8 of the second Object). Hence the mass m_{O} embedded in one Object is

1) The mass is formed by a 1:1:1 mixture of protons, helium nuclei and electrons

We may suppose now that the universe (in the time when the microwave background radiation began propagating) consisted of baryons (protons, helium nuclei, etc) and electrons, neutrinos, photons and dark matter particles. Supposing now that we have a mixture consisting of protons, helium nuclei and electrons in a relation 1:1:1, then the medium mass of a “particle” in this mixture is

(C6)

and the number of particles in one Object is in this case

(C7)

2) The mass is formed by a 1:1:10 mixture of protons, helium nuclei and electrons

Supposing now a mixture consisting of protons, helium nuclei and electrons in a relation 1:1:10, then the medium mass of a “particle” in this system is

. (C8)

and the number of particles in one Object is then

. (C9)

This section may be summarized by the statement that there are

particles in one Object (C10)

C2. Estimates from Cluster Distances

According our calculations the distance between Clusters is

12 [cm] = 1.2 × 10 –1 [m], see Table 3 and Figure 1 7 . Similarly as in the previous case we suppose again a relatively simple organization of Clusters, i.e. a cubic body-centred arrangement in which a Cluster in the centre has 8 “nearest neighbour” Clusters distant [m], where b_{C} is the half of the body diagonal in a cube with a side

(C11)

The volume V_{C} of this cube is therefore

. (C12)

Using now our result on the density of the matter, see already Equation (C3)

we are able to calculate for this model the mass m_{2C} of Clusters embedded in a cube having the volume V_{C}.

(C13)

Here again we have to take in account that there are two Clusters in the space of the cube (in each corner there is only 1/8 of the second Cluster). Hence the mass m_{C} embedded in one Cluster is

. (C14)

1) The mass is formed by a 1:1:1 mixture of protons, helium nuclei and electrons

Similarly as in the preceding Appendix C1 we suppose again a mixture of protons, helium nuclei and electrons in a relation 1:1:1, respectively. The medium mass of a “particle” in this mixture is (see Equation (C6)).

and the number of particles in one Cluster is then

particles. (C15)

2) The mass is formed by a 1:1:10 mixture of protons, helium nuclei and electrons

Identically as in the preceding Appendix C1 the medium mass of a “particle” is in this case, see equation (C8),

and the number of particles in one Cluster is then

. (C16)

This section can be summarized by the statement that there are

(C17)

C3. Consequences of Previous Calculations

We are now able to calculate easily the number of Clusters in one Object. Because there are particles in one Object (Equation (C10)) and there are particles in one Cluster, it follows that an Object should be composed from N_{C} Clusters, where

Supposing that densities in the Object and in the Cluster are equal then this value is independent on the value of the density and on the mass of the particle (e.g.) and depends only on the relation of the volumes V_{O}/V_{C}, because

(C19)

1 It is just the low-energy limit of Compton scattering: the particle kinetic energy and photon frequency are the same before and after the scattering, however this limit is valid as long as the photon energy is much less than the mass energy of the particle.

2 The term “scattering” (e.g. of radiation) is used throughout the Appendix, however, in the moment when in the quantity f_{m} (see equation (A6)) the atomic scattering factor is substituted by the relict radiation factor, see Subsections 2.1. and 5.2. we speak about an interaction of photons with the primordial matter.

3 In a conventional experiment the scattered intensity is corrected for scattering on “air”, absorption, divergency of the X-ray beam, Lorentz and polarization factor. During our calculations we have included only the polarization factor.