CMB and Random Flights: temperature and polarization in position space
aa r X i v : . [ a s t r o - ph . C O ] J un Prepared for submission to JCAP
CMB and Random Flights:temperature and polarization inposition space
Paulo H. F. Reimberg a,b
L. Raul Abramo a a Instituto de Física, Universidade de São Paulo, CP 66318, 05314-970, São Paulo, Brazil b Institut de Physique Théorique, CEA-Saclay, 91191 Gif-sur-Yvette, FranceE-mail: [email protected], [email protected]
Abstract.
The fluctuations in the temperature and polarization of the cosmic microwavebackground are described by a hierarchy of Boltzmann equations. In its integral form, thisBoltzmann hierarchy can be converted from the usual Fourier-space base into a position-space and causal description. We show that probability densities for random flights play akey role in this description. The integral system can be treated as a perturbative series inthe number of steps of the random flights, and the properties of random flight probabilitiesimpose constraints on the domains of dependence. We show that, as a result of these do-mains, a Fourier-Bessel decomposition can be employed in order to calculate the random flightprobability densities. We also illustrate how the H-theorem applies to the cosmic microwavebackground: by using analytical formulae for the asymptotic limits of these probability den-sities, we show that, as the photon distribution approaches a state of equilibrium, both thetemperature anisotropies and the net polarization must vanish.
Keywords:
Cosmic Microwave Background, Random Flights, Boltzmann Equations ontents
A.1 The case of G lL ( R, X ; r , . . . , r n ) when f ( k ) = k f ( k ) The cosmic microwave background (CMB) has been a prolific source of theoretical [4] andexperimental [2, 13, 15] results over the last couple of decades. Because it relies on low-energy interactions well inside the linear regime, the CMB now occupies a central part inour understanding of Cosmology, providing crucial information about the initial state of theUniverse and about many processes that distorted it since the surface of last scattering.A fundamental description of the temperature and polarization of the CMB starts withthe basic interaction dynamics (free propagation and scatterings with matter sources), and isfully realized through the Einstein-Boltzmann hierarchy of equations. We shall here describethe system in a backward fashion: starting with the integral version of Boltzmann’s equations,we will uncover details about the physical system described by that set of equations. Ourprocedure will be directed by the underlying structure of those integral equations, in particularby the way in which the hierarchy couples the equations. By removing and then reintroducingthe source terms we will be able to reveal some hidden details about the microscopic dynamics– 1 –f the physical processes that lead to the properties of the CMB which are observed today. Tobe clearer, we shall demonstrate that the Boltzmann equations preserve, in a very fundamentalway, the fact that the fluctuations in the temperature and polarization of the CMB wereimprinted through Thomson scatterings of low energy photons and free electrons during therecombination. Consequently, the hierarchy can be rewritten in terms of a series expansionover the number of scatterings that photons suffered before decoupling from matter. Inthis series expansion, besides the probability for each interaction, the number of scatteringsappears as a key ingredient, since it determines the probability density that mediates the wayin which the source terms imprint their contribution into the final signal.The argument which leads to these conclusions will take us through a treatment of theCMB in position space and, in many senses, the present work is a further development of[1]. When performing the conversion from Fourier space to position space, it is unavoidablethat a certain class of integrals over products of spherical Bessel functions appear in thedescription. The proper understanding and interpretation of these integrals constitute thekey barrier that we should overcome in the description of the system in position space. Aswe shall see, these integrals codify the memory of the fundamental scattering dynamics. Infact, we shall demonstrate that these integrals constitute a generalization of random flightprobability densities.Random flights are a classical problem in mathematical physics [18], with many appli-cations in physics and astronomy [3]. The problem was first proposed in the beginning of the20 th century in context of the study of bird migrations. Lord Rayleigh, soon after, appliedthe same ideas in acoustics. Further contributions on this subject are described in ref. [5].In very simple terms, a random flight (in a D -dimensional space) is the trajectory performedby a body which moves at constant speed and changes its direction of motion into anotherrandom direction at Poisson-distributed time intervals. If the movement has a fixed origin,we may ask, based on the length of the intermediate displacements, as well as on the numberof displacements, what is the probability for the moving body to reach a distance r from theorigin.The precise form of the visibility function also plays a fundamental role in the descriptionof the CMB in terms of random flights, since it models the probability of scatterings duringthe recombination — which changes as a function of time only. The spatial independence ofthe visibility function and the elasticity of Thomson scattering (which are, of course, alreadycodified in the usual Boltzmann hierarchy) are the fundamental ingredients which lead to arandom-flight-based description of the CMB.This paper is organized as follows: after introducing the Boltzmann hierarchy in theintegral form in section 2, we temporarily decouple the evolution of the temperature fromthe polarization, and show the consequences of this simplification for the expression of thetemperature and polarization in section 3. We then show, in section 4, how the family ofintegrals over spherical Bessel functions that appears in that description is related to randomflight probability densities functions. After clarifying the structure of this dynamical systemunder the approximation of uncoupled temperature evolution, we go back to the original formof the hierarchy in section 7, and show that the formalism introduced in the previous Sectionscan be extended to treat the general case. We present, in section 8, a computational toolbased on Fourier-Bessel expansions which is suitable for computing the probability functionsappearing in our description. Finally, in section 9 we use asymptotic arguments to show thathigh order terms should not contribute to the observables, which is expected from Boltzmann’sH-theorem. – 2 – Boltzmann’s equations
The hierarchy of Boltzmann’s equations describing CMB temperature fluctuations and po-larization can be written in terms of a set of coupled integral equations — see, e.g., [14]. Letthe temperature anisotropies observed at position x o and at (conformal) time η o along thedirection ˆ o be given in terms of its momenta as: Θ( x o , η o , ˆ o ) = Z d k (2 π ) / e i k · x o π X lm i l Θ l ( k , η o ) Y ∗ lm (ˆ k ) Y lm (ˆ o ) . (2.1)Similarly, the polarization, in terms of the usual Stokes parameters Q , U and I , is decomposedas: Q + iU I ( x o , η o , ˆ o ) = Z d k (2 π ) / e i k · x o π X lm i l α l ( k , η o ) Y ∗ lm (ˆ k ) Y lm (ˆ o ) , (2.2)where Y lm are the spin-weighted spherical harmonics [16].The momenta of the CMB temperature and polarization are then given by the integralequations: Θ l ( k , η o ) = Z η o dη e − µ ( η ) ( µ ′ ( η ) " θ SW ( k , η ) j l ( k ∆ η ) − k V b ( k , η ) j ′ l ( k ∆ η )+ 12 h Θ ( k , η ) − √ α ( k , η ) i (cid:20) j ′′ l ( k ∆ η ) + 12 j l ( k ∆ η ) (cid:21) +(Ψ ′ + Φ ′ )( k , η ) j l ( k ∆ η ) ) , (2.3)and α l ( k , η o ) = − s ( l + 2)!( l − Z η o dη µ ′ ( η )e − µ ( η ) h Θ ( k , η ) − √ α ( k , η ) i j l ( k ∆ η )( k ∆ η ) . (2.4)In eqs. (2.3) – (2.4) a prime denotes a derivative with respect to conformal time η , the opticaldepth to Thomson scattering is µ ( η ) , and we have defined the interval ∆ η = η o − η . It is some-times convenient to define the ubiquitous source term P ( k , η ) = (cid:2) Θ ( k , η ) − √ α ( k , η ) (cid:3) .The system above is closed once the perturbed Einstein equations are used to determine howthe linear scalar cosmological perturbations θ SW , V b , Φ and Ψ evolve with time. However,both the precise nature of the perturbed Einstein equations, or of the initial conditions thatare used to evolve those equations, are irrelevant for our results.Equations (2.3) – (2.4) show that the primary sources of temperature fluctuations arethe Sachs-Wolfe term, θ SW , the baryon velocity, V b , and the gravitational potentials Φ and Ψ (also known as the Bardeen potentials — we work in the conformal-Newtonian gauge). Theprimary source of the polarization of the CMB, on the other hand, is the quadrupole of thetemperature fluctuations. The integral equations then couple all the momenta of temperatureand polarization, mediated by the visibility function g ( η ) = µ ′ ( η )e − µ ( η ) .Henceforth we will take x o = 0 , i. e., the observer is taken to be at the origin of thecoordinate system employed for the description of the problem.– 3 – Uncoupling the temperature evolution
We can decouple eqs. (2.3) – (2.4) by neglecting the term α in eq. (2.3). This trunca-tion represents the approximation whereby deviations from the equilibrium temperature aredescribed by the Sachs-Wolfe (SW) and integrated Sachs-Wolfe (ISW) effects. Within thisapproximation, then eq. (2.3) becomes: Θ l ( k , η o ) = Z η o dη n g ( η ) (cid:2) θ SW ( k , η ) j l ( k ∆ η ) − k V b ( k , η ) j ′ l ( k ∆ η ) (cid:3) + e − µ ( η ) (Ψ ′ + Φ ′ )( k , η ) j l ( k ∆ η ) o . (3.1)Neglecting polarization as a source term for the temperature anisotropies is in fact a verygood approximation, and the reason for this underlies the argument presented in this paper.The visibility function g ( η ) should be regarded as the probability per unit (conformal) timethat photons will scatter with some free electron — and, in fact, g ( η ) is defined in such a waythat this probability is normalized, R ∞ dη g ( η ) = R ∞ dµ e − µ = 1 . This means that each timea factor of the visibility function intermediates a source term, that source term is damped bya factor ǫ , with < ǫ < . Since the lowest-order polarization term has at least one factor ofthe visibility function, it contributes as a source term to the temperature with two factors ofthe visibility function. Hence, the SW and ISW effects dominate the intensity of the signal,and eq. (3.1) accounts for the largest contribution to the temperature anisotropies. Let’s now take the lowest-order contribution to the temperature anisotropies, and express itin terms of position space. Expressing eq. (2.1) as: Θ( η o , ˆ o ) = X lm θ (0) lm ( η o )Y lm (ˆ o ) , (3.2)the coefficients θ (0) lm ( η o ) are given by: θ (0) lm ( η o ) = 2 i l Z η o dη Z dk (2 π ) / k Z d ˆ k e − µ ( η ) × (cid:26) µ ′ ( η ) (cid:20) θ SW ( k , η ) − V b ( k , η ) ∂∂η (cid:21) + (Ψ ′ + Φ ′ )( k , η ) (cid:27) Y ∗ lm (ˆ k ) j l ( k ∆ η o ) (3.3)We shall then define the primary source term operator as: S lm ( k, η ) = Z d ˆ k e − µ ( η ) (cid:26) µ ′ ( η ) (cid:20) θ SW ( k , η ) − V b ( k , η ) ∂∂η (cid:21) + (Ψ ′ + Φ ′ )( k , η ) (cid:27) Y ∗ lm (ˆ k ) , (3.4)where we stress the fact that we have included the optical depth to Thomson scattering inthe definition of the source term. In terms of eq. (3.4), eq. (3.3) can be written as: θ (0) lm = 2 i l Z η o dη Z dk (2 π ) / k S lm ( k, η ) j l ( k ∆ η o ) . (3.5)In order to obtain a position-space description, we will express the coefficients S lm ( k, η ) interms of their counterparts in position space, by means of a Hankel transform: S lm ( k, η ) = 2 ( − i ) l Z dX (2 π ) / X S lm ( X, η ) j l ( k X ) . (3.6)– 4 –sing now the orthogonality relation for spherical Bessel functions, Z dk k j L ( ak ) j L ( bk ) = π b L a L +2 δ ( a − b ) , we obtain: θ (0) lm = Z η o dη S lm ( X = ∆ η , η ) . (3.7)Eq. (3.7) has a straightforward interpretation: it states that, in order for a source term attime η contribute to the CMB signal at time η o , that source must be located at the sphericalshells of radius ∆ η = η o − η centered at the observer. The set of those spherical shells is ahypersurface which corresponds, of course, to the past light cone of the observer on { x o , η o } .Since the visibility function is highly peaked at the time of decoupling, the primary sourceterm contributes the most to the signal near the epochs when z ( η ) ≃ . We shall now decompose the polarization as: Q + iU I ( η o , ˆ o ) = X lm π lm ( η o ) Y lm (ˆ o ) , (3.8)with the aim of determining the coefficients π lm ( η o ) . The source terms in eq. (2.4) are Θ and α , which are built iteratively from an initial temperature quadrupole. We can, therefore,organize the iterative solution as a series into powers of the visibility function. As a first stepin the iterative solution, for example, α (which is of higher order in the visibility function)will not be taken into account — only the temperature quadrupole will contribute to generatepolarization at this order. The first iteration is, therefore: π (1) lm ( η o ) = −
34 2 i l s ( l + 2)!( l − Z η o dη g ( η ) Z η dη Z dk (2 π ) / k S lm ( k, η ) × j ( k ∆ η ) j l ( k ∆ η )( k ∆ η ) , (3.9)where the time intervals are defined as ∆ η = η − η and ∆ η = η o − η . The source term S lm is the same as was defined in eq. (3.4). Using once again the Hankel transform of eq.(3.6) we can recast eq. (3.9) as: π (1) lm ( η o ) = − π s ( l + 2)!( l − Z η o dη g ( η ) Z η dη Z ∞ dX X S lm ( X, η ) × Z ∞ dk k j l ( kX ) j l ( k ∆ η o )( k ∆ η o ) j ( k ∆ η ) . (3.10)The interpretation of the expression above is the following: at time η a source term generatesa temperature quadrupole. That quadrupole then generates, through a scattering at time η , the polarization which is finally observed at time η o . As we shall see, the integral of thesecond line of eq. (3.10) guarantees that the source term, at a distance X from the origin, islocated in the past lightcone of the observer, for all possible η and η . The variable X willalso be upper-bounded, and therefore the upper limit in the integration over the source terms– 5 –ill be replaced by a finite value that, as we shall see, corresponds the radius of the observer’spast lightcone.The next step in the iterative solution is to take the π (1)2 m just computed and use it asa source term for the polarization itself — this means that now the polarization piece of thesource term in eq. (2.4), α , is no longer assumed to vanish. This contribution, which we willcall π (2) lm , is therefore given by: π (2) lm ( η o ) = − s ( l + 2)!( l − Z η o dη g ( η ) Z d k (2 π ) / π i l " − √ α ( k , η ) Y ∗ lm (ˆ k ) j l ( k ∆ η o )( k ∆ η o ) = − s ( l + 2)!( l − Z η o dη g ( η ) 4 π i l π " − √ (cid:18) − (cid:19) √ Z η dη g ( η ) × Z dk (2 π ) / k Z d ˆ k Θ ( k , η ) Y ∗ lm (ˆ k ) j ( k ∆ η )( k ∆ η ) j l ( k ∆ η o )( k ∆ η o ) , where now the interval ∆ η = η − η . In terms of the primary sources in position space,after using eq. (3.7) we obtain: π (2) lm ( η o ) = − π s ( l + 2)!( l − Z η o dη g ( η ) Z η dη g ( η ) Z η dη × Z ∞ dXX S lm ( X, η ) " Z dkk j l ( kX ) j l ( k ∆ η o )( k ∆ η o ) j ( k ∆ η )( k ∆ η ) j ( k ∆ η ) . (3.11)The term π (2) lm ( η o ) is weighted twice by the visibility function and corresponds, as we willshow in details, to the contribution to the total polarization coming from photons that haveThomson scattered twice during the recombination.At this point it is important to clarify our notation. We will always count the photonscatterings backwards in time: the time the photons are observed is always taken to be η o ; thelast time that the photons scattered before being observed is η ; and so on. By convention,we will always evaluate the primary source term S lm at the instant η , so the sequence ofscatterings ends with η . Hence if, as in the case described by eq. (3.11), there are twoscatterings between the generation of the signal at η and its observation at η o , then we have η o ≥ η ≥ η ≥ η , and the time intervals always express the differences between one time andthe previous one, so in that case ∆ η = η o − η , ∆ η = η − η , and ∆ η = η − η .The same argument allows us to calculate the next order contribution, which takes intoaccount three intermediate scatterings: π (3) lm ( η o ) = − π s ( l + 2)!( l − Z η o dη g ( η ) Z η dη g ( η ) Z η dη g ( η ) × Z η dη Z ∞ dX X S lm ( X, η ) × Z dk k j l ( kX ) j l ( k ∆ η o )( k ∆ η o ) j ( k ∆ η )( k ∆ η ) j ( k ∆ η )( k ∆ η ) j ( k ∆ η ) . (3.12)– 6 –he general term in the iterative series expansion with n intermediate scatterings can bewritten as: π ( n ) lm ( η o ) = − π s ( l + 2)!( l − Z η o dη g ( η ) × n − Z η dη . . . dη n T { g ( η ) . . . g ( η n ) }× Z η n dη Z ∞ dX X S lm ( X, η ) × ( n − Z dk k j l ( kX ) j l ( k ∆ η o )( k ∆ η o ) j ( k ∆ η )( k ∆ η ) . . . j ( k ∆ η n − )( k ∆ η n − ) | {z } ( n − times j ( k ∆ η n ) (3.13)Here, T stands for the time-ordered product of the sub-intervals, whose purpose is to repro-duce the chain of integrations mediated by visibility functions shown in, e.g., eq. (3.12).The coefficients π lm appearing in eq. (3.8) can be expressed, therefore, as: π lm ( η o ) = ∞ X n =0 π ( n ) lm ( η o ) , (3.14)with π ( n ) lm ( η o ) given by eq. (3.13).An important condition for the validity of this perturbative expansion of the CMBtemperature and polarization is that all the terms in the expansion of eq. (3.8), with anarbitrary number n of intermediate scatterings, must be expressed in position space. However,this can only be true if the k integrals over products of spherical Bessel functions appearingin eq. (3.13) can be in fact performed, and are well-behaved. In the next Section we willshow that, in fact, these integrals are probability densities for random flights with n steps, ina space of suitable dimensionality. From the previous Section — specially from eq. (3.13) — it is evident that a completetreatment of the CMB in position space requires that some specific integrals of products ofspherical Bessel functions should be computed. In order to fulfill this requirement, we willproceed in the following way: first, we will present a simplified version of those integrals, andwe will show that they give rise to probability densities associated with random flights. Next,we will show how the integrals we have to solve can be expressed in terms of the randomflight integrals.Let’s recall the well-known identities satisfied by spherical Bessel functions: z L + J L + ( z ) = ddz h z L + J L + ( z ) i . Since the spherical Bessel functions are defined as: j n ( z ) = r π z J n + ( z ) , we are able to write: – 7 – ∞ dk k j L ( k r ) n − Y i =1 j L ( k r i )( k r i ) L j L ( k r n ) = (cid:0) π (cid:1) ( n +1) / (cid:2) Γ (cid:0) L + (cid:1)(cid:3) n − r Ln r L +2 × ddr ((cid:20) Γ (cid:18) L + 32 (cid:19)(cid:21) n − Z ∞ dk r (cid:18) k r (cid:19) L + J L + ( k r ) n Y i =1 J L + ( k r i )( k r i ) L + ) . (4.1)The derivative of the second line in eq. (4.1) is, in fact, the probability density associatedwith a random flight — see, e.g., ref. [18]. This is the probability density that a particle whichmoves with a constant (and finite) speed, and which starts from a given position in space,will be at a distance r from the point of origin, after changing randomly directions n timesduring its trajectory. The length of the intermediate steps are denoted by r i , i = 1 , . . . , n .The order L of the spherical Bessel functions in these integrals is related to the dimensionalityof the space where the flight takes place: namely, the dimension D of that space is given by D = 2 L + 3 .Following the notation employed by ref. [18] we shall denote: p n ( r ; r , . . . , r n | L + 3) := ddr ( (cid:20) Γ (cid:18) L + 32 (cid:19)(cid:21) n − × Z ∞ dk r (cid:18) kr (cid:19) L + J L + ( kr ) n Y i =1 J L + ( kr i )( kr i ) L + ) . (4.2)However, the integral (4.1) is still not what we need in order to solve the momentum integralsthat appear in our iterations — see, e.g., the Fourier integration of eq. (3.13).We will now extend the random flight integrals to include the scenario that appears inthe context of the CMB. If l ≥ L (which is always the case in our iterative solutions), thenthe product of two spherical Bessel functions of order l can be written in terms of a singlespherical Bessel function of order L . This is a consequence of Gegenbauer’s relation [17, 18]and of the orthogonality of associated Legendre polynomials — see [1] for a derivation: j l ( kX ) j l ( kR ) = ( − L Z X + R | X − R | dr k L (cid:18) XRr (cid:19) L − P − Ll (cos α ) (sin α ) L j L ( kr ) (4.3)where r , R and X must form a triangle, with the angle α being given implicitly in terms ofthe relation r = R + X − RX cos α , and P − Ll (cos α ) is an associated Legendre polynomial.Applying eq. (4.3) and eq. (4.1) with L = 2 we can recast the Fourier integral in eq. (3.13)as: F l (∆ η , X ; ∆ η , . . . , ∆ η n ) := Z ∞ dk k j l ( kX ) j l ( k ∆ η )( k ∆ η ) n − Y i =1 j ( k ∆ η i )( k ∆ η i ) j ( k ∆ η n )= 12 (cid:0) π (cid:1) ( n +1) / (cid:2) Γ (cid:0) (cid:1)(cid:3) n − Z d (cos α ) (cid:18) X ∆ η n r (cid:19) P − l (cos α ) sin α × p n ( r ; ∆ η , . . . , ∆ η n | , (4.4)where we used the notation introduced in eq. (4.2).– 8 –n eq. (4.4) we have introduced the function F l (∆ η , X ; ∆ η , . . . , ∆ η n ) , which denoteswhat we shall call extended random flight integrals . In general, F lL is defined by eq. (A.1).We should examine eq. (4.4) more carefully. As anticipated, the presence of the term p n ( r ; ∆ η , . . . , ∆ η n | should not be surprising, due to the interpretation of a random flightprocess and its validity with respect to the physics of the recombination. The dimensionality( D = 2 × ) of the space associated with the random flight, however, is not yet fullyunderstood. That dimension is determined by the order of the spherical Bessel functions whichmediate the sources of anisotropies and the final CMB signal, but since only the quadrupoleof the temperature fluctuation contributes to the polarization, the spherical Bessel functionof order is the one that characterizes the random flight for the CMB. A possible explanationfor this dimension is that, after separating the angular dependence of the CMB from its radialand time dependence through the spherical harmonic decomposition, the light cone has onlytwo dimensions left. Since the multipole L has L + 1 degrees of freedom, we end up with L + 3 dimensions where our relevant variables can perform random flights. However, a morerefined argument to explain the dimension 7 is not yet known.Looking back now at eq. (3.13), we recognize that for the π ( n ) lm term in the polarizationexpansion, the probability density p n ( r ; ∆ η , . . . , ∆ η n | appears clearly. The interpretationof an expansion in the number of interactions during recombination is therefore strengthened.We should remark, however, that explicit formulae for p n ( r ; ∆ η , . . . , ∆ η n | are not knownin the general case [9, 10].As discussed above, the intervals ∆ η , . . . , ∆ η n express the times elapsed between con-secutive scatterings. All these subintervals are elements of a partition of the time interval η − η , which is therefore the total time elapsed since the instant the photon leaves equilibriumwith matter, at time η , until the instant η when the photon has last scattered prior to itsobservation. This time interval represents, therefore, the effective duration of recombinationfor a given photon. The lengths of each subinterval are weighted by the visibility functions,and integrated in order to contemplate all possible histories for photons during recombination.The interval η o − η expresses the time elapsed since the photon scattered for the last time,before it is observed at the time η o (we are assuming that no further scatterings take placeduring this interval). We can represent a photon’s history by means of figure 1: on the leftwe show the photon’s interactions prior to observation at the vertex of the cone, while on theright we show a diagrammatic representation of that history, with the relevant elements thatappear in eq. (4.4).We call special attention to the diagram represented on the right of figure 1. Thisdiagram represents the extended random flight performed by a photon during recombination.The steps ∆ η n , . . . , ∆ η (going forward in time) belong to a standard random flight, anddescribe the trajectory of a photon that has left equilibrium with matter at an instant η ,then propagated freely for a distance (or a time interval) ∆ η n , then Thomson-scattered withan electron at time η n , then propagated freely for a distance ∆ η n − , and so on until theinstant η , when it scattered for the last time. The standard random flight ends at theinstant η . The photon, at that moment, is a radius r away from the point where the flightstarted. The steps indicated by ∆ η and X do not belong to the standard random flight, butare present in the extended random flight, and are introduced through the spherical Besselfunctions of different order in the k integral of eq. (4.4). This is necessary because thosetwo steps are not associated with any movement between successive scatterings, but are infact associated with the distance from the observer to the origin of the photon, and to theend-point of the random flight. It should be strengthened that ∆ η , X and r are related– 9 – η∆η∆η α x r ∆ η ∆η ∆η Figure 1 . Connection between scatterings of a photon during recombination (left) with a diagrammaticrepresentation of the random flight (right). The steps ∆ η n , . . . , ∆ η correspond to the lengths of the photon’strajectory between successive scatterings. ∆ η corresponds to the propagation since the photon’s last scat-tering during recombination, at time η , and the observation at time η o . In both diagrams the observationtakes place at the upper vertex. by r = ∆ η + X − η X cos α . Since ≤ r ≤ ∆ η + ∆ η + . . . + ∆ η n , it follows that ≤ X ≤ ∆ η + ∆ η + ∆ η + . . . + ∆ η n , which then determines the domain of dependence ofthe problem. We shall now go back to eq. (3.13). In terms of the extended random flight just introduced,the polarization coefficient to n th order, π ( n ) lm ( η o ) , can be written as: π ( n ) lm = − π s ( l + 2)!( l − Z η o dη g ( η ) 1( n − Z η dη . . . dη n T { g ( η ) . . . g ( η n ) }× n − (cid:0) π (cid:1) ( n +1) / (cid:2) Γ (cid:0) (cid:1)(cid:3) n − Z η n dη Z η o − η dX X S lm ( X, η ) × Z d (cos α ) (cid:18) X ∆ η n r (cid:19) P − l (cos α ) sin α p n ( r ; ∆ η , . . . , ∆ η n | , (5.1)where we have already used the aforementioned upper bound for the variable X .– 10 – x α∆η Figure 2 . Marginalization over all paths with n steps, composed of the intermediate displacements ∆ η , . . . , ∆ η n , which lead to a fixed displacement r with respect to the origin of the flight. The distance r is determined by X and ∆ η for all possible angles α . The equation (5.1) can be understood as the combination of three procedures: • The integration over α corresponds to a marginalization over all possible paths composedof n steps of lengths ∆ η + . . . + ∆ η n that have a net displacement r determined by X and ∆ η , as shown in figure 2. This “average over paths” is a function of X , ∆ η , ∆ η , . . . , ∆ η n . • The contribution from the primary source term, S lm ( X, η ) , is then mediated by this“average over paths” that was just described, for all possible values of X . The maximumvalue that X may reach is ∆ η +∆ η + . . . +∆ η n = η o − η , which is nothing but the radiusof the observer’s past light cone up to the time η . After computing the contribution ofthe source terms, we end up with an expression that is a function of ∆ η , ∆ η , . . . , ∆ η n . • The last step is to let the intervals ∆ η , ∆ η , . . . , ∆ η n assume any values through theintegrations, each one weighted by its corresponding factor of the visibility functionto take into account the probability that the photon will scatter at that instant oftime. This accomplishes the goal of accounting for the contribution from sources at alldistances, and over any possible number of intermediate steps of the extended randomflights. – 11 –dding the contributions from all π ( n ) lm we obtain: π lm = − π s ( l + 2)!( l − Z η o dη g ( η ) ∞ X n =1 n − Z η dη . . . dη n T { g ( η ) . . . g ( η n ) }× n − (cid:0) π (cid:1) ( n +1) / (cid:2) Γ (cid:0) (cid:1)(cid:3) n − Z η n dη Z ∞ dX X S lm ( X, η ) × Z d (cos α ) (cid:18) X ∆ η n r (cid:19) P − l (cos α ) sin α p n ( r ; ∆ η , . . . , ∆ η n | . (5.2)In terms of the extended random flight integrals, eq. (4.4), we can write: π lm = − π s ( l + 2)!( l − Z η o dη g ( η ) ∞ X n =1 n − ( n − Z η dη . . . dη n T { µ ′ ( η ′ )e − µ ( η ′ ) . . . g ( η n ) }× Z η n dη Z ∞ dX X S lm ( X, η ) F l (∆ η , X ; ∆ η , . . . , ∆ η n ) . (5.3) We are now able to introduce a diagrammatic representation for all the terms that appear inthe series expansion of the temperature and polarization at n -th order. The graphic elementsare: • Solid lines mean polarization, and dashed lines mean temperature; • The vertical lines to the right of the diagrams represent the observable being calculated(time runs upward); • The interception of two lines determine what sources are being considered for a givenobservable.The rules for constructing these diagrams will be detailed in the next Section, but it is alreadyeasy to represent diagrammatically the contribution to the CMB polarization computed ineq. (5.2): (cid:1) + (cid:2) + (cid:3) + . . . + (cid:4) (6.1)– 12 – Full coupled temperature evolution
We will now show that the formalism developed in the previous Sections is not spoiled whenall the contributions to the temperature fluctuations are included. Let’s go back to eqs. (2.1)and (2.3), and write: θ lm = 2 i l Z η o dη Z dk (2 π ) / k Z d ˆ k ( g ( η ) " θ SW ( k , η ) − V b ( k , η ) ∂∂η e − µ ( η ) (Ψ ′ + Φ ′ )( k , η )+ 12 g ( η ) P ( k , η ) (cid:20) ∂ ∂ ( k ∆ η o ) (cid:21) ) Y ∗ lm (ˆ k ) j l ( k ∆ η o ) . The first two terms in this equation have already been treated in Section III. We nowproceed to analyzing the remaining term, i. e., that which contains the source P ( k , η ) = (cid:2) Θ ( k , η ) − √ α ( k , η ) (cid:3) . We write, therefore: θ lm = θ (0) lm + i l Z η o dη Z dk (2 π ) / k Z d ˆ k g ( η ) P ( k , η )Y ∗ lm (ˆ k ) (cid:20) ∂ ∂ ( k ∆ η o ) (cid:21) j l ( k ∆ η o )=: θ (0) lm + ∞ X n =1 θ ( n ) lm , (7.1)where we have defined the θ ( n ) lm in terms of the integral over sources. We shall now calculate the lowest-order contribution described in eq. (7.1) by taking P ( k , η ) to be Θ ( k , η ) — i. e., only the temperature quadrupole is taken as a source for thetemperature fluctuations, since the polarization (which is of higher order, since its source isthe temperature quadrupole) is at least a second-order contribution to the temperature. Thefirst order contribution to the temperature, θ (1) lm , is therefore computed as: θ (1) lm = i l Z η o dη g ( η ) Z dk (2 π ) / k Z d ˆ k Θ ( k , η ) Y ∗ lm (ˆ k ) (cid:20) ∂ ∂ ( k ∆ η o ) (cid:21) j l ( k ∆ η o )= i l Z η o dη g ( η ) Z dk (2 π ) / k Z η dη S lm ( k, η ) j ( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η o ) (cid:21) j l ( k ∆ η o )= 12 π Z η o dη g ( η ) Z η dη Z ∞ dX X S lm ( X, η ) × Z dk k j l ( kX ) (cid:20) ∂ ∂ ( k ∆ η o ) (cid:21) j l ( k ∆ η o ) j ( k ∆ η ) (7.2)where we have once again transformed coefficients into the position-space description usingHankel transformation. We must now treat the integral in the last line of the equation above: I l = Z dk k j l ( kX ) (cid:20) ∂ ∂ ( k ∆ η o ) (cid:21) j l ( k ∆ η o ) j ( k ∆ η ) . Clearly, we can write: I l = 3 (cid:20) ∂ ∂ (∆ η o ) − η o ∂∂ (∆ η o ) (cid:21) (cid:20) (∆ η o ) Z dk k j l ( kX ) j l ( k ∆ η o )( k ∆ η o ) j ( k ∆ η ) (cid:21) + (∆ η o ) Z dk k j l ( kX ) j l ( k ∆ η o )( k ∆ η o ) j ( k ∆ η ) , (7.3)– 13 –hich we can then express as: I l = 3 (cid:20) ∂ ∂ (∆ η o ) − η o ∂∂ ∆ η (cid:21) (cid:2) (∆ η o ) F l (∆ η o , X ; ∆ η ) (cid:3) + (∆ η o ) G (2) l (∆ η o , X ; ∆ η ) (7.4)where the integral corresponding to G (2) l is related to the already introduced function F l —see eq. (4.4). In fact, using an identity that is proved in the appendix, eq. (A.7), we obtain: θ (1) lm = 12 π Z η o dη g ( η ) Z η dη Z ∞ dX X S lm ( X, η ) × ( (cid:20) ∂ ∂ (∆ η o ) − η o ∂∂ ∆ η o (cid:21) (cid:2) (∆ η o ) F l (∆ η o , X ; ∆ η ) (cid:3) + 1(∆ η o ) l X l +2 ∂∂X (cid:18) ∂∂ ∆ η o + 2∆ η o (cid:19) h (∆ η o X ) l +2 F ( l +1) 2 (∆ η o , X ; ∆ η ) i ) . (7.5)In conclusion, first-order corrections to the temperature can also be written in terms theprobability densities of extended random flights. If we want to calculate further contributions to the temperature, then we must include thecontributions to Θ that come from the temperature quadrupole, given in eq. (7.4), and thosefrom α , given in eq. (3.10). Combining those two contributions leads to: θ (2) lm = 14 π Z η o dη g ( η ) Z η dη g ( η ) Z η dη Z ∞ dX X S lm ( X, η ) × Z dk k j l ( kX ) j ( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η o ) (cid:21) j l ( k ∆ η )+ 9 π Z η o dη g ( η ) Z η dη g ( η ) Z η dη Z ∞ dX X S lm ( X, η ) × Z dk k j l ( kX ) j ( k ∆ η ) j ( k ∆ η )( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j l ( k ∆ η ) . (7.6)The first term comes from the contribution of Θ to θ (2) lm , and the second term takes intoaccount the contribution of α to θ (2) lm . These contributions can also be expressed in termsof the extended random flight integrals: terms with two derivatives can be expressed interms of F l (∆ η , X ; ∆ η , ∆ η ) ; terms with one derivative can be expressed in terms of F ( l +1) 2 (∆ η , X ; ∆ η , ∆ η ) ; and terms with no derivatives can be expressed, as indicated insection A.2, in terms of F ( l +2) 2 (∆ η , X ; ∆ η , ∆ η ) .The complete series expansions for the temperature coefficients, ¯ θ lm , is represented dia-grammatically, up to second order corrections, as: ¯ θ (2) lm = (cid:1) + (cid:2) + (cid:3) + (cid:4) (7.7)– 14 –he first diagram represents θ (0) lm , the second, θ (1) lm — see eq. (7.4) — and the two lastdiagrams represent θ (2) lm , as shown in eq. (7.6).Even if we do not calculate here explicitly the contributions at all orders, the resultspresented in section A.2 show that all corrections can be expressed in terms of extendedrandom flight integrals. In general, ¯ θ ( n ) lm will be given by: ¯ θ ( n ) lm = n X q =0 θ ( n ) lmq (∆ η , ∆ η , . . . , ∆ η n , X ) F ( l + q )2 (∆ η , X ; ∆ η , . . . , ∆ η n ) , where we represent by θ ( n ) lmq the coefficients that appear in the expansion. As was seen explicitlyin the case n = 1 , these coefficients are differential operators combined with powers of thesubintervals of ( η o − η ) and of X . The complete expression for ¯ θ ( n ) lm will also depend onknowledge of the expression for polarization up to the order n − — which is the reason forleaving the superscript n explicit in θ ( n ) lmq . The last step in our construction of the series expansions of the CMB in terms of extendedrandom flight probabilities is to treat the contribution of the temperature quadrupole to thepolarization, allowing for the corrections to the temperature that come from higher ordercontributions of the source terms. We shall write, then: ¯ π (2) lm = − s ( l + 2)!( l − Z η o dη g ( η ) Z d k (2 π ) / π i l
12 Θ ( k , η ) Y ∗ lm (ˆ k ) j l ( k ∆ η )( k ∆ η ) = − i l s ( l + 2)!( l − Z η o dη g ( η ) ( Z η dη Z dk (2 π ) / k S lm ( k, η ) j ( k ∆ η )+ 12 Z η dη g ( η ) Z dk (2 π ) / k Z d ˆ k P ( k , η ) Y ∗ lm (ˆ k ) × (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) ) j l ( k ∆ η )( k ∆ η ) (7.8)where the bar over ¯ π (2) lm means that we are taking all corrections up to second order.At this point we can separate the first-order and the second-order terms, and focus on– 15 –he contribution from Θ to P : ¯ π (2) lm = π (1) lm − i l s ( l + 2)!( l − Z η o dη g ( η ) Z η dη g ( η ) Z dk (2 π ) / k × Z d ˆ k Θ ( k , η ) Y ∗ lm (ˆ k ) (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) j l ( k ∆ η )( k ∆ η ) = π (1) lm − i l s ( l + 2)!( l − Z η o dη g ( η ) Z η dη g ( η ) Z dk (2 π ) / k × Z η dη S lm ( k, η ) j ( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) j l ( k ∆ η )( k ∆ η ) = π (1) lm −
12 32 π s ( l + 2)!( l − Z η o dη g ( η ) Z η dη g ( η ) Z η dη Z dX X S lm ( X, η ) × ( Z dk k j l ( kX ) j l ( k ∆ η )( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) j ( k ∆ η ) ) . (7.9)The last integral of the equation above, between curly brackets, can be expressed, after usingeq. (A.7), as follows: Z dk k j l ( kX ) j l ( k ∆ η )( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) j ( k ∆ η )= (∆ η ) (∆ η X ) l +2 ∂∂X (cid:18) ∂∂ ∆ η + 2∆ η (cid:19) h (∆ η X ) l +2 F ( l +1)2 (∆ η , X ; ∆ η , ∆ η ) i + 3 (cid:18) ∂ ∂ (∆ η ) − η ∂∂ ∆ η (cid:19) h (∆ η ) F l (∆ η , X ; ∆ η , ∆ η ) i . (7.10)As we can see, despite the somewhat cumbersome coefficients, the final result can be expressed,again, as a combination of the extended random flight functions F l (∆ η , X ; ∆ η , ∆ η ) and F ( l +1)2 (∆ η , X ; ∆ η , ∆ η ) .The last term in eq. (7.9) has the same order as π (2) lm . Therefore, to second order intemperature corrections, the polarization results from the following three diagrams: ¯ π (2) lm = (cid:1) + (cid:2) + (cid:3) (7.11)As we showed above, this contribution can be expressed fully in terms of extendedrandom flight probabilities, therefore, up to second order the polarization coefficients can bewritten as: ¯ π (2) lm = π lm (∆ η , ∆ η , ∆ η , X ) F l (∆ η , X ; ∆ η , ∆ η ) (7.12) + π lm (∆ η , ∆ η , ∆ η ; X ) F ( l +1) 2 (∆ η , X ; ∆ η , ∆ η ) . – 16 –he coefficients π lmq ( q = 0 , ) are differential operators, and depend on the combinationsof the subintervals to some power. The coefficient π lm takes into account the contributionsfrom the π (1) lm term appearing in eq. (7.9) and also the contribution coming from the secondterm in the right hand side of eq. (7.10). The coefficient π lm can be obtained by workingout the first term in the right hand side of eq. (7.10).If we examine eqs. (7.9) and (7.11), it can be seen that ¯ π (2) lm is given by the sum of π (1) lm and the contribution from the temperature corrected by the polarization. We can write eq.(7.11) in a more symmetric way if we note that the following diagrams are equivalent: (cid:1) = (cid:2) (7.13)This equivalence is due to the fact that both diagrams have the same vertical lines at the right(we are calculating contribution to the polarization); they have the same number of vertices;they have the same kinds of lines connecting the vertices (continuous lines); and they havethe same initial source, i. e., the temperature fluctuations (the dashed lines). Hence, it isbetter to write eq. (7.11) in terms of the diagram on the left-hand side of eq. (7.13), sinceit is easier to draw these types of diagrams when there is a large number of scatterings, asshown in eq. (6.1).In general, to order n : ¯ π ( n ) lm = n − X q =0 π ( n ) lmq (∆ η , ∆ η , . . . , ∆ η n , X ) F ( l + q )2 (∆ η , X ; ∆ η , . . . , ∆ η n ) . (7.14)Explicit formulae for all the coefficients π ( n ) lmq are not known explicitly, but can be computediteratively after the identification of the diagrams that contribute to some order, and byemploying the techniques presented in section A.2. Naturally, the computation of ¯ π ( n ) lm requiresthe knowledge of ¯ θ ( n − lm , the computation of ¯ θ ( n − lm requires the knowledge of both ¯ θ ( n − lm and ¯ π ( n − lm , and so on and so forth –0 therefore, in this sense these coefficients form a hierarchy. If we accept that Boltzmann’s equations, as presented in eqs. (2.1), (2.3), (2.2) and (2.4),are accurate enough to allow the computation of the CMB temperature and polarizationto all orders, then we can ask how the diagrams for our series expansion should be built.Clearly, when further precision is required, quadratic and even higher-order corrections forthe metric perturbations may have to be included in Boltzmann’s equation, and at some pointthe diagrams presented here may no longer improve the accuracy of the result. However, ifwe assume that the Einstein-Boltzmann system of equations is exact, then all contributionscan be written as a sequence diagrams, whose building blocks are presented in table 1.As a general rule of thumb, the functions j l ( kX ) and j ( k ∆ η n ) must appear as the firstand last elements inside the k integral. Here, η n stands for the last index in the partition of– 17 –iag. segment numerical factor Bessel function (cid:1) − π q ( l +2)!( l − j l ( k ∆)( k ∆) (cid:2) h ∂ ∂ ( k ∆) i j l ( k ∆) (cid:3) j ( k ∆)( k ∆) (cid:4) j ( k ∆)( k ∆) (cid:5) h ∂ ∂ ( k ∆) i j ( k ∆) (cid:6) j ( k ∆)( k ∆) Table 1 . Building blocks for general diagramms. the time interval ( η o − η ) . The intervals ∆ must be replaced by the convenient difference ofconsecutive conformal times which represent the initial and the final instants of that segmentof the random flight. Outside the k integral there should be the integral over the spatial posi-tion of the primary source, R dX X S lm ( X, η ) . Finally, we have the integrals over conformaltime weighted by visibility functions — except the integral over η , which does not carry afactor of the visibility function.For example, the following diagram represents the third-order contribution to polariza-tion that arises from the polarization at second-order, when that second-order polarization is– 18 –ue to the first-order correction to the temperature: (cid:1) = −
92 32 π s ( l + 2)!( l − Z η o dη g ( η ) Z η dη g ( η ) Z η dη g ( η ) Z η dη × Z dX X S lm ( X, η ) ( Z dk k j l ( kX ) j l ( k ∆ η )( k ∆ η ) j ( k ∆ η )( k ∆ η ) × (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) j ( k ∆ η ) ) , (7.15)The diagram below, on the other hand, represents a third-order contribution to polar-ization arising from the second-order correction to the temperature, when that second-ordercorrection itself comes from the first-order correction to the temperature: (cid:2) = −
12 12 32 π s ( l + 2)!( l − Z η o dη g ( η ) Z η dη g ( η ) Z η dη g ( η ) Z η dη × Z dX X S lm ( X, η ) ( Z dk k j l ( kX ) j l ( k ∆ η ) ( k ∆ η ) × (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) (cid:20) ∂ ∂ ( k ∆ η ) (cid:21) j ( k ∆ η ) j ( k ∆ η ) ) . (7.16) From the previous arguments it is obvious that a full description of CMB temperature andpolarization depends upon the evaluation of the extended random flight integrals F l . Aswe showed in eq. (4.4), the extended random flights can be expressed as marginalizationsover random flight probability densities. Unfortunately, explicit formulae for random flightprobability densities in seven dimensions are not known, which means that at orders higherthan three the computations start to become highly complex. As a computational tool to dealwith the extended random flight integral, we will introduce their series expansion in terms ofa Fourier-Bessel series — this was also suggested by [12].We will expand the function F lL ( R, X ; r , . . . , r n ) , which is ubiquitous in our previ-ous discussions (just substitute ∆ η → R , ∆ η → r , etc.), and which was defined ineq. (A.1), in terms of a Fourier-Bessel series. Given the behavior of this function andits interpretation, the segments R, r , . . . , r n are fixed for any particular flight. Since r = √ R + X − R X cos α ≤ r + r + . . . + r n =: S n , it follows that the variable X must haveitself an upper limit — see figure 1 for the geometry involved in these limits. In fact, in thelimit where all flight steps r , . . . , r n are collinear, we have that r = S n , and in that case X ,– 19 – and S n form the sides of a triangle. Hence, the maximum value of X is determined jointlyby both R and S n , as ≤ X ≤ R + r + r + . . . + r n := ˜ S n .Now, we know that well-behaved functions with a limited domain can be expressed interms of a Fourier-Bessel series [8]. In fact, since F lL is one such well-behaved function, wehave: F lL ( R, X ; r , . . . , r n ) = ∞ X i =1 f ( n ) lLi j l (cid:16) x ( n ) li X (cid:17) , (8.1)where x ( n ) li = z li / ˜ S n , and z li is the i -th root of the spherical Bessel function j l ( z ) . Thecoefficients f ( n ) lLi are given by f ( n ) lLi = 2˜ S n j l +1 ( z li ) Z ˜ S n dX X j l (cid:16) x ( n ) li X (cid:17) F lL ( R, X ; r , . . . , r n ) . (8.2)Using eq. (4.4) we obtain: F lL ( R, X ; r , . . . , r n ) = ∞ X i =1 π ˜ S n j l +1 ( z li ) j l (cid:16) x ( n ) li R (cid:17)(cid:16) x ( n ) li R (cid:17) L j l (cid:16) x ( n ) li X (cid:17) n − Y q =1 j L (cid:16) x ( n ) li r q (cid:17)(cid:16) x ( n ) li r q (cid:17) L j L (cid:16) x ( n ) li r n (cid:17) . (8.3)Now, eq. (4.3) allows us to write (8.3) in the form: F lL ( R, X ; r , . . . , r n ) = ( − L π ˜ S n Z ˜ S n dr (cid:18) RXr (cid:19) L − R L P − Ll (cos α ) (sin α ) L × ∞ X i =1 j l +1 ( z li ) j m (cid:16) x ( n ) li r (cid:17) n − Y q =1 j L (cid:16) x ( n ) li r q (cid:17)(cid:16) x ( n ) li r q (cid:17) L j L (cid:16) x ( n ) li r n (cid:17) , (8.4)where r = R + X − R X cos α .The extended random flight integrals can, therefore, be much more easily computed withthe help of eq. (8.4), since in that representation only a discrete number of modes need to beadded, instead of the full-fledged integral that defines the random flight probability density. In this Section we shall address a more fundamental question. Up to now we have presented aformalism that allows us to express the CMB polarization and temperature in position space,in terms of an expansion over the number of interactions the photons have suffered duringrecombination. However, at this point we can ask: up to what order the contributions shouldto be taken into account in order to yield an accurate expression for the physical observables?But this question can only be answered if a concrete problem is given, so there is no generalanswer. Hence, we will invert this question, and ask: how important is it, for the observables,that we reach a certain order n in the expansion? To be more specific, we will study thecontributions arising from very high n terms in eq. (5.2).Since all computations depend on the probability densities for random flights with n steps, we will analyze the behavior of p n ( r ; ∆ η , . . . , ∆ η n | , for large n . Although explicitformulae for these probability functions are not known, an asymptotic expression is known in– 20 –he case where all steps have the same lengths, ∆ η = ∆ η = . . . = ∆ η n =: ∆ , and n → ∞ [18]. In that case we have: p n ( r ; ∆ , . . . , ∆ |
7) = 1Γ(7 / (cid:18) r n ∆ (cid:19) / (cid:18) r n ∆ (cid:19) × " F (cid:18)
72 ; 92 ; − r n ∆ (cid:19) − (cid:18) r n ∆ (cid:19) F (cid:18)
92 ; 112 ; − r n ∆ (cid:19) , (9.1)where: F ( a ; c ; z ) = Γ( c )Γ( a ) ∞ X k =0 Γ( a + k )Γ( c + k ) z k k ! is the confluent hypergeometric function [8]. Expanding F in a power series we obtain: p n ( r ; ∆ , . . . , ∆ |
7) = 7 / / (cid:18) r n ∆ (cid:19) / (cid:18) r n ∆ (cid:19) × ∞ X k =0 k ! (cid:16) − r n ∆ (cid:17) + k (cid:16) − r n ∆ (cid:17)(cid:0) + k (cid:1) (cid:0) + k (cid:1) (cid:18) − r n ∆ (cid:19) k . (9.2)Now, consider the sum: ∞ X k =0 k ! (cid:0) − z (cid:1) + k (cid:0) − z (cid:1)(cid:0) + k (cid:1) (cid:0) + k (cid:1) ( − z ) k ≤ ∞ X k =0 k ! 1 (cid:0) + k (cid:1) ( − z ) k ≤ e − z . It follows, therefore, that: p n ( r ; ∆ , . . . , ∆ | ≤ r (cid:18) r n ∆ (cid:19) / e − r n ∆2 . (9.3)Consequently, for fixed r > , ∆ > , and r ≤ n ∆ , lim n →∞ p n ( r ; ∆ , . . . , ∆ |
7) = 0 . Since p n is a normalized probability density, we conclude that, in the limit above, the prob-ability distribution collapses into a Dirac delta-function centered at the origin, which is inagreement with the general features attributed to the random flight probability densities [7].In other words, increasing the number of changes of directions (i.e., scatterings) in a randomflight leads to trajectories with increasingly smaller displacements from the origin.Let’s investigate the consequences of this fact for the polarization through eq. (5.1).Consider, in that respect, the integral: S lm = lim n →∞ Z η o − η dXX S lm ( X, η ) Z d (cos α ) (cid:18) X ∆ r (cid:19) P − l (cos α )(sin α ) p n ( r ; ∆ , . . . , ∆ | Z η o − η dX X S lm ( X, η ) Z dr rX ∆ η (cid:18) X ∆ r (cid:19) P − l (cos α ) (sin α ) δ ( r )= Z η o − η dX X S lm ( X, η ) ∆ (∆ η ) lim r → (cid:20) X ∆ η r P − l (cos α ) (sin α ) (cid:21) . (9.4)– 21 –e know, however, from either [18] or [1], that: Z ∞ dk j l ( kr ) j l ( kr ) j ( kr ) = π r r r P − l (cos α ) (sin α ) , (9.5)and also that [1]: lim r → Z ∞ dk j l ( kr ) j l ( kr ) j ( kr ) = r r r π δ ( r − r ) . (9.6)Therefore, we have that: S lm ( η , η ) = Z η o − η dX X S lm ( X, η ) π
30 ∆ X δ (∆ η − X )= π
30 ∆ (∆ η ) S lm (∆ η , η ) . (9.7)The final lesson is that, in the limit of an infinite number of scatterings, we basically endwith instantaneous recombination — with the crucial caveat that the source term is dampedby (∆ / ∆ η ) → . The conclusion is, therefore, that no net polarization is generated atthe limit of infinite number of scatterings. Similarly, for the temperature fluctuations thederivatives of the integral which appears in θ ( n ) lm ensure that the signal vanishes as well.The large n limit, therefore, cannot contribute significantly to the final observables.If we regard this order as the number of scatterings that a photon has experienced duringrecombination, then the collapse of the probability density associated with random flightsexpresses a fundamental physical fact: increasing the number of scatterings takes the systemcloser to thermodynamical equilibrium. As we know, in thermodynamical equilibrium theCMB temperature is given simply by the Planck distribution, with no distortions (apart fromthe dipole which is induced by the velocity of baryons). Maximizing the von Neumann entropyalso ensures that, in equilibrium, no net polarization is generated. These two facts, together,constitute the expression of Boltzmann’s H-theorem to the CMB [6].Describing the CMB by means of random flight probability density functions provides,therefore, an illustration of the statement of the H-theorem: when few scatterings take placeduring recombination (low n ), photons and electrons are surely out of equilibrium, and eachinteraction generates temperature fluctuations and polarization that are not sufficiently erasedby the subsequent scatterings. As a consequence, the low- n terms in the expansions accountfor the largest contributions for the signal of the physical observables. For a large number osscatterings, the signal is washed out by further scatterings, which can be expressed in termsof the collapse of the probability density in the large- n limit.All that is left to discuss is the physical grounds for assuming (as we did in this Section)that all the steps performed in the random flights have the same length. Despite the factthat this working hypothesis was not proposed because of any physical reason, but purelybecause of mathematical convenience, we can nevertheless argue that, when the electron-photon system was close to the tight-coupling regime, the assumption that the visibilityfunction is step-shaped is not completely crude. Continuity with respect to the arguments ofthe probability density distribution makes our working hypothesis less unnatural.
10 Conclusions
The Boltzmann hierarchy for the problem of the evolution of temperature and polarizationfluctuations of the CMB is equivalent to a system of coupled integral equations. We showed– 22 –hat this system can be written in position space, and that the objects that appear in thisdescription are the same probability density functions that appear in random flight problems.The emergence of random flight probability functions from Boltzmann’s equations clarifiesthe physical interpretation that CMB temperature fluctuations and polarization are generatedfrom scatterings of photons by low-energy electrons during the recombination. More impor-tantly, we showed that the number of scatterings during recombination is a key ingredientin the description of this problem — in fact, a perturbative expansion can be performed interms of the number of scatterings.We showed, using asymptotic formulae for the random flight probability distribution,that contributions coming from high-order terms (i.e., many scatterings) should indeed benegligible. Since high-order terms represent the past history of photons that have scatteredmany times during recombination, we concluded that their vanishing contribution to the CMBsignal is an illustration of Boltzmann’s H-theorem. In the context of the CMB, this theoremstates that, in thermodynamical equilibrium, the CMB temperature should be given by aPlanck spectrum, and that the net polarization should vanish.Using the fact that the random flight distribution functions vanish identically if onecannot form a closed polygon from its intermediate steps, we presented Fourier-Bessel seriesexpansions for the associated probability distribution function. These expressions lead tosimple numerical recipes for the computation of these distributions, since explicit formulaefor them are not presently known for any dimension. Another very important result whichderives from the emergence of random flights is the fact that, at each given time, the domainsof dependence of the problem are compact sets. This is not obvious from the usual treatmentof Boltzmann’s equations in Fourier space.From a more formal perspective, it would be important to understand which classes ofBoltzmann’s equations are amenable to a treatment in terms of random flight probabilitydensities. Many of those problems could be then examined in light of the formalism that hasbeen developed for the study of the CMB.Finally, we can foresee some possible applications of this work. The series expansion interms of the number of scatterings can be used for numerical simulations of constrained mapsof temperature and polarization. Due to the general vanishing property of the probabilitydensity functions for the extended random flight if intermediate displacements do not form apolygon, and the decreasing of the visibility function for z >> , we can in practice takeall the sources to vanish outside of a sphere of radius R sufficiently large, and calculate thetemperature and polarization corrections using Fourier-Bessel expansions, as shown in ref.[1]. In Fourier-Bessel basis only a discretized tower of modes contribute to each observable ateach multipole, and the computational advantages of this approach are described in ref. [11].In what concerns the convergence of the iterative process, depending on the desired accuracy,application or the angular scale that one wishes to examine, it may be sufficient to consideronly the first couple of scatterings of the photons, since going further in the expansion wouldbring only contributions from terms highly suppressed by powers of the visibility function.We should emphasize that in the limit of large number of interactions the signal will be doublysuppressed: the first suppression comes from the high order power of the visibility function,and the second from the vanishing of the temperature and polarization corrections due to thecollapse of the random flight probability density function. Also corrections of higher orderscould be comparable to the corrections coming from second order perturbations in space-timemetric, or more detailed models for the dynamics of interactions during the recombination.The partial summation of some classes of diagrams (in the spirit of a renormalization group)– 23 –an also be performed, which would lead to more accurate results without increasing thecomplexity. In particular, the position-space analysis is especially suited for introducinggravitational lensing (work in progress). A Extended random flight integrals times even functions
We shall show the possibility of expressing a larger set of integrals in term of random flightprobability densities. Specifically, we will concentrate in expressions with the form: F lL ( R, X ; r , . . . , r n ) = Z ∞ dk k j l ( kX ) j l ( kR )( kR ) L n − Y q =1 j L ( kr q )( kr q ) L j m ( kr n ) . (A.1)Our aim is to express integrals like: G lL ( R, X ; r , . . . , r n ) = Z ∞ dk k f ( k ) j l ( kX ) j l ( kR )( kR ) L n − Y q =1 j L ( kr q )( kr q ) L j L ( kr n ) (A.2)in terms of combinations of F l ′ L ′ ( R, X ; r , . . . , r n ) , for some adequate l ′ and L ′ . In G lL , f ( k ) is taken to be a real function such that its Taylor series carries only terms with even powersof k . The simplest such function is a k , for a ∈ R . Naturally, in that case the connection weare looking for is quite obvious. We will, therefore, study monomials in the Taylor expansion,namely, we will take f ( k ) = k . A.1 The case of G lL ( R, X ; r , . . . , r n ) when f ( k ) = k The following will be the object of our attention: G (2) lL ( R, X ; r , . . . , r n ) = Z ∞ dk k k j l ( kX ) j l ( kR )( kR ) L n − Y q =1 j L ( kr q )( kr q ) L j L ( kr n ) , (A.3)where the superscript indicates that we are treating the power k in the expansion of some f ( k ) .Recall the recurrence relations obeyed by spherical Bessel functions: ddx (cid:2) x ν +1 j ν ( x ) (cid:3) = x ν +1 j ν − ( x ) . (A.4)It then follows that, for some integer l , that: x l +2 j l ( x ) = ddx ( x l +2 j l +1 ( x )) . The chain rule gives: ddx n y ( x ) l +2 j l +1 [ y ( x )] o = ddy n y ( x ) l +2 j l +1 [ y ( x )] o dydx . If y ( x ) = kx , j l ( kx ) = 1 k kx ) l +2 ddx h ( kx ) l +2 j l +1 ( kx ) i . (A.5)– 24 –nserting (A.5) into (A.3) we obtain: G (2) lL ( R, X ; r , . . . , r n ) = Z ∞ dk k kR ) L k kR ) l +2 ddR h ( kR ) l +2 j l +1 ( kR ) i × k kX ) l +2 ddX h ( kX ) l +2 j l +1 ( kX ) i n − Y q =1 j L ( kr q )( kr q ) L j L ( kr n )= 1( RX ) l +2 Z ∞ dk k kR ) L ddR h R l +2 j l +1 ( kR ) i × ddX h ( kX ) l +2 j l +1 ( kX ) i n − Y q =1 j m ( kr q )( kr q ) L j L ( kr n ) . However, kR ) L ddR u ( R ) = ddR (cid:20) u ( R )( kR ) L (cid:21) + LR u ( R )( kR ) L for any function u ( R ) . Hence, G (2) lL ( R, X ; r , . . . , r n ) = 1( RX ) l +2 ddX ( ddR " ( RX ) l +2 Z ∞ dk k j l +1 ( kR )( kR ) L j l +1 ( kX ) × n − Y q =1 j L ( kr q )( kr q ) L j m ( kr n ) ! + mR ( RX ) l +2 × Z ∞ dk k j l +1 ( kR )( kR ) L j l +1 ( kX ) n − Y q =1 j L ( kr q )( kr q ) L j L ( kr n ) !) . (A.6)We obtained, therefore, a relation: G (2) lL ( R, X ; r , . . . , r n ) = 1( RX ) l +2 ∂∂X (cid:18) ∂∂R + LR (cid:19) h ( RX ) l +2 F ( l +1) L ( R, X ; r , . . . , r n ) i , (A.7)i. e., G (2) lL ( R, X ; r , . . . , r n ) can be expressed in terms of F ( l +1) L ( R, X ; r , . . . , r n ) , which isthe result we were looking for. A.2 Higher powers of f ( k ) We will not derive a closed-form expression for G ( q ) lL ( R, X ; r , . . . , r n ) in the case of any even q ,however, we will illustrate how to generalize the argument presented in the previous subsectionto these higher powers.First, we point out that higher powers in the expansion of f ( k ) must be canceled outby corresponding factors /k when eq. (A.5) is iterated for the appropriate number of times.For the fourth degree monomial in the expansion of f ( k ) we would have: j l ( kx ) = 1 k x l +2 ddx (cid:20) x l +2 (cid:18) k x l +3 ddx h x l +3 j l +2 ( kx ) i(cid:19)(cid:21) = 1 k (cid:20) x l +3 d dx ( x l +3 j l +2 ( kx )) − x l +4 ddx ( x l +3 j l +2 ( kx )) (cid:21) . (A.8)– 25 –his holds for j l ( kR ) , as well as for j l ( kX ) . We must bear in mind that, in the originalequations, terms like j l ( kR )( kR ) L appear, and therefore we must change the orders of the derivativeswith respect to R with a term like ( kR ) − L . This can be performed easily with the aid of theLeibniz formula for the n -th derivative of products of functions: kR ) L d q dR q [ u ( R )] = d q dR q (cid:18) u ( R )( kR ) L (cid:19) − q − X r =0 (cid:18) q − r (cid:19) d r dR r [ u ( R )] ( − q − r ( kR ) L L ( L − . . . ( L − q − r ) R q − r . This process must be iterated until all derivatives with respect to R have been interchangedwith terms like ( kR ) − L .As a final remark, we note that, even if the analytical problem is quite cumbersome, G ( q ) lL ( R, X ; r , . . . , r n ) can always be obtained, for even q , in terms of derivatives of the function F ( l + q/ L ( R, X ; r , . . . , r n ) . Acknowledgments
The authors would like to thank Walter Wreszinski for useful comments. This work wassupported by FAPESP, and P.R. acknowledges the support of the European Commissionunder the contract PITN-GA-2009-237920.
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