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Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O rd e r N u m b e r 9501513 C osm ic m icrow ave background p olarization Kosowsky, Arthur Brian, Ph.D. The University of Chicago, 1994 UMI 300 N. Zeeb Rd. Ann Arbor, M I 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T H E U N IV E R S IT Y OF CHICAG O COSMIC M IC R O W AVE BA C KG R O U N D P O LA R IZ A T IO N A DISSERTATION S U B M IT T E D TO T H E FA C U LTY OF T H E D IV IS IO N OF PH Y S IC A L SCIENCES IN C A N D ID A C Y FO R T H E DEGREE OF D O C TO R OF PH ILO SO PHY D E P A R TM E N T OF PHYSICS BY A R TH U R KOSOW SKY C HICAG O, ILLIN O IS AUGUST. 1994 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C ontents List of Figures iii A bstract iv A cknow ledgm ents v 1 Introduction 1 2 Physics of Polarization 4 3 E volution Equation for the N um ber Operator 12 4 A pplication to Com pton Scattering o f Photons 16 5 T he G eneral-R elativistic Liouville Equation 26 6 C om plete Polarization Equations 30 7 Power Spectra 35 8 D iscussion 39 R eferences 41 ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures 1 Definition of vectors and angles for Thomson scattering of a light beam or photon........................................................................................................ 2 7 The quadrupolar components of the intensity distribution. Any orien tation of a quadrupolar distribution can be w ritten as the sum of these two distributions. The small arrows indicate the corresponding fluid velocity in a tig h tly coupled flu id ............................................................... 3 10 Angles and directions for determining the orientation of different spher ical coordinate bases at a given point........................................................ iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 A bstract Polarization of the cosmic microwave background, though not yet detected, provides a source of inform ation about cosmological parameters complementary to temperature fluctuations. This paper provides a complete theoretical treatment of polarization fluctuations. A fter a discussion of the physics of polarization, the Boltzm ann equa tion governing the evolution of the photon density m a trix is derived from first princi ples and applied to microwave background fluctuations, resulting in a complete set of transport equations for the Stokes parameters from both scalar and tensor m etric per turbations. This approach is equivalent at lowest order to classical radiative transfer, and provides a systematic framework for investigating higher-order effects. Expres sions for the power spectra for various correlations between polarization components are derived. Experimental considerations and detection prospects for polarization are briefly discussed. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A cknow ledgm ents I would like to thank Scott Dodelson for teaching me the Boltzmann equation ap proach to the microwave background, Gunter Sigl for patient explanations of his work, and Lloyd Knox for penetrating questions. I would also like to acknowledge helpful conversations w ith Stephan Meyer and Robert Crittenden. Michael Turner has con tin u a lly provided support and encouragement. Thanks also to Edward Kolb, Tom W itte n , and Rene Ong for their tim e and energy. This work was supported prim arily by the NASA Graduate Student Researchers Program, and in part by the DO E (at Chicago and Fermilab) and by NASA through grant No. NAGW 2381 (at Fermilab). v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To Theresa — we made it! vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 In trod u ction Since the in itia l announcement by the COBE team of the detection of cosmic m i crowave background temperature anisotropies [1], a great deal of experimental ac t iv ity has resulted in nearly a dozen more anisotropy detections on a wide range of angular scales [2 , 3]. Simultaneously, detailed numerical analysis has sharpened the oretical expectations for the temperature anisotropy and its dependence on a variety of cosmological parameters, p rim arily in the context of the Cold Dark M atter (CDM ) scenario [4-6] but also in cosmological defect models [7]. W hile much work remains to be done, the focus of microwave background research has shifted from sim ply detect ing anisotropies to creating a detailed picture, both experimental and theoretical, of the anisotropies on all angular scale, and using this picture to constrain cosmological models [8]. The cosmological inform ation in the microwave background is encoded not only in temperature fluctuations but also in its polarization. Since, as discussed below in Sec. 2, the source term for generating polarization is fluctuations in radiation intensity, generally polarization fluctuations are expected to be somewhat smaller than tem perature fluctuations. Numerical calculations have confirmed this rough expectation, giving polarization fluctuations no larger than 10% of the temperature anisotropies [9, 10]. Greater experimental sensitivity is required to measure polarization than tem perature fluctuations, and so far only upper lim its have been established. However, polarization fluctuation measurements also have certain experimental advantages over temperature fluctuation measurements, making first detection of polarization w ithin the next few years a reasonable possibility. C urrently the best polarization lim it comes from the Saskatoon experiment [2], 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w ith a 95% confidence level upper lim it of 25 n K between two orthogonal linear polarizations at angular scales of about a degree, corresponding to 9 x 10~6 of the mean temperature. A n earlier experiment mapped a large portion of the sky w ith a 7° beam to place upper lim its of 6 x 10-5 in linear polarization and 6 x 10-4 in circular polarization for quadrupole and octupole variations [ 11], while a later measurement put lim its of around 5 x 10~5 on both linear and circular polarization at arcminute scales [12]. The CO BE satellite made polarized measurements and can in principle achieve a lim it of around 10-5 at angular scales from 7° to quadrupole, although doing so would require reanalysis of the entire data set [13]. W hile the sensitivity of the Saskatoon experiment is very good, it is designed p rim arily to measure temperature fluctuations. A new experiment planned for 1995 is optimized to measure polarization at 1° and 7° scales and aims for a sensitivity at or below 10-6 [14]. On the theory side, microwave background polarization has been discussed for many years [15-19]. Detailed calculations for CDM models have given linear po larization as large as 10% of the temperature anisotropy at medium angular scales, w ith a strong dependence on the ionization history of the universe [9, 10]. Another recent calculation has mapped expected correlation patterns between temperature fluctuations and polarization [20]. The aim of this paper is a detailed investigation of the theory of microwave back ground polarization. In contrast to previous work employing classical radiative trans fer theory [21], the evolution of polarization is derived from a photon description; this approach has previously been applied to temperature fluctuations in systematic investigations of second-order effects [6 , 22]. The usual classical Boltzm ann equation, adequate for describing temperature fluctuations, must be generalized to a density m a trix describing the photon polarization state. The formalism for this generalized 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Boltzm ann equation was recently developed in the context of neutrino-flavor evolu tion [23]. Here the appropriate equation is derived beginning w ith the fundamental description of the relevant Compton scattering process; the techniques easily gener alize to give a Boltzm ann equation for any particular density m atrix. The advantage of this approach is it provides a systematic perturbative expansion in the relevant small quantities, and thus provides the framework for an investigation of all secondorder polarization effects, which may be of particular interest in the case of early reionization for which the polarization contribution is the largest [24, 25]. Section I I of the paper provides an overview of the physics of polarization and its application to the microwave background. A simple calculation demonstrates that only quadrupolar variations in radiation intensity on a scatterer produce polarization. The Stokes parameters are defined and their connection to the photon density m atrix made explicit. Section I I I derives the general form ula for the tim e evolution of a density m a trix in terms of an interaction Hamiltonian. This section is rather formal; the relevant result is Eq. (25). Section IV specializes this result to the evolution of the photon density m a trix including Compton scattering. The calculation in this section is straightforward but long; the ultim ate result is Eq. (49), which is equivalent to the usual classical equations of radiative transfer [21]. The particular application to a cosmological context begins in Section V, which de rives the general relativistic Liouville equation for a perturbed Friedmann-RobertsonWalker spacetime. This collisionless part of the Boltzmann equation describes the photon geodesics in a homogeneous and isotropic universe w ith sm all scalar and ten sor m etric perturbations. The scalar perturbations are described w ith the physically appealing longitudinal gauge [26] instead of the more common synchronous gauge. Section V I gives the final evolution equations for the Stokes parameters describing the 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. microwave background. Once these equations are solved numerically, the results must be expressed in terms of statistical properties of the resulting radiation distributions to compare w ith experimental results; Section V II derives expressions for the power spectra of various correlations and cross-correlations between temperature and polar ization. This section includes a discussion of how to express the polarization direction as an expansion in vector spherical harmonics. Finally, the concluding section briefly considers experimental issues and discusses what detailed polarization measurements may eventually reveal about cosmology. This paper employs natural units throughout, in which fi = c = G = ks — I; Section I I uses gaussian units for electromagnetic quantities. The m etric signature is (H ----- ). Spinor normalizations and other field theory conventions in Section IV conform to M andl and Shaw [27]. 2 P h ysics o f P olarization This section gives a qualitative overview of the physics of polarization in the context of the microwave background. We begin w ith a review of Stokes parameters, the conventional method for describing polarized light. Then we show how polarization can be generated by scattering; application to processes on the last scattering surface predict distinctive correlations between hot and cold spots and polarization direction. The equivalent polarization description in terms of the photon density m a trix is then presented, w ith the connection to the conventional Stokes parameters made explicit. 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1 R e v ie w o f S tok es P a ram eters Polarized lig h t is conventionally described in terms of the Stokes parameters, which are presented in any optics text [28]. Consider a nearly monochromatic plane elec tromagnetic wave propogating in the ^-direction; nearly monochromatic here means that its frequency components are closely distributed around its mean frequency oj0. The components of the wave’s electric field vector at a given point in space can be w ritte n as E x = ax(t) cos [co0t - 0x{ t ) ] , E y = ay(t) cos [uj0t - 0 y(t)]. ( 1) The requirement that the wave is nearly monochromatic guarantees th a t the am pli tudes axand ay and the phase angles 9X and 6y w ill be slowly varying functions of tim e relative to the inverse frequency of the wave. I f some correlation exists between the two components in Eq. (1), then the wave is polarized. The Stokes parameters are defined as the following tim e averages: I = <«£> + <*;>; <3 = (a l)-(a 2 yy, U = (2axay cos(6x — 6y)); V = (2axay sm(9x - 9 y)). (2) The parameter I gives the intensity of the radiation which is always positive. The other three parameters define the polarization state of the wave and can have either sign. Unpolarized radiation, or “ natural lig h t,” is described by Q = U = V = 0. One im portant property of the Stokes parameters is that they are additive for incoherent superpositions of waves. The four parameters can be measured w ith a polarizer and a quarter-wave plate; the first three can be measured w ith only a polarizer. The V 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. parameter can also be measured as the intensity difference between le ft and right circular polarizations. The parameters I and V are physical observables independent of the coordinate system, but Q and U depend on the orientation of the x and y axes. I f a given wave is described by the parameters Q and U for a certain orientation of the coordinate system, then after a rotation of the x —y plane through an angle <j>, the same wave is now described by the parameters Q' = Qcos{2(j)) + Usin{2(j>), U' —Qsm(2<j)) + Ucos(2<j)). = (3) From this transformation it is easy to see that the quantity Q2+ U 2 is invariant under rotation of the axes, and the angle 1 °= 2 tan -1 U — Q (4) transforms to a — (j> under a rotation by <j> and thus defines a constant direction. The physically observable polarization vector P is here defined as orthogonal to the direction of wave propogation, having magnitude (Q2 - f t / 2)1/ 2 and polar angle a. For a wave w ith linear polarization, the vector P lies along the constant orientation of the electric field. Note that since the definition (4) is degenerate for a and a + 7r, only the orientation of P is defined and not the direction. We take the range of a to be —7t /2 < a < 7t /2 w ith the sign of a the same as the sign of U. W hile the radiation transport equations below are most conveniently formulated in terms o f the Stokes parameters, their interpretation must be in terms of the physical observables I , V, and P . 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / I Figure 1: Definition of vectors and angles for Thomson scattering of a lig h t beam or photon. 2.2 P o la riza tio n and th e Last S ca tterin g Surface In the early universe, at redshifts greater than about z fa 1100, the baryons, electrons, and photons comprise a tig h tly coupled fluid. Small m etric perturbations induce bulk velocities of the fluid, and the resulting anisotropies in the photon distribu tion w ill induce polarization when the photons scatter off charged particles. A fte r recombi nation, the photons freely propogate along geodesics, and any polarization produced before recombination w ill remain fixed. A sufficiently early reionization can of course generate further polarization. A simple idealization of the last scattering surface elucidates the process of po larization generation. Consider in itia lly unpolarized light which undergoes Thomson scattering at a given point and is then viewed by an observer. I f the intensity of the lig h t incident on the scattering point is uniform in every direction, then obviously no 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. polarization can result; however, if the incident intensity varies w ith direction then polarization can be generated. Choose the z-axis to lie in the direction of the outgo ing light, which is described by the Stokes parameters I , Q , U, and V\ represent the lig h t incident on the scattering point by the intensity I'(6, <j>). Define the polarization vectors for the outgoing beam of light so that ex is perpendicular to the scattering plane and ey is in the scattering plane, and likewise w ith the incoming polarization vectors e'x and e! (see Fig. 1). Also, instead of dealing w ith I and Q , it is convenient to describe the scattering process in terms of I x = ( I + Q)/2 and I y = ( I — Q)/2. The Thomson scattering cross-section for an incident wave w ith linear polarization e' into a scattered wave w ith linear polarization e is given by d<7 dO 3<7x | a/ ; |2 Sir where o r is the to ta l Thomson cross section. The incoming wave is unpolarized by assumption, and thus satisfies I'x = / ' = /'/2 . The scattered intensities are I7' - * • '-‘ >2 + ’ K h ■ = = (6) Thus the scattered wave has the Stokes parameters / = Ix + I = ^ I / ' ( l + cos2 d), lD7T Q = h - I y = ^ I ' s m 29. (7) This calculation gives no inform ation about the U or V parameters. As w ill be shown later, the V parameter remains zero after scattering and w ill not be considered further [21]. The U parameter can be determined by using Eq. (3). Simply rotate the outgoing basis vectors in the above calculation by ir/A and recalculate Q , which w ill be equal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to U in the original coordinate system. The result is U = 0. These results can alternatively be obtained from the physical description of the polarization state in Rayleigh scattering [21]. Note that Eq. (7) gives the well-known result that sunlight from the horizon at midday is linearly polarized parallel to the horizon. The to ta l scattered intensities are determined by integrating over all incoming intensities. Note th a t the outgoing U and Q flux from a given incoming direction must always be rotated into a common coordinate system, using Eq. (3). The result is 3<7j I = Q = U = + c o s 2 0) /'( M ) , 167T 3(7x ^ [ dSl sin20cos(2< £)/'(M ), 167T J f dn sin 2 ^ s in (2 ^ )/,(0 ,^). (8 ) 10 7T J The outgoing polarization state depends only on the intensity distribution of the un polarized incident radiation. Expanding the incident intensity in spherical harmonics, (9) lm leads to the following expressions for the outgoing Stokes parameters: 3(77’ 8 / 4/7r ■ - V 7Taoo + - y — a 20 1C>7r L3 Q = TT U - 3<7r /2tt — \/ — Rea22, 47t V 15 3<7t [2tt t — -— \ — Im a 22. 47T V 15 , ( 10 ) Thus scattering generates polarization from in itia lly unpolarized radiation if the ra diation intensity at a given point as a function of direction has a non-zero component of Y22 - 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Angular Dependence Coefficient cos 2<p Re a22 oc Q sin 2(p -Im a22 a U 1J / / Figure 2: The quadrupolar components of the intensity distribution. Any orientation of a quadrupolar distribution can be w ritten as the sum of these two distributions. The small arrows indicate the corresponding fluid velocity in a tig h tly coupled fluid. This particular form for the source of polarization leads to a correlation of the direction of the polarization vector P w ith hot and cold spots on the cosmic microwave sky [20]. Consider a given region on the last scattering surface w ith a spherical mass overdensity; the electron-photon fluid w ill have a bulk velocity towards the center of the overdense region w ith a velocity gradient away from the center (m aterial further from the center w ill be falling inwards more quickly). In the frame of some particular scattering point away from the center, the flu id velocity in towards the point is greater along the radial direction than perpendicular to it, resulting in a quadripolar radiation intensity variation w ith the largest intensity along the radial 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. direction (see Figure 2). Choose an observation direction at a right angle to the radial direction and take this direction to be the polar axis. Then the radiation intensity at the scattering point w ill have a component proportional to cos(2 ?i> — 2/?) w ith a positive coefficient, where /? is the radial direction. The scattered Q intensity is proportional to the cos(2 ^ ) dependence of the incident intensity and the scattered U intensity is proportional to the sin( 2 ?i) piece, by Eqs. (9) and (10). Thus a, the direction of P in Eq. (4), lies along /?, the radial direction. For the opposite situation, that of a mass underdensity, all the velocities change sign, so both Q and U change sign and the direction of P changes by 7t/2. When the dominant contribution to the temperature fluctuations is a gravitational potential difference (the Sachs-Wolfe effect, Ref. [29]), a mass overdensity corresponds to a cold spot in the microwave background; in this case cold spots w ill have radially correlated polarization and hot spots tangentially correlated polarization, in agreement w ith the result of Ref. [20]. For adiabatic acoustic oscillations, the density and velocity perturbations are out of phase so no specific correlation results. 2.3 P h o to n D escrip tio n The Stokes parameters can be defined equivalently in terms of a quantum-mechanical description. The polarization state space of a photon is spanned by a pair of basis vectors, which we take to be the orthogonal linear polarizations |ei) and |e2). For a photon propogating in the z-direction, the basis states |ei) and [62) are oriented like the x and y axes, respectively. An arbitrary state is given by |e) = d ie 101 |ei) + a < i ^ 2 162 ) - (11) The quantum-mechanical operators in the linear basis corresponding to each 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Stokes parameter are given by I = |ei)(ei | + |e2)<e2|; Q = l^i) <ei | — |e2) <^2 !; U = |e i)(e 2| + |e2)(e i|; V = i|e 2)(e i| — *|ei)<e2 |. (12 ) The single-particle state expectation values of these operators reproduce the defini tions (2). For photons in a general mixed state defined by a density m a trix p, the expectation value for the I Stokes parameter is given by f pn P12\ ( I) = tv p i = tr /I II \P2i P22) \0 0\ I = pn + P22 1/ (13) and sim ilarly for the other three parameters. These relations thus give the density m a trix in the linear polarization basis in terms of the Stokes parameters as I + Q U + iV U -iV \ I - Q ) — —( f l + Q<T3 + U(Ti + F<72) (14) where 1 is the id entity m a trix and cr,- are the Pauli spin matrices. Thus the density m a trix for a system of photons contains the same inform ation as the four Stokes parameters, and the tim e evolution of the density m atrix gives the tim e evolution of the system’s polarization. 3 E volu tion Equation for th e N um ber O perator This section considers the quantum number operator for a system of particles and derives its evolution equation, including local particle interactions. Taking the expec ta tion value of the operator equation gives the Boltzmann equation for the system’s 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. density m atrix, which is a generalization of the usual classical Boltzm ann equation for particle occupation numbers (the diagonal elements of the density m a trix). The derivation here applies techniques previously developed in the context of neutrino m ixing [23, 30]. We adopt second-quantized formalism w ith creation and annihilation operators for photons and electrons obeying the canonical commutation relations K ( p ) , 4 ( p ')] = (27r)32p°<53(p - p % a ', {br (q),bl(q')} = (27r)3^-6 3(q - q')<5rr,, (15) where s labels the photon polarization and r labels the electron spin; bold momentum variables represent three-momenta while plain momentum variables represent fourmomenta. The density operator describing a system of photons is given by P=j (P'KV), where pij is the density m atrix. (16) The particular operator for which we want the equation of m otion is the photon number operator ^ •(k J s a ftk M k ). (17) The expectation value of T> is proportional to the density m atrix, as seen by direct calculation: (2>y(k)) = tr[^ (k )] = (27r)3<5(0)2& % (k). (18) The last equality results from repeated application of the commutation relation Eq. (15); the in finite delta function results from the infinite quantization volume 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. necessary w ith continuous momentum variables, and cancels out of all physical re sults. The tim e evolution of the operator P ,j, considered in the Heisenberg picture, is = ![# ,!> « ] (19) where H is the fu ll Ham iltonian. We w rite the Ham iltonian as a sum of the free field piece plus an interaction term: H = H0+ H i (20) where the interaction piece is a functional of the fu ll fields in the problem. O ur goal is to express the right side of Eq. (19) as a perturbation series in the interaction Ham iltonian H j . We make the usual assumption of scattering theory th a t interaction the fields in a given begin as free fields and end as other free fields,and the in ter actions are isolated from each other. Consider the evolution of an operator through a single interaction beginning at t = 0 : before this tim e, the fields can be taken as free to a good approximation; at t = 0 the interaction Ham iltonian begins to tu rn on, and the interaction finishes at some later tim e, after which the fields can be taken as free once again. Then the tim e dependence of an arbitrary operator £ to first order in the interaction Ham iltonian can be expressed as [23] m = e ( t ) + i j ‘ dt' [ « ? ( < - ( ') , ( “ ( ( ) ]. ( 2 i) where £°(t) is the free-field operator w ith in itia l condition £°(0) = £(0), and H ° is the interaction Ham iltonian as a functional of the free fields. Equation (21) can be proven as follows. The tim e derivative of both sides gives j tm = j te m + ; { ” (<) ]+> J ‘ < w ± [« ? (t - o - f M ] • 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 22 ) The tim e derivatives in the first and th ird term on the right side can be replaced by commutators w ith H ( t ) using the Heisenberg equation. B ut these two terms depend only on free fields which are evolved w ith the free Ham iltonian H 0( t ) = Ho{0). Equation (22) becomes j t m = i K °(o ), «"(<)]+*• m o ) , m , w and so to first order in H i this just gives the Heisenberg equation for the operator £. Now we can express the tim e evolution of Z>,j in terms of free field operators. A p plying Eq. (21) to the com mutator on the right side of Eq. (19) gives | © „ ( k ) = i [//?((), D °(k)] - J ‘ dt' - <'), [# ? « , ®&(k)]] . (24) The integral on the right side can be cast in a more practical form by making the following physical assumption: the duration of each collision (the tim e interval over which the interaction Ham iltonian is non-negligible, on the order of the inverse energy transfer) is small compared to the tim e scale for variation of the density m a trix (on the order of the inverse collision frequency). The collision process relevant to the microwave background is Compton scattering off electrons, and for the cosmological epoch of interest, the electron density is always low enough for this condition to be easily satisfied. Then the tim e step t in Eq. (24) can be chosen large compared w ith a single collision and small compared to the tim e scale for density m a trix evolution. A fte r extending the tim e integral to in fin ity and taking the expectation value of both sides, we find [23] (2* f s ( 0)2k ° j t f,i:i(k) = ; ( ! » ? ( ( ) , I * ( k ) ] ) - \ /_“ [tf?(0) , i * ( k ) ] ] ) . (25) Here the integral from zero to in fin ity has been replaced by an integral over all time; 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the difference is a principle part integral which is second-order in the interaction H am iltonian and thus ignored. Equation (25) is the Boltzmann equation for the density m atrix pij. The first term on the right side is a forward scattering term which is responsible for the M SW effect in a neutrino ensemble [30]; for photons this term is zero, as w ill be shown below. The second term on the right side is the usual collision term. The tim e integral over the free field tim e dependence enforces energy conservation in each collision. The interaction H am iltonian w ill in most cases depend on background fields; for example, in the case of the microwave background the Compton scattering collisions are essentially four-point interactions quadratic in both the photon field and the electron field. In principle, a second coupled equation for the electron density m atrix must be solved simultaneously. However, in many physical situations, the background fields may be assumed to have a fixed distribution, generally thermal. In the early universe, the electrons m aintain a thermal distribution to a very high approximation and the evolution of their density m a trix becomes trivia l. The derivation of Eq. (25) has been completely general. In the appropriate lim it the classical equations of radiative transfer are reproduced; the advantage to the current approach is that it gives a systematic method for analyzing all higher-order effects. This Boltzmann equation has previously been applied to neutrinos interacting through both charged and neutral current processes in supernovae [31]. 4 A p p lication to C om pton Scattering o f P h o to n s In principle, the complete evolution of the cosmic microwave background is deter mined by Eq. (25), generalized slightly to include spatial dependence of all quantities. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In this work this space dependence w ill sim ply be put in by hand when taking expecta tion values and assumed im p licitly; more form ally it can be included through W igner functions, describing a jo in t space-momentum distribution [32]. A ll th a t remains to be done is substitution of the correct interaction Ham iltonian and sim plification of the right side. General relativistic terms emerge from the total tim e derivative on the left side; these w ill be treated in detail in Sec. 5. Microwave background photons interact w ith all charged particles. However, the rate of scattering varies w ith the mass of the charged particle as the inverse mass squared; thus it is an excellent approximation to consider only Compton scattering off electrons and ignore baryons. This section proceeds w ith evaluation of the right side of Eq. (25) for Compton scattering. 4.1 In tera ctio n H a m ilto n ia n The interaction Ham iltonian density for the fundamental three-point interaction of QED is given by [27] Wqed(jc)=-e:i>(x)JL(x)il>(x) : where ifr is the electron field operator, (26) is the photon field operator, a slash indicates contraction w ith 7 ^, and the colons signify normal ordering of the operator product. The interaction Ham iltonian is the density integrated over all of space: #qed(t) =j d3x7iQEu(x). (27) The scattering m a trix describing all scattering processes in QED is given in terms of the interaction Ham iltonian by 00 00 S = Y L S [n) = E n=0 n=0 ( —i ) n r W " / d4x1 • • • n ‘ W Q E D ^ l ) . . . «QED(*« )} J 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (28) where T signifies a time-ordered product. The nth term in the series represents all scattering processes w ith n interaction vertices. Compton scattering is thus contained in the n = 2 term of the scattering m atrix. Comparing the n = 2 term w ith the n = 1 term gives the interaction Ham iltonian for second-order scattering processes: 1 S (2) = -o / = -i r L J to o ro o dt — oo J — ood t 'T { H qED( t) H QED(t')} d tH ^ it). (29) J — CO Using W ick’s theorem to sim plify the time-ordered product and denoting the piece of ifU ) describing Compton scattering by H j yields H i { t ) = e2| ~ d t ' J d3x V - ( . r ) 7 "S F(* - [ A : ( x ) A $ ( x ') + A ^ x ' ) A t ( x ) ] (30) where Sf is the Feynman propagator for the electron, and ?/>+ {4>~) and A + (j4_) are linear in absorption (creation) operators of electrons and photons respectively. Fourier transforms of the fields and propagator are defined using the following conventions: A“ M = J + 4 ( * ) e r (*)=“ ■*]. c [ F^ d ‘i k $ + m J (2tt)4 k2 - m2 + tO .-ik .x (oU ’ ^ ^ where ur (k) is a spinor solution to the Dirac equation w ith spin index r = 1,2 and £g{k) are photon polarization four-vectors, chosen to be real, w ith index s = 1,2 labeling the physical transverse polarizations of the photon. The summation convention over repeated spin and polarization indices is always im plied. The Fourierspace interaction Ham iltonian is obtained by substituting Eqs. (31) in to Eq. (30). The IS Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. functional identity J d'xe *-* = ( 2 tr) 4 8 \ k ) (32) allows triv ia l integration over the four-momentum of the electron propagator. The resulting interaction Ham iltonian is H i CO = J dq'dp dp'(27r)363(q ' + p ' - q - p) exp [ i i (^/0 + p'° - q° - p°)] x y . ( « ') 4 ( p ') ( M i + M i ) a,(p)b,(q) (33) M. = M .\ + M . 2 , M i(q 'r',p's',qr,ps) ee e^ V A m + 4 + ’n H , ( p ) M ‘l ) 2( p ■ q ) M 2(qv , p ' s W , Ps) - + (34) 2(p • q) w ith the abbreviations , d3q m dcl = ( 2 n) 3 q0’ , d3p dP = ( 2 ir) 3 2 p° for electrons and photons respectively. A ll of the operators in Eq. (33) are free-held operators, so this is the proper expression to substitute into the left side of Eq. (25). 4.2 Forward S ca tterin g Term We now proceed to evaluate the first term on the left side of Eq. (25). First we display operator expectation values needed here and in the following subsection, using operator definitions and the commutation relations, Eq. (15): (<1x0,2 ■■■M 2 ■■■) = (ai«2 • • •) (6162 • • •), ( p 'K ( p ) ) = 2 p°( 2 n) 3 6 3(p - p > mn(p), (b t(q ')bn{q)) = ^ (2 7 r)3<53( q - q ' ) ^ „ ^ n e(q), 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = ^ ( 2 x ) 6<53(qi - q'1)(53(q2 - q 2)V '< V 24 ™e(qi)ne(q2) — ZO„\6c3z„ „/\c3/„ „/\ c c 1 / vL 1 ( 2^ ) (5 (q i - q 2)(5 (q 2 - q i ) ^ r 1r ' ^ 2r ' 2 rae(q2) 1 - 2 n*(q i) t t («i; { p \ ) a s M as'2 (P'i)as2 {P2 )) = 4p?P2(2tt)6<53(p i - pi)<53(p 2 - P 2K 4 ( P iW 'M +4p?p^(27r)6(53(p i - p'2) 63(p 2 - p i K 4 (p 2) [«5SlS' + psis'( p i) ] . (35) The last relationship neglects the correlation term between all four operators when P i = P i = P2 = p 2• The expectation values for electron operators assumes a particular form for the electron density m a trix appropriate to therm al equilibrium , w ith equal populations in each spin state and no correlations between the states; n e{q) represents the number density of electrons of momentum q per unit volume. Using the definitions (17) and (33) and the commutation relations (15) the com m utator in the forward scattering term becomes #°(°)>Z%(k)] = / cMq'dpdp'(27r)3<$3(q' + p' - q - p) { M i + M 2) x [bl{ql)br {q)al{p')aj {k)2p0{27r)36i363{p - k) -bl,{q')bT{q)a}{k)as{p)2pl0{27r)36js'63{p' - k)]. (36) On using the above expectation values, it follows that i { [tf?(0), s?.(k)]} = x u r (q) [ j M W ne{q)_ '-J -J iq ^ + // + <1 (Sup,j (*) - «,•,*.(*)) - t s{ k ) ( j - $ + m)fis,{k)\ ur (q), (37) where the integrals have been performed w ith the delta functionals. A ll of the terms involving jt cancel out on using the gamma-matrix identity filf} = 2A • B — 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the polarization vector properties k ■£i(k) = 0 and £; • £j = —6 {j. For the remaining terms we use the identity « r( ? ) { + ™ ) t a,ur (q) = tir(q)(2q ■e9 - = 2q ■£suT(q ) ts,ur (q) + m t a) j a,ur (q) 2 m = « r( 9 )^ /(rf + m )^ w r (?), (38) where the second equality follows from the Dirac equation and the th ird equality uses the Gordon identity. Thus we have ;([fl? (0 ),X > P .(k )]) = 0 (39) and the forward scattering term does not contribute to the photon density m a trix evolution. 4.3 S ca tterin g Term The scattering term isconsiderably more cumbersome to evaluate,being quadratic in the interaction Ham iltonian. After substituting the expressionsfor H i and T>ij and taking the expectation value, the scattering term reads \ L M 0) ’ = / d q i d P i 4 > i dq 2 dq'2 dp 2 dp '2 x (27r)7<53(q i + p i - q i - Pi)<$4(<?i + p'2 - q2 ~ p2) M (l) M {2 ) 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (40) The energy delta functional comes from the tim e integral on using Eq. (32). The ar guments of the m a trix element indicates the subscript to be attached to all dependent variables in Eq. (34), and of course summation over all spin and polarization indices is implied. Substitution of the expectation values Eq. (35) into the above expression and performing the integrals over q 2, p 2, q 2 and p 2 using the various delta functionals yields 1 f °o 1 ± 0),23?.(k)]]> = ^ t t W O ) x dqdq'dp'(27r)484(q' + p' — q — k )M (q 'r ',p 's [,q r , k s \ ) M ( q r , ks2,q 'r ' ,p's2) x [ne(q)<5,-Si Ps'2j( k ) — ne(q )<5:Sl5jS'/>s'S2(p )] + J dqdc^dp(2ir)464(q' + k - q - p ) M { q ' r \ ks[, q r ,p s i) M (q r ,p s 2, q'r', ks2) x [n e( q % s;<5SlS'/>;S2(k) - n e(q ) 5iJ-,j5i, 2/3,»,1( p ) ] j. (41) The subscript “ 1” on all momentum variables has been dropped for notational sim plicity. A ll terms quadratic in the electron phase-space density have been dropped since for all cosmological scenarios this number is negligible compared to unity; all terms quadratic in the photon density m a trix cancel exactly. By relabeling the in tegration variables and spin indices (im p lic itly summed over) in the second integral, Eq. (41) reduces to 1 r °o 1 = ~(27t)363(0) X J dqdq'dp'(2ir)464(q' + p' — q — k )M (q 'r ',p 's [,q r , k s \ ) M { q r , ks'2,q'r',p's2) x [n e( q ) ^ 4j ( ^ Sl^ ' j ( k ) + Sjsl2 Pis1 (k )) - 2ne(q % -, 1tf,-,»/v1«2(p')]- (42) This equation is an essentially exact expression for the collision term in the case of the microwave background: the approximation that the duration of the Compton 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. scattering be small compared to the tim e between scatterings is em inently satisfied for any cosmological scenario, and assuming the electrons to be unpolarized is essentially exact unless magnetic fields become im portant at some epoch. Evaluating the m a trix elements and performing the integrals in Eq. (42) is a straightforward process. This paper is concerned w ith the first-order perturbations away from a perfectly homogeneous and isotropic universe, and the scattering term w ill be exp licitly calculated to first order. Evaluating the m a trix elements involves standard techniques of quantum field theory and yields the fam iliar Compton crosssection to lowest order: M W r \ Ps1> k s i) M ( q r , ks'2i q’r', ps'2) (43) r r t = 2e4 ( j T j + ) (£*. (k) • £4 (p H 2(p) • £4 (k) - £ Sl(k) • eS2(p)es;(p) • £s-(k) + 8sA 8s>iS2) +2(£Sj(k) • £s-(p)£S2(p) • £s-(k) + £ Si(k) • £S2(p)es;(p) • ea»(k) - . (44) The following subsection then obtains the general Boltzmann equation for the photon density m a trix to first order in terms of the photon energy and polarization vectors. 4 .4 S ca tterin g Term to F irst O rder Now we proceed to evaluate Eq. (42) to lowest order in scattering kinematics. After substituting the m a trix element from Eq. 43 into Eq. (42), the Boltzmann equation (25), now e xp licitly including spatial dependence, becomes i + k - p ) + » - *«•> ^ - x (n e(x,q)5v j (<5iSips-j(x,k) + 5is#/aiSi(x,k)) - 2ne(x,q')6lSi^ s/ps-S2(x,p/)) 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. X q -p q -k Kq - k q -p (£s; (p) • £s, (k) £4 (k) • £ 32 (p) - e , , ( k ) -e S2(p )e s;(p ) ■£4 (k) + 6 4 ^ 4) + 2 (es;(p ) • e4i (k ) es-(k ) • eS2(p) + eSj(k ) • eS2(p )e s;(p ) • es,(k ) - 64 ^ 4) (45) where jE'(q) = (q 2 + m 2)1/ 2 is the energy of an electron w ith momentum q. The electrons are described by an unpolarized therm al M axwell-Boltzmann distribution: ( n, ;( x ,q ) = n e(x) 2w \ 3/ 2 J exp (q ~ m v (x ))2 2m TP (46) w ith Te the electron temperature and v (x ) the electron bulk velocity. Useful integrals of the electron d istribu tion are d3 q f d3 q J 7( 2 k^ ) p ^ ( X’ q) = mui(x K ( x )- (47) For relevant cosmological situations, the kinetic energies of the electrons and pho tons are negligible compared to the electron mass, im plying that the energy transfer in a Compton scattering event is small compared to the characteristic photon energy: p <C m, q <C m (w ith obvious abbreviations p = p° = |p| and q = |q|). Furthermore, if the electron and photon temperatures are comparable, p -C q. We expand the various functions in Eq. (45) in terms of p/q and q / m , using the following asymptotic expansions: E ( q + Q) ~ m 1 4 q* +, q -Q +, Q2 m' 2m 2 o n e(q + Q) ~ ne(q) 8 1 _ Q • (q - mv) _ ( k - p + E (c O -E (q + k - p ) ) ~ m T, 6 ( k - p ) + (k + Q2 + 2m T e P ) 'q g ^ P) + m dp 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (48) where in the last expression the derivative of the delta functional is defined through integration by parts [33]. W ritin g out the polarization sums e xp licitly yields the equation d , dtPi^ e4n e(x) f °° , k) = i S f c L fd S l\ ^d 8 (k - p) e f1 dpPJ 17 [6{k - + (k - p) • VW - ^ ~ (49) x{ -2 (fc + p) ^'(x>k) + 4P-e»'(k)p-fi(k)Pvfaty +4p • £,-(k) p • £2(k)p2i(x,k) + ^ + (* + p) ^ ~ %(pn(x,p) + p22 (x,p)) ’£?' ^ ~ £^k^’£^P'£j^^ ’£l^P^ x(/?i2(x,p) - p 2i(x,p)) +2(e,-(k) • ei(p)gj(k) • e2(p) + e,-(k) • e2(p) £j(k) • e^p)) x (p i 2 (x,p) + p2i(x,p)) +4e,-(k) -£i(p)£j(k) •£1(p)pn (x,p) +4e,-(k) -e2(p)e,-(k) • e2(p)p22(x,p) Here the photon momentum integral has been rew ritten as an energy integral and an angular integral over the momentum direction. This is the basic equation describing the evolution of the photon density m a trix to first order in the kinem atic variables. B y rew riting the momentum and polarization vectors in a spherical coordinate basis and incorporating the velocity-dependent term into the left-hand side, the equation becomes equivalent to Chandrasekhar’s radiative transfer formalism; cf. Chapter 1, Eq. (212) of Ref. [21]. Before performing the final angular integrals, we must consider the left side of the equation for the particular space-time geometry in which we are interested, which determines the azimuthal dependence of p. The left side of the equation w ill be analyzed in the following section, and then in Section 6 we perform the remaining momentum integrals to complete the evaluation of the right side. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 T he G en eral-R elativistic L iouville E q u ation The last section has analyzed the right side of the Boltzmann equation, Eq. (25); we now turn to the left side, describing the propogation of photons in the back ground space-time. The Boltzmann equation w ith no collision term on the rig h t side is the Liouville equation, describing the free evolution of a system’s phase space dis trib u tio n . W ritin g the equation has already assumed definition of a set of space-like hypersurfaces; th a t is, the equation contains an explicit tim e derivative. The back ground space-time here w ill be the canonical Friedmann-Robertson-Walker (FRW ) space-time. In this paper only the flat case w ill be considered; techniques pertaining to open universes have also been extensively developed [34]. Scalar and tensor m etric perturbations are added to the flat background space-time; we neglect vector per turbations, which kinem atically decay and are unim portant unless a source of vector perturbations, such as topological defects, exists. The m etric we consider is fif00 = 1 + 2 $ (x, t), g0i = 0, gij = - a 2 { t) [{l - 2'F(x, t))<5,j + Ay(x,<)]. (50) (51) In this section, Greek subscripts refer to space-time indices running from 0 to 3 while Roman subscripts refer to spatial indices running from 1 to 3. The function a(t) is the usual cosmological scale factor. The scalar perturbations, defined in the longitudinal gauge, are given by the two scalar function $ and ^ [26]. The m etric perturbations hij are defined in the transverse-traceless gauge and are subject to the constraints h? = 0, dj hij = 0. (52) W ith the perturbations defined in this way, no residual gauge freedom remains, in contrast to the more conventional synchronous gauge condition. A second advantage 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the present definition is that in the Newtonian lim it, the m etric perturbation $ sim ply corresponds to the Newtonian potential. The inverse m etric to first order in the perturbations is S’00 = 1 - 2 S(x,i), i" =o, 9" = - ^ [ ( 1 + 2 * ( x ,!))«" - A«(x,i)]. (53) (54) In this section we w ill consider the Liouville equation to first order in the m etric perturbations; second-order treatments have been undertaken in Ref. [35]. Photons are described by space-time coordinate x 11 and four-momentum k w ith k° = E. Our coordinate system has x° = t; the photon momentum satisfies - - — (55) The photons obey the geodesic equation d2 x )1 -W dxa dx 13 + Y* ° * l x l x = 0' . (56) dx 11 dxu 9 ftu~ d \l\= ° ' (57) where A is an affine parameter along the photon geodesic, which may be defined so th a t dt/dX = dx°/dX — k°\ thus dxl /dX = k' using Eq. (55). Therefore, using the definition of the Christoffel symbol T, dk» ■* u u ( l d g ai} =s dgva\ kak@ U ( 58) w ith the geodesic condition k^k^ = 0 . For convenience we also display the same equation w ith index lowered [36]: dkn ^ 1 dgaP kakp dt ~ 2 dx» k° ' 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 j L io u ville ’s equation for any phase-space distribution function / is df_ _ d£_dx^_ df_d&_ = df_ df_ df_dk^_ _ dt dxv- dt dk ** dt dt dx 1k° dk ^ dt We change to a convenient choice of momentum variables, the photon energy k° = k and u n it vector k \ The wave vector is then given by V = h k l ( l + $ + f - ^ k mknhmn^j , (61) where the norm alization is determined by k^k^ = 0, and the Liouville equation in the new variables is = df_ dt dt df_ y_ d£dk dx 1 a dk dt df_ dti_ dt Now it is a straightforward m atter to calculate the derivatives dk°jdt and dk'/dt. Equation (58) to first order gives dk _ , — dt K a + d * _ M + a dt dt 2 ! i d * + l j it y dhii' a dx* 2 dt (63) The derivative of k 1 is most easily computed by differentiating Eq. (61) and equating this result w ith Eq. (58); the result to first order is ^ = 2- F ( $ + $ ) dt a adx1 + <p) + -& '‘P A ( $ + $ ) + ^ a d x d t - - k rkmknhmn + ^ - k mkndhmnd x x a la ~ 2a 6 i j kmkndhjmd xn - 6 i j km dt + krkmkn^ ^ . dt (64) Since this expression contains no lowest-order terms, and d f / d k l is itself linear in the m etric perturbations, the final term of the Liouville equation drops out to first order. Expanding the distribu tion function as / ( x , k, k, t) = /<°>(fc, t) + f W ( x , k , k, t) 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (65) leads to the zeroth-order equation g / m _ a k 8f m = o dt whose solution is sim ply f^ °\k ,t) = a (66) dk f(°\ka), which is ju st the uniform redshift of a spctrum w ith cosmic expansion. The first-order Liouville equation is d fW , d fM fr + dt —~ ' 1 dx' Note that k a ,d fW — a K “ a " dk d fW , — “ dk a * A rC dt dt a dx1 2 dt = 0. (67) in this equation is the physical, not the comoving, photon wave number. The terms in the Liouville equation depending directly on the m etric perturbations determine the form of the directional dependence of the distribution function. A Fourier transform over the spatial dependence of the equation gives for the Boltzmann equation ^ ( K Ot , A ,k) + - ( K • k ) f W ( K , k , k ) - - & - i r / (1)(K , & ,k) a a ok d fW (k ) dk = C { K ,k , U ) (68) where C represents the collision term on the right side. For scalar perturbations, the previous section shows the right side contains source terms proportional to k • v, where v is the local velocity of the electrons. B ut for scalar perturbations, v oc K [26], so if we choose spherical coordinates for k w ith axis in the K direction, then is manifestly independent of the azimuthal angle qJ; in other words, (69) for scalar perturbations. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Tensor perturbations do depend on </>, but in a simple manner. We neglect any electron velocity v arising from tensor perturbations as corrections to the scalarinduced velocity. The </>dependence of the distribution function is determined by the perturbation term , which can be w ritten as = H J ( A + ( K ,i) e £ ( K ) + h * { K , i) e * ( K ) ) , where (70) efjand efj arepolarization tensors for the plus and cross polarizations of the gravity wave. Again, choose spherical coordinates w ith the z-axis pointing in the direction of K . In this coordinate system the polarization tensors are given by e+. = —e+y = 1 and e*y = e*x = 1, w ith the other components zero.Contraction of the u n it vectors w ith the polarization tensors gives y & e fj = sin2 0 cos2 (j) — sin 2 6 sin 2 (j) = sin 2 0 cos 2 <j), k'Wefj = 2 sin 2 0 cos ^ s in <j> = sin 2 6 sin 2^. (71) Therefore, for a given plane wave component of a m etric tensor perturbation, / ( 1](K , k, k) = / (1)(K , Jfc,9) cos 2<t> (72) for the plus polarization of the gravity wave and / (1)(K ,fc ,k ) = / (1)(K,&,0)sin2<^ (73) for the cross polarization. 6 C om plete P olarization E quations Now we have assembled all the ingredients for deriving the final polarization evolution equations: Eq. (49) w ith the perturbation expansion for pij given by Eq. (65) and the 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. angular dependence of p,j given by Eqs. (69) and (5). Note the unperturbed photon density m a trix satisfies p^ = p^2 and p $ = P21 = 0, since it represents a uniform unpolarized blackbody spectrum. The lowest-order term on the right side of Eq. (49) is zero; the resulting equation for p gives the uniform shift of the photon spectrum w ith scale factor in an expanding universe, Eq. ( 66 ). The first order term gives k)= r dpT I I f -■ p)h (r+f) k) + 4 p • § i ( k ) p • e i ( k ) / ^ ( K , k ) + 4 p • £t-(k ) p • e2( k ) p ^ K , k ) + (& + p ) (£i^ ' £ l^p ^ef(k) - e *'(k) • £2^P)ej(k) ’ £i (p )) x ( / 4 12)( K , p ) - / 4 11)( K , p ) ) + 2 ( e i ( k ) • e i ( p ) £ j ( k ) • £2( p ) + £<(k) • £2( p ) £ ; ( k ) • £ i ( p ) ) x (/® (K ,p ) + /$ (K ,p j) + 4 £ ,(k ) • ^ ( p j f i ^ k ) • ^ (p J p S V O ^ p ) + 4 £ , ( k ) • £2( p ) £ j ( k ) • £2( p ) / 4 2}( K , p ) + (k -p ).v (K ) ' kr + -p + (& p ~ 2) dp + ^22 (p)) + 4 p • £ j ( k ) p • £X( k ) p {$ { k ) + 4 p • £ , ( k ) p • £2( k ) p £ f(& ) + 4 £ ,-(k ) • £ i ( p ) £ j ( k ) • £ i ( p ) p ft^ p ) + 4 £ i( k ) • £2( p ) £ j ( k ) • £2( p ) p g ^ p ) .(7 4 ) As in the previous section, K is the Fourier conjugate of x , and he is the mean electron density which is constant to lowest order. The remainder of this section evaluates the remaining angular integrals in this expression and converts the equations for the density m a trix elements to equations for the brightness of each Stokes parameter. To evaluate the angular integrals most conveniently, choose the 2-axis of the spher ical coordinate system to coincide w ith K , independently for each Fourier mode. Note 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. th a t on transforming back to real-space coordinates, care must be taken because the density m a trix is not invariant under a change of basis (see Sec. 2). The basis for the photon direction and polarization vectors is taken to be kx = sin9cos(j) £ ii( k ) = cos 0 cos ^ £2z(k) = — sm<j) = sin 0 sin S i^ k ) = cos 0 sin<^> ^ 2y(k) = cos (j) kz = c o s 0 £ i z( k ) = - s in 0 (75) £22( k ) = 0. The same definition is used for p and its associated polarization vectors, w ith 0 —> O' and <j> —> <j)'. The angular integral in Eq. (74) is over O' and </>', and the various dot products are given by p/ -£ i(k ) = s in 0' cos 0 cos(</>/ — (j)) —co s^sin # , p ' • £2( k ) = s in 0' s in (^ — <f>), £ i(k )-£ i(p ) = cos 0 cos O'cos(<f>' — <j>) + sin# sin £ i ( k ) • £2( p ) = — cos 0 sin(</>' — <f>), £2( k ) • £ i ( p ) = cos 0 'sm((j)' — <j>), £2( k ) - £ 2( p ) = cos{<(>'-(j>). (76) Two additional convenient abbreviations are /x' = v • p = cos O' and /« = v • k = cos 0. Now the angular integrals are straightforward, resulting in expressions like — o{1)(k d tP n { k , u) p )~ e4ne - ( I - 2/ £ £ | - ( 4 /x2 - 1 ) £ ^ - P n \ k , p') - %p 2 £ ^ w v iftfc y ) ^ - p {22 ( k , p ' ) , (8 „ 2\ dp ii n1 3 dp 22 + k v p ( - - 2/1 J — + 2 kvp — (77) for scalar perturbations, where dependence on the Fourier mode K is im p licit. In solving the evolution equations, it is convenient to split the density-m atrix pertur32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. bation into two parts, one due to the scalar m etric perturbations and one due to the tensor m etric perturbation. In making this split, the bulk velocity v is entirely attributed to the scalar perturbations. For the final set of evolution equations, we change variables to comoving wave number p = ka, and convert the density m atrix elements to Stokes parameter bright ness distributions: -1 = p dp u(p) 2 dp = P d p u (p ) 2 dp A £ /(K ,p , p) -- pdpu(p) 2 dp A V (K ,p ,p ) = A /( K ,p , p) (/» n (K iP>^) + A>22 ( K ,p ,/i) ) , -l A g ( K ,p ,^ ) ( p n ( K , p ,/0 - M V (K ,p ,/i)) , -l (p iV (K ,p ,/« ) + P a V (K ,p,/i)) , -- i ( M V (K .P .rt-rfd K .P .rt), where the superscript i stands for s,+, or x , representing the three types of pos sible m etric perturbations: scalar and two polarizations of tensor. For blackbody perturbations, 4 A / = A T / T . We also define moments of these variables: A //(p ) = J ^ d p ' W ) A j( p , / 0 (79) where Pi is the Legendre polynomial of order I. For scalar perturbations, the brightness is governed by the set of equations [17] <9$ ~TT, at d^i 2 . XT k at a _ 1 -crTne A / — A j 0 + i v p — P2 { p ) ( A S[2 + ^ Q 2 — Ago) dA Q 1. -\— i K p A q dt a d A I, 1. 3 xr— I— i K p A jj at a 1 dA + -iK p A y at a — —a r n e A sq + -(1 — P 2 ( p ) ) ( A sI2 + A q 2 — A q 0) _ — —<jfneAu^ = v — -pA y-t -c r Tn e A s 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (80) The evolution of the brightness thus does not depend on the direction of K , only on its magnitude; the brightness depends on the direction of K only through in itia l conditions, which factor out of the linear evolution equations. The equations for U and V have no source terms, so for each K the evolution leaves U = 0 and V = 0. The coordinate dependence of Eq. (3) gives a non-zero U on transform ing back to x space, but V remains zero. For tensor perturbations, the evolution equations take their simplest form after the coordinate transformation [37] Af = (1 — fi2) cos(2<^)A+, A * = (1 — fi2) sin(2$i>)Aj, Aq = (1 + /f 2)cos(2<£)A£, A q = (1 + fi2) s in (2^)A g , A = —2 /is in (2 ^ )A y , A y = 2fi cos(2^)A ^. (81) The (j> dependence is determined by Eqs. (72) and (73), and the fi dependence is chosen to sim plify the final equations. A fter this change of variables, the brightness equations become [38] dAt 1 -^ r + - , A ^ r , j + = -^ A + . n = -< rr n . ( A j - A+) , ^ ¥ - + ^ - iK fiA $ A+ dh+ /-. 2 - ? r = - £rTn , ( A j + A + ). - = -e rTn eA £ , + -l K n ~ The definition of h+ is given in Eq. + ^ A +4 + ^ A q 2 + | a $ 0 (70). (82) (83) The x tensor perturbation gives same equations. Again, V = 0 since it has no source term. For a given cosmological scenario, which determines the m etric perturbations, Eqs. (80) and (84) must be evolved numerically. This can be done efficiently by 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. expanding the brightness functions in terms of Legendre polynomials, Eq. (79), giving a large set of coupled ordinary differential equations [17]. Several detailed codes to calculate temperature fluctuations have been implemented using this scheme [4, 5, 6]. 7 P ow er Sp ectra Numerical solution of the above transport equations gives the Fourier space brightness functions A f;( K ) and A q;(K ) for scalar perturbations and A / ((K ) and A [/;(K ) for tensor perturbations, where e represents the two gravity wave polarization states + and x . The temperature fluctuations in real space are 1 = i + 1 £ £ ( 2 '+ u a t c o s o y * K / o x [A J ,(K ) + sin2 O' cos 2<£,A /](K ) + sin 2 O' sin 2 ^ A £ ( K ) ] , (84) where To is the mean temperature of the microwave background, and (O', <f>') represents the same direction as (0, <j>) except in the coordinate system defined by the K direction. The polarization is more complicated, because for each K mode the coordinate system in the direction (0, </>) has a different orientation; when the Q and U bright nesses are summed up, the axes must be rotated to the orientation in the x coordinate system, using Eq. (3). To determine this rotation angle for each K mode, refer to Fig. 3; the needed angle is labelled direction of K be denoted by the angle between the vectors 0 and O'. Let the (O k i^ k )- On the unit sphere, the lengths of the sides of the spherical triangle A B C are ju st the angles they subtend. The angle O' is given by the law of cosines, cos O' = cos 0 cos O k + sin 0 sin O k cos(<j>K — <j>), 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (85) At x Figure 3: Angles and directions for determining the orientation of different spherical coordinate bases at a given point. and the rotation angle is sin £' = sin Ok sin(4 k — 4) csc 9'. ( 86 ) Then the Q and U brightnesses are given by = ) E E ( 2 ' + l)fl(c o s 9 ')e iK -* 1o x [(a * ,(K ) 4 K / + (1 + cos2 O') cos2^/A q ;(K ) + (1 + cos2 O') sin2^'A Q /(K )^ cos 2£' + ( —2 cos O' cos 2^/A g i (K ) + 2 cos O' sin 2<^A q;(K )) sin 2£'J, (87) and U the same except for the replacements cos £' —> — sin £' and sin £' —> cos 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. From these two quantities, the polarization vector P follows from Eq. ( 4 ) as P (x ,M ) 4 \/2 0 ^ /A p (A p + A q ) + c j ) j ^ - ^ A p ( A p - A q ) Ap = \ / + Ay. (88) (89) The predictions of a given cosmological scenario are only statistical. The tra d i tional statistical measure of temperature fluctuations is the angular power spectrum C ( 6 ), defined by CTTm = 1 + ^ , % • q2 = cos0, (90) where the angle brackets represent an ensemble average over in itia l conditions; this average can be replaced in calculations w ith an average over space, assuming ergodicity. We further define correlation functions for polarization and cross-correlations between temperature and polarization: CPP(0} = / P(qi)P(q2)' T0 C t p {6 ) = C v v {6 ) = C” W To / ’ r ( q O P ( q 2) \ To To / ’ P(qi) P(q2) \ To S To / P(qi) V T ( q 2) To To (91) Correlation functions are commonly characterized by the coefficients C\ of an expan sion in Legendre polynomials: CTTW = ± 2 - i ~ C r TP,(cos 9) (92) 1=2 and likewise for the others. The I = 1 term, corresponding to the dipole from proper m otion w ith respect to the rest frame of the microwave background, is ignored. 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. An equivalent definition of Ci is obtained by expanding the fluctuations in spher ical harmonics: T ( x ,M ) ^ rp — ® ^ 1—2 = 0 ,x 2_^ a / m ( x ) V / m ( 0 , (j)) , m ——l £ £ 1—2 oo = J-Q y s ,„(x )x ,„ ( « ,« , m ——l I . E i>ta(x)x,„ (« ,« , 1=2 m = —l P ( X , 0 , (j)) _ J-0 oo _ I _ - (v \Y (ft rh \ (93) 1=2 m = —l where X /m are vector spherical harmonics, defined by [39] v _ "* * / 1 dYimz — /---------- ( • /i o i ® ^ / ( z + i ) \ sm^ ^ . . . n v y ijT + i) dYtm 2\ aa $ I i ^ y fnA^ (^4) J d n X llm, • X ;m — dll'dmm1! (95) J d n X t m, - ( r x X /m) = 0. (96) The vector spherical harmonics form a complete orthonorm al basis for vector functions on a sphere. In Eqs. (95), a;m and bim can be obtained from aim and 6/m by using the orthogonality properties of the harmonics; num erically they can be obtained directly w ith the relations used as a check. The harmonics obey the addition theorems <*<W = = Aiir 57- T ^ E "* m = —l An r ^ Y L(t> u < h)Y U K h ), a 7 T E x U (. . « ' X i . f c « , (97) (98) m = -l where 0*2 is the angle between vectors in the and the ( 6 2 , ^ 2 ) directions. Substituting these relations into Eq. (94) for the various correlations and using the 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. orthogonality of the harmonics gives (|«/m|2) = C j T (a*mbim) = C f p 6 w 6 mmi; (99) and sim ila rly w ith the other correlation functions. The values for C/ in a given the ory can be obtained either by calculating the values of the coefficients in Eqs. (95) using the orthogonality of the spherical harmonics and then calculating Ci from Eqs. (101), or by substituting the temperature and polarization fluctuations directly into Eqs. (92) and (93), and then extracting the moments [?]. For temperature fluc tuations from only scalar perturbations, it is simple to derive the usual form ula T/ roc SiJo K d K \A h m 2' (100) where the sum over K has been approximated by the integral V f d3K /(2 7 r)3 w ith V the sample volume. 8 D iscu ssion Since the in itia l detection of cosmic microwave background temperature anisotropies, the physics of the anisotropies has been studied in great detail. W ith in the context of cold dark m atter models, the shape of the fluctuation power spectrum depends on cosmological parameters such as f ie , & 0 -, H , the spectral index of the in itia l perturbation spectrum, the am plitude of prim ordial gravity waves, and the ionization history of the universe. Polarization remains undetected and may stay so for the next few years, so not as much theoretical effort has been focussed on this aspect of the microwave background. B u t w ith detection on the experimental horizon, it is not too early to begin refining polarization estimates and providing clearer targets for experiments. 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The detection of microwave background polarization is very difficult. The most o p tim istic scenarios predict polarization no larger than 10% of the temperature fluctu ations, or a few parts in a m illio n of the temperature. Needless to say, this sensitivity is hard to attain. The same backgrounds which affect the temperature measurements w ill also affect polarization measurements. One advantage of measuring polarization which has been realized for a long tim e is that the experiment can chop between two orthogonal polarizations on the same patch of sky, which involves rotating a po larizer, instead of mechanically redirecting the telescope. In practice, temperature measurements can only chop at a few Hz, while polarization measurements can chop at hundreds of Hz. A ny atmospheric noise is thus supressed much more effectively. A m itigatin g effect is th a t since the orientation of the horn is im portant, side lobes from the ground and diffraction effects can add noise differently to the two polariza tio n channels. As w ith temperature measurements, the ultim ate experimental lim it is astrophysical foreground sources, particularly our own galaxy. U nfortunately the dependence of the temperature fluctuations on various cosmo logical parameters is degenerate: different sets of parameters may result in virtu a lly identical fluctuation spectra [5]. As some of the parameters are better determined through other methods, this degeneracy w ill become less vexing. The inform ation encoded in the polarization fluctuations is complementary to that in the temperature fluctuations and may help to lift the degeneracy for some parameters. For example, it is already known that the ratio of the polarization power spectrum to the temper ature power spectrum depends sensitively on the ionization history of the universe [9]. W hether the polarization correlation functions in Eqs. (93) contain substantially different inform ation than the temperature correlation, Eq. (92), awaits numerical simulations [40]. 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R eferences [1] G.F. Smoot et al., Ap. J. 396, L I (1992). [2] E.J. Wollack et a l, Ap. J. 419, L49 (1993) (The Saskatoon or Big Plate experi ment). [3] A reasonably complete and current list of temperature anisotropy detections and upper lim its (aside from Saskatoon): C.L. Bennett et a l, subm itted to Ap. J. (1994) (CO BE D M R two year data); K. Ganga et al., Ap. J. 410, L57 (1993) (FIRS); S. Hancock et al., Nature in press (1994) (Tenerife); T. Gaier et al., Ap. J. 398, L I (1992) and J. Schuster et a l, Ap. J. 412, L47 (1993) (SP91); P. Meinhold and P. Lubin, Ap. J. 370, L l l (1991) (SP89); M. Dragovan et al., subm itted to Ap. J.(1994) (P Y T H O N ); P. Meinhold et al., Ap. J. 409, L I (1993) and J. Gunderson et al., Ap. J. 413, L I (1993) (M A X 3); A. Clapp et al. and M. Devlin et al. submitted to Ap. J.(1994) (M A X 4); E.S. Cheng et al., Ap. J. 420, L37 (1993) (M S AM ); G.S. Tucker et al., Ap. J. 419, L45 (1993) (W hite Dish); P. de Bernardis et al., Ap. J. 422, L33 (1994) (ARG O ); A.C.S. Readhead et al., Ap. J. 346, L56 (1989) (Owens Valley). [4] N. Sugiyama and N. Gouda, Prog. Theor. Phys. 8 8 , 803 (1992). [5] J.R. Bond et al., Phys. Rev. Lett. 72, 13 (1994). [6] S. Dodelson and J.M. Jubas, Fermilab preprint Pub-93/242-A (1993). [7] A small sampling of recent papers includes A. Stebbins and S. Veeraraghavan, Fermilab preprint Pub-94/047-A (1994); B. Allen et a l, Fermilab preprint Conf- 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94-197-A (1994), to appear in [8]; D. Coulson et al., Princeton preprint PUPTH-93/1429 (1994); D.N. Spergel, Ap. J. 412, L12 (1993). [8] For an up-to-date review of current results and ideas related to the microwave background, see Proceedings of the Case CM B Workshop, ed. L. Krauss, in press (1994). [9] R. Crittenden, R. Davis, and P. Steinhardt, Ap. J. 417, L13 (1993). [10] R.A. Frewin, A.G. Polnarev and P. Coles, subm itted to Mon. Not. Roy. Ast. Soc. (1993); D.D. Harari and M. Zaldarriaga, subm itted to Phys. Lett. B (1993). [11] P.M. Lubin, P. Melese and G.F. Smoot, Ap. J. 273, L51 (1983). [12] R.B. Partridge, J. Nowakowski and H.M . M artin, Nature 331, 146 (1988). [13] G.F. Smoot, private communication (1994). [14] P. Tim bie et al.; S. Meyer, private communication (1994). [15] M .J. Rees, Ap. J. 153, L I (1968). [16] C.J. Hogan, N. Kaiser and M .J. Rees, Phil. Trans. R. Soc. London, A 307, 97 (1982). [17] J.R. Bond and G. Efstathiou, Ap. J. 285, L47 (1984). [18] J.R. Bond and G. Efstathiou, Mon. Not. R. Ast. Soc. 226, 655 (1987). [19] N. Kaiser, Mon. Not. R. Ast. Soc. 20 2 , 1169 (1983). [20] D. Coulson, R.G. Crittenden, and N. Turok, Princeton University report P U P T 94-1473 (1994), unpublished. 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [21] S. Chandrasekhar. Radiative Transfer (New York: Dover, 1960), chapter 1. [22] W . Hu, D. Scott, and J. Silk, Phys. Rev. D 49, 648 (1994). [23] G. Sigl and G. Raffelt, Nuc. Phys. B 4 0 6 , 423 (1993). [24] J.P. O striker and E.T. Vishniac, Ap. J. 306, L51 (1986); E.T. Vishniac, Ap. J. 322, 597 (1987). [25] G. Efstathiou, in Large-Scale Motions in the Universe, ed. V.C. Rubin and G.V. Coyne (Princeton: Princeton University Press, 1988). [26] V .F. Mukhanov, H .A. Feldman, and R.H. Brandenberger, Phys. Rep. 215, 203 (1992). [27] F. M andl and G. Shaw, Quantum Field Theory (New York: W iley & Sons, 1984). [28] See, e.g., J.M . Stone, Radiation and Optics, (New York: M cG raw -H ill, 1963); M. Born and E. Wolf, Principles o f Optics, 6th ed. (New York: Pergamon, 1980). [29] R .K. Sachs and A .M . Wolfe, Ap. J. 147, 73 (1967). [30] G. Raffelt, G. Sigl, and L. Stodolsky, Phys. Rev. Lett. 70, 2363 (1993). [31] G. Raffelt and G. Sigl, Astropart. Phys. 1 , 165 (1993). [32] For reviews of W igner functions and related mock phase space distributions, see N.L. Balazs and B .K . Jennings, Phys. Rep. 106, 123 (1984); M. H ille ry et al., Phys. Rep. 106, 121 (1984). [33] J. Bernstein, Relativistic Kinetic Theory (Cambridge: Cambridge University Press, 1988). 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [34] L.F. Abbot and R.K. Schaefer. Ap. J. 308 . 546 (1986): M. Kamionkowski and D.N. Spergel. submitted to Ap. J. (1994): M. Kamionkowski. D.N.Spersel. and N. Sugiyama, submitted to Ap. J. Letters (1994). [35] E. Martinez-Gonzalez, J.L. Sanz, and J. Silk, Phys. Rev. D 46, 4193 (1992). [36] P.J.E. Peebles, Principles of Physical Cosmology (Princeton, NJ: Princeton U n i versity Press, 1993). [37] A.G. Polnarev, Sov. Astron. 29, 607 (1985). [38] R.G. Crittenden et al., Phys. Rev. Lett. 71, 324 (1993). [39] J.D. Jackson, Classical Electrodynamics, Second Edition (New York: W iley & Sons, 1975), chapter 7. [40] For a more detailed treatm ent of the correlation functions, see the paper based on this thesis, to be subm itted to Ann. Phys. (1994). [41] A. Kosowsky, in preparation (1994). 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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