Student Friendly Guide. Cosmic Microwave Background

Transcription

1 Student Friendy Guide to the Cosmic Microwave Background Robert D. Kauber Juy 5, 05 Abstract This document provides what is hopefuy a reativey easy-to-understand introduction, suitabe for someone with undergraduate physics training, to the power spectra anaysis of the cosmic microwave background (CMB, as unveied by the COBE, WMAP, Panck and other experiments of the past two decades. It proceeds step-by-sma-step at a moderate (by physics standards eve of mathematica sophistication, and can serve as a simpified initia foundation for those wishing to enter the fied, or for others who simpy desire a more in-depth understanding of the CMB than the onine popuar accounts offer. It is haf way between those onine popuarizations and typica cosmoogy textbooks. Fu discosure I am not a researcher/expert in this topic, but fee I can offer a fair pedagogic introduction to some areas of the subject for others ike me who have physics backgrounds, but are not working in the fied. Suggestions and corrections are aways wecome via the feedback ink on the home page above (which aso contains inks to pedagogic treatments of other physics topics. RDK

2 Two Parts to This Document This document is divided into two main parts, the basic introduction and the appendices. The basic materia covers the physica mechanisms giving rise to the cosmic microwave background (CMB and how we anayze the CMB using spherica harmonics. The appendices provide a fairy in depth deveopment of spherica harmonic anaysis and power spectra anaysis, for those not competey at home with them. The first part comprises 3 pages; the appendices, 7 pages. Tabe of Contents Part. The Cosmic Microwave Background Power Spectrum.... Spatia Temperature Variations from Eary Quantum Fuctuations Pane Standing Waves in a Spherica Anaysis Spherica Harmonic Anaysis The CMB and the Ampitude a of Each Spherica Harmonic Y Converting Between Cartesian Fourier and Spherica Harmonics Anayses Potting C vs What the CMB Power Spectrum Tes Us Summary Further Reading... 3 Part. Appendix A. Summary of Spherica Harmonics Appendix B. Introduction to Power Spectra Anaysis 38

3 The Cosmic Microwave Background Power Spectrum If you are reading this, you have no doubt seen the diagram of Fig.. It shows the now famous cosmic microwave background (CMB power spectra anaysis of temperature variations across our sky. In this document, I try to present an introduction to it (where it comes from, what is the math behind it, and what it means that is reasonaby transparent to anyone with an undergraduate background in physics. Figure. Power Spectrum of the Cosmic Microwave Background Variations Spatia Temperature Variations from Eary Quantum Fuctuations In this section, we do a brief overview of where the temperature variations of the cosmic microwave background (CMB come from.. Traveing Pressure (Acoustic Waves from Quantum Fuctuations Tiny spatia variations in one or more quantum fieds right after the big bang ead to spatia variations in mass-energy density. These variations grow to macroscopic scaes via infation and the subsequent expansion of the universe. Such variations comprise regions of greater mass-energy density and regions of esser massenergy density. The denser regions attract other matter and initiay the dense regions grow denser and the rarefied regions grow more rarefied. This process woud simpy continue except that a pressure deveops in the denser regions that begins to repe the matter moving toward them. The pressure arises as foows. For the first 380,000 or so years after the big bang, temperature was so high that atoms coud not form, and the universe consisted of a soup (a pasma reay of eementary partices interacting with one another. The primary interactions were eectromagnetic ones between charged baryons and photons, which were continuay and freneticay bouncing off one another. Thus, the photons exert a pressure on the baryons. If mass-energy were distributed competey eveny, the pressure, as in a static perfect fuid, woud be the same everywhere.

4 However, as the denser regions attract mass-energy (baryons and photons from the ess dense regions, the e/m interactions in the denser region increase, thus increasing the pressure in that region. This pressure buids unti it utimatey repes the incoming mass-energy and causes incoming baryons to rebound. The rebounding baryons form a pressure wave that propagates. This is just what happens with sound (acoustic waves, athough the pressure there is due to gas moecues interacting with other gas moecues, not photons. With everyday sound waves, high and ow density regions propagate due to the pressure variations between those regions. The speed of those waves for an idea gas (which the pasma approximates is where p is pressure, ρ is density, and k is an appropriate constant. csound = k p / ρ ( Thus, the baryon-photon pasma ends up with a whoe spectrum (a different waveengths of essentiay acoustic waves propagating through it in a directions. Note that photons interact with eectrons as we as baryons, but the mass of the eectrons is so much smaer than that of the baryons that they are effectivey saves to the motion of the baryons and carried aong with them. Their effect is insignificant, so they can be ignored.. Traveing Waves to Standing Waves From eementary physics we know that two oppositey moving waves (aong a string for exampe, wi form a standing wave if they have the same waveength. See Fig.. Note that if traveing waves have different waveengths, they wi destructivey interfere and not produce a standing wave. On average, waves of differing waveengths wi cance one another out, but those of the same waveength wi tend to produce standing waves. Over a given ength (say of string between two was L, the standing waves formed must have engths of λ n = L/n, where n =,,. The fundamenta wave (n = woud have λ = L, i.e., it woud have nodes at its ends (as in no motion where the string is pinned to the was on either end but none intermediate. (See the LH side of Fig. 3 beow. It woud be haf of a spatia sine wave osciating up and down in time. The first harmonic (n = woud be a fu sine wave with one node in the midde. Each higher harmonic woud have one more node. Higher n means shorter waveength and higher osciating frequency in time ν, since c sound = νλ. two traveing waves (same waveength one standing wave (different times shown Fig. Standing Waves Forming from Traveing Ones The sound waves in the baryon-photon pasma interact with one another, destructivey interfering in some regions and forming standing waves, via constructive interference, in others. These standing waves are referred to as baryonic acoustic osciations (BAOs.

5 .3 Which Waveengths Get Preferred (Form Standing Waves 3 Standing waves in the universe (at the 380,000 year mark can ony form over engths equa to the distance an acoustic wave coud trave in 380,000 years. At greater distances, traveing waves (traveing at the speed of sound for the medium woud not have time to competey overap and form a standing wave. If we ca the distance sound can trave in that 380,000 years L, then at that time, we woud expect to find standing waves of waveengths λ n = L/n. Ony those, in the time avaiabe since the creation event, woud be abe to neaty overap in a constructive way. Other waveength waves woud randomy cance one another through destructive interference. We can think heuristicay of L = (c sound (t recombin, where t recombin is 380,000 years in appropriate units, but note that due to pressure and density changes as the universe evoves, c sound varies over time, so we woud need to integrate to find L, rather than use a simpe mutipication. Additionay, the universe continuay expands, so any ength L one might determine at one point in time gets stretched at ater times. Cosmoogists can readiy make the appropriate cacuations to get the distance L sound coud trave from the first instant of creation up to the 380,000 year mark. compressed Standing Waves (times shown when at peak higher temp L L L rarefied fundamenta n = st harmonic n = nd harmonic n = 3 ower temp Figure 3. Fundamenta Standing Wave and First Two Harmonics A ese being equa, the standing waves of waveengths λ n woud be spread randomy at different ampitudes and in different directions..4 Caveat on Wave Shape To keep things simpe and iustrate a point, we have depicted the acoustic wave shapes as sinusoids, but they are actuay more compicated. Certainy, the usua sound waves with which we are famiiar typicay have more diverse shapes, though they can be anayzed as summations of sinusoids via Fourier series anaysis. In fact, from both theory and anaysis of the CMB, it is beieved that via infation, the arge scae BAOs emerged, at east in arge part, from primordia quantum harmonic osciator fuctuations. Further, it appears the fundamenta (owest state of those osciators was, by far, the argest contributor to the fuctuations. From eementary QM, we know (ook it up if you don t reca that the fundamenta mode shape in space of a QM osciator is Gaussian. See Fig. 4. The wave function ψ, and hence its probabiity density ψ., are norma (be shape curves in space. Thus, after infation, the mass density woud be expected to have simiar shape. ψ ψ x x 4 = e σ ψ = e σ σ π σ π x Figure 4. Gaussian Wave Form in Space for Ground State of Quantum Harmonic Osciator

6 4 The growth of pressure (sound waves from density fuctuations described in Sect.. woud then, via the appropriate dynamics, ead to sound waves (BAOs of simiar, Gaussian shape. So, the n = wave of Fig. 3 shoud be more Gaussian shaped than sinusoida, though a Fourier anaysis of it woud have a arge (haf wave sinusoida component. Simiary, the peaks and vaeys between nodes of the other modes (such as n = and 3 of Fig. 3 woud actuay be quite cose to Gaussian shape. But again, a Fourier anaysis woud have arge components corresponding to the sinusoids shown in Fig. 3. Even though the wave forms are more Gaussian than sinusoida, we can sti consider them to have modes of particuar waveengths (shorter for higher mode number n with n + nodes. We may, esewhere in this document, depict the sound waves graphicay as sinusoida for simpicity, but keep the points of this section in mind as you contempate that materia..5 Phases of the Various Acoustic Waves Note that a the wave formation started at the same time. So, for exampe, a of the various fundamenta modes (corresponding to n = in Fig. 3 woud have their corresponding traveing waves begin to propagate at the same time. They a propagate at the same speed and are subject to the same dynamics. So the many different fundamenta standing waves set up a over the universe woud a be in phase in time and remain so as time progresses. Ditto for higher modes. Hence, a BAO standing waves throughout the universe having the same number of nodes remain in phase in time. That is, a having the same mode number n peak at the same time..6 Pressure (Acoustic Waves to Photons of Various Frequencies (Temperatures Wherever the standing waves are denser, the pressure is higher. Higher pressure means hotter baryons (moving faster = higher energy. In interactions with photons, the baryons woud impart more energy to the photons, resuting in higher frequency ight emanating from the denser regions. Measuring the ight frequency at a given point in space woud then be equivaent to measuring temperature at that point. In short, for a oca region of baryons, higher ρ higher p higher T higher frequency of γ scattered..7 Freezing the Temperature (Photon Frequency Patterns at 380,000 Years At around 380,000 years after the big bang, average temperature of the universe had faen to the point where baryons coud capture eectrons and form atoms (mosty protons and eectrons to form hydrogen, but sma amounts of heium and ithium atoms were aso formed. This point in time is, strangey enough, caed recombination, even though it was the first time partices combined to form atoms, not a recombining, which impies it had happened before. (And some think physics can be confusing. Once recombination occurred, there was no onger a charged baryonic pasma scattering photons widy and reguary. The atoms, with exceptions so rare as to not matter, did not scatter the photons, so the photons were effectivey reeased from their eectromagnetic bondage to baryonic matter. The photons streamed forth and spread outward through the rest of the universe. The key point is that at that moment ( cosmic moment, the photon frequencies refected the temperatures of the oca regions they ast were in contact with. These frequencies were, in effect, ocked in unti the present day. Those variations in frequency are what COBE, WMAP, Panck, and other experiments have recenty measured. That, incuding the average background temperature/frequency at.75 o K, is the cosmic microwave background radiation (CMBR. Note however, that due to the expansion of the universe since recombination, the frequencies of a photons have been reduced (waveength stretched by expansion. This is taken into account when the photon frequencies data is anayzed. The reative differences remain as a photons are stretched by the same ratio by cosmic expansion (subject to some subteties that we won t get into.

7 .8 Phases of Different Modes at Recombination 5 As noted in Sect..5, pg. 4, for a given mode number, the various individua oca standing waves of that mode are a in phase in time (but scattered a over randomy in space at different ampitudes and peak at the same time. However, the different modes osciate at different frequencies, so waves of different mode numbers peak at different times. This is iustrated for the owest three modes in Fig. 5. Note that for fundamenta (n = mode osciation frequency f., the nth mode has osciation frequency equa to n.f.. So in the time it takes the n = mode to osciate ¼ of a fu cyce, the n = mode has osciated through ½ of a fu cyce. And the n = 3 mode has osciated through ¾ of a fu cyce. Further, the fundamenta standing wave at recombination (one hump as for n = in Fig. 5 resuts from a oca high density region acting as a gravitationa attractor puing in baryons. At recombination time, there has just been enough time to pu in baryons, but not enough time for pressure to have started them moving away from the center of the region. In other words, the n = mode at recombination is a purey compressive hump. It is ¼ of the way through a fu cyce (90 o in phase. We wi see that this resuts in the first peak shown in Fig., pg.. compressed, higher temp L L L t recomb fundamenta n = rarefied, ower temp t = 0 L L t = 0 t = t recomb L L t = t recomb L t = of fu cyce 4 L t = t recomb of fu cyce L st harmonic n = nd harmonic n = 3 t = 0 t = 3 t recomb t = 3 t recomb t recomb t = 3 of fu cyce 4 Figure 5. Evoution of the First Three Standing Wave Modes to Recombination Time The n = mode, on the other hand, with a compressive hump and a rarefied hump (three nodes, osciates twice as fast as the n = mode. So by the time of recombination, it has osciated ½ of a fu cyce (80 o in phase. But that is simpy a straight ine. So we woud not measure any rea pressure/temperature increase in that region at that particuar time. It woud, effectivey, not show up in our data. This resuts in the first trough after the first peak in Fig.. The n = 3 mode, with four nodes, osciates three times as fast as the n = mode. So by the time of recombination, it has osciated ¾ of a fu cyce (70 o in phase. This resuts in two rarefied humps and one compressive hump (an overa regiona average of rarefied. This shows up as the second peak in Fig.. Higher modes ead in simiar fashion to subsequent troughs and peaks in Fig..

8 .9 The Last Scattering Surface 6 For us, as we ook out into the universe, we ook into the past, due to the great distances ight must trave to reach us over the history of the universe. Looking in any direction, the ight from recombination (CMB reaches us from the same distance away, from a directions, at the same time. Thus, it is effectivey coming, from our perspective, from a vast she at a distance a but severa hundred thousand ight years from the edge of the visibe universe. Hence, that she ooks to us ike the surface of a giant sphere. Since this spherica she represents the ast ight to be scattered from primordia free baryons, it is ca the ast scattering surface (LSS. The modes of Fig. 5, and higher modes, we see now are those from the LSS. But the LSS is best treated in a spherica coordinate system and we have been working with waves in a Cartesian system. The next two Sects. 3 and 4 show how we go from that Cartesian system to the spherica system in which our measurement of the CMB is carried out. 3 Pane Standing Waves in a Spherica Anaysis 3. Graphicay for Simpified Case Consider one pane wave of the fundamenta baryon acoustic mode (simiar to n = in Fig. 3. Pretend, for the sake of this discussion, that at the time of recombination, the wave extends in its direction of trave across the universe. That is, over each ength L of Sect..3, the fundamenta standing wave is repeated. Things are actuay far more diverse than this with various modes overapping in various directions a over the pace. However, we simpify here to make a point. We represent this simpification in Fig. 5. Fig. 5 is aost sef-expanatory. The fundamenta wave of the figure is frozen after recombination. The baryons were most compressed at the peaks right before recombination, so photons that emanated from those regions are hottest (higher frequency. The noda regions were not so compressed, and the photons from those are coder (ower frequency. Hotter (peaks Coder (nodes L Us Source ocation of ight we see shorty after recombination No nodes. Aost uniform sky seen. Source ocation of ight we see somewhat ater Four nodes. Rippes seen in CMB temp Source ocation of ight we see even ater Eight nodes. More rippes (waviness seen in CMB temp Figure 6. Over Time, Pane Waves Appear As Higher and Higher Modes on Sphere from which Light Reaching Us Left Just after recombination (eft most part of Fig. 6 ight from the time of recombination had not traveed very far to reach us, so we can t yet see much variation in temperature of the photons reaching us. Somewhat ater (midde part of Fig. 6, the ight reaching us from that time is from further away, so we start to see some nodes.

9 7 At a ater time, we woud see more nodes, as in the right hand part of Fig. 5. As time goes on, we woud see more and more nodes (higher modes appearing on the spherica she of the ast scattering surface (LSS. Note that if the midde figure were just a bit earier, with the circe just touching the dark ines (noda ines, we woud see a quadrupoe ( coder regions and hotter regions in 360 o. Higher and higher n-poes appear as time goes on. At the present time, a fundamenta standing wave from recombination of the idea wave shown in Fig. 6 (which is not the case in nature woud appear to us to have hundreds of nodes, spaced a itte ess than one degree apart. 3. The Rea, Not Idea, Case Of course, we don t have the idea situation as in Fig. 6, where a singe wave form repeats itsef perfecty, over and over, across the universe. What we actuay have are a whoe sew of sma waves of various engths between humps (see Fig. 3 a over the pace, depicted schematicay in Fig. 7. Ony waves on ast scattering surface shown. Others spread everywhere. Symboized as Figure 7. Some Loca Gaussian Shaped Temperature Standing Waves on Last Scattering Surface (Ony ower modes shown For a oca fundamenta wave (one hump, n = the space between nodes at our present time woud be a itte ess than one degree. 3.3 Converting Pane Waves to Spherica Coordinates Because a pane wave, such as those of Figs. 5 and 6, appears to us as rippes on a spherica surface, our preferred method of anayzing data from those rippes is with a spherica coordinate system. To do that, one needs to convert a pane wave expressed in Cartesian coordinates (which is the easiest way to express a pane wave into spherica coordinates. This is done via a mathematica reation known as Rayeigh s pane wave expansion, which we wi present after discussing spherica harmonics in Sect Spherica Harmonic Anaysis 4. Background Our data is a map of the temperature variations of the cosmic microwave background (CMB radiation in the sky, and thus is taken effectivey over a spherica she. So, if we want to break down the temperature signa we see via a moda anaysis, it is best to use the spherica anaog of a Fourier spectra anaysis, i.e., spherica

10 8 harmonic spectra anaysis. The centra concept behind spherica harmonic anaysis is expained briefy in the foowing paragraphs, with a more in depth treatment in Appendix A. A spherica harmonic series paraes the Fourier series, but we use spherica coordinates θ and φ. rather than Cartesian ones. or in fat D, ( θ φ Y, and a are generay compex, though they f ( θ, φ = ay ( θ, φ ( = 0 m = can resut in a rea function f ( θ, φ, as in our case. inkx f ( x = An ( cos nkx + i sin nkx = An e n= 0 n= 0 simiar to nxny x x y y nxny nx = 0 ny = 0 nx = 0 ny = 0 anaogous anaogous to Y to a ( θ, φ in in xkx y f x, y = A cos n kx + i sin n kx cos n ky + i sin n ky = A e e ky (3 Unfortunatey, the spherica harmonics Y ( θ, φ are not of as simpe a form as the sines and cosines of the inear dimension case. (See Appendix A for an extensive deveopment of spherica harmonics. However, just as the Fourier components (the harmonics, one for each n vaue in D are orthogona over a given spatia Y θ, φ are mutuay orthogona on the surface of a sphere (over the intervas interva, the spherica harmonics 0 < θ < π and 0 < φ < π. ( is what is used in quantum mechanics for the hydrogen atom soution to the Schroedinger equation in spherica coordinates, so you have probaby seen a reation ike this before (with a compex wave function ψ, rather than a rea f... Each a represents the ampitude of a particuar spherica harmonic component in the summation. In the hydrogen atom we were aso concerned about the r direction and had a separate soution which was ony a function of r (the Shroedinger equation was separabe into a purey r dependent part and a part dependent on θ and φ. Here we ignore the r dependence since we are ony concerned with describing a function [the temperature variations] of anguar ocation, i.e., effectivey on the surface of a sphere. The "" index, an integer, is associated with the number of spatia osciations (nodes, to be precise, equa to + in the θ direction in spherica coordinates, and "m", the number of spatia osciations (nodes in the φ direction. m, aso an integer, varies from " " to "" (reca the hydrogen atom. For exampe, the. =. 0 mode has zero osciation in the θ direction, i.e., a monopoe, or constant vaue over the whoe sphere, that sets the overa scae. = is a dipoe: one fu osciation over the sphere in the θ direction. Go to higher and higher vaues, and you get more spatia osciations (and therefore smaer waveength over the sphere surface. The and m integers of spherica coordinate anaysis are anaogous to the n x and n y integers of (3 for Cartesian coordinate anaysis. The a are anaogous to the Fourier ampitudes A nxn. y 4. Finding the a Note that since the Y are orthonorma and compete over a sphere at fixed radius r (see Appendix A, Sect..4 pg. 30 the a can be determined (Sect..5. pg.3, eq. (8 as 4.3 CMB Spherica Harmonics π π * = ( θ φ ( θ φ sinθ θ φ. (4 a Y, f, d d 0 0 Note with the CMB that we are deaing with a scaar function over the sphere surface, i.e., the temperature T. For us, ( θ φ ( θ φ f, = T, (5

11 9 The = 0 component (zero spatia osciation spherica harmonic in ( is Y 00 =, so a 00 woud equa the mean temperature T =.75 o K, which is the constant background over the entire sky. The variation from the mean, T, as a function of θ and φ, are anayzed in terms of its spherica harmonic components Y ( θ, φ for > 0. Because T is a scaar, there are no issues of what direction our harmonic spatia waves are osciating in. That is, we don t have to worry about ongitudina vs transverse directions for a scaar. Our temperature waves are scaar waves, so variations are merey numbers. 4.4 Expressing Pane Standing Waves in Spherica Coordinates 4.4. A Simpe Compex Sinusoid The reation auded to in Sect. 3. for expressing a pane standing wave in spherica coordinates is known as Rayeigh s pane wave expansion. (See We state it here without proof as (6. For a singe (compex sinusoida wave of unit ampitude with wave vector k (having wave number k pointing aong unit vector n k, where r is the vector from the origin to the point in question (having ength r pointing aong unit vector n, where θ k, φ k and θ, φ are the spherica coordinate anges for n k and n, respectivey, and where j (k. r are spherica Besse functions, ik r ikxx iky y ikzz * e e e e 4π i j ( kr Y ( θ, φ Y ( θ, φ = = k k coordinate Cartesian = 0 m = independent coordinates spherica coordinates = Rayeigh expansion form ( Arbitrary Pane Wave Shapes Any wave shape other than a sinusoid, such as a sum of oca Gaussian waves over the whoe sky (Fig. 7. pg. 7, can be expressed, in the spirit of Fourier harmonic anaysis, in Cartesian coordinates (bounded by a box L x by L y by L z in the discrete soution case as ( r = f f x, y,z arbitrary wave form Cartesian coordinates for discrete case for continuous case ( r f coud be temperature pattern at a r in a of space at the present time of photons scattered at recombination, i.e., a sum of a Gaussian oca temperature variations π π π inxk xx inyk y y inzk zz nxnynz x y z Lx Ly Lz = A e e e k = k = k = nx = ny = nz = compex sinusoids, Cartesian coordinates ik 3 xx iky y ikz z ik 3 = A e e e d k = A e r d k. 3 k 3 k ( π ( π The same wave shape can be expressed, in the spirit of spherica harmonic anaysis, as (7 f ( r, θ, φ a j ( kr Y ( θ, φ = same wave form, spher coords = 0 m = (8 By using (6 in (7 and equating the resut to the RHS of (8, one can obtain reationships between the a coefficients of spherica harmonic anaysis and the A nxnyn (or A(k of Cartesian harmonic anaysis. We wi z keep things simpe at this point and hod off unti Sect. 7.4 to do this. For now, we simpy note the impications of doing so. Bottom ine: If we have a theory deveoped for standing wave formation in Fourier components in Cartesian coordinates, we can deduce the a we woud expect to measure for that formation in a spherica coordinate

12 0 system. Conversey, if we have data, i.e., the a, taken in a spherica coordinate system, we can determine the corresponding Fourier components A nxnyn (or A(k for theoretica anaysis. This is done in anayzing the z CMB. 5 The CMB and the Ampitude a of Each Spherica Harmonic Y 5. Measuring for the CMB The spherica harmonics Y, individuay, are not pure sines or cosines, though they are osciating spatia wave forms. (See Appendix A, pg. 9, eq. (65 for the θ dependence of severa of the owest modes. Showing them graphicay in what foows woud not be easy, so we won t do that. However, the a ampitudes of ( and (4 are anaogous to ampitudes of sine waves. 5. Measuring the CMB Ampitude and Variance 5.. Measurement in Our Universe Reca the CMB radiation dispays the acoustic standing wave patterns that existed right at recombination. In other words, we have a cosmic snapshot at one time (recombination, and for any given mode (such as the fundamenta with n =, a ocaized pane waves everywhere woud be in phase. (See Fig. 5, pg. 5. But the heights of the individua standing waves of the same mode woud vary from individua wave to individua wave. As shown in Sect. 4.4., pg. 9, the sum of many pane waves of many modes at many different ocations can be converted to spherica harmonic representation with a given ampitude coefficient a associated with each Y. Thus, the particuar a we measure woud be those at the time of recombination in our universe. In other universes with the same aws but different initia conditions (different set of random density fuctuations, at their recombination times, we woud have different a. 5.. Measurement Over an Ensembe of Universes Consider having an ensembe of universes we coud coect data from. Each is structured in the same manner as ours, with the same physicas aws, partices, and parameters (same curvature, baryon density, dark matter, dark energy, photon density, etc. However, each has different initia conditions, i.e., random QM fuctuations at the onset of infation, and these occur randomy throughout members of the ensembe. If we did have a ot of universes simiar to ours and coud make many measurements (one in each such universe, a given a woud vary in vaue between them. It woud be positive in approximatey haf of them and negative in haf, with the average a over a universes tending to zero as the number of universes got arge. Since our mean vaue for a ( > 0 woud tend to zero, it is not of great use to us as a characterizing parameter of the CMB. The variance (or, equivaenty its square root, the standard deviation of a ( > 0, however, is reated to the squares of the a and thus must be a positive number, and not tend to zero. It woud give us a measure of how great a contribution a particuar spherica mode (given and m makes to the physics of universes ike ours. If we coud somehow get an experimenta vaue for that variance of a over an ensembe of universes, we woud, as the saying goes, be in business. There actuay is a trick that does something cose to this, and we discuss it in the next section.

13 5..3 A Trick to Get Many Measurements from Our One Universe We don t have any extra universes simiar to ours we can access to make extra measurements of a in each, so what do we do? If we consider that for given, each m is equay ikey to have the same measured a vaue, then measuring those m modes effectivey gives us what we woud get from extra measurements in different universes. If we average a ( > 0 over a m for a given, we wi tend to get zero, just as we noted in Sect. 5.. for an average over different universes. So, mean vaue for a ( > 0 over a m isn t of much use to us. However, if we average the square of a over a m for a given we wi tend to get a non-zero vaue, such as we noted in Sect. 5.. for different measurements in different universes. Since < m <, we have + vaues of m to use. For fairy arge, this essentiay means good statistica vaues approaching the true vaues to high confidence. Thus, we want to compute, for each vaue, the variance, which is abeed C, as + * * = = = = m= a a a a a a C no sum on + (9 where the average symbo is considered to be for a m vaues (given a singe vaue in our universe now. Caution that in the iterature, C is sometimes taken as the average over a m vaues as above, and sometimes as just the vaue for a singe m, as beow. Sometimes one sees * * m = m δ δm m = a a a a C where the average symbo is considered to be for a singe m vaue (and singe vaue over many universes. (0 The connection between the two versions ies in the presumption that (0 for given, over a arge number of universes, woud be the same for any m (each m mode woud tend to have the same power. Thus, the average over a m, for given, as in (9, woud tend to equa (0. (0 is theoretica (we can never measure other universes, but (9 is something we can measure experimentay with the CMB data. We can, in fact, get + separate measurements, impying, for arge, considerabe confidence the computed vaue (9 is an accurate refection of what is reay going on (and not a statistica aberration. Since power of any wave is proportiona to the square of the ampitude of that wave, the variance can be thought of as a measurement of the power of the mode (incuding a m vaues for that for (9. Potting C vs is caed a power spectra anaysis. (See Appendix B for an introduction to [review of] the theory of power spectra anaysis. Subject to some further considerations to be discussed beow, for the CMB, this gives us the power spectra anaysis of Fig. that has become so famous. Note, in passing, that a system dispaying the same statistics over many measurements in time or space as it does over many measurements from different simiar systems in an ensembe is caed ergodic. Here we are averaging over different measurements in space (spherica coordinate space with the presumption that gives us the same statistica resuts as measuring over different universes. So our system is an ergodic one. 5.3 The Key Reationship Note that if we use (4 with f.. (θ,φ = T(θ,φ in (9, we get

14 + * * = = m = m no sum on + m= + π π π π C a a a a a * = Y ( θ, φ T ( θ, φ sinθ dθ dφ Y ( θ, φ T ( θ, φ sinθ dθ dφ m= π π π π + * = Y θ, φ Y θ, φ T + ( θ, φ T ( θ, φ sinθ dθ dφ sinθ dθ d φ. m= dω dω Ω Ω To streamine notation, we can use unit vector notation, where n is the unit vector pointing in the θ, φ direction and n is the unit vector pointing in the θ., φ. direction. Thus, ( is re-written as ( + * C = Y ( θ, φ Y ( θ, φ T ( θ, φ T ( θ, φ dω dω + Ω Ω m= + * = Y ( Y T ( T dω d Ω. + Ω Ω n n n n m=. ( 5.4 Finding C Experimentay We have a data set of discrete points of the CMB where each point has a T vaue and an associated θ, φ (or equivaenty an n. That is, we have T(θ,φ = T(n in digita format. We can process this data via computer in either of two equivaent ways. The second is simper and is preferred. First way Step. Use f.. (θ,φ = T(θ,φ = T(n in (4 to determine a. Step. Use a in (9. Second way An equivaent, and one step way, is simpy to use ( (which is simpified beow in ( Things to Know about C 5.5. Nomencature and Units C is the variance of the spherica harmonic ampitudes a for fixed. It is aso known as the component of the power spectrum of the CMB (square of wave ampitude is proportiona to power in a wave. Further, in its dependence on, it is known as a correation function, for reasons discussed in Sect If f.. (θ,φ = T(θ,φ in (4 and the spherica harmonics Y are unitess (which they are, then the a have units of temperature. Hence, C, via (9, has units of (temp C as a Correation Function Consider (, where for two different ocations on the sphere of the CMB (directions we see in space n and n, the vaues of T are particuary high, i.e., oca maxima. There exists a certain Y at a given which aso has peaks (oca maxima at these ocations. Thus, in the integra of ( this particuar contribution, from the T vaues at the two points, and the Y vaues at those points, wi make a arger contribution to the vaue of C than if the same pair of points did not dispay oca maxima. Thus we wi get arger vaues of C for a given if there are peaks in the sky correated with the peaks we find in Y for that. Thus, we say C represents a correation between points a certain distance apart in the

15 3 CMB. Actuay it is a measure of the correation between a points integrated over the whoe sphere for given. The more such points peak at the between peaks distance of the spherica harmonic, the greater C. Thus, as a function of, C is a correation function Independence of C from Choice of Coordinate Reference Frame One might question, from (, if we woud get different C for different choices for our coordinate axes. (See Fig. A-, pg. 7. For exampe, for a different aignment of our z axis, the vaues for θ and φ woud change for the two different points we integrate over, so one might expect the tota integra to be different. However, the C actuay are independent of choice of coordinate axes, as shown beow. Showing C independence from coordinate axes chosen We wi need something caed the addition theorem for spherica harmonics, stated without proof beow in (3. Note that P is the Legendre poynomia of order. (See Appendix A Sect.., pg P 4π 4 Y, Y, π + m Y Y = + m= cosθn n * * ( n n = ( θ φ ( θ φ = ( n ( n θ n n is the ange between n (aigned with θ.., φ.. and n (aigned with θ., φ... Using (3 in (, we have C = P T T dω dω 4 Ω Ω (3 ( n n ( n ( n. (4 π Reation (4 ony depends on the ange between two points in the sky and the temperature at each of those points. Hence, it is independent of the particuar coordinate axes aignment chosen. Aso, (4 seems simper to evauate than (, so my guess (not being a practitioner in the fied woud be that it is the preferred way to anayze the data. That is, use (4 instead of ( in Sect An Exampe: The Dipoe ( = One can best foow the foowing exampe by first becoming famiiar with Appendix A, Sect..5.5 pg. 35. Fig. A-3 there shows one of the simpest spherica harmonics, the one for =, i.e., the dipoe. In the figure, three different dipoe aignments aong with their particuar spherica harmonic coefficients a (for different m are shown in the same coordinate system. However, an equivaent way to ook at the figure is to consider the dipoe fixed reative to the sky, but with different coordinate systems in each case. The a for each coordinate system (for the same dipoe are different (and are shown in the referenced section. However, in the sum of (9 over m, we shoud get the same C in each case. The respective a contributions for each coordinate system in Fig. A-3 are shown summed together in the foowing chart. 3 Comparison Chart for Fig. A-3, pg. 36 ( = /( + = /3 in (9 Fig. A-3 a Fig. A-3 b Fig. A-3 c ( 0 a + a + a = C X X = ( 0 a + a + a = C X X X = ( 0 a + a + a = C X X X = X The concusion: The C for the same pattern in the sky is the same as found in any coordinate system.

16 Loca Variation Impications for C as a Correation Function We coud consider either ( or (4 and what happens for oca maxima or minima temperature variations, though for simpicity, we wi focus on (4. Fig. 8 depicts two oca maxima schematicay, projecting 3D to D for simpicity. temp represented by radia distance z θ higher temp sice at constant φ n' n mean (constant temperature over θ = dashed circe Figure 8. Two (Hypothetica Loca Maxima in CMB In Fig. 8, we have two oca soid ange regions where the temperature is higher than surrounding regions. That is, over a region of the integration of (4 (or ( Ω T T n and n Ω both arge for n and n aigned as in figure. (5 There is some Legendre poynomia P., for some, that peaks reguary at the anguar distance between n and n. Thus, in the region of higher temperatures, we woud get a greater contribution to the integra (4 from P. than from other P.. ( Ω Ω high for regions where and at temp peaks P n n T n T n n n. (6 For other regions, P. may or may not peak, but the variation in T does not, so the other regions woud not contribute so much to the integra (4 for =.. ( Ω Ω ow for regions where and not at temp peaks P n n T n T n n n (7 For other, and thus other P., the peaks in the P. woud not correspond to the peak temperature regions in Fig. 8, so we woud not get as great a contribution. ( ( Ω Ω ow for even in regions where and at temp peaks P n n T n T n n n (8 Note aso that for., if there were more regions of oca maxima in other parts of the sky separated by the same ange as for those maxima in Fig. 8, we woud have more contributions ike that of (6 to (4 and a higher vaue for C.. Bottom ine: For given, C is greater if there are more oca extrema in the sky with anguar separation of that dispayed by the extrema in P. Thus, C provides a measure of the correation between number and eve of variations in CMB temperature the sky for a particuar vaue of. Since is inversey reated to anguar separation (ange = 80 o /., C is aso represents a correation between oca extrema and their prevaence at a given anguar separation.

17 5 5.6 The Variance of the Variance C Note that C is measured ony for our universe, so it wi have some error in it even for arge, but finite, (where we have many m to sampe and sum over in (9 or (. Of course, we expect the error to be ess, the arger the, since there are more sampes taken (more m. But there is some error nonetheess. If we had a neary infinite number of universes in an ensembe, we coud make a neary infinite number of measurements, and pin C down precisey. We are imited, instead, to our + measurements for given in the one universe we do have. We express our unknown error in C as a confidence eve, based on the usua reation for experimenta standard deviation (9. (It is aso common to use N in the denominator of (9, but cosmoogists seem to use N. i= σ x Standard Deviation of x = = N ( x i x N N measurements. (9 For us, x = C, our average over a m vaues ( + in number, as in (9. We have N = measurements. So (9 becomes Standard Deviation of a = Define as C + m= ( a C + no sum on. (0 So, + a C m C = = C + no sum on For some reason I have not been abe to uncover, cosmoogists assume the summation in the numerator of ( is approximatey. Thus, C Cosmic variance ( no sum on standard deviation C C. ( C + + Strangey, though the LHS of ( represents a standard deviation (divided by C, it is caed the cosmic variance (one more source for confusion on this issue, as variance is typicay the square of the standard deviation. For one standard deviation at given (= C, our confidence eve is 68%; for two, 95%. These are the error bars one sees in a typica pot of the C vs. Note they are much wider for ow, than for higher vaues. Keep in mind the difference between the variance of the a, which is C, and the cosmic variance C which is the standard deviation of C (divided by C. C is a variance (not reay of a variance (C. 6 Converting Between Cartesian Fourier and Spherica Harmonics Anayses 6. Express a in Terms of Fourier Ampitude A(k As foreshadowed in Sect. 3.3, we wi now express the spherica harmonic coefficients a for a given CMB temperature distribution in terms of the Fourier component coefficients for the same distribution. We do this for the genera case where a can be at any radius r from the origin, not just at the radius of the LSS. (

18 6 We start by repeating (4 for any sphere of radius r, with some aternate notation empoying the unit vector n from the origin to the unit radius sphere, and with the arbitrary function f repaced with CMB temperature T. π π 0 0 at radius r * * ( θ φ ( θ φ sinθ θ φ ( n ( r r n a = Y, T, d d = Y T dω = r. We then use the owest row of (7, repeated beow, which describes the CMB temperature everywhere (not just on the spherica surface of the LLS. ( π genera Cartesian form form ikxx iky y ikz z 3 ik r 3 T ( r = A e e e d k A e d k 3 k = 3 k. (7 Now use the Rayeigh pane wave expansion (6 k r ( π * * m k k m m ( k m k = 0 m = = 0 m = i e = 4π i j kr Y θ, φ Y θ, φ = 4π i j kr Y n Y n k = kn (6 to express (7 in terms of spherica harmonics/coordinates. * 3 T ( r = T ( r n = 4 3 A k π i j kr Y m Y m d k ( π nk n. (3 = 0 m = Now put (3 into (4, where we use (78 of Appendix A in the second ine, (4 * * 3 a = Y A 4π i j 3 kr Y m Y m d k n dω k ( π nk n = 0 m = ( 4π * * 3 = 3 A i j kr Y m Y Y m dω d k ( π k nk n n = 0 m = δ δ mm (4 = 4π ( π 3 * 3 where a here is at radius r from the origin, A( k i j ( kr Y ( nk d k, A k is Fourier coefficient for k wave. * 6. Express C = a a in Terms of Fourier Ampitudes A(k m Now we wi express the variance C in terms of A(k instead of T(n (as we did in ( and (4, and do so for any radius r. First, we start with (9 or (0 (each uses a different way to take the mean, repeated beow, * m and insert the ast row of (4 into it twice. C = a = a a no sum on, (9 or (0

19 * 4π * 3 4π * 3 C = a m a = A ( ( i j 3 k r Y m d k A i j 3 kr Y d k k nk ( π k nk ( π ( 4π ( π 3 3 = i i A A j k r j kr Y Y d k d k 6 7 * * ( k ( k m ( nk ( nk * * 3 3 = ( i i 4 A k A k j k r j kr Y m nk Y nk d k d k. 4π Second, we want to eiminate the primed quantities and simpify. To do this, we note in the ast ine of (5 that * A A 0 for k k k k, (6 since A(k can be positive or negative, and in sums over many universes (or many times or many m, its vaue and its negative wi at times pair with the same A * (k and those two terms wi cance. Over arge sums, enough pairings arise to give a tota vaue approaching zero as the number of universes (number of times, number of m vaues approaches infinity. However, * A k A k some factor A k 0 for k = k, (7 where A k is a variance in k space, simiar in some ways to C in spherica harmonic space. This is some justification for the foowing, which is proven in Abramo and Pereira, With (8, the ast ine of (5 turns into 3 ( = ( ( * A k A k A k π δ k k. (8 * 3 = ( m ( k ( k C i i A k j kr j kr Y n Y n d k. (9 π Consider that things are isotropic in that waves varying in direction about the origin, but aways having the same k = k, wi have the same A( k (5. And our integra in (9 is effectivey an integra in k space of a continuum of spherica surfaces, where sub-integration is carried out over each surface and then those surface integras are integrated in the radia direction. For each of these spherica surfaces, we can take A = A( k k and use the same orthogonaity reation in k space as we used in (4 in r space, to get C i i A k j kr j kr Y Y d k * 3 = ( m π nk nk k dωdk * = ( i i A( k j ( kr j ( kr Y m Y d k dk π Ω nk nk Ω δ δ mm (30 Abramo, L. Rau, and Pereira, Thiago S., Testing gaussiantity, homogeneity and isotropy with the cosmic microwave background, Advances in Astronomy, Vo. 00, Artice ID 37803, 5 pages. See ( and the ine after (7.

20 Or 8 C = i i A k j kr k dk i A( k j ( kr k dk π = π. (3 = π ( A k j kr k dk. = and thus, (3 is the reationship we sought between C and A(k. Actuay, it is a reationship between the variance in our spherica harmonic system and the variance in the Cartesian Fourier system. We wi have a itte more to say about this in Sect Potting C vs 7. Anguar Variation As shown in Appendix A, Sect..5.7, pg. 38, the anguar separation between variations, such as those shown in Fig. 8, for a given harmonic mode., is anguar variation o 80 = = π in radians (3 7. Peaks in Power Spectrum It turns out that the fundamenta standing wave (see Figs. 3 and 4, pgs. 3 and 3 of ength L, via the mechanism depicted in Fig. 4, pg. 6, becomes a spherica harmonic with nodes separated by a itte under o. In other words, the C for the CMB (see Fig., pg. peaks at about = 00 or so. We wi discuss the meaning of this and the other peaks ater on. 7.3 Background in Power Spectra Anaysis Needed for Subsequent Materia In Appendix B, pg. 38, we present a simpified, introductory overview of power spectra and how they are derived from the wave ampitude (and wave variance. This information wi be needed as a prerequisite for the foowing section. Some readers may aready know that materia. Some may not, or may be a bit rusty on it. 7.4 Why CMB Pots Have ( + C/π on the Vertica Axis Rather than C The reason we see ( + C /π on the vertica axis of typica CMB power spectrum pots such as Fig., pg., has to do with our conversion from Fourier anayses in Cartesian coordinates to spherica harmonic anaysis in spherica coordinates (Sect.6., pg.6 pus our desire to work with spectra power (rather than variance pots having og scaing on the axis. The theory of infation and subsequent expansion posit an effectivey uniform (subject to subteties distribution of initia fuctuations in power across a waveengths k, i.e., scae invariance. On average, no fuctuation waveength has an intrinsic, enhanced power over any others. The dynamics of acoustic wave formation change this, of course, but initiay, before that got started, the whoe thing was egaitarian. Representing this initiay democratic situation is rather straightforward in Cartesian space, where each Fourier component of wave vector k woud be independent of a others. But this becomes ess straightforward for spherica harmonics where each different spherica harmonic ampitude a is a funky combination of the Fourier ampitude A(k, as shown in (4. As we wi see, this funkiness of spherica harmonics can be modified in an advantageous way by mutipying each C by ( + /π. For the case of a universe with no acoustic wave formation, this shoud give

21 9 us a curve in Fig. of a straight horizonta ine, subject to some subteties beyond the scope of this discussion. So any changes via acoustic wave formation seen in our universe woud show up in the figure as deviations from the horizonta ine and jump out at us. They woud not be submerged in other, unreated variations with due to the funkiness of spherica harmonics Reate C to the Temperature Power Spectrum in k From Appendix B, pg. 38, reation (4 the Fourier temperature power spectrum P T (k is, with a conventiona choice of constant to turn the proportiona sign to an equa sign, k PT k A k π =. (33 As aso shown in Appendix B, reation (5, with symbos changed to k space beow, shows how, given P T (k, we can determine the power over a given interva of a power spectrum, where P T (k is the power per unit k. k k power between k and k = P k dk, Soving (33 for A k and inserting into (3 gives us T C = 4π P k j kr dk, (34 T which reates the Fourier power spectrum P T (k, i.e., the power per unit k, in terms of C. It is often desirabe (and it is in CMB anaysis to work with the power per unit ogk, instead of the power per unit k. If we pot such a quantity and have a og scae for k on the horizonta axis, one can eyeba the power in an interva as simpy the area one sees under the curve inside that interva. To do this, we need to find an integrand for a reation parae to (34 that is integrated with respect to d(ogk. To find this integrand, reconfigure (34 as dk C = 4π P k j kr k = 4 j kr k P k d ogk π k d( ogk T ( k T T, (35 where we introduce the new symbo ( k. For this way of writing C, ( k = k P ( k is power per unit ogk. T 7.4. Evauate the Integra for C with the Spherica Besse Function in It From integra tabes, we find From (36, therefore, ( + ( og z z d = j. (36 ( + d ( kr ( og rdk = j kr d kr = j kr = j ( kr kr kr d k = j ( kr = j ( kr d ( og k. k = ( + It is aso known that a spherica Besse function of order peaks when its argument is approximatey given by, i.e., j ( kr peaks at about kr =. Further, this peak is very narrow with j effectivey zero esewhere. Hence, T T (37

22 0 over the integra of (35, ony a narrow range of k contributes (around k =. /. r, and over this range T ( k approximatey constant (and equa to ( =. Thus (35 becomes effectivey With the ast part of (37, we have T k k / r π T C 4 k = / r j kr d ogk. (38 ( Back to the Power Spectrum Pot T ( k ony region near k= / r contributes π C T k = / r. (39 is is the power spectrum per unit og k, for Fourier components in Cartesian coordinates. It is the number we reay want to compare to our theory, which is grounded in Fourier anaysis in Cartesian coordinates. So a og pot (or any other pot of C (a spherica harmonics based number vs distorts our perspective on what the rea physics is doing (which we grasp more readiy in Cartesian coordinate form. Thus, if we mutipy (39 by. ( + /π, we get (approximatey ( k fee more at home with. ( + π T T, our Fourier anaysis friend that we C k = / r the number typicay potted on CMB vertica axis vs. (40 8 What the CMB Power Spectrum Tes Us 8. The First Peak The first peak in Fig., pg., is at about 00 (a itte ess than o, which is reated to the width L of a fundamenta mode. (See Fig. 5, pg. 5. The ength L is the distance sound traves from just after the big bang to recombination. Cosmoogists can cacuate what that ength shoud be. Knowing the distance to the LSS, they can then cacuate what the anguar separation (degrees, or equivaenty the spherica mode number they woud see. Note, from Fig. 9, that this anguar measurement varies depending upon whether the universe is curved or not. α α' L L Fat Surface Positive Curvature Surface Figure 9. Measuring Anges Seen to Ends of a Length L A viewer ooking at the ends of a known ength at a given distance away in a positivey curved space measures a arger ange between the ends than a viewer of the same ength and given distance away in a fat space. In Fig. 9, α < α. A negativey curved space woud have a smaer ange than a fat space.

23 For the CMB data, the anguar measurement of L, the width of our fundamenta mode, shows up as the ocation of the first peak, i.e., the, vaue or equivaenty the ange associated with that vaue via (3. If the cacuated vaue of that ange ( vaue for a fat universe matches the CMB data ocation, we have a fat universe. If the measured anguar vaue is arger (peak moves to the eft, to ower, we ive in a positivey curved universe; if smaer (peak moves to the right, to higher, in a negativey curved one. The data match for the peak ocation is, to fairy high accuracy, dead on for a fat universe. Note that subteties and uncertainties exist in the cacuation and the measurement, and other parameters affect peak ocation to some degree as we, but higher peaks dispay the same dependence on curvature, so they can be used to check and refine the concusions from the first peak. And for curvature, they check pretty we. 8. The Second Peak From Fig., we can see that the second acoustic peak is much ower than the first. To understand this, we need first to recognize, from Fig. 5, pg. 5, and reated discussion, that the odd numbered peaks (first, third, etc in the CMB power spectrum (n =, 5, 9, in Fig. 5 are, overa, compressive in nature. Even number peaks (second, fourth, etc. in the spectrum (n = 3, 7, in Fig. 5 are, overa, rarefied in nature. The osciations resut from massive baryons in more dense regions attracting more baryons with the resuting increase in baryons heating up the baryons and the photons they are in contact with. The photon pressure acts ike spring to repe the baryons, so they spread and become ess dense. After a time of expansion/rarefication the photon pressure decreases and gravity begins to pu baryons back coser together again. The whoe process repeats eading to an osciation that is anaogous to a spring mass system. (See Fig. 0. The mass in that system corresponds to baryon mass, the spring to photon pressure. same photon pressure max rarefaction max rarefaction even peaks Gravity force Initia cond Resuting osciation Less Mass (fewer baryons max compression Osciation range Initia cond Resuting osciation Greater Mass (more baryons max compression odd peaks Figure 0. Anaogy of Spring Mass Osciation to Baryon Acoustic Wave Osciation Consider a given rarefied region, represented by the initia condition ocation in Fig. 0. With that same rarefied region, if there are more baryons in the adjacent denser region, the attraction wi be greater, resuting in greater compression. A spring-mass system, ike our BOAs, rebounds back to its initia condition (rarefied for BAOs regardess of the extension (compressed fro BAOs. But the extension (compression is greater for greater mass. We know the odd numbered peaks represent overa compression; the even ones, overa rarefication. If we have greater mass (more baryons the compression eve wi be greater, so the odd numbered peaks wi be more pronounced (higher in our spectra density pot, as in Fig.. Thus, increasing baryon number in our universe wi increase the height of the odd numbered peaks reative to the even numbered peaks. We see this effect in Fig., where there is a genera trend for peak height to reduce with increasing. The first peak (compressive is much higher than the second (rarefied, and the third peak is about as high as the second.

24 The bottom ine: The reative heights of the odd and even numbered peaks refect the baryon density. Cosmoogists can, of course, quantify this to give us a precise vaue. 8.3 Higher Numbered Peaks and Parameter Determination The heights, ocations ( vaues, and shapes of the peaks in the CMB spectra density te us a whoe ot about the various parameters of our universe, such as curvature, baryon density, dark matter density, dark energy density, etc. In genera, a given parameter can affect more than one peak. For exampe, curvature affects the ocation of a peaks. Conversey, characteristics of a given peak can be affected by more than one parameter. For exampe, different baryon density means a different osciation frequency (sower for more mass for a standing waves. This wi change the vaue (sighty, as it turns out of for each peak, incuding the first. So, what we said before about the first peak determining curvature has to be modified a bit. We aso have to take into account the shift in vaue due to baryon density. Again, cosmoogists do this as part of the anaysis of the CMB spectrum. The above is an exampe of degeneracy in cosmoogica parameters, where we need additiona information from other parts of the CMB spectrum to pin down a particuar parameter. Note that cosmoogists use computer programs (see Sect. 0 to generate mode CMB spectra. One inputs various parameters (dark energy percentage, baryon density, etc. and the program spits out a power spectrum. By varying the parameters one gets a fit to the data and pins down which parameters yied that fit. See Fig. for a graphic of what the CMB spectrum woud ook ike for various vaues of certain parameters. Figure. Pots of the CMB Spectrum for Different Vaues of Key Parameters From webpage of Professors Hu and Dodeson

25 3 From graphs such as those of Fig., cosmoogists have determined our universe is fat to high accuracy and has vaues for dark energy of about 69%, dark matter of about 6%, and ordinary (baryonic matter of about 5%. The resuting mode is caed the concordance mode, or the ΛCDM mode. (for ambda cod dark matter, where ambda represents dark energy in cosmoogica constant form. 8.4 The Damping Tai Note in Fig. that the curve is damped at high. The reason for this is that recombination did not happen instanty, but took some time, some thousands of years, to transpire. In other anguage, the LSS is not infinitesima in thickness. So, the photons wi scatter a few times during this transition period, rather than be reeased unscathed in one grand, feeting moment to forever hod the patterns impressed on them by the BAOs. If the average distance a photon traves is arger than the waveength of the BAO at a given, then the scattering wi tend to smooth out the temperature contrasts at that vaue, and thus any peaks at that wi be dampened. So, at shorter BAO waveengths (higher power spectrum vaues wi drop dramaticay. And that is what we see in Fig.. Other factors such as the baryon density (higher means more scattering of photons affect the degree of damping. So precise determination of the degree of damping heps in pinning down key parameters. 9 Summary We have expored the physica basis of baryonic acoustic osciations in the eary universe, and from that, deveoped the appropriate mathematics, at a reativey eementary eve, that ead to the CMB power spectrum. We then showed, again at an eementary eve, how that spectrum can be used to determine key cosmoogica parameters. The presentation has been greater in depth than that of typica CMB popuarization websites, yet markedy simper and more rudimentary than that of typica cosmoogica textbooks. It has been intended to serve those with undergraduate, or better, backgrounds in physics, who are not (yet, or ever wi be speciaists in the fied. 0 Further Reading Professor Whu at the University of Chicago does an exceent job of describing much of the materia herein, with ots of graphics, on his website See aso, Max Tegmark's page The math on those pages is not as extensive as that found herein, however. For a next step, after this document, I recommend Testing gaussianity, homogeneity and isotropy with the cosmic microwave background by L. Rau Abarmo and Thiago S. Pereira ( The math in that document is more advanced and more tersey presented than that herein. Textbooks on the subject incude Modern Cosmoogy by Scott Dodeson (Academic Press, 003, Physica Foundations of Cosmoogy by Viatchesav Mukhanov (Cambridge 005, The Cosmic Microwave Background by Ruth Durrer (Cambridge, 008 Primordia Cosmoogy by Patrick Peter and Jean-Phiippe Uzan (Oxford, 009. These books are quite advanced compared to the present document and represent entire graduate courses on the CMB. The present document is a good preiminary introduction to those texts. To generate CMB power spectra pots yoursef, with different parameters (for dark energy, dark matter, baryon density, curvature, etc go to NASA s

26 4 Appendix A. Summary of Spherica Harmonics A good reference for this materia is Cahi. What foows is ony a summary and be usefu primariy for those who have studied this before, but are a itte rusty. To get to spherica harmonics, we have to start with Legendre poynomias.. Legendre Poynomias Introduction We start by noting that any function f. (x can be expressed as a Tayor power series expansion, s= 0 ( s ( 0 ( s s f th f ( x = csx cs = f is s derivative of f. (4 s! Legendre poynomias are of simiar form, n n i n 0 n i i= 0 P x = a + a x + a x a x = a x, (4 but are cut off after n terms and have the foowing normaization (which may seem strange at first but heps in the ong run 0 P x = = P = a + a + a a =, (43 n n n and orthogonaity condition (note the interva of integration < x <, which again is chosen because it eventuay heps, m Pn ( x x dx = 0 m < n. (44 (44 says that each Legendre poynomia of order n is orthogona (over the interva shown to any monomia x m, provided m < n. Thus, the first few Legendre poynomias (you can check if they meet the criteria of (43 and (44 are 3 = = ( = = ( = ( 3 = ( P x P x x 0 3 P x x P x x x 4 P x x P5 x x x x 8 In a pot, they ook ike Fig. A-.. (45 Cahi, Kevin, Physica Mathematics, (Cambridge 03, pgs

27 5 Figure A-. Pots of the First Six Legendre Poynomias There are shortcuts to finding Legendre poynomias caed the Rodrigues formua ((8.8 pg. 306 of Cahi and a recurrence reation ((8.38 of Cahi. The Rodrigues formua is P ( x n ( x n d =, (46 n! dx n n n which you can check to see gives the first few poynomias in (45. Hepfu Way to Express Legendre Poynomias It turns out to hep in many things in physics (as we wi see shorty for one exampe, if we take the argument x for Legendre poynomias to be cos θ. That is, in (45 and (46, P P P Take x cosθ ( cosθ = 3 P ( cosθ = ( 5cos θ 3 ( cosθ = cosθ 4 P ( cosθ = ( 35cos θ 30 cos θ ( cosθ = ( 3cos θ 5 3 P ( cosθ = ( 63cos θ 70cos θ + 5cosθ P ( cosθ n ( cos θ ( θ n (47 d =. (48 n! d cos n n n With this substitution x cosθ, the horizonta axis in Fig. A- woud go from θ = π (x = to θ = 0 (x = with the vertica axis ocated at θ = π/ (x = 0. P (cosθ vs θ woud then ook ike a haf cyce of a cosine curve. Higher order Legendre poynomias woud then be trigonometric functions of θ (though a bit more compicated that the ones we usuay dea with. The integration imits of the orthogonaity reations (44 were taken knowing in advance that we woud be using the substitution x cosθ. The to integration range for x corresponds to 0 to π (which, since that is the range of variation of the θ coordinate in poar coordinates, we are foreshadowing where we might eventuay use these reations. See Fig. A- beow.

28 . Legendre Poynomias Invoved in Soutions to Key Physics Equations 6 Legendre poynomias hep us in soving some key probems in physics, but we have a few more steps to go to see how... Specia Cases of a Genera Case Type Equation Genera Case Type Equation A genera from of second order equation one runs into in physics is ( f + g x f = 0 (49 where is the Lapacian and g(x is a genera function of space. One exampe is the time independent Schroedinger equation with f = ψ, h µ Specia Case, g = constant µ h g ψ + V ( x ψ = Eψ ψ + ( E V ψ = ( x 0. (50 A specia case of (49, where g(x is a constant (equa to k beow, is referred to as the Hehotz equation, f + k f = 0. (5 The soutions to the Hehotz equation are spatia (static waves. So one shoud (rightfuy consider the spatia osciations in the CMB coud be described mathematicay by the soutions to it. More on this ater. Note that (5 can be expressed in Cartesian or spherica coordinates (or any other coordinates, though we won t be concerned with those others. In Cartesian coordinates, (5 has soutions of form Specia Case g = 0 ikx or f x sin kx, cos kx, e = cos kx + i sin kx D case. (5 For g = 0, we have the Lapace equation. For f = Φ, the eectric potentia in a region free of charge, is a key reation for eectrostatics. For Newtonian gravity, f = Φ woud represent the gravitationa fied in a region free of mass. ( g Lapace eq (e.g., eectric potentia in region of zero charge 0 where 0 Specia Case, Spherica Coordinates with g = g(r Φ = = (53 Cases where (49 has g dependence ony on the radia distance r from the origin of the coordinate system used, f + g r f = 0, (54 are common. One exampe is the Schroedinger equation appied to the hydrogen atom, comprising an eectron in the fied of a proton s couomb potentia V(r = e /4πr. h µ µ h g r Schroedinger eq, H atom ψ + V ( r ψ = Eψ ψ + ( E V ψ = 0 For probems of this type, spherica coordinates work best, and we express in terms of r, θ, and φ. (55

29 7 z θ r dθ r da = r d Ω d Ω = sinθ dφ dθ r sinθ dφ y φ x Figure A-. Spherica Coordinates with Differentia Soid Ange dω Shown Specia Case, Spherica Coordinates on D Surface of Sphere If we wish to examine behavior of something on the surface of a sphere (such as the CMB radiation, we can fix r in (54, and examine the resuting soution. That soution woud have functiona dependence ony on θ and φ. In effect, g(r = const in (54, and we woud essentiay be soving the Hehotz equation (the equation for spatiay varying static waves on the surface of the sphere. We shoud expect f in this case to parae the soutions to the Schroedinger equation for the hydrogen atom with respect to behavior with respect to θ and φ. This shoud hep most readers foow the remaining parts of this appendix, since most have some famiiarity with that probem... Soving the D Surface of a Sphere Case Separation of Variabes to Sove g(r Case In cases where g is a symmetric function of r (purey a function of r and not θ or φ, then the probem of soving (49 simpifies. We try a separation of variabes approach, i.e., we separate f into functions dependent on r, θ, and φ, as foows. f ( r, θ, φ = R( r Θ θ Φ φ = R( r Y ( θ, φ works for g = ony a function of r Y ( θ, φ We then substitute (56 into (54, where is expressed in spherica coordinates. When we do this (again, this is a summary review, we aren t going through a the steps, we find the foowing three separated equations, each ony invoving one coordinate. d dr r + g ( r r = ( + ( an integer. See why beow. Radia equation R dr dr sin d dφ m an integer and m. See why beow. This is actuay Hehotz eq with x φ. Φ m 0 + Φ = d sin d Θ ( m θ + + Θ = 0 associated Legendre equation d d sin θ θ θ θ Using separation of variabes we have converted a partia differentia equation in three variabes, which is generay very difficut to sove, into three separate ordinary differentia equations, each in ony a singe function dependent on a singe independent variabe. (56 (57 (58 (59

30 Radia Equation (not needed for CMBR anaysis 8 Whie we wi ignore (57, as we are ony deaing with functions on a spherica surface (at constant r, we note that it gives rise in QM to the radia part of the hydrogen atom wave function and in doing so, the fundamenta energy eigenvaues. 3 Φ Equation The Hehoz equation in φ, (58, is readiy soved as imφ e Φ m ( φ = One soution for each vaue of integer m., (60 π where the constant of the denominator is chosen so Φ is appropriatey normaized (over 0 φ π. Note (60 expresses a (compex wave osciation as one progresses in the φ direction. We stated in (58 that m must be an integer. The reason is that its soution (60 can ony be singe vaued (as one progresses around φ past 360 o, if m is an integer. The number of nodes in 360 o increases directy with m. Associated Legendre Equation There is a fair amount of agebra invoved in determining Θ(θ of (59, the associated Legendre equation, but we cut to the chase here. For each different and each different m, there is a separate soution to (59, which we abe as Θ. These soutions are proportiona to entities caed the associated Legendre poynomias, abeed as P, which are cosey reated to Legendre poynomias. Thus, where we wi ater choose the proportionaity constant to suit out needs, (59 can be re-written d for Θ = constant P, as dp is associated Legendre poynomia m P.(6 sinθ + ( + P = 0 sinθ dθ dθ sin θ that soves associated Legendre equation If one goes through the math (see Cahi, we are ony summarizing, one finds P in (6 to be (where we use the symbo in pace of n in the Legendre poynomia symbo associated Legendre Legendre poynomia poynomia m m d P = sin θ P cos and m integers with m m dcos θ x cosθ ( θ. (6 The soutions to (6 are reated to our earier discussed Legendre poynomias via (6 where we take x cosθ, as discussed in Sect... We note that it is common in the iterature to write the argument of P as cosθ instead of θ. In that case, (6 becomes m m/ d P ( cosθ = ( cos θ P ( cos θ ( and m integers with m. (63 m dcos θ You can check that the first few Legendre poynomias in (47, used in (63, actuay sove (6. 3 Note that for the specia case g(r = constant (54 becomes the Hehotz equation, and we have a different radia equation (57. For that constant equa to zero, g(r = 0, (54 becomes the Lapace equation, with yet a different radia equation. However the anguar dependent equations (58 and (59 are the same for g(r isotopic or constant (whether that constant is zero or not. Thus, in the discussion that foows regarding the soutions to (58 and (59, a concusions are the same for the 3D spherica forms of each of the Hehotz equation (5, the Lapace equation (53, the hydrogen atom Schroedinger equation (55, any case with isotropic g(x (54, and any case on the D surface of a sphere (ike the CMB anaysis. In the iterature, one sometimes reads that the soution forms we are about to expore (associated Legendre poynomias, in particuar are derived from the Lapace equation, and sometimes from the Hehotz equation or genera isotropic form for g. Hopefuy, this footnote wi avoid confusion when you run into such statements.

31 9 Using Rodrigues formua (46 (with x cosθ in (63, we get P ( cosθ = ( cos θ m/ m dcos θ! m/ d = ( cos θ! dcos d m m+ d ( cos θ d ( cosθ ( cos θ The bottom row of (64 ony makes sense if m + 0 with m + an integer. Since m is an integer (see above, then must be. Further, if m exceeds, then any derivative in that bottom row wi yied zero for P. Thus, we justify what was stated in (57, (58, (6, and (63 about and m. When a is said and done, the first severa associated Legendre poynomias, found via (64, ook ike ( m ( + m m+ θ 3 P00 = P0 = cosθ P0 = 3cos θ P30 = cosθ 5cos θ 3 P = sinθ P = 3sinθ cosθ P3 = sinθ 5cos θ m! where P m = ( P.! = 3sin P3 = 5cos sin 3 P33 = 5sin θ P θ θ θ To check, and strengthen one s understanding, any of (65 can be inserted into (6 to verify that they are indeed soutions to the associated Legendre equation..3 Competeness and Orthonormaity of P m.3. Competeness Other than arbitrary constants by which they coud be mutipied, a possibe soutions to the associated Legendre equation are incuded in P of (63, so the set of those soutions shown there spans the space of a possibe soutions over θ. This is simiar to the D spatia (static wave (Hehotz equation (5 where the set of a ik x n e, where k n = nπ/l spans the space of a soutions to that equation over the interva treated. ik x Further, we know from the Fourier theorem that the soutions e n span the space of a possibe functions of x over the interva, not just functions that are specific soutions to the Hehotz equation. Simiary, the associated Legendre poynomias P (cosθ span the space of a possibe functions of θ, not simpy those that are soutions to the associated Legendre equation. Thus, the set P (cosθ is compete over θ. Any possibe function of θ can be shown equa to a sum of a P (cosθ over a and m, where the constant coefficient a m represents how much of P m is present. For exampe, in the m = 0 case (no variation in the φ direction since e imφ = for m = 0, any function purey of θ (and not φ can be represented as f ( θ = a P ( cosθ + a P ( cosθ + a P ( cosθ +... = a P ( cosθ m = 0. ( = 0. (64 (65

32 .3. Orthonormaity For same m 30 We first state, then shorty after that prove, the orthogonaity of P which have the same m, but different. Note that since P is rea (see (65, for exampes, we do not need to take the compex conjugate of P times P, as is done, for exampe, with quantum wave functions. P m x P x dx = 0, same m, x = cosθ. (67 The inner product of P with itsef (same m and same represents the normaization and is (reference to proof given beow ( + m! P same, same cos m x P x dx = m, x = θ. (68 + m! We can combine (67 and (68, and substitute cosθ for x, to yied the orthonormaity condition for P ( + m! 0 0 m = = m = m + ( m! π π π ( + m! = P ( cos ( cos sin same 0 m θ P θ θd θ = δ. m + ( m! P x P x dx δ P cosθ P cosθ d cosθ P cosθ P cosθ sinθdθ For different m On pg. 39, Cahi derives the orthonormaity reations for the P, for different m, but, as we wi see, we wi not need these. Proof of (67 To prove the orthogonaity condition (67, we start by rearranging (6, ( cosθ dp = + sinθ dθ dθ sin θ d m associated Legendre sinθ P ( cosθ ( P ( cosθ equation, rearranged. If we consider m fixed, then (70 can be considered an eigenvaue probem with eigenfunctions P and eigenvaues. ( +. Eigenfunctions are aways orthogona. Thus, for integration over the stated interva, we have (67. Proof of (68 If I have time, I may someday write out the proof of (68, which I never actuay ooked at mysef before writing this. (I had simpy accepted the resut, ike one accepts integra formuas in a tabe without working them through onesef. For now I simpy note that it is ony agebra, comprising substitution of the ast ine of (64 with cosθ x into the LHS of (68 and working it a out. The fu proof can be found in Arfken and Weber 4. Cahi merey states the resut (69 without proof..4 Competeness and Orthonormaity of Y m (θ,φ.4. Competeness We have reaized that P is compete over θ for the interva 0 to π. Using simiar ogic as we had with the D Hehotz equation (5 and its soutions being compete (spanning the space of a functions over x, the (70 (69 4 Arfken, George B., and Weber, Hans J., Mathematica Methods for Physicists (Academic Press, San Diego 995, pgs

33 3 soutions e imφ to equation (58 (identica in form to the Hehotz equation are compete over φ for the interva 0 to π. Thus, from (56, where we now reaize we need the and m sub/superscripts, imφ e Y ( θ, φ = Θ ( θ Φ m ( φ = ( constant P ( cosθ no sum on m (7 π is compete over the D space defined by 0 θ π and 0 φ π. Any function of θ and φ can be described by a sum over and m of terms a Y, where the a are appropriate constants..4. Orthonormaity Normaization Using (7, integrating Y (, θ φ over a θ and φ (effectivey integrating over the surface of a sphere with radius r = in Fig. A- pg. 7, and normaizing by setting the resut equa to unity, we have π π ( θ φ ( θ φ ( θ φ ( θ φ θ θ φ * * =,, Ω =,, sin no sum on 0 0 π Y Y d Y Y d d m π 0 0 = Θ θ Φ φ Θ θ Φ φ sinθdθdφ = * * m m π π 0 0 imφ imφ e e constant P ( cosθ P ( cosθ sinθdθdφ π π π π = constant P ( cosθ P ( cosθ sinθdθ dφ 0 π 0 = π = constant P cosθ P cosθ sin θdθ. 0 From the ast ine of (69, with the ast ine of (7 equa to one, we see constant ( m ( + m! Thus, from (7 we see the spherica harmonics are ( m ( + m (7 + =. (73! imφ +! e Y ( θ, φ = P ( cosθ no sum on or m.! π Our normaization constant is the quantity inside the square root sign above. Orthogonaity For the same m and. (74 From (69, we see that for the same m and any, P m and P are orthogona over θ. Thus, Y m and Y must aso be orthogona over an entire sphere, i.e.,

34 = π 0 0 π π π 0 0 * m ( m ( + m 3 ( θ, φ ( θ, φ sinθ θ φ ( no sum on, Y Y d d m ( m ( + m imφ imφ +! e e P m ( cosθ P ( cosθ sinθdθdφ! π π +! π = ( cos ( cos sin 0.! P 0 m θ P θ θdθ = = 0 for (75 tes us that for the same m and, the spherica harmonics are orthogona. For different m and m and any and = = For different m and m, regardess of and, we have π 0 0 π = 0. π π 0 0 * m ( θ, φ ( θ, φ sin θ θ φ (, and may be same or different Y Y d d m m ( ( ( ( ( im m π ( π P ( cos ( cos sin 0 m θ P θ θd ( θ m! m! φ φ + + e e P cosθ P cosθ sinθ dθdφ + m! + m! π + m! + m! + m! + m! im π π φ φ im im e e d 0 = 0 for m m (76 tes us that any two spherica harmonics with different m vaues are orthogona. For the same m and = = π 0 0 =. π π π 0 0 * ( θ, φ ( θ, φ sinθ θ φ ( and the same, no sum on them Y Y d d m ( m! imφ + ( + m π ( m! π ( m cos cos sin + m imφ e e P cosθ P cosθ sinθdθdφ! π + π imφ imφ P θ P θ θdθ e e dφ! 0 π 0 ( + m! = + ( m! (77 tes us that the inner product of a spherica harmonic with itsef equas one. This is our normaization for spherica harmonics. Bottom ine: Two spherica harmonics are orthogona (inner product over a spherica surface is zero uness both have the same and m. In that case the normaization is given by (78 with m = m and =. π π * Y m ( θ, φ Y ( θ, φ sinθ dθ dφ = δ δm m ( Spherica harmonics orthonormaity condition φ (75 (76 (77.5 Expanding a Function of θ and φ in Y.5. The Expansion Since the spherica harmonics are compete, we can expand any smooth function f.. (θ,. φ. in terms of them.

35 33 f, = a Y,. (79 ( θ φ ( θ φ = 0 m= Noting that Y θ, φ f θ, φ θ dθ dφ = Y θ, φ a Y θ, φ sinθ dθ dφ = 0 m= π π π π * * m sin m π π * sin 0 0 m θ φ θ φ θ θ φ δ δm m m = 0 m= = 0 m= = a Y, Y, d d = a = a, (80 we reaize that we can find a a for any f.. (θ,. φ. via π π 0 0 m = ( θ φ ( θ φ sinθ θ φ (8 a Y, f, d d..5. Note for m m Note from (65, ( ( m ( + m! P! m m = P. (8 Using (74 with m m, incorporating (8, and recaing that P is rea, we find ( + m ( m imφ +! e Y m ( θ, φ = P m ( cosθ no sum on or m! π m * Y ( θ φ imφ imφ + + m! m m! e m + m! e = ( P ( cosθ = ( P ( cosθ m! + m! π + m! π =,..5.3 Ramifications of m m Resut In our summation (79, for every spherica harmonic term a Y there is a corresponding term a Y = a Y. (84 * * m m m m (83 So, with appropriate choice of reation between a and a m a, i.e., m * a m rea number the m and m terms summed woud be rea for a given. That is, = =, (85 m * * * ( reafunction ( no sum on or a Y + a Y = a Y + a Y = a Y + Y = m m m m a Aso, for m = 0, Y 0 is rea. See (74 and recognize that P is rea for any m. Bottom ine: f in (79 can represent a rea number (such as temperature of the CMB even though the Y are compex..(86

36 .5.4 Comparing to Fourier Anaysis in the Cartesian Case 34 We can compare the representation of a function of θ in terms of a series of associated Legendre poynomias to the representation of a function of the Cartesian coordinate x in terms of as series of sines and/or cosines (or equivaenty, the rea and/or imaginary part of e ikx. Before doing so, we reca some trigonometric reations we wi use aong the way. ( a ( b ( c sin θ = cos θ cos θ = + cos θ sinθ cosθ = sin θ (87 ( a ( b sin θ = 3sinθ sin 3θ cos θ = 3cosθ + cos 3θ ( c ( d cosθcosθ = cos θ +cos3θ sinθ cosθ = sin3θ sinθ Now take (79 with φ = 0, i.e., consider f = f.. (θ, and insert (74.! + m 0 ( cos θ = θ φ = = θ = ( cosθ ( + m! f a Y, a P a P = 0 m= = 0 m= = 0 m= a Compare this to a typica Fourier series expression for a function h(x, = ( cos + sin (88.(89 n n (90 n= 0 h x a nx b nx Now ook at (65 (repeated beow for convenience to see what some of the associated Legendre poynomias used in (89 ook ike. ( m ( + m 3 P00 = P0 = cosθ P0 = 3cos θ P30 = cosθ 5cos θ 3 P = sinθ P = 3sinθ cosθ P3 = sinθ 5cos θ m! where P m = ( P.! We then convert (65 using (87 and (88 to get ( ( = 3sin P3 = 5cos sin 3 P33 = 5sin θ P θ θ θ ( ( P00 = P0 = cosθ P 3 0 = 3cos θ = 3 + cos θ = + cos θ 4 4 P 3 = sinθ P = 3sinθ cosθ = ( sin θ P 3 3 = 3sin θ = cos θ P30 = cosθ 5cos θ 3 = cosθ 5 + cos θ 3 = 7 cosθ + 5 cosθ cos θ 4 4 = 4 cosθ + 5 ( cos θ +cos3θ == 9 cosθ + 5 cos3θ P,P,P simiary yied expressions in cos θ, sinθ, cos3θ, and sin3θ The point is that terms in P of order have highest order exponents in both sine and cosine factors combined of as shown in (65. These can be converted, as shown in (9, into expressions having terms with (65 (9

37 35 ony sine and/or cosine of anges θ ( times θ.. not raised to any power. But those are exacty the kinds of terms we find in (90, for n and θ x. In essence, the summation (89 is the same summation as (90. For exampe, for the a3 cos3 x term in (90, we woud have a comparabe term in (89 c 3 cos3θ, where the c 3 is a combination of different a. Each term in (90 has a corresponding term (when we add together a the appropriate a in the right way in (89. Bottom Line: The sum of a spherica harmonics in θ to yied a given function is effectivey the same thing as a Fourier series summation of sines and cosines to yied the same function. nd Bottom Line: From (9, we see that the order of a spherica harmonic indicates the number of osciations (in space across a constant radius sphere from θ = 0 to π, just as the order n of a term in a Fourier anaysis indicates the number of osciations over the given interva from x = 0 to L. Higher n means more waveengths inside L. Higher means more waveengths inside 0 to π. (We use quotes on waveengths in the spherica case, because (as we can see from the = 3 case in (9 a given harmonic incudes sub waves that osciate ess rapidy (e.g., the =3, m = 0 case of (9 has both a cos 3θ and a cosθ term, not just a cos 3θ term. Question. So, one might ask, if we coud use either the compicated spherica harmonics or the simper sines and cosines, why not anayze a spherica surface, ike the LSS (ast scattering surface using the atter, rather than the former? Answer. I beieve it is for the foowing reasons.. We can convert readiy between Fourier harmonic components in Cartesian coordinates (used to deduce concusions about pane waves from infation, etc. and spherica harmonic components in spherica coordinates (in which our measurements are done via the methodoogy of Sect. 3.3, pg. 7.. It is conventiona. 3. Spherica harmonics are used esewhere in physics, particuary in QM for the hydrogen atom wave function, and physicists are famiiar with them. In the H atom case, we had no rea choice but to use spherica harmonics, because they are eigenfunctions of the Schroedinger equation with the isotropic potentia energy of the Couomb potentia of the nuceus. The eigenvaues of Y are m and (actuay (+, which gave us respectivey, the z direction anguar momentum and the square of tota anguar momentum. 4. Spherica harmonics are eigenfunctions of any system having isotropic g(x in a governing equation ike (54. Since one of the foundationa postuates of modern cosmoogy is isotropy, in the ong run, it shoud prove advantageous to represent quantities of interest in the universe (ike wave patterns in the LSS via eigenfunctions to equation (54 with isotropic g = g(r. This woud be particuary reevant for other cases invoving 3D (unike the D spherica surface of constant radius of the LSS. In such cases, we woud have a non-trivia equation in r (see (59, which is an eigenvaue probem with eigenvaue ( +. Thus, for each (denoting a given eigenfunctions of r, there is a reated eigenfunction of θ, P (for given m. However, any given sine or cosine function, such as cos 3θ, is not an eigenfunction for given. So, if we want to have the advantages of working in an eigenfunction basis for 3D isotropic cases, we need to use spherica harmonics, not sine/cosine functions..5.5 An Exampe: A Dipoe Given that we aign our spherica coordinate system (See Fig. A-, pg. 7 in a given direction (say the z axis is aigned perpendicuar to the pane of the gaactic disk. One might reaize that if there were a particuar

38 36 mutipoe with respect to θ (particuar vaue aigned with a node right on the z axis, then we coud readiy determine a suitabe a. For exampe, consider the dipoe of Fig. A-3, part a, aigned with the z axis. It is described, at east in arge part, by a temperature variation with θ of cosθ. Note from (74, and (79, + ( m! imφ ( θ φ = ( θ φ = ( θ φ == ( cosθ. (9 ( + m! T, f, a Y, a P e = 0 m= = 0 m= From (65, we see the = terms incude P0 = cosθ P = sinθ P = sinθ. (93 Higher order ( > terms represent higher mutipoes and = 0 represents a constant (no mutipoe vaue of temperature with respect to θ equa to.75 o K for the CMB (see dashed circe in Fig. A-3. So we woud expect the P 0 term in the summation (9 to refect the dipoe of Fig. A-3a, i.e., a 0 woud be much arger than any other a. higher temp z θ sice at constant φ z z temp represented by radia distance ower temp mean (constant temp over θ = dashed circe a Aigned with z axis, a0 >> other ampitudes b Perpendicuar to z axis, a and a- >> a0 c Aigned with some θ in between, bba a a significant Figure A-3. Temperature Dipoe Aigned in Three Different Directions Simiary, in Fig. A-3b, the dipoe is aigned such that the temperature variation, at east in arge part, is described by sinθ. Thus, in that case, P and P woud be the dominant contributing modes and be much arger than a 0. For the case of Fig. A-3c, we woud have significant contributions from a three P. We now derive the a for each of the cases in Fig. A-3 and see if our intuitive statements above regarding them are correct. To do this, we first note what the = part of the spherica harmonics summation ooks ike in genera, where we ook ony at the sice where φ = 0, to keep things simpe for now. Genera Reation for Dipoe with Any Aignment ( m ( + m +! imφ T = ( θ, φ = am Y m ( θ, φ = a m P m ( cosθ e m= m=! take φ=0 3! 3 0! 3! = a 0 cosθ a sinθ + a sinθ!! 0! = a0 cosθ a sinθ + a sinθ. (94

39 37 Case of Fig A-3 a For the first case in Fig. A-3, we have a simpe cosine dependence on θ, ( θ φ T =, = X cos θ X = max difference from mean temp. (95 So, in (94, we readiy see that a0 = X a = 0 a = 0. (96 3 Case of Fig A-3 b For the second case in Fig. A-3, we have a simpe sine dependence on θ, ( θ φ T =, = X sinθ. (97 The foowing satisfy (94 and are the a vaues for the dipoe as shown in Fig. A-3b. So, a0 = 0 a = X a = X. ( X 3 X 3 T = ( θ, φ = ( 0 cosθ + sinθ + sinθ = X sinθ. ( Case of Fig A-3 c For the third case of Fig. A-3, we have the foowing vaues, which we prove are correct beow. X X X a0 = a = a =, ( Note that the soid ine of Fig. A-3c is described by X. sin. (θ + 45 o. From the trig reation o ( α ± β = α β ± α β α = θ β =, (0 sin sin cos cos sin with and 45 ( o X ( o o X X X sin θ + 45 = sinθ cos45 + cosθ sin45 = sinθ + cosθ. (0 Using our stated vaues for the a from (00 in (94, we see it equas (0. That is, A Comment X 3 X 3 X 3 X X T = ( θ, φ = cosθ + sinθ + sinθ = sinθ + cosθ, ( I ve tried to make this simpe for iustrative purposes, but the astute reader may have noticed that the a and a are not uniquey defined above. Any of an infinite number of combinations of them coud make (94 equa to (97 in Case b, or to (0 in Case c. Had we not imited this case to φ = 0, in order to keep things simpe, we coud have shown they were actuay unique for the more genera case. See Sect..5.3 pg. 33, where we show that, in genera, for T to be rea (not compex, we must have a m * a rea m = =. This is a further constraint that pins down the a and a vaues. Note that (98 and (00 obey this constraint..5.6 Other Mutipoes In simiar fashion for a soe higher mutipoe (more spatia osciations in temperature with θ = more highs and ows = more nodes = higher vaue =, for this exampe, the vaues for a m in the case woud dominate over the a vaues for other.

40 38 In a genera case with many mutipoes superimposed (as in the CMB, we get a spectrum of a, the vaues for each and m refecting how strong that particuar mode is reative to a others in the summation (9. The difficut thing about spherica harmonics, as seen by the exampes of (65, is that it is not easy to visuaize the various modes and what physicay, exacty, their ampitudes a represent. For a Fourier series ike (90, we can envision what each mode ooks ike, and even draw figures representing the summations of the various modes. For spherica harmonics, on the other hand, this is probematic..5.7 Anguar Separation and From Fig. 3-A, pg. 36, one can see that for =, there are 80 o degrees between nodes, i.e., there is a singe variation ( bump, as it were over the range of 0 θ π. For =, (think variation over θ, as in (9 we woud have three nodes (and two bumps, making a singe variation (one bump over 90 o degrees. For = 3 (variation over 3θ as in (9 we woud have four nodes (three bumps, and a singe variation (one bump over 60 o. We can readiy generaize to anguar variation o 80 π = in radians =. (04 Appendix B. Introduction to Power Spectra Anaysis. The Basis in Eectronics for Power Considerations Consider instantaneous power in an eectric circuit Pinst = VI, (05 where, for charge q(t. a sinusoid, ω the frequency in radians/sec, and for simpicity impedance Z = pure resistance R, Thus, instantaneous power is = sin ω ( = q t A t A q peak. (06 I = qɺ t = ω Acosωt V = RI = Rω Acosωt inst ω sin ω peaksin ω peak ω peak P = R A t = P t P = R A A = q. (07 Since, for T = /f = π/ω the period of the osciation, T T T π A variance of q = ( q( t dt A sin t dt A sin t dt T = T ω = T = T A standard deviation of q = variance = rms vaue of q = q rms =, the average power (time average of the instantaneous power is T T Ppeak A P = Pmean = P sin 0 inst dt = Rω A ωt dt = = Rω T T 0 symbo def Ppeak A ( = peak is variance of q t A q (08. (09 The point is the average power P is proportiona to ω times the ampitude of the charge squared A. Now, for q(t a more genera shape (non sinusoid, we can find the Fourier components of q, which when added together (or integrated in the continuous case, give us q(t. The ampitude of each such component is, in

41 39 genera, different at each vaue of ω, so we abe such ampitudes as A n (discrete case or A(ω (continuous case. That is, = ( nsinωn + ncosωn or = ( ω sinω + ( ω cosω (0 q t A t B t q t A t B t dt n= 0 ( ω take B = B = 0 for simpicity, n where we take the cosine parts to be zero, for simpicity, though to be compete for handing any shape q(t, they woud need to be incuded. So, we woud, in genera, find a different average power P in (09 for each ω, and we define Pn = mean power for nth Fourier component in discrete case P ω = mean power per unit ω at ω in continuous case, In genera, when peope use the word power in the context of spectra anaysis, they mean mean power over a fu cyce as in (09 and (, not instantaneous power, as in (07. From (09, we can see that if our Fourier components A n (or A(ω of q(t. had the same A n (or A(ω at each ω n (or ω, the P n data points (or the P(ω curve woud rise quadraticay with ω. A pot of P(ω/ω vs ω, on the other hand woud be a straight ine. See Fig. B-. ( P(ω α ω P(ω A(ω ω ω Figure B-. Constant Ampitude A(ω Intuitivey, the behavior shown in Fig. B- coud be expected. A sinusoida wave of the same ampitude as another sinusoida wave, but osciating faster shoud have more power in it. Power depends on both the ampitude of the wave and the rate at which it osciates. White noise is defined as a straight ine P(ω vs ω curve, as shown in Fig. B- beow. Each sinusoida component at each ω has the same power density (power per unit ω, but as can be seen in the RHS of the figure, different ampitude A(ω. P(ω A(ω α P(ω ω ω ω Figure B-. White Noise = Constant P(ω Fig. B-3 shows what white noise woud ook ike in time, where the instantaneous contributions of every component of Fig. B- are summed as in (0 (actuay, integrated for the continuous case of Fig. B- at each point in time.

42 40 Figure B-3. White Noise Instantaneous Power from A Components vs Time More genera type systems show varying behaviors, with pots that are more compicated than the ones above, but the reationship ( A( ω A P( ω ω A( ω = ampitude of ω component wave is variance of same wave ( aways hods at each ω for any system.. Extrapoating to Other Types of Systems The reations we deveoped in the prior section are commony used for other fieds outside of eectronics. For spatiay varying waves, for exampe, a of (05 to ( can hod with different physica quantities, e.g., ( ω S ( x = x k = t x T λ q t S x ω k A ω A k P ω P k λ = waveength any function of wave number. For the CMB, we woud take S = T, temperature (not period T as the symbo was used for above. And thus, we can immediatey deduce, in parae with (09, that ( A( k = ( A( k P k k = k A k is variance of wave component of temperature A k k T x ( = peak for component A k T k. In this case, P(k vs k woud be the temperature power spectrum. (Power is now being used in a more genera, mathematica, sense, rather than being the kind of power one is accustomed to deaing with in mechanics and eectronics. That is, power in the present sense does not have to be the time rate of change of work energy. In the CMB anaysis the above (Cartesian coordinates based Fourier harmonics reationships are converted to (spherica coordinates based spherica harmonics reationships, as shown in Sect. 6. Note that units of P(k are power per unit k. When we integrate P(k over an interva k, we get the tota power in that interva (a very usefu reationship in many areas of physics and engineering. power between k and k P k dk k (3 (4 = (5 k Again, power in other contexts can mean something other than power as known in typica physics courses, and with other systems, such as that of (3, (5 hods with the surrogate symbos of (3 repacing those shown in (5.