C osm ic A cceleration and th e T heory of th e M icrowave Background By Benjam in M ark Gold B.S. (Michigan S tate University) 1997 M.S. (University of California, Davis) 2001 DISSERTATION Subm itted in partial satisfaction of the requirem ents for th e degree of D octor of Philosophy in Physics in the O FFIC E OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Com m ittee in Charge 2005 -i- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3191126 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3191126 Copyright 2006 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract Cosmic Acceleration and the Theory of the Microwave Background by Benjamin M ark Gold Doctor of Philosophy in Physics University of California, Davis Professor Andreas Albrecht, Chair This work represents an investigation of several details of the anisotropies of th e cosmic microwave background. The focus is on the response of the microwave background to the conditions of the universe at very early and late times, during both of which the expansion rate is thought to accelerate. During the radiation era, any perturbations to th e otherwise smooth background will oscillate due to the immense density and pressure at these early times. After th e universe cools and m atter becomes dom inant, the photon perturbations travel freely to us while the m atter perturbations continue to collapse under gravity to form galaxies and stars. The initial conditions for these perturbations are set by an early period of acceleration called inflation. The increasing precision of cosmological datasets is opening up new opportuni ties to test predictions from cosmic inflation. In this work I study the impact of high precision constraints on the prim ordial power spectrum and show how a new -ii- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. generation of observations can provide impressive new tests of the slow-roll infla tion paradigm , as well as produce significant discrim inating power among different slow-roll models. In particular, I consider next-generation measurements of the Cosmic Microwave Background (CMB) tem perature anisotropies and (especially) polarization. I emphasize relationships between the slope of the power spectrum and its first derivative th a t are nearly universal among existing slow-roll inflation ary models, and show how these relationships can be tested on several scales with new observations. Among other things, the results give additional motivation for an all-out effort to measure CMB polarization. W hile photons continue to travel almost freely during th e m atter era, changes in the expansion rate due to acceleration at late tim es can subtly affect their dis tribution. Such acceleration is posited to be due to the effects of an otherwise unobserved dark energy. Also in this work I discuss several issues th a t arise when trying to constrain the dark energy equation of state w = P / p using correlations of the integrated Sachs-Wolfe effect w ith galaxy counts and lensing of th e cosmic microwave background. These techniques are com plem entary to others such as galaxy shear surveys, and can use d a ta th a t will already be obtained from cur rently planned observations. In regimes where cosmic variance and shot noise are the dom inant sources of error, constraints could be made on the mean equation of state to w ithin 0.33 and its first derivative to w ithin 1.0. Perhaps more interesting is th a t the determ ination of dark energy param eters by these types of experiments depends strongly on the presence or absence of perturbations in the dark energy fluid. -n i- Reproduced with permission of the copyright owner. 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C ontents C ontents iv List o f Figures vii List o f Tables ix 1 Introd u ction 1 2 4 M odels o f Inflation in th e Early U niverse 2.1 In tro d u c tio n .................................................................................................... 4 2.2 Definition of in fla tio n .................................................................................... 5 2.3 A scalar field in general r e l a t i v i t y ............................................................. 6 2.4 Slow r o l l ........................................................................................................... 8 2.5 Power spectrum flu c tu a tio n s ....................................................................... 9 2.6 Power spectrum am plitude and s h a p e ........................................................ 12 2.7 Cosmic acceleration t o d a y .......................................................................... 13 3 T heory o f th e M icrowave Background 14 3.1 In tro d u c tio n ..................................................................................................... 14 3.2 An overview of the m e th o d .......................................................................... 15 3.3 Choice of g a u g e .............................................................................................. 18 3.3.1 Synchronous g a u g e ......................................................................... 18 3.3.2 Conformal Newtonian g a u g e ....................... 20 -iv- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 3.5 Evolution e q u a tio n s....................................................................................... 20 3.4.1 M e t r i c .................................................................................................. 20 3.4.2 General p a r t i c l e s .............................................................................. 21 3.4.3 P h o t o n s ............................................................................................... 24 3.4.4 N e u tr in o s ........................................................................................... 25 3.4.5 Cold dark m a t t e r ............................................................................. 26 3.4.6 B a r y o n s ............................................................................................... 26 3.4.7 Dark e n e r g y ....................................................................................... 27 Line of sight e f f e c ts ....................................................................................... 28 3.5.1 Integrated Sachs-Wolfe e ffe c t......................................................... 29 ............................................................................................... 29 3.5.2 Lensing 4 T esting Inflation w ith th e M icrowave Background 32 4.1 In tro d u c tio n ................................................................................................... 32 4.2 Scalar Field I n f la tio n ................................................................................... 34 4.2.1 Slow roll, ns, and n s ....................................................................... 36 4.2.2 Model s p a c e ........................................................................................ 38 ........................................... 39 4.3.1 C M B .................................................................................................. 42 4.3.2 Lym an-ct............................................................................................... 44 4.3.3 Error contours for current and future d a t a ................................ 45 Testing In flatio n ............................................................................................. 50 4.4.1 R e s u lts ................................................................................................. 50 4.4.2 Projected errors and c ro s s-c o rre la tio n s ...................................... 55 4.4.3 The Fisher m atrix a p p ro x im a tio n ................................................ 57 4.3 4.4 4.5 D etermining how well experim ents can do C onclusion ................................................................................ 5 Lim its o f Dark Energy from C ross-C orrelations 62 64 5.1 In tro d u c tio n .................................................................................................... 64 5.2 Background t h e o r y ....................................................................................... 65 -v- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.3 5.4 5.5 5.2.1 The ISW effect and dark e n e r g y ................................................... 65 5.2.2 Lensing co rrelatio n s.......................................................................... 66 5.2.3 Galaxy c o rre la tio n s ........................................................................... 67 Calculating the sensitivity to dark e n e rg y ................................................ 69 5.3.1 Choosing a form for dark e n e rg y ................................................... 71 5.3.2 Galaxy survey characteristics ....................................................... 77 R e s u lts .............................................................................................................. 79 5.4.1 Dark energy p e r tu r b a tio n s ............................................................. 81 C onclusion....................................................................................................... 83 Bibliography 85 A S elected Inflationary M odels 92 B 94 Full-sky expressions for angular power spectra C U seful M athem atical D erivations C .l 97 Param eter m arginalization for a m ultinorm al distribution in m atrix f o r m ................................................................................................................. 97 C .2 A lternate derivation of the evolution of the density of a general cos mological f lu id ................................................................................................. 100 C.3 Connection between two forms of the Fisher m a t r i x ............................ 103 C.4 Specific lensing noise e s ti m a to r s ............................................................... 105 D Code 108 -vi- Reproduced with permission of the copyright owner. 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List of Figures 4.1 Values of 77.5 and n s 'a t CMB scales for several inflation models 4.2 Error in the prim ordial power spectrum from a WMAP-like experiment. 40 4.3 Error in the prim ordial power spectrum from an after-Planck-like experim ent......................................................................................................... 41 4.4 68 % confidence regions in th e n s - n s ' plane for various experiments. 46 4.5 68 % confidence regions in th e ns~ns' plane for various experiments w ith a different prior...................................................................................... 47 4.6 Combining CMB and Lyman-cr experim ents............................................ 48 4.7 Combined constraints in the n s ~ n s plane from Lyman-cr and CMB e x p e r im e n ts .................................................................................................... 51 4.8 M arginalized (one-sigma) errors of n$ (solid line) and n s (dashed line) as a function of pixel noise and beam w id th .................................. 56 4.9 Cross-correlation coefficients for all the param eter pairings of four different e x p e r im e n ts .................................................................................... 58 4.10 Plots of the angular power spectrum as a function of cosmological param eters......................................................................................................... 60 4.11 Plots of the angular power spectrum as a function of power spectrum param eters......................................................................................................... 61 5.1 Noise levels for CMB lensing reconstruction ......................................... 68 5.2 Response of the tem perature power spectrum to dark energy bin param eters......................................................................................................... 72 5.3 Response of the ISW -lensing power spectrum to dark energy bin param eters......................................................................................................... 73 -vii- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. . . 37 5.4 Response of the ISW -galaxy power spectrum to dark energy bin param eters........................................................................................................ 74 5.5 Three dark energy eigenvectors for an ideal ISW -lensing-galaxy count correlation experim ent..................................................... 75 5.6 E rror in wo and wa as a function of to tal galaxy num ber..................... 76 5.7 E rror contours (68.3%) in the w0~wa plane.............................................. 80 5.8 The effect of dark energy perturbations on error contours (68.3%) in the w0-vja plane............................................................................................... 82 -viii- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables 4.1 Noise param eters used for sim ulating various experim ents................. 45 4.2 M arginalized errors in n s and n s for simulated experiments with different priors on the Taylor expansion cutoff........................................ 50 M arginalized errors in n s and n s ' for sim ulated experiments with different priors on the consistency relation............................................... 52 Inflationary model p a ra m e te rs ................................................................. 93 4.3 A .l -ix- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -x- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 C hapter 1 Introduction The first direct observation of the cosmic microwave background was made by chance by Penzias and W ilson [54] in 1964. By then the theory of the microwave background had already begun to be worked out, most recently before then by Dicke, Peebles, Roll and Wilkinson[16], which added to previous work by Alpher, Herman, and Bethe. T he next m ajor step in observation came w ith th e launch of the Cosmic Background Explorer (COBE) satellite in 1989, which m easured both the frequency spectrum[48] and the anisotropy[5] (angular power spectrum ) of the microwave background. The first was im portant in confirming th a t the microwave background was indeed black-body as theorized nearly from the beginning, and the latter measurement was the first to confirm th a t anisotropies beyond the dipole were present. These anisotropies continue to be of great interest today because of how much inform ation they contain about th e physics of th e early universe. Meanwhile, the theory of the microwave background has continued to develop. The general theory of linear perturbations in a homogeneous and anisotropic back- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 ground cosmology was first worked out by Liftshitz[41] in 1946. Later this work was used to develop a theory of the anisotropies in the microwave background, first for baryons and photons alone by Peebles & Yu [52] in 1970, w ith later additions by many authors, continuing even today. This work represents an investigation of several details of the anisotropies of the cosmic microwave background. In partic ular, the focus is on the response of the microwave background to the conditions of the universe at very early and late times, during b o th of which the expansion rate is thought to accelerate. The acceleration at very early times sets up the initial fluctuations th a t form the initial conditions for all the rest of the physics of the microwave background. These initial conditions m ay contain information about physics a t high energy scales, possibly as high as 1017 GeV. The acceleration at late times affects the microwave background by changing the expansion rate of the universe, which in tu rn alters th e rate of collapse of large-scale structure. This structure can induce secondary effects in the microwave background, and measuring them could give some clues as to w hat is causing this late acceleration. The microwave background is now understood to have its origins in the early universe. During the radiation era, any perturbations to th e otherwise smooth background will oscillate due to the immense density and pressure at these early times. After the universe cools and m atter becomes dom inant, the photon p ertu r bations travel freely to us while the m atter perturbations continue to collapse under gravity to form galaxies and stars. The initial conditions for these perturbations are set by inflation, which is introduced in chapter 2. T he effect of these initial conditions on the microwave background is investigated in chapter 4. The work contained in th a t chapter first appeared in [24], which uses numerical techniques to determine in detail how well future microwave background m easurem ents will be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 able to determ ine inflation param eters. This leads to some new ideas about how to distinguish different inflationary models. T he evolution during the radiation era of the photon perturbations from their initial state to w hat we see today is governed by a complex set of differential equa tions discussed in chapter 3. W hile the photons continue to travel almost freely during the m atter era, changes in the expansion rate can subtly affect their distri bution. These effects and how to observe them are discussed in chapter 5, which contains work originally published in [23]. T h at work uncovers some fundamental limits to how well correlations between the microwave background and other tracers of structure can determ ine th e expansion history of th e universe. This m ethod is one of many th a t can be used to determ ine the nature of dark energy, and chapter 5 includes some discussion of how feasible it will be. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 2 M odels of Inflation in th e Early U niverse 2.1 In tro d u ctio n Inflation is a way of getting around the flatness and horizon problems in cos mology. The flatness problem is simply th a t a flat universe does not seem to be stable: any deviation in energy density away from precisely th a t needed for zero m ean curvature should cause the significance of curvature to monotonically increase in tim e relative to m atter or radiation. Since we observe a nearly flat universe, the origin of this flatness is a puzzle w ithout inflation. The horizon problem is th a t for a conventional universe dom inated by m atter or radiation, the maximum distance a causal signal may have traveled since the initial singularity increases faster th an the expansion of the universe. This means th a t for any pair of comoving points today, one can always find an early tim e before which those points were not in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 causal contact. In particular, the points of origin for the microwave background on opposite sides of the sky should not have been in causal contact at th e tim e th a t the photons were em itted, which means th e extremely sm ooth nature of the background is also a mystery w ithout inflation. Inflation as a solution to these problems and others was first proposed by Guth[25] and soon after improved and refined by several other authors[ 2 , 42, 43, 26, 4], In this chapter I will present an overview of inflation and of the models typically used to generate it, as well as a derivation of inflation’s most successful prediction: a nearly scale-invariant spectrum of prim ordial density perturbations. Much of this treatm ent is similar to th a t of Lyth [45] and can be found in detail in Liddle & Lyth [40]; I have attem p ted to collect together the pieces essential for finding the density perturbations. 2.2 D efin itio n o f in flation T he general spacetime m etric which has the property of being b o th homogeneous and isotropic is known as the Priedm ann-Lem aitre-Robertson-W alker m etric ds 2 = dtl - a (t) dr 2 1 + kr2 ( 2 . 1) where a is the tim e-dependent (but not space-dependent) scale factor, and k = —1 , 0,1 describes the type of spatial curvature (closed, flat, or open, respectively). Prom this m etric and Einstein’s equations come two differential expressions for how Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 the scale factor evolves w ith time 2 H2 87tG k ( 2 .2 ) (2.3) where G is N ewton’s constant, and p and P are the to tal average energy density and pressure of the universe. I have not explicitly included a cosmological constant in the above expressions, but one can be incorporated by including a component of the universe w ith energy density A / ( 8 itG) and pressure —A /(87tG). Inflation can be very broadly defined as any period during which the expansion of the universe accelerates, i.e. an era during which a is positive. The requirement for this is for the dom inant form of energy density to have an equation of state w = P / p less th a n —1/3. Such an equation of state is sufficient to solve both the flatness and horizon problems. More narrowly, inflation usually refers to a period of accelerated expansion during the early universe, typically well before nucleosyn thesis but after the creation of undesirable relic particles. Also, inflation typically refers to a period during which the dom inant energy density has an equation of state w = —1 or very close to it. 2.3 A scalar field in gen eral r e la tiv ity The action for a scalar field 4>w ith a potential V {</>) in a generic spacetime w ith m etric is (2.4) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 where g refers to the determ inant of the metric. The energy density and pressure of this field are f = i r w + vw , (2.5) p = \ g “w9 ^ d r4, - V(<t>). (2.6) Prom this it is clear th a t if the field becomes dom inated by its potential energy it will have the behavior of a cosmological constant, and if this energy density is dom inant the universe will experience inflationary expansion. The Euler-Lagrange equation of motion for the scalar field th en works out to '< j > - ^ V 2<f>+ Z-i> + V'{<i>) = 0 . cr a (2.7) During an early period of inflation, two simplifying assum ptions are typically made. First is to assume th a t the spatial geometry of th e universe is flat. Referring back to equation 2 .2 , as long as the energy density dilutes w ith expansion more slowly th a n 1/ a 2, it will dom inate th e behavior of th e universe and curvature will rapidly become irrelevant. The second assum ption is th a t th e gradient term in the equation of m otion vanishes, which will be true at later times even if it is comparable to the other term s at earlier times. Since inflation is expected to last for at least 60 e-foldings (representing an expansion of the scale factor by roughly 10 26), in typical models these two assum ptions are true to a high degree of accuracy for all but the first few e-folds of inflation. Under these assum ptions along w ith the assumption th a t the scalar field energy dominates, we can rew rite equation 2.2 and the equation of m otion as .\ a\ a) 2 , 1 3 Mj, \4? + v w ] , 4>+ 3 - ^ + V ,{(t)) = 0, a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 2 .8 ) (2.9) where I have now defined the Planck mass scale to be M p = (87xG)~1^2. These two equations, together w ith a choice of potential, completely determ ine the (classical) behavior of inflation. 2.4 Slow roll “Slow roll” is the approxim ation where th e equation of m otion is taken to be in the strongly dam ped regime w ith negligible <f> and small 0 relative to II, so the equation of m otion simplifies to 3H<j) = —V'. This also implies th a t V((f>) dom inates the energy density, so in the slow roll approxim ation H 2 = P /3 Mp and H is taken to be roughly constant. This leads to an exponentially increasing scale factor a ~ exp(Ht), leading to a definition for the number of e-foldings A N = H A t between two points of tim e during inflation. There are several conventions for defining slow roll param eters. Here I will use the convention of [40]: MpV" M tV 'V'" ( 2 . 10 ) T he slow roll approxim ation holds as long as these param eters are small. The motivation of this convention is so th a t e, 77, and £ will all be of similar order, which makes collecting lowest order term s easier in complex expressions. Some authors define slow roll param eters in term s of the Hubble param eter these definitions are equivalent to ones in term s of V (<f>) as long as the field is potential-dom inated, because in th a t case H 2 ~ V. However, when the field has non-negligible kinetic energy (such as very near the end of inflation) H 2 ~ + V and the definitions diverge. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 It will be useful later to express the derivatives of th e slow roll param eters with respect to 4>in term s of other slow roll param eters MP^ = r /V '2 i- ( 2 e ) I , M p ~T7 = — r]V2e. d<f> \[2e (2.11) ( 2 . 12) The derivative w ith respect to (f>is related to scale factor during slow roll by # d in a 2.5 = H = {2. 13) V Pow er sp ectru m flu ctu a tio n s In the standard picture of inflation, small quantum fluctuations in the scalar field have their length scale greatly increased by th e expansion of spacetime, so much so th a t they rapidly become larger th a n the causal length scale and “freeze out” . A fter inflation ends, the causal horizon can eventually catch up to these length scales and the initial density fluctuations th en can grow into galaxies and all the other structure th a t we see today. If we separate the field into a homogeneous background piece and a perturbation <p(x,t) = <j)(t) + 8<f>(x,t) and use th e equation of motion, then to linear order the fluctuations themselves follow the equation (w ritten in Fourier space), 6(f) + 3-5<f> + 6(f>+ V"(4>)8<f) = 0. a \a J (2.14) Working through the details is made somewhat simpler by switching to conformal time (defined by dr = d t / a ), and examining th e quantity u = a8(f). Also, we are going to be interested in the fluctuations as generated on scales where th e potential Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 is smooth, so the final term above will be dropped. After some m ath, the equation of motion then can be rew ritten as which has also made use of the fact th a t during inflationary expansion, r ~ (aH)-'. Now th a t the equation of m otion has the form of a harmonic oscillator, quan tizing u(k, t ) proceeds in the norm al way; it is w ritten in term s of creation and annihilation operators u(k, t ) (2.16) — w(k, r)a(k) + w*(k, r ) a ) ( —k), where w ( k , r ) m ust satisfy th e same equation of m otion th a t u ( k , r ) does. This leads to a solution for w(k, r), valid as long as the potential V (and hence H ) are slowly varying, w(k, t ) = — . (2-17) (2.17) Finally, w hat we’re after is the power spectrum of the fluctuations e_ 2 (I'M 2ir 2 2 27 + 1 . (2.18) The am plitude of the fluctuations freezes out once their physical scale is much larger than the horizon distance, and this am plitude is then w hat sets the initial conditions for the universe after inflation. In this lim it k <C aH, so th e power spectrum simply becomes ( H / 2 n ) 2. The field perturbation 5<p is defined in a spatially flat slicing of spacetime. In the comoving frame, the field pertu rb atio n vanishes by definition, b u t this frame has a non-zero curvature perturbation. This curvature pertu rb atio n 77 is very generally Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 related to the difference in tim e coordinate between the two frames St via the simple relation 7Z = HSt. Using p S t = Sep, we can write the power spectrum P-ji of the curvature fluctuations as The field fluctuations originate during inflation, and as long as th e field is slowly rolling during th a t tim e we can rew rite the power spectrum of curvature fluctuations using the slow-roll param eters H2 Pn = 8tr2M |e ’ (2'20) where H and e are to evaluated a t th e tim e when the scale of interest exited the horizon during inflation. After inflation ends, the horizon grows faster th an physical scales and eventually a curvature perturbation of a given wavelength will fall back into the horizon. W hen this happens th e density pertu rb atio n of m atter or radiation at th a t scale will depend on the curvature pertu rb atio n 1Z through its contribution to the effective gravitational potential <f>. Density and the gravitational potential are related by the Poisson equation 4 TrGSp = V 2$ , (2.21) so in Fourier space 8 = 5p/p is given by k\ 2_ 4 = - U j 1 2 f k x2 *■ (222) The gravitational potential itself is simply proportional to th e curvature p ertu r bation with a coefficient of order unity th a t depends on the details of the am ount of various relativistic and non-relativistic species present. Since each different Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 wavenumber for the curvature perturbations falls back w ithin th e horizon at dif ferent times, this coefficient will be dependent on wavenumber. Thus the density fluctuation power spectrum P$ of some species of particle after inflation is p‘ = ow(ii),T2{k)J k i ’ (2'23) where T ( k ) is a transfer function normalized to unity on large scales th a t contains the wavenumber dependence. The remaining constant of order unity is absorbed into th e definition of the prim ordial power spectrum am plitude at horizon entry 5H (sometimes w ritten as As) , defined by the expression Pi = 2.6 (A) T>(k)&l (2.24) Pow er sp ectru m am p litu d e and sh ap e We now have the first im portant expression for connecting observations to th e ory, which is th a t the prim ordial power spectrum am plitude (observed through density fluctuations at large scales) should be related to quantities related to the potential energy density during inflation, ■>H H Mpy/e (2.25) Prom this several more shape param eters can be found. T he “tilt” or spectral index of the prim ordial power spectrum of perturbations is defined to be di n 5% , ns = 1 + i ^ k A given wavenumber k is k/(aH) ( 2 ' 2 6 ) said to have left th e horizon during inflation = 1, sod i n k = d in a. W ith this in mind, . when th e spectral index can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 rew ritten n s = 1 + —— In a in a 1 —V 2 e — M p (2.27) = 1 + 277 —6 e. The second derivative (first derivative of th e tilt) is = —2£2 + 16ct7 - 24e2 2.7 (2.28) C osm ic acceleration to d a y Recent observations have led to the idea th a t th e expansion of th e universe may have begun accelerating again sometime in th e recent (cosmologically speaking) past, due to the influence of some hitherto unknown substance referred to as dark energy. Once again, the idea th a t this effect might be due to a scalar field dom inated by its potential energy is used to try and explain th e acceleration. For recent acceleration, this field is usually referred to as the quintessence field. There are theories th a t unify the inflaton and quintessence fields, b u t in general this is difficult to do since the energy scales involved are so far apart. For quintessence, the quantum fluctuations would be unobservable and so the development of section 2.5 is uninteresting. W hat is of interest and potentially measurable is the equation of state for dark energy, and possibly even th e evolution of large-scale fluctuations in its density. These ideas are developed in the next chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 C hapter 3 T heory o f th e M icrowave Background 3.1 In tro d u ctio n The microwave background has become one of the m ajor sources for cosmo logical information in modern physics. In this chapter, I will provide background m aterial for understanding th e theory of how the microwave background came to look like it does today. Most of th e physics takes place during the radiation era, when photons and baryons are strongly coupled together. The initial conditions are a fairly simple set of param eters which can be obtained from an inflation model (see chapter 2 for details), but because the microwave background’s deviations from smoothness rem ain small even today most of the theory is completely linear w ith respect to the initial conditions and can be discussed w ithout reference to them . Large portions of this chapter essentially follow th e treatm en t of M a & Bert- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 schinger [47], with additional portions from Dodelson[17] and Hu[30]. However, I have om itted m aterial not relevant to th e later chapters of this work, and have tried to fill in some details and unify th e presentation to be consistent w ith the notation used elsewhere in this work. 3.2 A n overview o f th e m eth o d All techniques for solving for th e evolution of small perturbations in the uni verse sta rt w ith the Boltzm ann equation. The full phase space occupation function f ( x , q , h , r ) for each species of particle is broken up into a background term fo(q) and a first order perturbation f ^ ( x , q, h, r ) , (3.1) where q is the m agnitude of the comoving particle m omentum , h is a unit vector describing the direction, and r is conformal tim e1. The background distribution fo(q) depends only on q because of th e assum ption th a t th e background is both homogeneous and isotropic. It is also generally assumed th a t each particle species started out at equilibrium in the early universe, and so the background distribution has the form (3.2) where gs is the num ber of spin degrees of freedom, h is Planck’s constant, m is the mass of the particle, k s is B oltzm ann’s constant, and T0 is th e tem perature of the particles today (+ is for fermions, — for bosons). 1For th is chapter an d th is chapter only, for convenience overdots (such as in a) will represent derivatives w ith respect to conform al tim e. T his is different from th e n o ta tio n of m ost of th e rest of th is work. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Because the photon perturbation is always small, we can define the tem perature fluctuations of the photons A T /T w ith the expression f { x , g, n, t ) = f 0 ^ > (3 -3) which means the photons have a therm al distribution w ith tem perature T + AT. Combined w ith equation 3.1, some m anipulation reveals th a t as long as the tem perature fluctuations are small T- = - ( ^ f ) <3-4> It will tu rn out later th a t the tem perature fluctuations are in fact independent of g, so we can shift to Fourier space and expand n in Legendre polynomials to define A e(k,r) as f AT — (x,h,r) .° ° , -> = / d3k e lk'x ]P (-i)* (2 f! + l ) A i (h,T)Pi (k ■ n). (3.5) e=o Because the evolution of small perturbations is linear, different wavenumber modes should evolve independently, and we can separate A e (k ,r) into a piece representing the initial prim ordial perturbations 5a and a transfer function which depends only on k = \k\, A t (k, t ) = SH( k ) A e(k, r ) (3.6) is the photon tem perature at th e origin (x = 0), today ( r = W hat we observe r 0), as a function only of the direction n. This function is th en usually expanded in term s of the spherical harmonics AT, (n = ^ 2 ^ 2 aemYem(n), ( 3 . 7) T r=o m=—i which, together with the previous expressions implies the following expression for the spherical harmonic coefficients aim = ( - 0 ^471- [ cPkY?m (k)6H( k ) A e ( k , T = T0). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 3.8) 17 Finally, the results of observations are usually reported using the angular power spectrum Ce of these spherical harmonics, which is defined by the expression (O'imQ'l1m') In theory, the angle brackets indicate an ensemble average over the probability distribution of the prim ordial power spectrum (no theory gives exactly where the density fluctuations occurred, only a probability distribution). In reality, however, we can only observe one universe and the only averaging possible is over the 21 + 1 values of m for each multipole moment I. Thus especially at low 7, there will always be some uncertainty as to whether the m easured Ce is close to the “tru e” average Ce for our universe. This uncertainty goes by th e nam e of cosmic variance. Finally, combining the definition of the angular power spectrum w ith the relation of the spherical harm onics coefficients to the photon distribution gives th e im portant result Ce = 4n J d3k P s ( k ) A j ( k , r = r 0), (3.10) where Ps(k) is th e power spectrum of prim ordial fluctuations. T he basic procedure for calculating the Ce from a theoretical model, then, is to obtain a set of coupled differential equations from the B oltzm ann equation for each species of particle, and integrate the differential equations until today to solve for A e ( k , t) . This, combined with a model for the prim ordial power spectrum (usually given by inflation), gives the angular power spectrum CeHistorically, th e first numerical codes used the system of coupled differential equations to solve for the photon distribution and hence A e(k, r ) , evolving all multipoles of interest forward in tim e until today. However, after recom bination the universe becomes largely transparent to photons, and to lowest order they stream Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 freely to us today. This means th a t the small angular scale (large £) fluctuations we see today were really just larger scale (smaller £) fluctuations at the last scat tering surface th a t we see projected onto th e sky. Modern codes (starting with C M B F A S T [59]) take advantage of this realization to evolve a much smaller set of differential equations (because only lower multipoles are needed) and then project the anisotropies onto the sky to find the angular power spectrum today. 3.3 C hoice o f gauge To consider the evolution of perturbations during th e radiation era we need to pick a coordinate system in which to m easure those perturbations. There are two systems commonly used for tracking perturbations, each w ith advantages and disadvantages. For this section and most of this chapter I will presume flatness of the universe. Including curvature makes only minor changes to a few equations; typically only equations for the m etric are directly affected, and I will note where this occurs. Otherwise the most significant changes are in the expansions in spher ical harmonics; these functions simply change to their ultra-spherical counterparts, which modifies the geometric factors in their recurrence relations. 3.3 .1 S y n ch ro n o u s g a u g e The synchronous gauge is defined by requiring th a t all th e clocks of freely falling observers agree (i.e. are synchronized, hence th e name). This is equivalent to requir ing th a t the tim e-tim e and time-space com ponents of the p ertu rb ed m etric rem ain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 identically zero. Hence the line pertu rb ed line element is ds 2 = o 2(r) [—d r 2 + (<5y + /ip-(x, r ) ) dx ld x ^ , (3.11) where r is conformal time, and the sym metric tensor h y ( x ,r ) is the perturbed p art of the metric. This tensor has six degrees of freedom, and these are typically divided up according to there spatial transform ation properties. Two degrees of freedom transform as scalars, two as vectors, and two as tensors. The inclusion of vector and tensor degrees of freedom is one of the strengths of the synchronous gauge; i t ’s weakness is th a t the gauge conditions do not completely fix the gauge degrees of freedom and so there will be some unphysical solutions for the density perturbations. Almost all numerical codes work in th e synchronous gauge. R ather th a n work in real space, i t ’s easier to use th e linearity of perturbations to evolve each wavenumber k independently. The scalar p art of th e metric p ertu r bation can be Fourier transform ed h-®caiar)(£, t ) = J d3k el*'x k i k j h ( k , r ) + ^ Qrj(k,r) where h(k, r ) and r](k, r ) now represent th e two scalar degrees of freedom in Fourier space. Vector perturbations decay for an expanding universe and are only sourced by exotic forms of m atter (such as cosmic strings), so they will be ignored. Tensor perturbations can propagate (these are essentially gravitational waves), bu t are only sourced by prim ordial tensor fluctuations. Their contribution to the observed microwave background tem perature anisotropy is mostly at low £, and even there it may be impossible to separate from other effects, so I will not discuss them in detail. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 3 .3 .2 C on form a l N e w t o n ia n g a u ge T he conformal Newtonian gauge is m otivated by the Newtonian limit of general relativity. The perturbed line element is ds2 = a2( r ) [—(1 + 2 T ( x ,r ) ) d r 2 + (1 + 2 $ (T ,r ) ) d x ldxi\ , where the two fields (3.13) r ) and T (x, r ) are scalar potentials where in the New tonian limit 'h is simply the gravitational potential and $ = —T. Typically the Fourier transform s of these fields are used directly in the evolution equations. The advantages of this gauge are th a t the m etric remains diagonal and th a t the degrees of freedom have a simple physical interpretation. The gauge degrees of freedom are also fixed by this gauge so there are no unphysical solutions. The disadvan tage is th a t tensor degrees of freedom are not easily added to this formulation, and there are some numerical issues which arise during com putation th a t make the synchronous gauge somewhat easier to use. 3.4 E volu tion eq u ation s 3.4.1 M e tr ic The background metric is still th a t of a Friedm ann-Lem aitre-Robertson-W alker cosmology, so the evolution of the scale factor is given by the two equations (overdot indicates a derivative with respect to conformal time) a' 2 aJ 8-jrGa2 p, 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.14) 21 where the overbar indicates the background average of density p or pressure P. Curvature can be included in th e first equation above as a component of the energy density which scales as a -2 ; w ith this scaling the equation of state is P = —(1 /3 )p and so curvature does not contribute to the second equation. Plugging the perturbed m etric of equation 3.11 into E instein’s equation R ^ u — \gnvR = T)W and carrying through to linear order gives a coupled system of equa tions for the evolution of h ( k , r ) and r](k,T) k 2rj — Z CL = ^ k G oPST q , (3.16) k 2f] = 4tt G aH V ST f, (3.17) h + 2 - h — 2k2r] = —87rG a 2hT/, h + 677 + 2 ^ (h + 677) - 2k2rj = 24ttG o2 (kCkj - ^ 3.4.2 (3.18) [T] - ^ T fcfe), (3.19) G e n e ra l p a rticles The evolution of the particle distribution for each species follows the Boltzmann equation df df d f dx d f dq d f dn . . T r = T P T iT r + i T r + A T r = C^ ^ where C represents any collisional term s th a t may be present. Recalling th a t we split the distribution function into a background piece and a first-order term , we wish to find the first-order part of th e Boltzm ann equation. T he first two term s are easy, the fourth consists of two first-order pieces m ultiplied and so can be neglected. The tricky one is the th ird term . The comoving m om entum q we have been using does not include the metric perturbations and is thus not exactly equal to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 conjugate m om entum to x. The true conjugate m om entum is Pi = V™3 - (3 -2 1 ) Prom the requirem ent th a t the time derivative of the conjugate mom entum be equal to zero we can find the first order expression for d q / d r , = -\qhijniU j. clt (3.22) Z Then the first order Boltzm ann equation (after dividing through by fo(q)) is 1 i d l n f 0(q) _ l n [f l 2 q lj7llUj dq ~ f0 ^ ’ <9/(1) , dr + e f (0 oqN ( ^ or, after a Fourier transform ation ^ or + i «(jU )/<» + d in g e = 1 C[/]. f0 (3.24) M assless particles For massless or extremely non-relativistic particles, either th e simplification q —>e or q —►0 can be made. In either case, it is then simple to integrate out the g-dependence of the Boltzm ann equation to obtain a new variable g S S t d e M t m U n , T) (3.25) J g dq g /0(g) Neglecting the collision term , the B oltzm ann equation in this variable is dF + ik fiF = 4 rj — |( f i + 6fi)/i2 or (massless), (3.26) (3.27) where f i = k - n , and the factor of 4 comes from th e integral of d l n / 0/d ln g . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 Even further, the directional dependence can be decomposed into spherical har monics. A further assum ption th a t the m om entum distribution depends only on the angle between k and h (i.e. th a t there is no externally specified preferred an-—+ gle for the particle distribution) means th a t F( k, h, r ) can be w ritten as a sum of Legendre polynomials OO F(ic, h, r ) = 2£ + 1)Fe(k, r ) P e{k ■h). e=o (3.28) The Fe(k,r) have been defined in such a way th a t they are simply the harm on ics of the (Fourier transform of) the fractional particle density, thus the density perturbation S = 8p/p is simply equal to F0. The fluid velocity and shear term s simply correspond to the dipole and quadrapole pieces of F as well. Using the recursion relation for the Legendre polynomials, their orthogonality, and the fact th a t P^{p) = |(3^r2 — 1), the Boltzm ann equation can th en be further reduced to a hierarchy of multipole equations dF 2 = —kFi — - h f = ^ ( F o - 2F2), f)F (massless), (3.29) k a 7 = 2 7 T T ^ - ‘ - < < + 1>F« ! - (<>2) for massless particles. Ideally, massive particles are am enable to a similar treatm ent. In practice the two im portant non-relativistic fluids are cold dark m atter, which can be dealt w ith even more simply, and baryons, which couple to photons and have significant corrections due to collision term s and finite (even if small) sound speed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 3 .4 .3 P h oton s The starting point for the photon equations is the hierarchy of equations already derived for ideal massless particles. The photons only differ through th e addition of scattering term s proportional to aneaT (derived in [7]), where n e is the electron density and aT the Thom son scattering cross-section. The scattering depends on the photon polarization; in order to take this account properly we can introduce another variable G(k, h, r ) which is the difference of th e two polarization components. The original F ( k , h , r ) simply represents the sum over polarization states'. The two multipole hierarchies of equations are then 2 —— = —kF\ — - h or 3 (photons), (3.30) ? I ± = h F o - 2F2) + a n ea r (F1baryon - Fx), or o f)P k 4 1 —— = —(2Fi — 3F3) + — (h + 6fj) + — aneOT(Go + G 2 — 9 F2), or 5 15 10 ^o r = IdI T+T T1 [ ^ - 1 - ~ ane ^ F e, (£ > 2) for the sum of polarizations, and 8G 1 —— = —kG\ + - a n eOT(F2 — Go + Gq) or 2 (photons), — — = —(Go — 2 G 2) —aneoTGi, or 3 <9G k 1 —— = —(2Gi — 3G3) + — arz.e<Jr(Go + F 2 — 9 G 2), or 5 10 = 2Z T 1 {lGl- x ~ {£ + 1)G' +l] " an*UTG^ ^ > 2) for the difference. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.31) 25 3 .4 .4 N e u tr in o s Massless neutrinos follow exactly th e equations derived earlier for generic mass less particles. Massive neutrinos species, however, cause a bit of a problem since interesting mass ranges (roughly .1-10 eV) correspond to a regime where the neu trinos are transitioning from relativistic to non-relativistic during th e radiation and recombination era. Thus neither the massless nor extremely massive approxima tions are appropriate. The traditional solution is to take the original distribution function of all vari ables f ^ ( k , q, n, t ) and define a new set of variables dq by expanding in the Leg endre polynomials 00 f {1\ k , q, n, t ) = ^ 2 ( - i ) e(2£ + 1 q, r ) P t (k ■n) (3.32) t=o This results in a hierarchy of differential equations th a t depend not only on wavenumber (and tim e), but on particle m om entum as well dTo = —fc -T i + e 6 dr dr = ^-(T O6 o- am q (massive neutrinos), (3.33) 2 T 2), d^2 dr cMe d r = 2 ? y r f ( » ' - ■ - (« + 1)'f<+l1' (0 2) To relate back to things like the density or velocity pertu rb atio n one needs to solve these equations and then integrate th e appropriate multipole moment over q. Computationally, this can become cumbersome because not only does one need to grid over wavenumber k bu t also over particle m om entum q, using enough grid points th a t the numeric integration over q needed at the end can be done accurately. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 3 .4 .5 C o ld dark m a tte r In the synchronous gauge, coordinates are defined by freely falling observers. Since cold dark m atter m atter is by definition a pressureless fluid of particles th a t interact only gravitationally, th e dark m atter particle coordinates can be used as the definition of the coordinates in the synchronous gauge. Thus in the synchronous gauge, the cold dark m atter particle velocities are effectively absorbed into the coordinates, and the fluid can be described by a single equation flT? 1 ^ = -± h (cdm), (3.34) where the factor of 1/2 is related to the equation of state. 3 .4 .6 Baryons Baryons are massive particles, but their velocities do not necessarily match those of the cold dark m atter and so may be non-zero in th e synchronous gauge. They also interact strongly w ith photons before recom bination and thus a collision term is present. It is a reasonable approxim ation, however, to neglect the higher moments (t > 1) of the m omentum distribution. The two equations describing the baryon distribution are = -kF i - -h or (baryons), (3.35) = _ “ F 1 + c2A;Fo + ^ a n et7T(F 1 - F 1photoIl) ) a 3 where R is the background photon-to-baryon energy density ratio and is the square of the baryon sound speed. For an isolated non-relativistic fluid the speed would effectively be zero, bu t interaction w ith th e photons results in an Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. sound 27 im portant correction. The final term of the differential equation for F\ comes from Thom son scattering; it is the m irror of th e corresponding term in the equation for ^photon aiKj ^ ie coefficieilt is the consequence of m om entum conservation. 3 .4 .7 D a rk e n er g y If the observed cosmological acceleration is due to a cosmological fluid, th en it too can be described with a distribution function and may have perturbations. For a general fluid w ith a constant equation of state w which relates the background density and pressure P = wp, an alternate approach to deriving the distribution equations (see appendix C.2 for details) gives (3'36) - 3aa { % - W) S - { 1 + w ) ( e + > i ) ' Q = - - ( 1 - 3w)9 - - ^ — e + k 2 ( ay ’ 1+ w \ l + w5p - a] , J where 5 = Fo = 8p/p, 0 = T \ / ( l + w) and is th e divergence of the fluid velocity, and a = F2/2, sometimes called the shear p ertu rb atio n 2. The above pair of equations is not closed, and a different derivation is needed to provide differential equations for a and higher moments of the distribution. For dark energy, however, it is sufficient to take th e lim it k —►0 as dark energy models m eant to approxim ate the behavior of a cosmological constant tend not to fluctuate strongly on small scales. The above equations also illustrate some of th e problems faced when considering dark energy perturbations. For an isentropic fluid, 5P/5p is the square of the sound C o n v e n tio n s v ary on th e definition of th e shear p ertu rb atio n . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 speed cs, which in tu rn is equal to o Cs = dP dp — dw w + p— dp (3.37) As long as re is a fixed constant, c?s = |u>| and the above differential equations simplify to (for large scales) -(1 + w) \ 0 + h (3.38) 6 = — (1 — 3w)9. CL There would already seem to be causality problems for models w ith w < —1, how ever if w is slowly varying (as it does in many dark energy models) the sound speed as defined above formally diverges as w approaches —1, simply because for w = —1 the density p becomes constant and thus d w / d p diverges. This poses difficulty b o th in model building and numerical calculation of dark energy perturbations. 3.5 Line o f sigh t effects As stated in section 3.2, the most common technique to solve for the microwave background anisotropies today is to evolve the entire coupled system of equations described in the previous section up until the after the photons decouple, and then basically project the result onto th e sky we see today. After decoupling, photons travel almost completely freely until we observe them , experiencing only the cosmological redshift as the universe expands. However, there are two corrections to this picture due to secondary effects during this tim e th a t will be im portant in later chapters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 3 .5 .1 In te g r a te d S ach s-W o lfe effect Even if the photons have decoupled from other m atter, they still feel the effects of gravity. Once we have the distribution function F for photons after decoupling we could project it onto the sky to find the tem perature anisotropies today, except for the presence of the h term in th e evolution equations. Because of this, the full expression for the anisotropies today will contain an integral over h, or in the more familiar conformal Newtonian gauge, a term th a t looks like /■today /-today F dr = [other terms] + / / J decoupling (T — $ )e lfc^ T-Ttoday)d r (3.39) Jdecoupling After decoupling, $ = - $ to a high degree of accuracy and so this effect amounts to an integral over —2$. 3 .5 .2 L en sin g The integrated Sachs-Wolfe effect directly changes the tem perature of each pho ton as it travels through changing gravitational potentials. G ravity can also deflect the photon paths themselves, leading to a shift in the observed anisotropy power spectrum. This effect is most easily analyzed using w hat is called th e projected potential <f)(h), which is defined in such a way th a t the deflection angle of a photon observed in direction n is equal to V<p{n). W ith this definition, the projected poten tial in term s of the full three-dim ensional gravitational potential (in the Newtonian gauge) is /•today (j>(n) = - 2 d r ^ ( r ) $ ( f ,r ) , (3.40) J decoupling where the integral over r is equivalent to an integral over distance for th e photon traveling in direction n, and is a geometrical factor th a t expresses the lensing Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 efficiency. The factor of two is the same factor of two th a t always arises when com paring photon deflections in general relativity to th a t of the Newtonian prediction. T he relationship between the lensed and unlensed tem perature perturbations is therefore A y le n s e d A T — y — (ft) = - jr (ft+ W ) (3.41) AT AT = — (ft) + V ^(ft) ■V — (ft) + i v . « f t ) V ^ ( n ) V iV ' ^ ( f t ) + ■■. , including up to second order term s. The full analysis using spherical harmonics involves some difficult m athem atics, but it is possible to capture the physics by using a Fourier transform instead. This is equivalent to treatin g th e sky as a plane (the fiat-sky approxim ation), and is valid as long as the angles of interest are small. The transform is defined by f(i) = J dhf(n)e-'l \ (3.42) so the lensed tem perature perturbations are - A £ ■k - A 1 ff 2 JJ where = l\ + « , A T (- } (2tt)4 T j ^ <3-43) ^^ — I. The power spectra are defined by ( 2 7 (2tt)2<5(£-- i)CjT = ( h f , . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.44) 31 Plugging in the expression for the lensing tem perature anisotropies to find the lensed tem perature Ci is made easier by sticking to second order or lower terms, in which case there are only four combinations of the term s from equation 3.43. Also, the term th a t has the form AT1* f H2f AT1 J \ 4) 4 • ( « - « ) , (3.45) will be equal to zero, since the delta functions in the definition of the spectra force £x = £ = £' for this expression. This leaves three term s, which after accounting for all the delta functions leads to lensed C j T = C j T + f£ x £x ■ ( £ - £ ( ) ( 2t t ) 2 2 c r ^<ir«i c t! (3.46) n2r TT f d£2 p4_n 4><i> -2 The final term simply acts to remove power added by th e second term , as a conse quence of th e familiar notion th a t lensing should conserve photons over the whole sky. Thus given a relatively sm ooth lens power spectrum , lensing effectively smears out features in the tem perature angular power spectrum . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 C hapter 4 T esting Inflation w ith the M icrowave Background 4.1 In tro d u ctio n Over the last several years, extraordinary observational support has emerged for the idea th a t key features of our universe were formed by a period of cosmic in flation. During inflation, the universe enters a period of “superlum inal expansion” which im prints certain features on the universe. The physical degree of freedom responsible for inflation, generically called th e “inflaton” , has yet to find a com fortable home in fundam ental theory, and there are many competing ideas for how fundam ental aspects of inflation could play out. None th e less, at the phenomeno logical level a standard picture of inflation has emerged. For details, please refer to chapter 2 of this work. From the observational point of view, th e standard picture is defined by a set Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 of observable characteristics th a t are the same across virtually all proposed models for th e inflaton. The most well known predictions from the standard picture of in flation are th a t the universe has critical density (to w ithin roughly one part in 105), th a t the prim ordial perturbations are coherent, (leading, for example, to acoustic peaks in the microwave background power spectrum ), and th a t the power spec tru m of prim ordial perturbations is nearly scale invariant, w ith th e tilt param eter n s constrained to be close to unity. A unique spectrum of coherent gravitational waves is also predicted, which could eventually come w ithin range of direct gravi tational wave detectors, and which could also be observed indirectly via signals in the microwave background polarization. B ut inflation makes many more predictions th an these. Specifically, a given model for the inflaton will predict a detailed shape for the prim ordial power spec tru m th a t goes way beyond w hat can be described simply by a single tilt param eter. The detailed shape of the power spectrum is a reflection of the particular evolu tion of the inflaton during inflation, som ething th a t is precisely specified in a given model. In this chapter we show how th e next generation of experim ents could bring studies of the power spectrum shape to a whole new level. These studies present two kinds of opportunities: One opportunity is to make additional tests of the standard picture of inflation. To this end, we focus on particular power spectrum features th a t are known to exist across essentially all inflation models. The search for these features could either confirm or falsify the stan d ard picture of inflation. The second opportunity is to go beyond broad tests of the standard picture. The next generation of experiments which we consider here will provide im portant additional information. This inform ation could actually distinguish among different specific inflaton models, assuming th e stan d ard model is not falsified, or it could Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 provide very useful constraints on the alternatives if th e standard model is ruled out. Our approach here is similar to and inspired by a large body of earlier work on this subject [13, 36, 39, 33, 12, 18, 34, 27, 28]. Our emphasis here is identifying w hat useful inform ation about the prim ordial power spectrum and inflation might be revealed by a new generation of experiments. For this work we assume th a t the prim ordial perturbations are adiabatic. As emphasized in [9], relaxing this assum ption would result in more degeneracies and would lead to somewhat weaker constraints on param eters. The organization of this chapter is as follows: Section 4.2 gives background inform ation about slow roll inflation. Section 4.2.1 introduces the aspects of slow roll inflation we intend to test. Section 4.3 discusses the CMB and Lym an-a d a ta (existing and simulated) we use to test inflation. Section 4.4 gives our m ain results and 4.5 gives our conclusions. A ppendix A gives details of the inflation models we use for our plots. This work was first published in [24]. 4.2 Scalar F ield In flation In the standard picture, inflation occurs when the potential energy density V (</>) of a scalar field (j) (the inflaton) dom inates th e stress-energy [25, 2, 42, 43]. This scalar field may be a tru e scalar field or an effective field obtained from some more complicated theory. The period of potential dom ination is usually closely connected to very slow evolution of the inflaton field, the so-called “slow roll” behavior, and it is this slow evolution th a t produces a nearly scale invariant spectrum of p e rtu r Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 bations. In the slow roll inflationary scenario, however, 0, and therefore V(<j>), are not completely constant during inflation, and this leads to deviations from total scale invariance. The dynamic behavior of the potential is determ ined by the equation of motion for a scalar field in an expanding universe 0 + 3 H(f> + V'{4>) = 0. (4.1) The gradient of the field is ignored, as even if present it will be quickly damped by the inflationary expansion to th e degree th a t it is irrelevant for th e classical evolution of the background spacetime, which is w hat we determ ine from Eqn. 4.1. The field is considered to be in a slow roll regime if 0 is negligible. The Hubble constant H = a /a is related to the to ta l energy density of th e universe which if dom inated by th e scalar field is (4.2) where M P = 1/ x/SttG has been set to unity. It is custom ary to define slow roll param eters such as (4.3) although several other conventions also exist in the literature. Assuming th a t these param eters (and the higher derivatives of V ) are small leads to expressions for the primordial am plitude of density p ertu rb atio n s1 [3] (4.4) 1T he am plitude squared A $ 2{k) is sim ply k 3Ps(k) from c h ap te r 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 the spectral index of these perturbations d l n k A s 2(k) ns{k)= dink = 1 + 2’> - f e > <4 -5> and the derivative of the spectral index n s '{k) = flln l = —2£2 + 1677c - 24e2. (4.6) The right-hand side of each equation above is to be evaluated during inflation at the tim e when the scale of interest k exits th e Hubble radius. 4 .2 .1 S low roll, ns, an d n s A general feature of slow roll is th a t ns' is higher order in the slow roll param eters th an ns- Thus if we assume higher order term s become increasingly small, then barring a conspiracy of cancellation between term s Iris'| ~ (ns — l ) 2 or less. As pointed out in [19], while this assum ption is commonly m ade it is an addition to the common assum ption th a t the slow roll param eters are small, at least when formally considering “the space of all possible inflation models” . We emphasize here th a t in practice the slow roll hierarchy between n s and (ns)' is indeed realized in the vast m ajority of published models, so detection of a large ns', while not completely ruling out slow roll, would force a rethinking of the standard picture of inflation. This relation can be generalized to higher derivatives, resulting in a kind of consistency relation dnns |n + l < It s - 1 d in kn (4.7) which could be taken as defining a kind of ‘norm al’ class of inflationary models. For inflationary scenarios involving m ultiple fields (often called hybrid models) this condition is relaxed. W ith multiple fields the extra freedom introduced makes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 0.08 n 1 r j L i- . r l. “i r ~\ r J L 0.06 0.04 0.02 0.002 n “i r r J L n r j e r 0.001 □ ........ 0 0 J l 0.02 I L □ 0.04 ln s - J I 0.06 L 0.08 !| Figure 4.1: The values of n s and n s ' at CMB scales are plotted for several models. The lower graph is a zoomed view of th e b ottom of the upper graph. In both, th e upper line is ns' = n s — 1 and th e lower line is ns' = (ns — l ) 2. The thick line on the lower graph shows the evolution of n s and n s w ith scale for two orders of m agnitude in each direction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 it easier to remain on the edge of violating the slow roll conditions over many efoldings. Thus the tendency for models to lie under the curve |n s'| = (ns — l ) 2 is not as strong for hybrid models. 'Of course, theories which generate prim ordial density perturbations from something other th an inflation also have no need to obey the above consistency relation. To discuss observational constraints, we take the point of view of [63] th a t the prim ordial power spectrum is an unknown function, which may be sampled by experiments at one or more scales. Statem ents about th e slope of this function (and higher derivatives) then can only be tested by effectively sampling the function to high accuracy at several nearby scales. C urrent analysis tends to use all th e d ata to provide only lim ited inform ation about th e power spectrum. We wish to emphasize th a t higher quality d a ta over a range of scales will allow us extract significantly more inform ation about the prim ordial power spectrum , inform ation th a t can have a great impact on tests of the inflationary picture. 4.2 .2 M o d e l sp ace One of the simplest models to evaluate is a pure exponential. For V = V0 exp(A0) the spectral index n s — 1 —A2 for all scales, and thus ns' and all higher derivatives are zero. Thus for this model m easurem ents of n s simply m ap into constraints on A, w ithout presenting an opportunity to falsify the general model. B ut a measurem ent of ns' which excludes zero can rule this type of inflaton potential altogether. This model is special in th a t the potential is constructed to form th e simplest possible power law spectrum of perturbations. Most inflationary models have more complicated forms, but many proposed models approxim ate th e exponential behav Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 ior on the scales which cosmological measurements probe. A previous survey of models of inflation and their spectral index can be found in [46]. In figure 4.1 several different models taken from a sampling of the literature have been plotted. (Specific inform ation about each model plotted is given in the Appendix.) The types of models range from high-order polynomial to mass-term to brane-world inspired scenarios [40, 20, 26, 60, 43]. Despite the difference in the form of each m odel’s potential, almost all of these live on or below the line \ns '\ = (ns - l)2. C ertain models can exhibit more exotic behavior, such as the running-mass model described in [14] (an example this type of model is m arked by the star on Figure 1), or the interesting type of potentials given by Stew art and Lyth[19]. These can give |n s'| ~ \ns — 1| over a range of scales, resulting in a markedly different prim ordial power spectrum . These models form an im portant ‘altern ate’ class of models which will be easy for future d a ta to confirm or rule out. 4.3 D eterm in in g how w ell ex p erim en ts can do To find the possible im pact of CMB and Lyman-a: experim ents, we model the primordial power spectrum w ith a Taylor series expansion of the spectral index around a particular scale / 7 \ - l + n s + n s ' x + \ n s " x 2+... A s 2(k) = P ( p ) . (4.8) where x = In(k/k*), k* is th e pivot point, and P is an overall normalization. We then use Fisher m atrix techniques to jointly estim ate param eters for each experi ment. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 101 1 1 1 11111--------- 1— 1 1 1 11111 1 1 1 11 m i i 1— r I I 1IIII I I I I I I14- A r* 10 ° id " T t itI 10"1 s w < 10-2 10-3 10-4 10-4 I I I 10- 3 10"2 I I I 1 I I I 1 I 1 10- 1 1 I I I I I I I I I I I 10 ° 1 I I I I I 101 k ( h M p c -1) Figure 4.2: One-sigma error in the (binned) prim ordial power spectrum from a WMAP-like microwave anisotropy tem perature and polarization experim ent (squares w ith solid error bars), and from a Lyman-o: experim ent as in [15] (crosses with dashed error bars). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 10° io 02 -1 CO W 02 C < *0 10 ° k ( h M p c -1) Figure 4.3: One-sigma error in the (binned) prim ordial power spectrum from an after-Planck tem perature and polarization experim ent (squares), and from a Lyman-o: experiment w ith to tal uncertainties one hundred tim es smaller th a n cur rent experiments (triangles). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 Param eterizing the power spectrum a particular way has its own set of advan tages and disadvantages. One nice feature of the step-wise form of [64] is th a t it is easy for each bin am plitude to be a nearly statistically independent param eter in th e likelihood analysis. The coefficients of a Taylor series expansion generally have larger covariances. The disadvantage of th e step-wise form, however, is th a t th e quantities of interest to us (such as th e spectral index) are not simply related to th e shape param eters. 4 .3 .1 CMB We first consider CMB experim ents which measure b o th th e tem perature and polarization anisotropies. For scalar perturbations there are three power spectra ( T E C) described by C\ ' ' , where (T, E, C) indicate the tem perature, ZTmode polariza tion, and cross-correlation power spectra. These Q all have similar dependence on the prim ordial power spectrum , and are found by Cl = (4x)2 J " A s 2( k ) |A ,(k, t = To) | 2 , (4.9) where A ?(k,T = r 0)are transfer functions for th e CMB, and A s 2(k) is the squared am plitude of the prim ordial power spectrum . The functions A e ( k , r = To) depend on cosmological param eters, and can be conveniently calculated using th e CMBFAST code [59]. Error in a cosmological param eter Sj can be estim ated as y j (F -1 )^, where F is the Fisher m atrix *■«= £ I,J=(T,E,C) E g ( 0 - ' ) 4 i 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.10) 43 The covariance m atrix C has elements which can be approxim ated by [36, 65] CtT = ( U + l ) U y (Cj + N l ) 2 ' (4U) c “ - W (4 1 2 ) Ccc = ^ [ c f + N f f ’ (2£ + l)/.kj. x [ ( C f )2 + [Cj + Nj) (Cf + N f ) ] , (4.13) for the diagonal elements, and th e off-diagonal elements are ^ In equations 4 .1 1 C tc = c- = through 4 .1 6 , (4 1 4 ) ( 2 1 + 1 ) U y C‘ { C l + N ' ) ' (415) ( 4 ' 1 6 ) f sky is th e fractional sky coverage of the experiment, and we have defined a noise term n where <J(t ,e ) ( T, e ) 2 til (O.4250fwhmq 2 — 0(T,JS)Pfwhme i ( a 1?\ is the noise per pixel in th e tem perature and polarization measurements and #fwhm is the w idth of the beam. For experiments like W M A P which obtain tem perature and polarization d a ta by adding and differencing two polarization states, the noise per pixel for each is related by o r 2 = cte2/2. The derivatives dC t/ d s i are evaluated via finite difference using a numerical code derived from CMBFAST and DASh [37]. We consider only flat models, using as param eters 2the acoustic angular scale £&,£lmh2, exp(—2 r), the prim ordial power spectrum norm alization P, and the first seven coefficients in th e expansion 2 For an analytic expression for £a , see [29], These cosmological p aram e te rs correspond closely th e th e A , B, A4, and Z p aram eters of [38]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 of n$. We use 0.05h M pc-1 as the pivot. The Fisher m atrix calculation for the errors in the param eters also assumes a “tru e” model around which the derivatives are taken; for this we use a ACDM model with fh,h2 = 0.0222, Q,mh2 = 0.136, Qa = 0.73, r = 0.1, and a flat prim ordial power spectrum . 4.3.2 Lyman-ct For the Lyman-a: data, the error bars th a t were reported in [15] for the linear m atter power spectrum are used. The prim ordial power spectrum is related to the linear m atter power spectrum by PLu(k) = P0k A s 2( k ) T 2(k), (4.18) where T 2(k ) is the transfer function and contains the dependence upon cosmological param eters, and Pq is a normalizing constant. T hen th e error bars for the power spectrum param eters are calculated via standard error propagation techniques using the previous equation and the analytic form for the transfer function [51] = [q) ln (l + 2.34 q) 2.34q x [l + aq + (bq)2 + (c q f + (dq)4] , (4.19) where q= k / ilti2 exp(—2fh) and a , b, c, d are fit param eters which are irrelevant to the error analysis. Weuse the results of the CMB param eter estim ation as inputs for determ ining the errors in h, fl, and Q(,. The large system atic norm alization error reported in [15] is a problem for esti m ating prim ordial power spectrum am plitudes, bu t does not affect local estim ates of the slope or higher derivatives, so we do not include it. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 Table 4.1: Noise param eters used for simulating various experiments. Experim ent W MAP-like PLANCK-like SPT CMB-pol 4 .3 .3 aT 20 fxK 10 /rK 12 fjK 3 /iK 6>fwhm 18’ 6’ 0.9’ 3’ / sky 0.7 0.7 0.1 0.7 l max 1500 3000 4000 3000 Error co n to u rs for cu rren t and fu tu r e d a ta An illustration of CMB and Lyman-o: constraints on th e prim ordial power spec tru m is shown in figures 4.2 and 4.3. If we fit a function to all the d ata points, and assume th a t function to be linear th en of course th e slope will be tightly con strained. If we allow the function to have a more complicated shape, the slope at any point becomes less well constrained. Figure 4.2 roughly represents current experim ental limits. The main point of this chapter is th a t future d ata can become good enough to loosen the assumptions on the shape and still produce very tight constraints. Figure 4.3 gives an illustration, by showing constraints on the binned power spectrum form some future experim ents accurate enough to clearly distin guish a model w ith n s — 0.05 from a model w ith rts = (0.05)2, even using d ata spanning only one order of m agnitude in wavenumber. The ultim ate limiting factor in how precise all these measurem ents can be is due to cosmic variance. For the CMB, the fractional error from this effect is A C t/C t ~ 1/y/I , which m eans even for large i each individual Ci can only ever be known to w ithin a few percent. Thus the error in figure 4.3 is m ostly cosmic variance limited, and the only way to further reduce the error is to assume some smoothness for th e prim ordial power spectrum and bin the d ata. Figures 4.2 and 4.3 use a step-wise param etrization to constrain th e prim ordial power spectrum in bins in In k, which Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 0.05 P L A N C K -lik e ■SPT ■CMB-pol 0 - 0 .0 5 0.05 0 n0 - 0.05 n fidu cial Figure 4.4: 68% confidence regions in the n s ~ n s plane for various experiments of the sort described in table 4.1. The SPT constraints are shown dashed, PLANCKlike dotted, and CMB-pol constraints solid. We use prior 1-6 as described in section 4.4. The thin lines are constraints w ithout polarization information. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 0.05 ■PLANCK-like ■SPT ’f id u cial ■CMB-pol 0 - 0 .0 5 0.05 0 ns - 0.05 n£ducial Figure 4.5: 68% confidence regions in th e n s~ ns plane for the same experim ents as in figure 4.4, only now using prior lib as described in section 4.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 0.08 CMB ■p rojected 0.06 ^S SSW M A P ext 0.04 0.02 — f 0 0 0.02 - - i~r i 0.04 ns - 0.06 0.08 1 Figure 4.6: Solid contours show constraints in the ns ~n s' plane from the CMB d a ta of the “CMB-pol” experim ent of table 4.1 (using prior 1-6) for scales near k = 0.0565. Several different fiducial models are shown, two representative of the types shown in figure 4.1 and one large-ns' model of th e type described in [19]. The dotted contours are the result of assuming th e consistency relation of equation 4.7 and extrapolating from CMB scales to Lym an-a scales (k = 2.39). If the relation is valid, constraints from th e two types of d a ta will be correlated in this plane as illustrated by the pairs of adjacent solid and dashed contours. All contours are drawn enclosing regions of 68% confidence level. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 is meaningful as long as the power spectrum is a sm ooth function of wavenumber relative to the bin width. For the best CMB d a ta points, which here occur near scales corresponding to t ~ 1000, th e bin w idth is roughly equivalent to binning a few hundred Ci together. From figure 4.3 we can see th a t the CMB accurately probes the primordial power spectrum at somewhat larger scales th an those where th e Lym an-a d ata is most accurate. This means the two experiments provide constraints on n s and ns at different scales, allowing us to further test our models by looking at how they predict these quantities should change w ith wavenumber. We will explore this idea further in section 4.4. To then make error contours in th e n s - n s ' plane, we use the covariance m atrix from the param eter analysis to marginalize over other param eters and determine the covariance m atrix for just n s and n s ' ■ Of th e four hypothetical CMB experi ments listed in table 4.1, we show how well the last three place constraints in the n s - n s plane (for two different priors) in figures 4.4 and 4.5. We examine the fourth experiment (the best) in detail (using th e weakest prior) in figure 4.6. These results are discussed at length in the following section. For hypothetical Lym an-a experi ments, we do not understand the physics connecting th e prim ordial power spectrum to measurements as well as for the CMB. We therefore “sim ulate” improvements in experiments as an overall reduction in statistical uncertainty due to larger samples, and an improved knowledge of the transfer function due to decreased errors in h, Q, and f4. For Lym an-a experiments to provide constraints com petitive w ith those expected from Planck will require datasets roughly 100 tim es larger th an current ones, and more im portantly, an understanding of system atics (or at least those th a t affect estim ates of n s and ns') down to the percent level. Given th a t these sys- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 Table 4.2: M arginalized errors (xlOOO) in n s and ns' for sim ulated experiments w ith different priors on truncating the Taylor expansion of the spectral index. Power spectrum param eters are amplitude, tilt (ns), and for column I-ra, the first m derivatives of the tilt. Experim ent WMAP-like w polarization PLANCK-like w polarization SPT w polarization CMB-pol w polarization prior 1-6 Sns , Sns ' (xlOOO) prior 1-3 5ns , 5ns ' (xlOOO) prior 1-1 5nS) 5ns ' (xlOOO) 309, 203, 20.3, 16.0, 24.6, 15.2, 9.43, 5.42, 258, 195, 16.6, 12.4, 21.5, 10.7, 8.28, 4.08, 69.1, 67.8, 11.6, 6.90, 16.3, 6.37, 6.17, 2.32, 274 260 13.9 10.7 18.6 13.8 6.71 4.37 152 132 10.7 7.48 14.6 8.67 5.67 2.42 33.9 20.6 9.79 3.94 14.5 4.78 5.60 1.75 tem atics represent a lack of understanding of the (scale-dependent) light-to-mass and baryon-to-dark m atter ratios, such an improvement may not appear soon. We hope to study the error for future Lym an-a experim ents in more realistic detail in future work. 4.4 T estin g Inflation 4 .4 .1 R e su lts For completeness, we discuss a range of possible priors, each of which represents a different points of view on w hat one wants to take as an assum ption and w hat one is trying to test. In tables 4.2 and 4.3 we report the constraints th a t various CMB experiments can place on ns and ns'. Our weakest prior, which we refer to as 1-6, is simply to use the four cosmological param eters (I a , &bh2, Om/i2, r) and eight power spectrum param eters (P and the first seven coefficients in the expansion of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 0.08 ■ L y m a n -a •p r o je c te d 0.06 ■ com b in ed 0.04 0.02 0 0 0.02 0.04 ns - 0.06 0.08 1 Figure 4.7: C onstraints in the ns ~ns ' plane from the Lyman-ct d ata of figure 4.3 for scales near k = 2.39 (thin large ovals). The projected ovals from CMB scales (dotted ovals, from Fig. 4.6) are then combined with the Lym an-a d a ta to form improved constraints (thick small ovals). Again, 68% confidence level regions are plotted. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 Table 4.3: Marginalized errors (xlOOO) in n s and n s ' for sim ulated experiments w ith different priors on the higher derivatives of the spectral index. Prior 1-6 from Table 4.2 above is for reference. Priors Ha and lib use all six derivatives in the joint param eter estim ation, bu t Ila imposes the constraint th a t all higher (ns" and above) derivatives are w ithin 0.01 of scale invariant, and lib imposes the consistency relation (Eqn. 4.7) on the higher derivatives. Experim ent WMAP-like w polarization PLANCK-like w polarization SPT w polarization CMB-pol w polarization prior 1-6 5ns , Sns ' (xlOOO) prior Ila 5ns , 5ns ' (xlOOO) prior lib Sns , Sns ' (xlOOO) 309, 203, 20.3, 16.0, 24.6, 15.2, 9.43, 5.42, 97.8, 93.1, 12.3, 7.50, 16.6, 7.36, 6.81, 3.38, 70.6, 69.4, 11.6, 7.26, 16.5, 7.00, 6.36, 3.13, 274 260 13.9 10.7 18.6 13.8 6.71 4.37 38.9 34.1 10.1 5.14 14.5 5.19 5.72 2.26 34.6 22.4 9.79 4.60 14.5 4.80 5.61 1.82 ns). Having so many param eters for the power spectrum allows the shape to vary quite a bit, and loosens constraints on each term of the expansion. We include so many param eters not so much because constraints on all of them will be interesting (some, in fact, will probably always be unm easurable), bu t to show the effect various assumptions about them will have on th e constraints on n s and ns'. If the reader dislikes these param eters, our prior 1-1 is equivalent to not including them at all. To get b etter constraints requires either using a more restrictive prior or im proving the experiment. In table 4.2 we change th e prior by using fewer power spectrum param eters. Prior 1-3 uses only th e first four term s of the expansion of ns (i.e. up to third order), and prior 1-1 uses only the first two terms, such th a t the only power spectrum param eters for prior 1-1 are am plitude, ns, and ns'. In table 4.3 we change the prior by placing a priori constraints on th e higher derivative term s of the expansion, rather th a n dropping them completely. Prior Ila supposes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 th a t all the higher derivative term s (ns" and higher) are “small” , less th an 0.01. Prior lib also imposes the constraint th a t the higher derivatives be “small” , but supposes they fall off in the form of th e consistency relation of equation 4.7. Prior 1-6 represents the weakest assumptions, and prior 1-1 the tightest as sumptions. (Prior 1-1 basically adds ju st one new param eter, ns', to the canonical set.) Prior lib represents an assum ption of the standard inflationary picture for the higher derivatives, b u t places no prior constraint in th e n s — ns' plane so those param eters can be used to test th e standard picture. (Of course, if the the standard slow roll picture fails, prior lib may no longer be of interest) In the tim e since th e preprint of th e paper describing this work first appeared, the W M AP collaboration announced their results [6]. O ur predicted error for ns and ns' of 0.069 and 0.034 matches up quite well to their reported errors of 0.060 and 0.038, (using our prior 1-1, which m ost closely m atches the “W M A Pext” analysis of [62],)3 W hile the W M A P results are not statistically very significant for our purposes, we plot the error contour in figure 4.6 for comparison. (Note th a t if the central value does not change much as the d a ta improve th e implications for inflation will be very interesting.) We have sim ulated CMB experim ents w ith and w ithout polarization measure ments. Since the prim ordial power spectrum affects th e CMB tem perature and polarization in exactly the same way, naively polarization simply adds a second way of measuring the same thing and should only reduce uncertainty by a factor of V2. However, allowing joint estim ation of other cosmological param eters in ad dition to those describing the shape of the prim ordial power spectrum introduces 3O ur characterization of th e W M A P noise and beam size is som ew hat m ore pessim istic th a n th eir rep o rted num bers, b u t we suspect th is is com pensated by fu rth er experim ental details we do not include. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 confusion and near-degeneracies. The value of measuring polarization is not so much th a t it directly puts limits on the power spectrum , b u t in reducing the confu sion w ith other param eters. O ur analysis shows the increasing value of polarization as CMB experiments improve. In figures 4.6 and 4.7 we see the error contours in th e n s - n s ' plane for a hy pothetical super-Planck CMB tem perature and polarization experiment and for a “super” Ly-a; survey. We see th a t such d a ta would provide very significant con straints in this space. In particular, these experim ents are good enough to clearly distinguish points on the line n s — n s — 1 from the line n s' = (ns — l ) 2 for all but very small values of ns, and thus would offer significant tests of th e standard inflationary picture. Combining d a ta from bo th experiments will provide additional constraints and tests. Each experim ent provides constraints in the n s - n s ' plane, b u t on somewhat different scales. These am ount to providing constraints on th e inflaton potential V(4>) near a particular wavenumber k. There are several possible options for com bining d ata from several experiments. One approach is to use a single param etrization for the prim ordial power spectrum and then to jointly estim ate all param eters using the full dataset. If both experiments were at th e same scale, this would am ount to simply overlapping their individual error contours. To perform joint estim ation for experiments at different scales, we would want to find param eters th a t are “good” across different scales, b u t this conflicts w ith our aim to test how good (i.e. constant) a param eter n s really is across a large range of scales. Also, a more immediate concern, is th a t different experim ents often have a (sometimes poorly characterized) system atic error in th eir relative norm alization which causes problems for a joint param eter analysis. For a recent discussion of these issues, see Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 [21]. For simplicity, then, we choose a second option, which is to look at the two experim ents separately and then use th e consistency equation (Eqn. 4.7) to produce the inequality Ts(^Lya) — Ts(^CMb) / /, N 1\2 ■. , — 7—j------- < (n s (khya) - l ) z In k iya — In kcu B as an additional inflationary test. /, on\ (4.20) By looking at only derivatives of the power spectrum we avoid the relative norm alization problems. Visually, this am ounts to projecting the CMB contours for n s and ns' up to Lyman-a: scales (or vice-versa) and checking to see if the contours overlap, as shown in figure 4.7. This third test is not redundant because we use different pivot points {k\jya and /ccmb) for the different datasets. A potential w ith large higher derivatives could pass the ns' < (ns — l ) 2 test at a particular scale and yet fail it when d a ta from different scales is used. Finally, we would like to rem ind th e reader th a t for figures 4.6 and 4.7, th e real inform ation is in the size of the error contours rath er th a n their actual placement. In the absence of real data, we show contours from sim ulated d a ta for a small sample of models all of which show correlations between th e two experim ents consistent w ith the standard inflationary picture. U ltim ately n atu re will tell us if such correlations are really there. 4 .4 .2 P r o je c te d errors a n d c r o ss-co r rela tio n s To further show the value of b e tte r experim ents, we have investigated how the projected errors in n s and ns' should change as a result of improving both resolution and noise levels. Figure 4.8 shows the marginalized errors (with prior 1-6) w ith and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 0.03 0.02 1111»*LLi 0.01 0 4 2 £ 6 10 8 ctt ( f i K ) 0 .0 3 0.02 0.01 0 1 2 3 4 5 6 0 fwhm ( a r c m i n u t e s ) Figure 4.8: M arginalized (one-sigma) errors of n s (solid line) and ns' (dashed line) as a function of pixel noise and beam w idth. Values w ith(w ithout) polarization are shown as thick(thin) lines. For the lower plot, beam w idth has been fixed at 6', and for the upper noise has been fixed at 10/rK. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 w ithout polarization, as functions of beam w idth and pixel noise. A source of worry is possible cross-correlation between our power spectrum pa ram eters and other cosmological param eters. Figure 4.9 shows th e cross-correlation coefficient for the experiments of table 4.1 as defined by r - = _______ ( F ' I h _______ (A91f A “good” experiment will be able to distinguish param eters and ten d to have low correlations between them , however a very noisy experim ent will also have low cor relations between param eters (presuming th e noise is uncorrelated) so this figure m ust be interpreted w ith some care. Nevertheless, some interesting features can be pointed out. Confusion between tilt and optical depth is well-known, and polariza tion helps greatly a t reducing such confusion. We did find th a t there is generally a correlation between I a and our power spectrum param eters (of which n s is of the most interest). The correlation seems to arise because power spectrum param eters can combine to mimic a slight horizontal shift of a peak. M easuring th e locations of multiple peaks makes this conspiracy of power spectrum param eters more difficult, however, and for the best experim ents th e correlation disappears. For the near future, an accurate determ ination of th e angular scale of the sound horizon at last scattering will be im portant for placing constraints on inflationary param eters. 4 .4 .3 T h e F ish er m a tr ix a p p r o x im a tio n One possible source of error in our calculations stems from the approxim ations th a t go into the Fisher m atrix analysis technique. For low i, th e covariance m atrix C is dom inated by cosmic variance, and equation 4.10 can be rew ritten in term s of a new variable Zt = lnC^. As pointed out by Bond et. al. [8], these Zt are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 A§ n s n s ’ n j ^ n ^ n ^ n j p n j , 8) u\>um ** T # • • • • ■ • o «0 A§ n gn g’n lp J n ^ n ^ n ^ n ^ ^ b “ m 1A T 0 •#•••0000 • 0 0 • O o o • • 0 0 • O O o o ...............................................O 0o of o• O° o•O• o• O00• o° 0 * 1 8 OO O o • O « O 80 • • • • • O O • 0 0 0 - • ' ■• • 0 • • • • O O • 0 0 0 • • o o • 0 0 0 ■ • Oo • 0 0 0 ° ° • 0 0 0 . oo• # 0 0 • • 0 O0 ° • 0 • O• • • • 0O0O • OO....................O0O0 O • • • *• O 0O * 0 o o • •0O 0 ° • 0 • • • • 8 # O 0 O ■0000 Oo ° WMAP-like 0• ...............o o • • ■ 0 • O l • 0 » o0 • O • • 0 • • O •0 •o • • O O • • o O • P8la*n •c k -lik' 0e * 0 ^ O O o 0 . O • • 0 • • • O • • ° • O • 0 8OO o • • o • • 0 - 0 - o • • 8O • 0 0 o • • O o 0 0 • • 8O O 0 0 0 8 8 • o o • 0 0 o • o •• •o • ° • • O 0 O •00 • SPT ° o o O °00* • • • • . O • 8 • • o 8 • • O O O 80 • 0O • • O^ 8 8O• •0-0-00 • ^ o 0 O • o 0O • • O 0 ................. CMB-pol Figure 4.9: Cross-correlation coefficients for all th e param eter pairings of four different experiments. Filled circles represent positive correlations; open circles indicate negative correlations. The radius of each circle is proportional to the cross-correlation coefficient r^. For reference, note th a t = 1 for all diagonal elements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 b etter variables th a n the C\ for Fisher analysis in th e sense th a t they are Gaussian distributed. For large enough I, the difference between using Zi and Ct is minor, however. Ju st having variables th a t are Gaussian distributed is not enough; the variables should respond linearly to the param eters. Linearity in response to the cosmological param eters is a well-studied problem and has been discussed extensively in the literature (see [38] for a recent discussion). W hat remains is to check our power spectrum param eters. We have done this by examining Ct as a function of the power spectrum param eters. Figures 4.10 and 4.11 show Ce as a function of param eters for several choices of I. The vertical scale is arbitrary; th e thing to look for in these figures is th a t ideally dCe/dSi should be constant, so th e slope of each line should be as well. For the ranges we use for each param eter, deviations from linearity are small for most param eters. The least linear param eter is the angular scale £a , which is not surprising since varying this param eter prim arily shifts the power spectrum (which is oscillating) left or right. Even so, for smaller changes in the slope becomes close to linear, and the worst deviations are at high m ultipole moments where the signal-to-noise ratio drops. Only the region w ithin the central five points was used for the Fisher m atrix construction. For the power spectrum param eters we are m ost interested in (ns and ns'), deviations from linearity are less th a n one percent for param eter values w ithin all the errorbars th a t we report in tables 4.2 and 4.3. For th e higher order param eters, the worst deviations from linearity occur only for small £ (~ 10), for which we found the linear approxim ation to be valid w ithin one percent for param eter values Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 0.02 0.022 0.024 0.13 0.135 0.14 nm mh z 300 305 1. 310 0.78 0.8 0.82 0.84 0.86 e x p (-2 -r) Figure 4.10: Plots of the angular power spectrum as a function of cosmological param eters for several choices of t. The vertical axis of each plot is Ci in ar bitrary units; the different lines in each plot are for (from dark to light colors) I = 2 0 ,2 2 0 ,4 2 0 ,.... The horizontal axis for each plot is a different cosmological param eter: baryon density (upper left), m a tte r density (upper right), angular scale (lower left), and optical depth (lower right). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 96 - 98 0 . 0 2 - 0.01 100 102 104 0 n. iiIiii ih 0.01 0.02 - 0.96 0.98 1 0 . 0 2 - 0.01 0 1.02 1.04 0.01 0.02 na Figure 4.11: Plots of the angular power spectrum as a function of power specrum param eters for several choices of i. The vertical axis of each plot is Ce in ar bitrary units; the different lines in each plot are for (from dark to light colors) t = 20, 220,420,___ The horizontal axis for each plot is a different power spectrum param eter: am plitude (upper left), tilt (ns) (upper right), first derivative of the tilt (lower left), and second derivative of th e tilt (lower right). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 less th a n around 0.02. In our calculations, we varied the higher order param eters by am ounts several times smaller th an this, and while the resulting errorbars were sometimes larger th a n this, at these small i the error from cosmic variance is also large, so the effect on the overall param eter analysis is small. We checked this by redoing the param eter analysis w ithout including low t values at all, and the qualitative results of our work do not change. For higher t (~ 200 and above), the linear approxim ation remains good up to param eter values of order unity. 4.5 C on clusion We have shown th a t the next generation of cosmological experiments should determine the shape of the prim ordial power spectrum sufficiently to allow new tests of the the standard picture of inflation. If the stan d ard picture is upheld, a new level of differentiation among different inflaton potentials will be possible. We have investigated the potential im pact of new d a ta on b o th the CMB and the Lym an-a forest. The Lym an-a d a ta offers a promising route to testing slow-roll models both on its own, and in conjunction w ith CMB data. C urrently published d ata does not get too far w ith this enterprise, bu t next-generation observation could have considerable im pact4. For the CMB data, if we are interested in general constraints w ithout placing restrictive priors on th e prim ordial power spectrum , Planck-like CMB experiments 4As this work was com pleted we learned th a t th e Sloan D igital Sky Survey is preparing to re lease a new L y m an -a d a ta se t. W hile n o t as large as our survey which is “n ex t generation”L y m an -a d a tase t sim ulated in th is chap ter, it m ay have considerable im portance to th e issues raised in this chap ter [58] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 do not quite have the precision to pu t the strong limits on ns' th a t we desire. It is not until relatively high t th a t the uncertainty from cosmological variance is low enough for our requirements, and it is precisely at these high I th a t the Planck ex perim ent noise rapidly becomes dom inant so a further improvement beyond Planck is needed. Our hypothetical “CMB-pol” experiment should start placing interest ing constraints in the n g- ng plane. Also im portant is the measurement of the (E-mode) polarization channel, which is vital to reducing degeneracies th a t make the tests more challenging. For constraints on the power spectrum itself (as opposed to other cosmological param eters), inform ation from low multipole moments (£ < 500) contributes very little due to cosmic variance. However, coverage of a reasonable fraction of the sky is needed to retain high resolution in £, and simply to beat down statistical noise. The proposed South Pole Telescope (SPT) may do well in this regard. Higher multipole moments are useful up until Silk dam ping reduces th e overall CMB signal. As CMB experim ents improve, polarization will become more im portant as the key to breaking degeneracies between th e effects of the power spectrum shape, which affects tem perature and polarization identically, and other cosmological param eters, which generally do not. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 C hapter 5 Lim its o f Dark Energy from C ross-C orrelations 5.1 In trod u ction U nderstanding cosmic acceleration is one of the largest problems facing physics today. So far the most direct m easurem ents of acceleration have come from distanceredshift measurem ents [55, 56, 35]. This acceleration is thought to be due to the effect of dark energy, a new form of energy density th a t dominates the current universe. However, a universe w ith dark energy exhibits a different evolution for density perturbations, and this has observable consequences. In this chapter, I will discuss how well certain m easurem ents of growth can constrain the expansion history of the universe, and thus dark energy. In particular I will discuss the lensing of the cosmic microwave background (CMB) by intervening structure combined w ith measurements of galaxy num ber density correlations from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 surveys. These types of measurements are attractive since they can be obtained essentially “for free” from experiments already planned. I will also discuss the sensitivity of these measurements to the presence of perturbations in the dark energy fluid, which can have an im portant effect on the results [10]. In this chapter “lensing” refers specifically to lensing of the CMB by foreground structure, rather th an th e extremely productive research area of studying the shape distortion of background galaxies due to lensing from foreground galaxies (the theory of which is covered extensively in [32, 61]). This work was first published in [23]. 5.2 B ackground th eo r y 5 .2 .1 T h e IS W effect a n d d ark en er g y As photons travel from the last scattering surface (LSS) to us, they fall into and climb out of potential wells th a t lie along their path. If the gravitational potential $ does not change w ith time, th en the accompanying blueshifts and redshifts will cancel each other out, leaving no net effect. However, if the potential does change over tim e there may be some overall change in each p h o to n ’s wavelength, and hence the observed tem perature. This is the integrated Sachs-Wolfe (ISW) effect, w ith the change in tem perature A T ISW observed in a direction n expressed simply as (5.1) where the gravitational potential is w ritten as a function of position x and lookback distance D (used as a proxy for conformal tim e). For a flat, m atter-dom inated universe, th e potential rem ains constant over tim e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 even though the density perturbations themselves do not. This means th a t any ISW effect originates either from early times, when the density of radiation was still significant enough to affect the expansion rate, or from late tim es when dark energy became dom inant. Thus if th e late ISW signal can be separated out from others, it provides a clean measurement of dark energy. • Normally the ISW effect itself is buried by the prim ary tem perature anisotropies from the LSS. However, the prim ary anisotropies are set at the last scattering surface, at a completely different epoch and at different length scales th a n those of the structure growth responsible for the ISW effect. The prim ary anisotropy should thereby be uncorrelated w ith the ISW effect and other m easurem ents of growth. Thus cross-correlating other m easurem ents w ith CMB tem perature m aps can be a useful tool for bringing out inform ation about growth, as has been discussed recently by [1, 53] and dem onstrated by [57]. 5.2 .2 L en sin g co r re la tio n s The first measurement I will consider correlating w ith th e tem perature map is a measurement of gravitational lensing of the CMB. The microwave background is gravitationally lensed by m atter th a t lies between us and the LSS. The m ap of photon deflection angles over the sky can be w ritten as the gradient of a scalar field cj) called the projected potential, which depends on the 3D gravitational potential $ as (5.2) where Ds is the distance to th e last scattering surface. Just as w ith CMB tem perature maps, th e m ap of the projected potential can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 decomposed into spherical harmonics. Then a two-point function of the modes can be put together to construct an angular power spectrum. The expression for the power spectrum can be w ritten as a line-of-sight integral over th e 3D gravitational potential r / _ n\ 2 Cf * ~ f ddDDdPtk. DD<&2( kD) ,( T> DS )° p m (5,3) k-tZ# W hile this equation is only exact in the flat-sky (large £) limit, th e im portant feature is th a t it captures the physics of how the angular power spectrum depends on a line-of-sight integral of the gravitational potential m ultiplied by a kind of window function and the primordial (i.e. the growth function has been separated out) power spectrum P${k). The exact expressions and further details can be found in appendix B; they are used for the com putations in this chapter for multipole moments where the difference is im portant. However, for th e rem ainder of this section I will w rite only the Lim ber-approxim ated integrals for clarity. The cross correlation between the tem perature and lensing is due to th e presence of the ISW effect in the tem perature, and thus its power spectrum has th e simple flat-sky form Cj*~ 5 .2 .3 J d D D 9 (k ,D )i{ k ,D )^ P -P m ) (5.4) k=e2* G a la x y co rrela tio n s The second type of observation I will consider is counting galaxies projected on the sky. On large scales, fluctuations in the num ber density of galaxies should track fluctuations in the gravitational potential (possibly w ith some bias). Prom the map of num ber density over the sky, we can obtain th e auto-correlation power Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 100 nK c\2 t— H + < 1 0 nK < 1 0 nK E - m o d e 10 100 1000 I Figure 5.1: Noise levels for CMB lensing reconstruction. The dark curve is th e lensing power spectrum for a typical cosmology. The different do tted lines are noise levels for CMB tem perature experim ents w ith a 4' beam and 0.1 /iff noise per pixel (dotted line) or 0.01 / i K noise per pixel or less (dashed line). The lower dashed line is for an E-mode CMB polarization measurement. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 spectrum Cf ~ J dDD<S>2( k , D ) n ] { D ) P l ( k ) \ k_ t!g , (5.5) which depends on the potential as viewed by some window function n g(D) describ ing the distribution of galaxies th a t are actually observed. The galaxy spectrum should be correlated b oth w ith the ISW p a rt of the tem perature spectrum C j 9 ~ / d D D $ ( k , D ) $ { k , D ) n g(D)P%{k) k=eQ (5.6) and w ith th e lensing potential power spectrum C f- (5.7) dD D3 >\k, D ) n g( D ) ^ ^ - P f k ) k=et£ Again, in the calculations described later these Lim ber-approxim ated integrals are used only for high multipole moments. The exact expressions are used for low multipole m oments (£ < 100). 5.3 C alcu latin g th e s e n sitiv ity to dark energy The power spectra themselves are com puted numerically using the techniques described in the previous section and a version of the CMBFAST code [59] which I modified to ou tp u t lensing spectra and other information. The basis for all the analysis is the Fisher inform ation m atrix. Given a fiducial model, the Fisher m atrix describes how sensitive the model is to changes in its param eters. First, all the power spectra are p u t together into a covariance m atrix / c j T + NT C/ = cT* c 7g c f c f + Nf C f9 c f cf9 \ C f + JVf ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.8) 70 Experim ental noise can be modeled as a contribution to the diagonal term s of this m atrix. In our case I take the noise contribution to the prim ary tem perature anisotropy N j to be negligible. However, even a small am ount of noise is significant when reconstructing lensing information, and I take this into account using the derivations for lensing error as a function of tem perature noise found in [49] and shown in Fig. 5.1. I take the prim ary contribution to the noise for galaxy surveys N f to be shot noise (bias for each survey is treated as a free model param eter). Some other possible noise sources are discussed in section 5.3.2. W hen I consider combining three galaxy surveys at different redshifts, each galaxy survey gets its own row and column, resulting in a 5 x 5 covariance m atrix. The Fisher m atrix is then constructed as a sum over multipole moments, (5.9) where the s* are labels for the actual model param eters. T he model param eters actually varied are the dark energy param eters discussed in th e next section, plus an angular scale param eter £a , baryon density Qbh2, m atter density Qmh2, prim ordial power am plitude A s and tilt ns, optical depth r , and a bias param eter bi for each galaxy survey. All models were constrained to be flat. I investigated two fiducial models; b o th had param eters equivalent to a m odern concordance cosmology (fid.e. = 0.75, ns = 0.9). = 0.04, flcdm = 0.21, h — 0.65, r = 0.1, The difference between the two was in the dark energy evolution; one model had a “pure A” equation of state of w = —1, whereas the other used a quintessence-type model w ith w = —0.9 today running smoothly to w = —1 at high redshift. The latter has a somewhat enhanced sensitivity to dark energy param eters due to the evolution of dark energy at low redshifts, and is used for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 reporting the final constraints on dark energy param eters in section 5.4. Using the “pure A” model instead degrades the dark energy param eter constraints by about 20% overall. 5.3 .1 C h o o sin g a form for dark en erg y Different experim ents have differing sensitivities to dark energy. W ith no knowl edge of the fundam ental physics behind cosmic acceleration, there is little reason to favor one functional form for the equation of state w(z) over another. In th a t light, principal mode analysis is useful for revealing w hat the experim ents can actually constrain. In such an analysis one chooses some basis set of eigenfunctions (cut off to a finite set size), and then expresses the fundam ental modes for each experiment as combinations of basis functions. In our case the basis functions will simply be a set of 25 boxcar functions covering the redshift range from z = 0 to z = 5. Figures 5.2-5.4 show how the various power spectra respond to changing the equation of state w(z) w ithin a bin. The top panel of each figure shows th e fiducial Ce, and the different lines on th e lower panel are dC i/d w i where i labels the bin, and each line is for a different i. Lighter colored lines are for higher redshift bins, and the bin centers have a spacing of A z = 0.25. From th e figures it is possible to see when dCijdWi ~ dCe/dwj, which means th a t those two bin param eters are impossible to distinguish using th a t power spectrum alone. A few example eigenvectors are shown in Fig. 5.5. T he solid line connects the am plitude for each of th e 25 w(z) bin param eters. In general, dark energy information is mixed w ith other cosmological param eters so th a t the eigenmodes are complicated combinations of several param eters. However, w ith the Fisher m atrix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 O £o < 1x10 10 100 m ultipole 1000 Figure 5.2: Response of the tem perature power spectrum to variations in the dark energy equation of state. The top panel shows th e fiducial C f T , and the b ottom panel shows d C j T/ dwi where Wi labels the value of w(z) in each redshift bin. The bins have a width of A z = 0.25 and bins from z = 0 t oz = 2.5 are shown, with lighter (more red) color indicating higher redshift. In this figure it can be seen th a t other th an at low i, the effects of different dark energy param eters on the power spectrum are nearly degenerate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 7--- 1--- 1—TT T----1—I I I I I | ■e- E- — < 10 100 multipole Figure 5.3: Response of the ISW -lensing cross-power spectrum to variations in the dark energy equation of state. The top panel shows the fiducial C j ^ , and th e bottom panel shows d C j ^ / d w i where Wi labels the value of w(z) in each redshift bin. The bins have a w idth of A z = 0.25 and bins from z = 0 to z = 2.5 are shown, w ith lighter (more red) color indicating higher redshift. For this figure the effects of different dark energy param eters are much less degenerate th an for the tem perature power spectrum. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 o T----1—I I I I I TTTF £?_ o <1 10 100 multipole Figure 5.4: Response of the ISW -galaxy power spectrum to variations in the dark energy equation of state. T he to p panel shows the fiducial C j 9, and th e bottom panel shows d C j 9/d w i where Wi labels th e value of w(z) in each redshift bin. The bins have a w idth of A z — 0.25 and bins from z = 0 t oz — 2.5 are shown, with lighter (more red) color indicating higher redshift. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 i i i i i i i i i i" i i i -i | i i i i | i i i i i i i i i i i i i i O (% 0 0 'V 3 a a = 0.0584 1 m CO o i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i .t—p . . , | .r— i— i—i— -j™ ,—i— i- r■■ -1 i i i i 1i i i i o a*. Qw o = 0.334 .+-> c 0 Gfl •H 0 i \ S-H > i “ £ o 0 i ■1 ■i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i ©©0 _ o 111111111 1 0 a = 0.544 1 r e d s h if t Figure 5.5: Three dark energy eigenvectors for an ideal ISW -lensing-galaxy count correlation experiment. The left p art of each plot shows th e 25 w(z) param eters, and on the right are the 9 other param eters (in order): angular scale I a , baryon density m atter density Vtmh 2, optical depth r, prim ordial power spectrum tilt ns, primordial power am plitude A s , and the bias param eters for each galaxy survey bi,b2,bz. The error in each eigenmode for a cosmic-variance lim ited experim ent is shown for each mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 (5w 1 (5w 0.2 1 0 10 1 0 11 1 0 12 n u m b e r of g a la x ie s Figure 5.6: Error in wq and wa as a function of to ta l galaxy num ber with lensing noise fixed at the “< 10 nK E-mode” level from Fig. 5.1. The solid lines w ith squares are for a survey with three redshift bins, dashed lines w ith triangles for a survey with one high redshift bin, and d otted lines w ith stars for one w ith a single low redshift bin. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 in hand not only can one perform principal mode analysis, one can also marginalize over some param eters and switch to more familiar dark energy param eters with simple m atrix operations. For example, it is simple to switch from the 25 w(z) param eters to the uwan param etrization [11, 44], which takes the form % w(z) = w0 + Wa — — . (5.10) 1 “r Z The full eigenmode analysis shows th a t even for ideal experiments, no more th an two or three dark energy param eters will be well-constrained, so for the rest of this chapter the “wa” param eterization will be assumed. 5 .3 .2 G a la x y su rv ey c h a ra c te ristic s My intent is to approxim ate the behavior of galaxy surveys which do not include spectra, b u t are able to obtain approxim ate photom etric redshifts through color information. I therefore consider the situation where one has a large num ber of galaxies th a t can be assigned to one of three redshift bins, centered around z = 0.5, z = 1.0, and z = 1.5. Each of these bins is approxim ated as a rounded boxcar function similar to the technique of Hu and Scranton [31]. This approxim ation is best for survey slices th a t are not severely m agnitude or volume limited. For those cases the overall effect is to smear out the edges of the bins, weakening the advantage of having several redshift bins. Galaxy surveys contain numerous system atics. Stellar contam ination should not be a m ajor problem since for faint surveys the m ajority of observed objects are galaxies. M odern m ethods can reduce stellar contam ination to a few percent or sm aller[50], and stars tend to sm ooth out the galaxy clustering rath er th a n strongly bias it in a particular direction, so th e overall effect on th e angular correlation power Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 78 spectrum (which is already nearly smooth) should be weak. More worrisome is th a t the analysis presumes knowledge of the redshift distri bution of galaxies in the sample, which of course could be in error. We can estim ate how small the overall redshift calibration error needs to be by converting the red shift error into an error in the angular scale of galaxy correlations, and examining the change in the correlation power spectrum compared to th e shot noise. Because of the large angular scales we’re interested in, it tu rn s out th a t even for galaxy surveys with 1012 galaxies (corresponding to roughly 104 per square arcm inute), shot noise at these scales is still large enough th a t an overall redshift calibration error of up to a percent is tolerable. Another worry is th a t the distribution of dust, which can b o th obscure galaxies and have its own long-wavelength emission, likely has its own anisotropies which may dom inate the small galaxy num ber count anisotropies we are interested in measuring. For cross-correlating w ith the tem perature spectrum , however, the angular scales of interest are quite large (~ 10°) and one can be hopeful th a t dust anisotropies on such large scales will be relatively local and it will be possible to measure and correct for them , as in [57]. For the rem ainder of this chapter I will assume any systematics in the galaxy correlations are below th e shot noise level (which itself is quite lim iting), bu t new experiments may well reveal im portant new information about w hat system atics to consider. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 5.4 R esu lts In the absence of noise, over 40% of the improvement in dark energy constraints gained by adding CMB lensing d a ta to the tem perature inform ation comes from the CT f cross-correlation channel. Even though in the ideal case th e noise in the CMB and lensing potential anisotropies is uncorrelated, th e inversion of the covariance m atrix in equation 5.9 means th a t noise in the auto-correlation spectra will find its way into the cross-correlation channel. This then means th a t the noise in th e reconstructed C f ^ spectrum will be th e limiting factor as to how well CMB lensing can constrain dark energy. In Fig. 5.1 I have calculated the noise in th e reconstructed C f lensing power spectrum for several different types of experiments. At small enough tem perature noise levels (a bit above 10 n K per pixel for 4' pixels) all of the noise in the lensing C f is in fact coming from cosmic variance in th e tem perature C j T . Note th a t even at this limit, the noise in the lensing spectrum is still roughly th e same order of m agnitude as the signal. P a rt of th e low signal-to-noise ratio appears to be related to the choice of a red tilt (ns < 1) in the prim ordial spectrum of the fiducial model. This reduces CMB power at th e small scales from which th e lensing signal is reconstructed, leading to more relative noise in the lensing power spectrum . The to tal noise in the lensing power spectrum can also be reduced further by combining several polarization modes as per [49], which results in an improvement by a factor of a few, so it may not be necessary to drive th e detector tem perature all th e way down to 10n K to get the desired precision. In any case cosmic variance is the ultim ate limit th a t I will consider here, and we shall see th a t it is a fairly restrictive one, at least as far as w(z) constraints are concerned. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 2 1 0 1 2 - 0 .5 0 0 .5 <5w0 Figure 5.7: Error contours (68.3%) in th e w0- w a plane. From innerm ost to out ermost they represent (a) a “perfect” CMB lensing-galaxy count cross correlation measurement where cosmic variance is the only lim itation, (b) a more realistic mea surement w ith galaxy density of ~ 102 arcm in-2 and the “< 10 nK E-mode” lensing noise of Fig. 5.1, (c) an ISW-lensing only experiment (no galaxy counts). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 How about cross-correlating galaxies w ith th e ISW effect in order to augment our inform ation about dark energy? This can provide interesting results, especially w ith th e possibility of adding even very lim ited redshift inform ation into the mix. I show the results in Fig. 5.6. Galaxy information, especially th a t coming from high redshift galaxies, can tighten the dark energy constraints by more th an a factor of two. Finally, the constraint contours in the WQ-wa plane are shown in Fig. 5.7. These are constraints made on w{z) simultaneously w ith constraints on all the other cos mological param eters described above. In th e absence of any other experiments, CMB lensing and galaxy counts combined can realistically constrain (contour b) wo to a precision of about ±0.33 when all other param eters are marginalized over. 5 .4 .1 D a rk en er g y p e r tu r b a tio n s T he above results were obtained assuming th e absence of perturbations in the dark energy fluid. A tru e cosmological constant has w = —1 and is perfectly sm ooth w ith no perturbations. However, many quintessence models are built with some sort of scalar field which in general can have its own density fluctuations. These density fluctuations become im portant at late tim es when the structure formation respon sible for the ISW effect is occurring, and thus significantly affect the above results. This effect is shown in Fig. 5.8. Note th a t including perturbations mostly increases the uncertainties along the degeneracy direction, especially for experiments lim ited only by cosmic variance, w ith the result th a t limits on both dark energy param eters are degraded by nearly a factor of three. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 2 1 0 1 -2 - 0 .5 0 6 0 .5 w0 Figure 5.8: The effect of dark energy perturbations on error contours (68.3%) in the w 0- w a plane. Contours (a) and (b) are as before in Fig. 5.7. Contours (c) and (d) show the effect of including dark energy perturbations for experiments considered in (a) and (b), respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 5.5 C onclu sion Cross-correlating the ISW effect w ith CMB lensing and galaxy counts can in principle place limits on dark energy; ideally (if cosmic variance were the only lim itation) one could measure w0 to ±0.093 and wa to ±0.32, though in practice realistic limits are probably worse by about a factor of three. These measurements depend on dark energy’s effect on the growth of structure, and thus have a depen dence on the equation of state quite different from distance-redshift measurements. Observation of such growth effects would bolster the case for dark energy as the source of cosmic acceleration. Also, these measurements are strongly sensitive to perturbations in the dark energy fluid (which do not affect distance observations) and thus may ultim ately be more useful as a measurement of perturbations th an as a precise determ ination of the equation of state. There is certainly room for improvement for these kinds of m easurements. The am ount of lensing being measured is small, so cosmic variance and the finite reso lution of experiments become im portant. In this work I have used the CMB tem perature m ap to reconstruct the lensing spectrum ; this m ethod essentially makes use of four-point correlations in th e original tem perature map. The three-point correlation function (known as th e bispectrum ) also contains contributions from lensing which are known to be significant [22] and potentially less noisy, although how well dark energy inform ation can be extracted from realistic d a ta is still under investigation. The scales on which galaxy densities correlate w ith the ISW effect are very large (degree scale or larger), so the deviations from a sm ooth background are small and even for large numbers of galaxies shot noise is significant. One way of overcoming Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 this problem is simply to use more information from galaxies th a n merely their number density. D edicated telescopes th a t can find galaxy shapes and redshifts are predicted to put good constraints on dark energy [61]. In th e future a wide array of complementary observations will be available to determ ine the nature of cosmic acceleration, each w ith their own sensitivity and lim itations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 B ibliography [1] Niayesh Afshordi. Integrated sachs-wolfe effect in cross-correlation: The ob server’s manual. Phys. Rev., D70:083536, 2004. [2] Andreas Albrecht and Paul Steinhardt. Phys. Rev. Lett., 48:1220, 1982. [3] Jam es M. Bardeen. Gauge invariant cosmological perturbations. Phys. Rev., D22:1882-1905, 1980. [4] James M. Bardeen, Paul J. Steinhardt, and Michael S. Turner. Spontaneous creation of almost scale - free density perturbations in an inflationary universe. Phys. Rev., D28:679, 1983. [5] C. L. B ennett et al. Cosmic tem perature fluctuations from two years of cobe dmr observations. Astrophys. J., 436:423-442, 1994. [6] C. L. B ennett et al. First year wilkinson microwave anisotropy probe (wmap) observations: Maps and basic results. Astrophys. J., 2003. [7] J. R. Bond and G. Efstathiou. Cosmic background radiation anisotropies in universes dom inated by nonbaryonic dark m atter. Astrophys. J., 285:L45-L48, 1984. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 [8] J. R. Bond, A. H. Jaffe, and L. Knox. Phys. Rev. D , 57:2117, 1998. [9] M artin Bucher, Kavilan Moodley, and Neil Turok. Prim ordial isocurvature perturbations: Testing the adiabaticity of th e cmb anisotropy. 2000. [10] R. R. Caldwell, Rahul Dave, and Paul J. Steinhardt. 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Astrophys. J., 142:414-419, 1965. [17] Scott Dodelson. Modern Cosmology. Academic Press, 2003. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 [18] Scott Dodelson, W illiam H. Kinney, and Edward W. Kolb. Phys. Rev. D, 56:3207-3215, 1997. [19] Scott Dodelson and Ewan Stewart. Scale dependent spectral index in slow roll inflation. Phys. Rev., D65:101301, 2002. [20] Gia Dvali and S.-H. Henry Tye. Phys. Lett. B, 450:72-82, 1999. [21] K aplinghat et al. Proceedings of th e davis meeting on cosmic inflation. [22] F. Giovi, C. Baccigalupi, and F. P errotta. Constraining the dark energy dynamics with the cosmic microwave background bispectrum . Phys. Rev., D68:123002, 2003. [23] Benjamin Gold. Limits of dark energy m easurem ents from cmb lensing-iswgalaxy count correlations. Phys. Rev., D71:063522, 2005. [24] Benjamin M. Gold and Andreas Albrecht. N ext-generation test of cosmic inflation. Phys. Rev., D68:103518, 2003. [25] Alan H. G uth. Phys. Rev. D, 23:347, 1981. [26] Alan H. G uth and So-Young Pi. Physical Review Letters, 49:1110-1113, 1982. [27] S. H annestad, S. H. Hansen, F. L. Villante, and A. J. S. Hamilton. Con straints on inflation from cosmic microwave background and Lyman-/o: forest. Astroparticle Physics, 17:375-382, 2002. [28] Steen H. Hansen and M artin Kunz. Observational constraints on the inflaton potential. Mon. Not. Roy. Astron. Soc., 336:1007-1010, 2002. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 [29] W. Hu. CMB tem perature and polarization anisotropy fundamentals. Annals of Physics, 303:203-225, 2003. [30] Wayne Hu. Weak lensing of the cmb: A harmonic approach. Phys. Rev., D62:043007, 2000. [31] Wayne Hu and Ryan Scranton. M easuring dark energy clustering w ith cmbgalaxy correlations. 2004. [32] Bhuvnesh Jain and Andy Taylor. Cross-correlation tomography: Measuring dark energy evolution w ith weak lensing. Phys. Rev. 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Cosmological Inflation and Large-Scale Structure. Cambridge University Press, 2000. [41] E. Lifshitz. On the gravitational stability of the expanding universe. J. Phys. (USSR), 10:116, 1946. [42] A. D. Linde. Phys. Lett. B, 108:389, 1982. [43] A. D. Linde. Phys. Lett. B, 129:177, 1983. [44] Eric V. Linder. Exploring the expansion history of the universe. Phys. Rev. Lett., 90:091301, 2003. [45] David H. Lyth. Models of inflation and th e spectral index of the density perturbation. 1996. [46] David H. Lyth. Models of inflation and the spectral index of the density perturbation. 1996. [47] Chung-Pei M a and Edm und Bertschinger. Cosmological perturbation theory in the synchronous and conformal new tonian gauges. Astrophys. J., 455:7-25, 1995. [48] J. C. M ather et al. M easurem ent of the cosmic microwave background spectrum by the code firas instrum ent. Astrophys. J., 420:439-444, 1994. [49] Takemi Okamoto and Wayne Hu. Cmb lensing reconstruction on the full sky. Phys. Rev., D67:083002, 2003. [50] Seb Oliver et al. Angular clustering of galaxies at 3.6 micron from the swire survey. 2004. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 [51] J. A. Peacock and S. J. Dodds. Mon. Not. R. Astron. Soc., 267:1020-1034, 1994. [52] P. J. E. Peebles and J. T. Yu. Primeval adiabatic perturbation in an expanding universe. Astrophys. J., 162:815-836, 1970. [53] H iranya V. Peiris and David N. Spergel. Cross-correlating the sloan digital sky survey w ith the microwave sky. Astrophys. J., 540:605, 2000. [54] Arno A. Penzias and Robert Woodrow Wilson. A m easurement of excess antenna tem perature at 4080-mc/s. Astrophys. J., 142:419-421, 1965. [55] S. Perlm utter et al. M easurem ents of omega and lam bda from 42 high-redshift supernovae. Astrophys. J., 517:565-586, 1999. [56] Adam G. Riess et al. Observational evidence from supernovae for an acceler ating universe and a cosmological constant. Astron. J., 116:1009-1038, 1998. [57] Ryan Scranton et al. Physical evidence for dark energy. 2003. [58] Uros Seljak. Private communication, 2002. [59] Uros Seljak and M atias Zaldarriaga. A line of sight approach to cosmic mi crowave background anisotropies. Astrophys. J., 469:437-444, 1996. [60] Gary Shiu and S.-H. Henry Tye. Phys. Lett. B , 516:421-430, 2001. [61] Yong-Seon Song and Lloyd Knox. D ark energy tomography. 2003. [62], David N. Spergel et al. First year wilkinson microwave anisotropy probe (wmap) observations: D eterm ination of cosmological param eters. Astrophys. J., 2003. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 [63] M ax Tegmark and M atias Zaldarriaga. Phys. Rev. D , 66:103508, 2002. [64] Yun Wang, David N. Spergel, and Michael A. Strauss. Astrophys. J., 20:510, 1999. [65] M atias Zaldarriaga and Uros Seljak. An all-sky analysis of polarization in the microwave background. Phys. Rev., D55:1830-1840, 1997. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 A ppendix A Selected Inflationary M odels The potentials used in the models shown in figure 4.1, from left to right are (in units where M P = 1) M 4 [1 + A In <f> (A.l) M 4 [ l - e - > / 2] 2 (A.2) M 4 [1 - e“ *] (A.3) M 4 [1 + A<f)2] (A.4) M 4 cos2 -^, M4 (A.5) M 4 [1 - A(j)12] (A.6) M 4 [1 - Acj)2] (A.7) M 4 [1 - A 04] (A.8) + ^4? (A + 0.6) (A.9) other param eters were chosen as in table A .l to produce a dp/p of roughly 2 x 10 5. For m ost models above, the ending point for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 Table A .l: Inflationary model param eters 9/ 4 2 x 10~4 1CT3 10“ 6 1(T16 OCO IO - A 5 x lO-1 — — 10~3 100 5 x 10-' 10~2 X M 1(T3 ltr3 1(T3 1(T4 Mi— 1 Model # 1 2 3 4 5 6 7 8 9 0.139 inflation can be found by numerically evaluating where th e slow-roll param eters become equal to unity. A numerical code was w ritten to facilitate this, and to then numerically integrate the equation of motion from a suitable startin g point to deter mine the detailed shape of the prim ordial power spectrum of density perturbations. This code is made publicly available in its entirety in appendix D. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 A ppen d ix B Full-sky expressions for angular power spectra In chapter 5, th a t flat-sky approxim ations for power spectra were used for clarity. However, the full expressions as given by Okamoto and Hu[30] are necessary for accurate numerical calculation. Recall th a t th e projected potential is defined as « n ) = - 2 |< i r ) ^ ^ * ( x ( n ) ,D ) . (B .l) As w ith any function on the sky, this can be broken up into spherical harmonics «n) = ^ ^ y 7 " (n ), (B.2) and averaging over the ensemble of coefficients defines the power spectrum (■(ftmfa'm1) = (B.3) Since the projected potential depends only on the full gravitational potential, the power spectra will depend on the power spectrum of th e gravitational potential Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 A$(fc, z). This dependence will come w ith a window function closely related to the lensing efficiency = 47T l2 dk A|(fc,z) k (B.4) where the j( are spherical Bessel functions. For regimes where the growth of the potential rem ains linear, the power spectrum A $ ( k , z ) can be replaced by F 2(D) A%(k,0) where F(z) is the growth rate for density perturbations defined by 5{z) = The projected potential power spectrum is then dk Cf* = 4tt I y A | ( M ) dD2F(D) Ds - D j e ( k D / H 0) DD, (B.5) T he galaxy-galaxy correlation spectrum is similar to the projected potential power spectrum except th a t instead of the window function expressing th e lens ing efficiency, the window function for galaxies depends on th e number density of galaxies n g(D) as a function of distance or redshift for the particular survey one wishes to describe. Otherwise th e equation is th e same C f = 4tr dk k d D 2 F ( D ) n g(D )je( k D / H 0) 0) (B.6) Remembering th a t the 1SW portion of the tem perature power spectrum is given by A T ISW(h) = - 2 I dD <3>(x(n), D), (B.7) the cross-correlation between tem perature and lensing is given by a similar expres sion CJ* = 4tr j y A | ( M ) J dD2F{D) f d D 2 F ( D ) j e( k D / H 0) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (Bd 96 T he other two cross-correlation spectra follow in direct analogy C j9 = x 47t J ^A|(&,0) d D 2 F ( D ) n g( D) j e ( kD / H 0) (B.9) d D 2 F ( D ) j e ( k D / H 0) for tem perature-galaxy correlations, and d D 2 F ( D ) n g{ D) j e{ k D / H 0 x d D 2 F ( D ) D * n £ ) j t { k D / H 0) for galaxy-lensing correlations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (B.10) 97 A ppendix C U seful M athem atical D erivations C .l P aram eter m argin alization for a m u ltin or m al d istrib u tio n in m a trix form Often we have to deal w ith probability distributions th a t are Gaussian in N dimensions, b u t where there is much correlation between dimensions. Such a prob ability distribution can be w ritten com pactly as a function of an TV-dimensional vector of param eters x = { xi , X2, ■■■, x jy} using m atrix notation (C.l) where No is a norm alization (which we’ll be unconcerned w ith), and F is the inverse of the N x N symmetric m atrix of covariances C. Its is frequently desirable to integrate th e probability distribution over one or more of the param eters to obtain a marginalized probability distribution of a smaller set of param eters. For example, marginalizing over one of th e param eters Xi to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 obtain the distribution as a function of x which is an N — 1 dimensional vector / OO dxiPi*). (C.2) -OO If the m atrix F were diagonal, the integral would be easy to perform algebraically, however if N is large and there are many non-zero covariances between x, and the other param eters then the algebraic expression can become quite cumbersome. Fortunately, there is a convenient technique using the m atrix notation which I offer here w ithout rigorous proof. Taking advantage of the fact th a t the resultant probability distribution will still be a m ultivariate Gaussian, we can rewrite th e previous expression as iY0 exp ^ - i x r F x ^ = J dxi N 0 exp ^ ~ ^ x TF x j , (C.3) where F is an N —1 x N —1 m atrix. The im portant inform ation about the probability distribution P (x) is contained in this m atrix, and the question is to determine how F can be obtained from the N x N m atrix F . Define to be th e m atrix minor of F , i.e. the N —l x N —1 m atrix formed by removing the z-th row and j -th column of F. Also define to be the N — 1 dimensional vector formed by taking the z-th column of F and removing the entry in the z-th row. Then th e m atrix F describing the probability distribution obtained by marginalizing over th e param eter aq is F = - - L v (i)v (i)T, *a (C.4) where Fa is the z, z-th entry of th e m atrix F. For a simple concrete example, imagine a two-dimensional probability distribu tion P(x, y ) = N q exp[—(ax2+ 2bx y+cy2)/2}, thus th e m atrix F for this distribution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 is given by F = (C.5) \ bc/ M arginalizing over the param eter y results in th e integral p ( X) = r dyNo exp 1 (iax2 + 2bxy + cy2) (C.6) J —c which is easy enough to do analytically 2?r P ( x ) = N 0\ — exp —| a 2 V b2 | x~ c (C.7) We can check th a t the m atrix technique gives the same answer. In this case M-22-1 is simply the l x l m atrix with entry a, and a one-dimensional vector w ith entry b. T hen the ( l x l ) m atrix F can be found through the m atrix technique to be F = a c b x b, (C.8) which m atches the algebraic expression found earlier. This m atrix technique has been checked to m atch the algebraic result for up to 5 x 5 m atrices using com puter algebra, b u t a rigorous proof remains beyond th e author at this time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 C .2 A ltern a te d erivation o f th e ev o lu tio n o f th e d en sity o f a gen eral cosm ological fluid Consider a sm ooth fluid w ith background density p and pressure P , with small perturbations. One way of defining this fluid is through its stress-energy tensor Tg=-(i> + Sp), (C.9) Tf = ( p + P ) v u Tj = (P + dPjS] + a), where dp and 5P represent density and pressure perturbations, Uj represents veloc ity perturbations, and u* represents shear perturbations. We are working in the synchronous gauge and in conformal tim e, where the to tal m etric tensor can be w ritten as g^vd x ^d x 1' = a2 [—d r 2 + (5^ + hij)dxldx^1 \, (C.10) where the tensor hij represents the m etric perturbations. Conservation of energy and m om entum is represented in general relativity by the equations + r va0T af3 + where the Christoffel symbol = 0, (C.ll) 9a0- v) , (C .1 2 ) is defined as = 2 ^ (W q ;/3 + 9ug- a ~ and the semicolon subscript indicates th e derivative x-a = dax. Looking at the first-order p art of th e u = 0 component of th e conservation equation will give the equation for the evolution of density perturbations. To obtain Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 this, it is useful to explicitly break T IW, and g,iv into background and first-order pieces. The metric is easy to break up by inspection Qyuv — 9[iv T Sg^v = a [diag( 1 ,1 ,1 ,1 ) -f- h ^ ] , (C.13) but we will also need to make use of the inverse metric, which to first order in is ^ [d ia g (-l, 1,1,1) - h i j ] , a* where the lowered indices on The stress-energy tensor (C.14) are intentional. is also easy to break up by inspection. The unper tu rb ed tensor T ‘“ is simply diag(—p, P, P, P), and we will only need the following perturbed components 5T0° = - 5 p , (C.15) ST f = (p + P ) v u STi = 8P, w ith no implicit summing over i in the last expression. A slight complication is th a t we need to find the stress-energy tensor to first order w ith both indices raised. We can make use of the m etric to write T n» = = + 5 T »gW + f £ 5 g av. (C.16) The first term is the background tensor T ^ u = diag(p, P, P, P ) / a 2, and th e second two term s form the first order p ertu rb atio n S T ^ w ith the components we are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 interested in as (C.17) a2 a again w ith no implicit summation. W ith the perturbed metric in hand we can also com pute th e Christoffel symbols to first order and w rite them as a background term plus a perturbed piece = ^a0 + STU ' After some tedious com putation, th e only non-zero unperturbed pieces of the Christoffel symbols are -pO 00 pO ii _ (Jj pi Oi (C.18) There are also only two sets of nonzero perturbed com ponents which we will need 1• (C.19) (C.20) Finally we are prepared to find th e first order piece of th e energy conservation equation + sr°a0f af3+ y%5t^ + sr^f013+ f aP5T00= o. ( c . 21 ) We can examine this piece by piece. The first term is (C.22) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 T he portions proportional to the perturbed m etric cancel in the second and th ird term s, leaving only \ h - 2 + ~ ( 6 P + UP), 2 a2 a2 a where h = T (C.23) r = J T ha. The last two term s follow in a similar way M + 4 ^ . 2 a 2 a a2 (0-24) P u ttin g these all together and multiplying by a2 gives 5p + 3-(<5/o + SP) + (p + P ) ^ Vi + \ { p + P)h = 0. CL £ Last, wecan make the definitions 5 = 5p/p, 6 = dlVi and make use (C.25) of the identities w = P / p and d i n P / d r = —3(1 + w)a/a (the latter is strictly true only when dw/dr can be neglected). W ith these in place, dividing the previous equation by p reveals ^ + 3 a (<hP ~~ W) + ^ + ^ + 2^ = (C.26) U ndertaking a similar exercise with the other com ponents of the energy-momentum conservation equation gives the equivalent expression for 0. C .3 C o n n ectio n b etw een tw o form s o f th e Fisher m a trix The general form for the Fisher m atrix of m ultiple angular power spectra is given in 5.9 as (absorbing the coefficient factors of \ f l into CcnqT1) c w ^ c o v r '80' dsi dsj Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.27) 104 For the special case where the Cov^ are 2 x 2 matrices, the above equation can be rew ritten as in chapter 4 as nr.l I,J = (T ,E ,C ) AnJ dsi ds-; ‘ £ (C.28) ) i j -8 0 1 3 where now the covariance inform ation is contained in a 3 x 3 m atrix D . The form for specific entries of D was also given in chapter 4. In this appendix I will explicitly connect the two forms. Analogous to the process in chapter 5, th e covariance m atrix for tem perature and E-mode polarization spectra can be w ritten as / 1 Cov^ = yj C c Jt T‘ ++ N ij C c JrE r'TE C JE r~iEE _i_ N atef C fE + + ^/sky (C.29) / Noise is presumed to be uncorrelated between T and E modes, however this is not im portant to the derivation here. For brevity I will use the notation C xx = C'fx+ N f , and C xx = d C f x j d Using this notation, the inverse of the covariance m atrix is |) /s k y C ov„1 r i T T r< E E u£ u£ ^ Lx riEE o (C.30) r^T E 2 u£ c*TE ^ LxQ r iT E ue y fiT T '-'£ The full expression for the i , j - th entry of th e Fisher m atrix can be com puted w ith some algebra cr,cr1cr,+icr-pv'cr, /sky p iT T p iE E _ U£ Ui /o T £ 2V J rys~iTT P iE E s~iT E /~iT E U ^ £ ; ~ , s~tTT n T E 2 n E E + ° — rs/~iTE cy/^iEE z u e - j cy/~iTE ZL /£ , on T E +. P iT T r ^ T E r^iE E f i E E /~iTE yoT T ;i Ue U£ P iT T f i E E n T E ;iW + pp s~iEE ° £ L/ £ -j s~iTTz r ^ E E .i Lpi Kj q P iT T /~vTE /~iTE -i u £ u £ -j Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ;j 105 T he expression in square brackets can be rew ritten in m atrix form / \\ T / (jT T r*TE 9 r^E E riT E cf 9nE E -ZU£ r iE E \ \ C* *J c:t e nTE 9 ( r iT T (^ tE E , s t T E 2\ Zl U£ U£ -f-U£ I 2 \ (jT E 2 riT T riT E ~9 L Lx d \~yn ( (J T T n s^iTT/~<TE q U£ a <rprpZ \ TE riE E V6 * ; i / (C.32) T he 3 x 3 m atrix in the middle, along w ith th e coefficient from th e previous equation, is the m atrix D -1 from equation C.28. After inverting and canceling several factors of { C j T C f E - C J E2), / q TT2 fiT T riT E D = + ue |) /s k y a \ L'e n T T s~iTE \y o Kyo 1 (r^T T r iE E TE2 \ cilT E 2 , s~<TE2 \ r ^ T E /~iE E /-iT E r> EE J L'e (C.33) L/i C£ / which corresponds exactly to the expressions given in chapter 4. C .4 Specific len sin g n oise estim a to rs Okamoto and Hu [49] give the general expression for th e noise in a lensing m ap th a t has been reconstructed from cosmic microwave background tem perature and polarization maps in equation 39 of their paper. I will give a few specific expressions for the most useful cases. If one only has a tem perature map, th en the noise in the reconstructed lensing m ap is given by N (ee)(ee) = L ^L + 1^ 2L + ^ ^ 3^ -i x ^Pifo 2 C *f1e C «f2e Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 where C f e are the tem perature multipole moments as m easured by experiment (including noise), and the tilde indicates the unlensed multipoles w ithout noise. The expressions for the noise if reconstructed from th e E-mode polarization m ap or the T E cross-correlation map alone are quite similar, w ith the former being N (-E E ) ( E E ) — L{L + 1)(2L + 1) ( riEE jp _V_______ x (C.35) , r^EE jp /even 9 riEEr’EE L't2 and the latter N\ { OE) ( e E) — L ( L + 1)(2L + 1) fiQE jp E hl2 x (C.36) i -1 fiQE jp (^e1 2^t2Lh + (-'e2 0 ^ h U 2 ryriQEri&E where “even” indicates th a t the expression in parentheses is non-zero only when the sum L + £i +£ 2 is even. The 0 © and E E expressions are w hat was used to produce figure 5.1 of chapter 5. To fully determ ine the noise for a lensing m ap reconstructed from both tem perature and E-mode polarization d ata, one also needs to compute (@e ) ( EE) N-. , which is a significantly more complicated expression. The coefficient sF ^ l ^ appears often and is related to the W igner-3j symbol in the following way s F e 1i e 2 — [ L ( L / x (C.37) + 1) + -^2 (^ 2 + 1) —^1 (^ 1 + 1)] ( 2 L + l ) ( 2 l j + l ) ( 2 l 2 + 1) 167T h l t2 s 0 — s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 The W igner-3j symbol only has an exact analytic expression if s = 0, in which case ( ix L \ 0 0 0 .1 )5 /2 (5/2)! (S/2 — £i)\(S/2 — L ) \( S /2 — ^2)! (C.38) / x ' ( 5 - 2 ^ ) ! ( 5 - 2 L ) ! ( 5 - 2 £ 2)! (5 + 1)! where S = t \ + L + i 2. At high multipoles, however, there is an approxim ation for the s = 2 case when the sum S is even 2 0 -2 1 (L 2 - l \ - t 2 l\l\ 1 lx L 0 0 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (C.39) 108 A p p en d ix D C ode E volve This code numerically integrates the equation of m otion for a single scalar field in order to find the detailed shape of the prim ordial power spectrum of density perturbations. The program is w ritten in C and includes an interactive command line for choosing the potential, setting initial conditions, and producing plots of the output. Custom inflationary potentials are relatively easy to add. The code is released under the BSD license (see th e license file for details), and can be found at the URL h t t p : / /bubba.u c d a v i s . e d u / ~ g o l d / e v o l v e .t a r .gz Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 P erl scrip ts I have w ritten several scripts useful for processing the outp u t of C M B F A S T and other codes. These are periodically u p d ated and collectively made available for free and public use at the URL h t t p : //b u b b a . u c d a v is . e d u /~ g o ld /p e r l/in d e x .htm l Included are scripts for diagonalizing and sorting the eigenvectors of Fisher m atri ces (or any symmetric m atrix), and a script th a t implements th e marginalization technique described in appendix C .l. Also available is a module for treating d ata files of a certain form at as objects upon which various m athem atical operations can be performed, e.g. two d ata files can be added together w ith autom atic spline interpolation as necessary. Several of these scripts make use of th e Math: :M atrix R eal package, available from CPAN at h t t p : //www. c p a n . o rg / Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.

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