james p. zibin, adam moss and douglas scott- the evolution of the cosmic microwave background

8/3/2019 James P. Zibin, Adam Moss and Douglas Scott- The Evolution of the Cosmic Microwave Background

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arXiv:0706.4482v1

[astro-ph]

29Jun2007

The Evolution of the Cosmic Microwave Background

James P. Zibin, Adam Moss, and Douglas Scott

Department of Physics & AstronomyUniversity of British Columbia,

Vancouver, BC, V6T 1Z1 Canada(Dated: June 29, 2007)

We discuss the time dependence and future of the Cosmic Microwave Background (CMB) inthe context of the standard cosmological model, in which we are now entering a state of endlessaccelerated expansion. The mean temperature will simply decrease until it reaches the effectivetemperature of the de Sitter vacuum, while the dipole will oscillate as the Sun orbits the Galaxy.However, the higher CMB multipoles have a richer phenomenology. The CMB anisotropy powerspectrum will for the most part simply project to smaller scales, as the comoving distance to lastscattering increases, although there will also be a dramatic increase in the integrated Sachs-Wolfecontribution at low multipoles. We also discuss the effects of tensor modes and optical depth due toThomson scattering. We introduce a correlation function relating the sky maps at two times and theclosely related power spectrum of the difference map. We compute the evolution both analyticallyand numerically, and present simulated future sky maps.

PACS numbers: 98.70.Vc, 98.80.Cq, 98.80. Jk

I. INTRODUCTION

The Cosmic Microwave Background (CMB) radiationprovides us with a vital link to the epoch before theformation of distinct structures, when fluctuations werestill linear and carried in a very clean way informationabout their origin, presumably during a phase of infla-tion. The simple dynamics of the generation and prop-agation of CMB anisotropies (see e.g. Refs. [1, 2, 3] andreferences therein) depends on a handful of cosmological

parameters, Pi, such as the Hubble constant, the mat-ter density, and spatial curvature, in addition to the ini-tial conditions set through inflation. These dependencieshave been thoroughly investigated over the past couple ofdecades and form the basis for estimating the parametersfrom the observed anisotropy spectrum of the CMB.

However, there is one dimension in the parameter spaceof the CMB that has received little explicit attention. Forfixed matter content and curvature of the Universe today,we still have the freedom to evolve the CMB anisotropiesforwards or backwards in time. For the practical businessof performing CMB parameter estimation, it is natural ofcourse to suppress this freedom, since we are interested

in predicting the anisotropies today. The constraint totoday can be applied in at least two ways, which itis important to distinguish. From the set of parametersPi we can calculate the proper-time age of the Universe,t0. This quantity is only determined to an accuracy setby the parameters Pi (e.g. using WMAP 3-year data,Spergel et al. 2006 find that t0 = 13.73

+0.130.17 Gyr). How-

ever, the WMAP results constrain the redshift of lastscattering, zrec, (defined as the centre of the recombi-

Electronic address: [email protected] address: [email protected]

Electronic address: [email protected]

nation epoch) to much greater accuracy: zrec = 1088+12

[4]. This very high accuracy is the result of our accu-rate determination of the mean temperature of the CMB,T = 2.725 0.001 K [5, 6]. Thus, even though t0 is onlyknown to an accuracy comparable to the other parame-ters Pi, implicit in analyses of the CMB is the very tightconstraint on a different temporal coordinate, zrec or T.

Essentially, the constraint on t0 arises from our deter-mination of the expansion rate today, together with in-formation on the content and geometry of the Universe,

which affect its expansion history. The constraint on Tis entirely independent of the content or geometry, henceits superior accuracy. Popular CMB anisotropy numer-ical packages, such as cmbfast or camb [32] automati-cally impose the tight constraint arising from the meantemperature. This constraint on T is equivalent, via theStefan-Boltzmann law, to a constraint on the energy den-sity in CMB radiation, . Therefore it is impossible,without modifying the code, to generate spectra withthese packages that correspond to a given model evolvedinto the past or future, since necessarily evolves withtime. We can vary the proper age t0 of a model byvarying the expansion rate today, but this necessarily

changes the relative contributions of matter and radia-tion in the past, which affects the physics at recombina-tion and hence the shape of the CMB spectrum.

Fortunately the required code modifications are rela-tively straightforward, and in addition it is possible todescribe the temporal evolution of the CMB anisotropiesanalytically to very high precision. In this work we sys-tematically describe this evolution both numerically andanalytically, within the context of the standard CDM( Cold Dark Matter) model, in order to complete thestandard results on the parametric dependence of theCMB. The verification of our numerical work with ouranalytical results and conversely the characterization of

our analytical approximations with the full numerical cal-
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culations will be crucial in this novel study. We willfind that while the temporal behaviour of the CMBpower spectrum is determined mainly by a simple ge-ometrical scaling relationship, less trivial physics ariseswhen we consider the behaviour of correlations betweenanisotropies at different times.

It can certainly be argued that the standard calcula-tions of the CMB anisotropy spectrum implicity describeits time dependence in that the spectrum must be evolvedfrom the time of recombination to the present. Neverthe-less, there appear to have been very few explicit discus-sions of the time dependence, with the exceptions beingprimarily concerned with the distant future. Gott [7]points out that at extremely late times the typical wave-lengths of CMB radiation will exceed the Hubble radius,and so the CMB radiation will be lost in a de Sitter back-ground. Loeb [8] mentions that as the time of observationincreases, the radius of the last scattering sphere also in-creases, and approaches a maximum in a CDM model.This leads to the potential for reducing the cosmic vari-ance limitation on the determination of the anisotropyspectrum. Krauss and Scherrer [9] point out that wellbefore this final stage, the CMB will redshift below theplasma frequency of the interstellar medium and hencebe screened from view inside galaxies.

Importantly, when discussing the future evolution ofthe Universe it must be remembered that even the qual-itative details can depend very sensitively on the modeladopted. An example is the potential destruction of theUniverse in finite proper time in a big rip [10] whenthe dark energy violates the weak energy condition, with

equation of state w < 1. In the present work, forthe sake of definiteness and simplicity, we conservativelychoose a model in which the dark energy is a pure cosmo-logical constant. However, using the techniques we dis-cuss it is straightforward to extend our results to otherspecific dark energy models, a subject we will return toin future work [11].

An interesting question that naturally arises in thepresent context is: How long must we wait before wecould observe a change in the CMB? The formalism thatwe develop here will be necessary to answer this question,and we will address this explicit issue in future work [11].

We begin in Section II with a description of the time

dependence of the bulk properties of the CMB, namelythe mean temperature and the dipole. After a brief re-view of the formalism used to describe the anisotropypower spectrum, its evolution is described analytically inSection III, including the effects of the integrated Sachs-Wolfe effect, tensor modes, and reionization, and numer-ical calculations are presented using our modified versionof the line-of-sight Boltzmann code camb. In Section IVwe introduce the difference map power spectrum and as-sociated correlation function, and present analytical andnumerical calculations. Section V presents our conclu-sions, and in the Appendix a description of an impor-tant approximation method is presented. We set c = 1

throughout.

II. TIME EVOLUTION OF THE BULK CMB

The temperature fluctuations on the CMB sky can bedecomposed into a set of amplitudes of spherical harmon-

ics (see Section IIIA). The angular mean temperature(or monopole) and dipole have a special status. Themean temperature is just a measure of the local radia-tion energy density, while the value of the dipole dependson the observers reference frame at linear order (highermultipoles are independent of frame at this order).

A. The mean temperature

As time passes, the change in the CMB that is simplestto quantify is the cooling of its mean temperature T dueto the Universes expansion. The CMB radiation was

released from the matter at the time of last scattering,when T 3000 K. It later reached a comfortable 300 Kat an age of about t 15 Myr, and is now only a frigidfew Kelvin. Indeed, the monotonicity of the functionT(t) means that T itself can be used as a good timevariable. Thus we can consider measurements of T asdirect readings of a sort of cosmic clock.

Today the CMB radiation is essentially free stream-ing, i.e. non-interacting with the other components ofthe Universe. Therefore the energy density in the CMBevolves according to the energy conservation equation = 4H, where H is the Hubble parameter andthe overdot represents the proper time derivative. Since

T4, we have T =

H T. Evaluating this expression

today, using T0 = 2.725 K and H0 = 73 kms1 Mpc1(subscript 0 indicates values today), we find

T0 = 0.20 K kyr1. (1)Thus in 5000 yr the mean temperature will drop by 1 K.

The CMB radiation continues to redshift indefinitelyas the universe expands in the late -dominated de Sit-ter phase. However, this does not mean that as the CMBbecomes increasingly difficult to measure, clever experi-mentalists need only to ever refine their instruments inorder to keep up. Instead, a fundamental limit existsbelow which a CMB temperature cannot be sensibly de-fined. An object in an otherwise empty de Sitter phasewill see a thermal field with temperature [12]

TdS =H

2=

1

2

3. (2)

Therefore, after the CMB temperature redshifts to belowTdS, the CMB becomes lost in the thermal noise of thede Sitter background, as pointed out in [7] (see also [13]).

To see explicitly the difficulty with measuring the CMBat such extremely late times, consider the typical wave-lengths of radiation in the CMB. A thermal spectrum attemperature T consists of wavelengths on the char-acteristic scale T1 (the precise peak position of the

Planck spectrum depends on the measure used for the


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thermal distribution). Therefore when T = TdS we have H1, i.e. the typical CMB wavelengths become oforder the Hubble length. Alternatively, at late times inthe de Sitter phase the frequency of a mode of CMB ra-diation of fixed comoving wavelength redshifts accordingto

(t) = (t1)eHdS(tt1), (3)

where

HdS

H0 (4)

is the asymptotic value of the Hubble parameter and t1 issome late proper time. Therefore the accumulated phaseshift between time t1 and the infinite future is

t1

(t)dt =(t1)

HdS. (5)

This expression tells us that when the frequency becomesless than the Hubble parameter (i.e. the wavelength be-comes larger than the Hubble length), a full temporaloscillation cannot be observed, even if we observe intothe infinite future. In terms of conformal time, the oscil-lation rate remains constant in the de Sitter phase, butthere is only a finite amount of conformal time availablein the future. Indeed, considering the quantum natureof such a mode, this calculation provides insight into thenecessity of a residual de Sitter thermal spectrum at thisscale.

The energy density in the CMB at arbitrary scale fac-tor a is given by

=3H20m

2P

8

a0a

4, (6)

where mP is the Planck mass and = 5 105 is thefraction of the total density in the CMB today. Usingthis expression we can show that we must wait untila/a0 1030 before = dS T4dS and the CMB be-comes lost in the de Sitter background. This correspondsto an age of t = 1 Tyr. If we ask instead at what scalefactor would the radiation density be equal to the Planckdensity, P = m4P, we find a/a0

1032. It might ap-

pear, therefore, that we exist at a special time, in that theradiation density today is roughly 120 decades removedfrom both the Planck era and the final era when T = TdS.To understand the origin of this coincidence, note that byvirtue of the above expressions and the energy constraint(or Friedmann) equation, the three densities P, , anddS are in the geometrical ratio m4P : m

2P :

2, up to nu-merical factors. Therefore, the apparent coincidence justdescribed is actually equivalent to the standard coinci-dence problem, namely that tot today, given that differs from tot today by only a few decades. Anydensity that today even crudely approximates the darkenergy density will necessarily be separated by roughly

120 decades from both P and dS.

B. The dipole

The observed dipole anisotropy in the CMB can beattributed to the Doppler effect arising from our peculiar

velocity, v, with respect to the frame in which the CMBdipole vanishes. That peculiar velocity, and hence thedipole, is expected to evolve with time. The magnitudeof the dipole can be specified by the maximum CMBtemperature shift over the sky, Td, due to the velocityv. This is given by the lowest order Doppler expression,

TdT

= v. (7)

(In terms of the spherical harmonic expansion to be in-troduced in Eq. (16), we have Td/T = a10, when thepolar axis is aligned with v.)

The current best estimate of the magnitude of the

dipole comes from observations of the WMAP satelliteindeed, the annual modulation by the Earths motionaround the Sun is actually used to calibrate satellite ex-periments, so this aspect of the time-varying dipole is al-ready well measured. The measured value of the dipole,in Galactic polar coordinates, is (Td, l , b) = ( 3.358 0.0017 mK, 263.86 0.04, 48.24 0.10) [14]. Equiv-alently, the Cartesian velocity vector is v0 (26.3,244.6, 275.6) kms1, where the first component is to-wards the Galactic centre, and the third component isnormal to the Galactic plane. Therefore, in natural unitswe have v0 = 1.2 103, so the lowest order approxima-tion, Eq. (7), is valid.

In order to determine the evolution of the dipole, wemust specify the worldline with respect to which theCMB is measured. During matter or domination, forwhich pressure gradients vanish to very good approxima-tion, we can take that worldline to be a geodesic. Then itis straightforward to show that the peculiar velocity (andhence the dipole) evolves, at lowest order in perturbationtheory, according to

v = v0a0a

, (8)

where a is the background scale factor. This expressionfor the evolution of v describes a simple cosmological-time-scale decay of the dipole anisotropy. However, the

expression was derived on the assumption that lowestorder perturbation theory was valid, which is certainlynot a good approximation on sufficiently small scales to-day. To properly describe the evolution of the velocity vwe must take into account the presence of the nonlinearstructures we observe on small scales today.

This velocity vector can be considered as a sum of in-dividual vectors contributing to the overall motion of theSun with respect to the CMB. In the local neighborhood,the Sun moves with respect to the local standard ofrest, which in turn moves with respect to the Galacticcentre. However, the peculiar motions in the Solar neigh-borhood are subdominant compared with the motion of

the Sun around the Milky Way [15], so for the purposes


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of the simple calculation which follows we ignore thesecontributions. Also, we will consider here times scalesshort enough that the motion of the Milky Way withinthe Local Group, and the Local Group relative to Virgo,the Great Attractor, and other distant cosmic structuresis approximately constant.

Just as today we can detect the modulation of theEarths motion around the Sun, in the future, with in-creasing satellite sensitivity, we may be able to observethe Suns motion around the Galaxy. For the motion ofthe Sun around the Milky Way, we assume that this issimply a tangential speed of 220 km s1 at a distance of8.5 kpc. Using the current observed value of v to inferthe velocity of the Galactic centre with respect to theCMB rest frame, the time dependent Sun-CMB velocityvector is then

v(t) = 222sin2t

T 26.3,222cos

2t

T

466.6, 275.6

km s1 , (9)

where the Galactic orbital period is T = 2.35 108 yr.In order to ascertain when a change in the dipole is

detectable, one could compute a sky map of the dipole attwo times. If the temperature variance of the differencemap is greater than the experimental noise variance, thena detection is probable. In this case, the variance of thedifference map, which we denote by CS, is CS = [(vx)2 +(vy)

2 + (vz)2]/(4), where v is the difference of the

Sun-CMB dipole vector between the two observations.

Using Eq. 9, and converting to fractional temperaturevariations, the signal variance of the changing dipole isthen

CS = 8.7 108

1 cos

2t

T

. (10)

Later in this paper we compute signal variances involvinghigher order CMB multipoles. These variances are ofcourse much smaller than that of the dipole. In a followup paper [11] we will discuss in detail the prospects fordetecting a change in the CMB with future experiments.

Finally, we note that in this simple calculation it isassumed that we have an exterior frame of reference ex-

ternal to the Milky Way in order to construct our co-ordinate system. This could be provided, for example,by the International Celestial Reference Frame, based onthe positions of 212 extra-galactic sources [16].

III. THE ANISOTROPY POWER SPECTRUM

A. Review of the basic formalism

There is much more information encoded in theanisotropies of the CMB than in the mean temperature,since the anisotropies are determined by the details of the

matter and metric fluctuations near the last-scattering

surface (LSS) and all along our past light cone to to-day. Therefore it is much less trivial to determine thetime evolution of the anisotropies than the mean tem-perature (or dipole). However, in the approximation thatall of the CMB radiation was emitted from the LSS atsome instant tLS when electrons and photons decoupled,and then propagated freely, the evolution of the primarypower spectrum of the CMB is determined by a simplegeometrical scaling relation which is closely related to themain geometrical parameter degeneracy in CMB spectra.In order to derive this relation, and to describe the be-haviour of the correlation functions introduced in latersections, it will be helpful to first summarize the stan-dard description of CMB anisotropies in a form that willbe easy to generalize. This subsection may be skippedby readers familiar with the material. For detailed treat-ments of the generation of anisotropies see e.g. [17, 18].

At very early times, when each perturbation mode, la-

belled by comoving wavevector k, is outside of the Hub-ble radius, the fluctuations can be described by a singleperturbation function, for the case of adiabatic pertur-bations. It is very convenient to take this function to bethe curvature perturbation on comoving hypersurfaces,R, since this quantity is conserved on large scales in thiscase, and hence can be readily tied to the predictions ofa specific inflationary model. In the simplest models ofinflation, R is predicted to be a Gaussian random field tovery good approximation, fully described by the relation

R(k)R(k) = 223(k k)PR(k)k3

, (11)

with primordial power spectrum PR(k) and k |k|. Fora scale-invariant spectrum we have PR(k) = constant.

The fluctuations at last scattering can be described bya set of matter and metric perturbations, i(x, ), wherefor future convenience we have used comoving coordinatex and conformal time . Since linear perturbation theoryis a very good approximation at the scales sampled by theCMB, this set of perturbations is determined from theprimordial comoving curvature perturbation by transferfunctions Ai(k, ) via

i(k, ) = Ai(k, )R(k). (12)

In the approximation of abrupt recombination, so thatthe LSS has zero thickness, followed by free streaming ofradiation, and ignoring the effect of gravitational lensingby foreground structure, the observed primary tempera-ture anisotropy T(e)/T in direction e is determined bythe fluctuations at the corresponding point on the LSS,i.e.

T(e, )

T()= F(i(rLS, e, LS)), (13)

for some linear function F. Here rLS = LS is thecomoving radial coordinate to the LSS from the pointof observation, taken to be the origin of spherical coor-

dinates. In the approximation that photons are tightly


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coupled to baryons before LS, the function F can bewritten in terms of two perturbation functions as

F(i(rLS, e, LS)) = 1(rLS, e, LS)+

rLS

2(rLS, e, LS).

(14)Eqs. (13) a n d (14) describe the generation of CMBanisotropies through the Sachs-Wolfe effect [19], withthe first term on the right-hand side of (14) the so-called monopole contribution, and the second term thedipole or Doppler contribution.

The preceeding equations have the very simple inter-pretation that when we measure the CMB anisotropies atsome time we are seeing the primordial fluctuationsR on the comoving spherical shell r = rLS = LS, asprocessed by the linear transfer functions A1 and A2 tothe time LS. If we observe at a later time , we see thefluctuations on a larger shell of radius r = rLS

LS,

as illustrated in Fig. 1. The fluctuations at the LSS con-tain structure at various scales, encoded in the trans-fer functions, due to acoustic oscillations within the pre-recombination plasma. Assuming the statistical homo-geneity of space, that structure will occur at the samephysical scales on the shells r = rLS and r = rLS. There-fore we expect that structure visible at time on angularscale will also be visible at , but at the smaller angularscale

= rLSrLS

, (15)

at least for small scale structure,

1. To make this

rigorous, and to derive in addition the scaling law for theamplitude of angular structure, we need to next introducea spherical expansion of the CMB anisotropy.

We expand as usual the temperature fluctuation ob-served in some direction e in terms of spherical harmonicsYm as

T(e, )

T()=m

am()Ym(e). (16)

The expansion coefficients am determine all the detailsof the particular sky map of the CMB observed at time. However, the statistical properties of the ams are

determined through Eqs. (12) to (14) by the statistics ofR encoded in Eq. (11). To make this explicit, we need thespherical expansion of the perturbations i(rLS, e, LS),namely

i(rLS, e) =

2

kdkm

im(k)j(krLS)Ym(e).

(17)Here i = 1 or 2, j is the spherical Bessel function of thefirst kind, and we have dropped the understood argumentLS. Next the identity

fm(k) = ki dkf(k)Ym(k) (18)

rLS

k

rLS

FIG. 1: Our spheres of last scattering at time (inner) and (outer). The group of vertical lines indicates the crests ofa mode k, and hot spots will be observed at the intersectionsof those crests with the spheres. One wavelength of the modewill span angle at but a smaller angle at .

(see, e.g., Ref. [20]) combined with Eq. (12) allows us towrite

im(k) = Ai(k)Rm(k). (19)Now, combining Eqs. (14), (17), and (19), we have

F(rLS, e) =

2

kdkm

Rm(k)T(k, , rLS)Ym(e),

(20)where

T(k, , rLS) A1(k)j(krLS) + A2(k)j(krLS) (21)

and the prime denotes differentiation with resepect torLS. Finally, equating coefficients between Eq. (20) [with

Eq. (13)] and Eq. (16), we obtain

am() =

2

kdkRm(k)T(k, , rLS), (22)

where we have restored the argument = rLS + LS.This expression gives the CMB anisotropy in terms ofthe primordial perturbations R and a new linear transferfunction T(k, , rLS).

In order to determine the statistical properties of theams, we need the expression

Rm(k)

R

m(k)

= 22(k

k)

PR(k)k3

mm

, (23)


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which can be derived from Eq. (11). Using this expressionand Eq. (22) we find

am()am() = C()mm, (24)

where

C() 4

dk

kPR(k)T2(k, , rLS). (25)

That is, each coefficient am has variance C() (whichis called the anisotropy power spectrum) and coefficientsfor different spherical modes are uncorrelated.

Note that Eq. (22), and hence Eqs. (24) and (25), holdeven when we relax the tight coupling and free streamingapproximations, with some transfer function T(k, , rLS).However, in the general case the transfer function mustbe calculated numerically.

B. Analytical time evolution for primaryanisotropies

The formalism developed in the preceeding subsectioncan now be applied to describe the time evolution of theprimary CMB anisotropy spectrum, under the abrupt re-combination and free streaming approximations. To de-termine the time evolution ofC(), Eq. (25) tells us thatwe only need to consider the behaviour of T2(k, , rLS) asrLS increases (note that we will often adopt the coordi-nate rLS as an effective time coordinate). To do this,Eq. (21) tells us that we only require the behaviour ofthe products j2 (krLS), j

2 (krLS), and j(krLS)j

(krLS) as

functions of rLS. This can be done in the limit 1 us-ing asymptotic forms for the Bessel functions. For large we can write [see Ref. [21], Eq. (9.3.3)]

j(x) = (x4 x22)1/4[cos() + O(1/)], for x > ,

(26)where = (x) is a phase. For x < , j(x) decaysrapidly. This allows us to write a scaling relation for theenvelope of the Bessel oscillations, namely

j(x) j(x), (27)for positive such that 1 also applies. This expres-sion will allow us to obtain the time dependence of themonopole contribution to T2(k, , r

LS), which is pro-

portional to j2 (krLS). We can write the dipole partj2 (krLS) in terms of spherical Bessel functions using re-currence relations and again apply Eq. (27) to obtainthe time dependence. The cross term proportional to

j(krLS)j(krLS) can be shown to be negligible, i.e. themonopole and dipole contributions add incoherently. Ap-plying Eq. (27), then, we find that for large ,

T2(k, , rLS) r2LSr2LS

T2(k, , rLS), (28)

where we have defined

= rLS

rLS. (29)

Applying Eq. (25) we finally obtain the scaling relationfor the power spectrum,

2C() 2C(). (30)

Importantly, Eq. (30) holds independently of the formof the functions Ai(k) and PR(k), so the result appliesto the acoustic peak structure as well as to non-scale-invariant primordial spectra.

This result confirms our previous prediction, Eq. (15),for the dependence of angular scales on observation time.But the dependence of the amplitude of the spectrum en-coded in Eq. (30) is also not surprizing, since the quantity( + 1)C is independent of in the Sachs-Wolfe plateaufor a scale invariant spectrum, as is well known. But theheight of that plateau, calculated using A1 = const andA2 = 0 above, is independent of the observation time.(Indeed that height is, up to numerical factors, simply

PR. Recall that C is determined by the ratio T/T.

The absolute anisotropies T exhibit the same expan-sion redshift as does the mean temperature T.) Hence as increases, the quantity 2C() must remain constant(up to corrections of order 1/), which is precisely whatEq. (30) says. Of course, the result (30) is valid for theentire acoustic peak structure, not just the Sachs-Wolfeplateau.

The result (30) is derived in the Appendix much moredirectly, without resorting to properties of Bessel func-tions, using the flat sky approximation. In that approachwe consider anisotropies in a patch of sky small enoughthat it can be approximated as flat, and errors are againof order 1/.

In addition to the main temperature anisotropies wehave been considering here, there are also polarizationspectra present in the CMB radiation. The polarizationis sourced primarily near last scattering, so its spectrawill also scale according to Eq. (30). A small part ofthe largest-scale polarization is sourced near reionization,so we expect that that contribution will scale with thecomoving radius to the reionization redshift, rather thanto the last scattering surface.

Having found the scaling relation (30), we can nextderive some simple consequences from it. First, we canwrite the total power in the anisotropy spectrum as

m

C() =

(2 + 1)C() 2 C()d (31)in the large approximation. Then, using Eq. (30), wehave

(2 + 1)C()

(2 + 1)C(), (32)

where the approximation comes from ignoring terms oforder 1/. That is, the total power is constant in time, forthe free streaming of primary anisotropies. This result isequivalent to the conservation condition stated in [22].Implicit in this result is the assumption of statistical ho-mogeneity, so that no new anomalous power will be re-

vealed at the largest scales as rLS increases. As we will see


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in Section IIIC below, secondary anisotropies, in particu-lar those generated through the integrated Sachs-Wolfeeffect, are expected to grow dramatically at late timesand hence the total power will not in fact be conserved.

Another consequence of Eq. (30) follows from the na-ture of the asymptotic future in our CDM model. Ob-servers in a universe with positive have a future eventhorizon, i.e. the conformal time converges to a finite con-stant f as proper time t . Therefore the angularscaling relation (29) tells us that as proper time (or scalefactor) approaches infinity, the value for any particularfeature in the C spectrum, such as a peak position, willapproach a finite maximum, i.e. features will approacha non-zero minimum angular size. (Geometrically, theLSS sphere approaches a maximum comoving radius, sofeatures on it must approach a minimum size.) For ourfiducial model we chose = 0.77, and so Eq. (29) givesfor the limiting scaling relation

f = 0

aLS

(aa)1daa0aLS

(aa)1da= 1.310. (33)

For example, the first acoustic peak, which we observeto be at the position 0 = 221 today, will asymptote tof = 290 in the late de Sitter phase. This asymptotic be-haviour is in marked contrast to that of a purely matter-dominated Einstein-de Sitter model. In the vanishing case, the numerator in Eq. (33) diverges (no event hori-zon exists) and the structure in the C spectrum shiftsto ever smaller scales.

The geometrical scaling relation (30) is very closelyrelated to the well-known geometrical parameter degen-eracy in the CMB anisotropy spectrum between spatialcurvature and [23, 24, 25]: If two cosmological modelsshare the same primordial power spectrum PR(k), thesame physical baryon and CDM densities today, b andc, and finally the same angular diameter distance dAto the LSS, then they will exhibit essentially identicalprimary C spectra. The degeneracy can only be brokenby secondary sources of anisotropy, such as the integratedSachs-Wolfe effect, or by other cosmological observations.

To understand the origin of this degeneracy and its re-lation to the preceeding discussion, recall that the energydensity in the CMB today, , is fixed to very high ac-curacy by the measurement of the mean temperature, as

we mentioned in the Introduction. Therefore if we con-sider models with identical values of the densities b andc today, then the densities of baryons, CDM, and pho-tons at last scattering are the same for all such models,since the densities scale in a well-defined manner (for ex-ample, /c a0/a). Therefore, given the same initialconditions in the form of PR(k), models that have com-mon values of b and c today will have identical localphysics at least to the time of recombination, when anyspatial curvature or will have negligible effect. Thusthese models will produce identical primary anisotropies.

If the models have different values of K (spatial cur-vature) and then the dynamics, including the propaga-

tion of CMB anisotropies, will differ significantly at late

times as those components come to dominate. However,if the models share the same angular diameter distance,then their C spectra, which should be calculated usingEq. (25) with rLS replaced by dA (at least for small scaleswhere the effects of spatial curvature on the primordialspectra can be ignored), will be identical. Geometrically,models with identical PR(k), b, and c share the samelocal physics to recombination, and hence the same phys-ical scales for acoustic wave structures (in particular thesame sound horizon). For models which additionally haveidentical dA, observers see the anisotropies generated ona spherical shell at the time of last scattering of identi-cal physical surface area (given by 4d2A). Hence thosephysical acoustic scales are mapped to identical angu-lar scales in the sky for the different models. In short,the observed primary anisotropies in models with identi-cal PR(k), b, c, and dA are produced under the samelocal physical conditions on a sphere of identical phys-

ical size, and hence appear identical. The scaling rela-tion (30) describes how the observed anisotropies changeif we hold the local physics at recombination (togetherwith and K) constant, but allow the time of observa-tion to vary, which amounts to simply varying the size ofthe sphere at last scattering that generates the observedanisotropies. The parameter degeneracy states that thesame anisotropy spectrum can be produced even if andK vary, as long as the size of the last scattering sphereis held constant.

To close this discussion of the primary anisotropies,we introduce the power spectrum difference C() C() C() between the spectra observed at two dif-ferent times. This is a measurable quantity which wemight consider a candidate for detecting the evolution ofthe CMB. Given some spectrum C() at a single time we can readily calculate the difference C() using thescaling relation (30). For small , we have

C()

C(), (34)

so that the change in the CMB power spectrum at fixed is proportional to . As we will see in the next Section,this behaviour differs from that of the power spectrum ofthe difference am() am(). Using Eq. (30) for thetime dependence, we can write

C() = LS C()

+ 2C() , (35)

at first order in /LS. Note that C can have eithersign, and will equal zero whenever (2C)/ = 0, as forexample on a scale-invariant Sachs-Wolfe plateau or atan acoustic peak.

Recall that the quantity C() describes the relativeanisotropies T/T, and hence is insensitive to the bulkexpansion redshift. If we wish to consider instead theevolution of the absolute temperature anisotropies T,the relevant quantity to calculate is

[T2C()] = T2 2

(aH)1C() + C() (36)


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8

at lowest order in /(aH)1, where we have used the

relation T = HT. Since in standard CDM modelsLS is a few times the comoving Hubble radius (aH)

1,the first term on the left-hand side of Eq. (36) is of the

same order as the second term. That is, the expansioncooling effect is on the same order as the geometricalscaling effect, and so it will be important to distinguishthe two effects.

C. Integrated Sachs-Wolfe effect

The simple scaling relation derived in the previous sub-section determines the time evolution of the power spec-trum of anisotropies produced near the LSS. However,in the standard CDM model, significant anisotropiesare also produced at late times as a result of the chang-

ing equation of state as the Universe becomes cosmolog-ical constant dominated. This process is known as the(late) integrated Sachs-Wolfe (ISW) effect [26]. Sincethese anisotropies are produced relatively locally, theirtime dependence must be explicitly calculated. Notethat anisotropies are also generated by the early ISW ef-fect, during the time that radiation still significantly con-tributes to the dynamics. However, those anisotropies areproduced relatively close to the LSS, adding coherentlyto the primary Sachs-Wolfe contribution, and hence scaleas do the primary anisotropies, according to Eq. (30).

The contribution of the late ISW effect can be de-scribed by adding to the transfer function, Eq. (21), aterm TISW(k, , ) which is an integral over the line ofsight to the LSS. For the case of interest, for which theanisotropic stress is negligible, we have [18]

TISW(k, , ) = 2A3(k)

LS

dg()j[k( )]. (37)

Here g() dg/d, with g() being the growth func-tion which describes the temporal evolution of the zero-shear or longitudinal gauge curvature perturbation, (also called the Newtonian potential), via

(k, ) g()A3(k)R(k). (38)

The function A3(k) is defined such that g() 1 at earlytimes in matter domination. Then, a gauge transforma-tion between the comoving, R, and zero-shear, , curva-ture perturbations during the matter dominated periodgives A3(k) = 3/5.

To evaluate the ISW contribution, we need to first de-termine the evolution of the curvature perturbation .To do this, we only need to solve the space-space, or dy-namical, linearized Einstein equation. For the case wherepressure and anisotropic stress perturbations can be ig-nored, which holds in a universe containing only dust and, this equation becomes

+ 4H

+ (3H

2

+ 2

H) = 0 (39)

(see, e.g., Ref. [27]). There are no spatial gradients inthis equation, which confirms that the growth function isindependent ofk. It is straightforward to verify that thegrowing mode solution to this equation is

() Ha

d

a2H

H2. (40)

Next, using the relation a3H = const, which holds ex-actly for a universe consisting of dust and cosmolog-ical constant, employing Eq. (38), and matching thegrowing mode solution to the initial condition 0(k) =(3/5)R(k), we obtain

g() =5

2

0H

a

daH2

, (41)

where 0 is the density in matter today relative to the

critical density, and the scale factor is normalized to unitytoday.

Now that the evolution of the growth function has beendetermined, we can evaluate the ISW contribution to thepower spectrum. It can be shown that the cross termbetween the Sachs-Wolfe and (late) ISW terms is negli-gible, so the two add incoherently in C. For the ISWpart we have

CISW () = 4

dk

kPR(k)T2ISW(k, , ) (42)

=72

25

2PR3

0

dg2()( ). (43)

To obtain this result we have assumed a scale invariantprimordial spectrum, PR(k) = PR, and we have used therelation [28]

0

dk

kj[k( )]j[k( )]

23( )( ),

(44)which holds for large . Unfortunately it is at small that the ISW effect is greatest, so this approximation,which appears to be the best we can do analytically, isnot terribly accurate at very late times when the ISWcontribution becomes very large, as we shall see. Nev-ertheless, Eq. (43) will give a reasonable estimate of theISW contribution to a change C over short time inter-vals.

Given a primordial amplitude, PR, and a matter den-sity parameter today, 0, Eqs. (41) and (43) allow us tocalculate the ISW contribution to the power spectrumat any time. In addition, taking the time derivative ofEq. (43), we find for the rate of change of the ISW con-tribution

CISW () =

72

25

2PR3

0

dg2(). (45)

Therefore, combining this expression with Eq. (35) for

the change in the primary Sachs-Wolfe power spectrum


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9

over short time intervals , we find for the total contri-bution

C() = LS

C()

+ 2C()

7225

2PR3

LS0

dg2()

. (46)

D. Gravitational waves

Inflationary models generically predict a spectrum ofprimordial gravitational waves, although the relativecontribution of these tensor modes to the total CMBanisotropy ranges from substantial to very small, depend-ing on the model. The anisotropies arise through thetensor analogue of the scalar ISW line of sight integral,Eq. (37). In place of the time derivative of the scalar cur-vature perturbation , the tensor contribution involvesthe rate of change of the transverse and traceless partof the spatial metric perturbation, hij . The evolutionof this part of the metric perturbation is given by thedynamical Einstein equation, which becomes (see, e.g.,[27])

hij + 2aHhij 2hij = 0 (47)

in the absence of tensor anisotropic stress, which is a validapproximation in the matter- or -dominated regimes.

The dynamics of hij as dictated by this equation de-pends on the mode wavelength relative to the Hub-

ble scale. For k/(aH) 1, the tensor mode is over-damped, and the (growing) mode decays very slowly. Fork/(aH) 1, the mode undergoes underdamped oscilla-tions, decaying like hij eik/a in the adiabatic regime.Therefore, for a particular tensor mode k, the rate ofchange hij is peaked near the time that the mode crossesthe Hubble radius, k/(aH) 1, and is small at earlyand late times. This means that the contributions to theCMB anisotropies from a particular scale arise primarilywhen that scale enters the Hubble radius. In particular,scales that enter significantly before last scattering willhave decayed before they could source the CMB. Thisimposes a small scale cut-off on the tensor anisotropyspectrum, with negligible power for

c rLSaLS, (48)where 1/aLS is the comoving Hubble radius at last scat-tering. Since aLS is fixed for a particular model, the cut-off scales with time of observation according to

c() = c()

rLSrLS

, (49)

which is the same scaling as for primary features in thescalar CMB spectrum, Eq. (29).

For c, detailed calculations for a matter domi-nated universe [29] show that the tensor anisotropy spec-

trum is nearly flat, mimicing the Sachs-Wolfe plateau.

In this case the tensor spectrum does not evolve apartfrom the scaling of the cut-off, Eq. (49). However, as de-scribed above, the largest angular scales will be sourcedat the latest times, and so for a universe with cosmolog-ical constant this will lead to some dependence on ob-servation time for those scales as the equation of statechanges at late times. For example, for time of observa-tion the tensor quadrupole is sourced near very roughlythe conformal time /2 [29]. In the matter-dominatedera, the comoving Hubble radius 1/(aH) = 1/a increaseswith time, but as the Universe enters the -dominatedera, 1/(aH) starts to decrease (indeed this defines accel-eration). Therefore, the largest scale modes that con-tribute to the tensor anisotropies will enter the Hubbleradius somewhat later in the presence of a cosmologi-cal constant than without (sufficiently large-scale modeswill never enter the Hubble radius). The modes whichenter near time f/2, where f is the asymptotic final

conformal time, will have a significantly delayed entrytime, and therefore we expect the very largest scale ten-sor anisotropies to be somewhat reduced at the latesttimes.

E. Optical depth

One final line of sight effect on the CMB anisotropiesthat we will consider is the time dependent optical depthdue to Thomson scattering. Looking back, it is the rapidincrease in scattering near the time of recombination thatmakes it possible to speak of a surface of last scatter-ing, where the primary anisotropies are emitted. Atlater times, reionization results in a time dependent at-tenuation of the amplitude of the power spectrum, whichit will be important to quantify. In addition, the increasein optical depth for observers at later times will result in ashift in the time of last scattering as defined for those ob-servers, which we should estimate as a consistency checkon our previous calculations.

The first of these effects occurs at the epoch of reioniza-tion (aR = 0.083 in our fiducial model, with the scale fac-tor normalized to unity today). Here, we see a decreasein the amplitude of intermediate to small scale ( >

30)

anisotropies as photons are rescattered. On these scales,C() is reduced by a factor e

2R, where R is the opti-cal depth to reionization. To compute the suppression inC(), recall the definition of the optical depth (a, aobs)between scale factors a and aobs, given by

(a, aobs) = T

aobsa

dt

dane(a

) da , (50)

where T is the Thomson scattering cross-section andne(a) the electron number density. Assuming reioniza-tion is sharp and the energy density of radiation is sub-

dominant, the optical depth to reionization, R(aobs)


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10

(aR, aobs) is given by (see, e.g., [30])

R(aobs) = 0.046(1 Yp) bm

mh2a

3R + h

2

mh2a3obs + h2 , (51)with R(aobs) = 0 for aobs aR. In this expression Yp isthe primordial helium fraction (assumed to be 0.24).

Eq. (51) gives R(1) = 0.088, so reionization reducesC() by a factor of approximately 0.84 by today [ofcourse we must evaluate the spectrum at different timesat the values related by the scaling relation Eq. (29)].Much of the suppression of C() occurs before thepresent time. Since the time the scale factor was half itspresent value, for example, the optical depth to reion-ization has only increased by 0.003. As aobs ,Eq. (51) says that the optical depth will only increaseby R() R(1) = 5.7 104, so that C() willonly be reduced by 0.1% relative to the value today. TheUniverse is essentially transparent on cosmological scalestoday.

The second effect we wish to quantify is the shift inthe time of last scattering LS at late times. We can de-fine the time of last scattering to be the time from whichthe integrated optical depth to the time of observationobs reaches unity, so that (aLS, aobs) 1 if aLS is thescale factor corresponding to LS. Then LS will clearlydepend on obs, with later observation times leading tolater times of last scattering (see Fig. 2). We ignoredany such effect in describing the time dependence of the

power spectrum in Section IIIB: we assumed that LSwas independent of observation time, with the comov-ing position of the LSS, rLS, simply equal to the intervalobs LS. Therefore it will be important to check thatthe increase in the time of last scattering LS, as weconsider observations into the future, is negligible withrespect to the relevant length scales in the accoustic os-cillations at last scattering.

Using the analytic form of the free electron fractionnear recombination given in [31], the optical depth toscale factor a near recombination (ignoring the reioniza-tion contribution) is given by

LS(a) = 0.366(1 Yp) (1000a)14.25(1000aobs)14.25

. (52)

The total optical depth to recombination also includesthe contribution from reionization, but Eq. (52) is suf-ficient to calculate the change in redshift of last scat-tering aLS that corresponds to the increase in opticaldepth as aobs calculated above. The condi-tion LS = gives aLS = 4 108. This impliesthat LS/LS = 8 107, so that the relative change inLS is far smaller than the relative size of the smallestaccoustic features, which are at the 1/max 103 level.Therefore, our assumption that last scattering occurs at

the same LS regardless of observational time is entirely

LS

f

0

FIG. 2: Conformal spacetime diagram illustrating the pastlight cones (dashed lines) for an observer today and in thedistant future at f. Each cone reaches back to the LSS de-fined by unity optical depth. Because of the extra opticaldepth between 0 and f, the time of last scattering for theobserver at f is an interval LS later than for the observertoday.

justified for the current and future evolution of the CMB.Of course for extremely early observation times, such thatthe conformal time to last scattering is of order the thick-

ness of the LSS, this approximation will not be valid.

F. Time evolution from CAMB

In order to confirm and extend the analytic results ofprevious subsections we have modified the camb soft-ware to compute the CMB power spectrum at differ-ent observational times. To do this, we simply modifyall routines such that we can evaluate the transfer func-tions T(k, , rLS) at different rLS, using the set of best-fitting cosmological parameters as measured today. Thechanges required to camb are straightforwardfor the

most part all that is required is changing the tau0 vari-able to trick the code into using a different observationaltime. The parameters we use are those of a standardsix parameter CDM model, given by ch

2 = 0.104,bh2 = 0.0223, h = 0.734, nS = 0.951, AS = 2.02109,and zR = 11.1, where PR(k) = AS(k/k0)nS1 with pivotscale k0 = 0.05 Mpc

1, and zR is the redshift of reion-ization. Where necessary, we set nS = 1 to simplifycomparison with analytic results. We do not expect sig-nificant differences in our results if these parameters arevaried somewhat. Since the parameters are relative tothe present day value, the spectrum measured by an ob-server with z > zR, for example, will not be affected by

the epoch of reionization and a future observer will see a


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11

universe completely dominated by dark energy.In Fig. 3 we plot the power spectrum (+1)C()/(2)

calculated from our modified version of camb, parame-terizing the time dependence in terms of the observa-tional scale factor a

obs, where a

obs= 1 corresponds to

today. It is clear that the angular scale of features re-sulting from projection of inhomogeneities near the LSScan be understood from the scaling relation (30). Forexample, in Fig. 3 we show the predicted scaling of thefirst acoustic peak position (located at = 221 at thepresent time) and find extremely good agreement withthe predicted value. The existence of a future event hori-zon means that during domination drLS/daobs tends tozero and so the acoustic peak positions become frozenin. With our parameters we see that the first acous-tic peak becomes frozen in at the value f 290 as wepredicted in Eq. (33).

FIG. 3: CMB power spectrum (+1)C/(2) as a function ofmultipole and scale factor of observation aobs, as calculatedfrom our modified version of camb for our fiducial CDMmodel. We also plot the analytic scaling of the first acousticpeak (thick dashed line) predicted by Eq. (30).

Recall that the scaling relation Eq. (30) predicts notonly how the angular sizes of features in the spectrumscale with time, but also that the magnitude of the power,

( + 1)C(), remains constant into the future at, e.g.,any acoustic peak. For late times, aobs > 0.3, Fig. 3 in-deed confirms this prediction. Our fiducial model under-went reionization at aR = 0.083, and we predicted in Sec-tion IIIE that as a result the power spectrum should beattenuated by approximately 16% on all but the largestscales. Again, this is visible in Fig. 3. Recall that wepredicted a negligible 0.1% reduction in C() betweentoday and the distant future.

Also visible in Fig. 3 is a substantial increase in powerat the largest scales at late times due to the increas-ing ISW effect in our CDM model, which we discussedin Section IIIC. Indeed, for aobs >

5.0 the quadrupole

power actually exceeds the power at the first acoustic

peak. The ISW contribution converges as the observationscale factor aobs approaches infinity, since in the integralin Eq. (43), we have g() 0 and f as aobs ,where f is finite. The asymptotic form of the powerspectrum at late times is plotted in Fig. 4, together withthe current spectrum. The dramatic increase in the ISWcontribution as well as the shift in peak positions pre-dicted in Eq. (33) are clearly visible.

1 10 100 10000.0

2.0

4.0

6.0

8.0

10.0

12.0

1010

(

+1)

/(2

)

FIG. 4: Anisotropy power spectra for scalars (top pair ofcurves) and for tensors (bottom pair, arbitrary scale), calcu-lated using our modified version of camb for today (dashedcurves) and for the asymptotic future (solid).

Fig. 4 also presents the gravitational wave contribu-tion to the anisotropy spectra today and in the asymp-totic future. In Section IIID we described the expectedbehaviour of tensor modes in the future, which entailedthe same geometrical scaling of the small-scale cut-off inthe spectrum, as well as a decrease in power at the verylargest scales. Both of these features are visible in Fig. 4.

As a check on our custom modifications to camb, weplot in Fig. 5 the power spectrum from camb for aobs =2.0, as well as the corresponding spectrum calculatedfrom a power spectrum generated for today, aobs = 1,and transformed to aobs = 2.0 using the scaling relationEq. (30). Additionally, the spectrum calculated from thescaling relation included the increased ISW componentcalculated from the analytical approximation Eq. (43).

To facilitate the use of this analytical expression, thespectral index was set to nS = 1 for these calculations.Since the curves coincide at all but the largest scales, itis clear that the scaling relation has accurately capturedthe evolution ofC(). However, the ISW contribution issubstantially overestimated, indicating the limitations ofthe approximation Eq. (44) involved in deriving Eq. (43).

To make further contact with the analytic results inprevious subsections, in Fig. 6 we plot the differenceC C(aobs) C(aobs) calculated using our modifiedversion of camb between the power spectrum today, ataobs = 1, and at a future time, when aobs = 1+ a, for thecases a = 104, 0.001, and 0.01. These curves exhibit

very accurately the scaling with a predicted in Eq. (34),


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12

1 10 100 10000.0

2.0

4.0

6.0

8.0

10

10

(

+1)

/

(2

)

FIG. 5: Anisotropy power spectrum for aobs = 2.0 calculatedwith our modified version of camb (solid line) and using the

scaling relation Eq. (30) and analytical ISW approximationEq. (43) (dashed). We set nS = 1 for these calculations.

when we recall that a = H0 for small . Slight depar-tures from this simple scaling are evident at the largestscales, where the spectra are nearly flat and hence theirprecise shape sensitively influences the location of zerosin C. In Fig. 6 we also plot the analytical result cal-culated from the power spectrum today using Eq. (46).Again we find excellent agreement with camb at all butthe largest scales. We also find reasonable agreement atlow , indicating that our ISW approximation is quitegood for small time increments from today.

1 10 100 1000

10-1 7

10-1 6

10-1 5

10-1 4

10-1 3

10-1 2

10-1 1

(

+1)|

|/(2

)

FIG. 6: Absolute value of the difference in the CMB powerspectrum ( + 1)|C|/(2) between aobs = 1 and a

obs =

1 + a. The solid curves were calculted from our modifiedversion of camb, and from top to bottom denote a = 0.01,0.001, and 104. The dashed curve was calculated using theanalytical expression Eq. (46) for a = 104. We used nS = 1for these calculations.

IV. THE DIFFERENCE MAP POWER

SPECTRUM

A. Analytical time evolution

In the previous section we found a very simple scal-ing relation to describe the time dependence of the CMBpower spectrum, which, together with the ISW effect,thoroughly describes the evolution of the spectrum. How-ever, if we are interested in the best way to observea change in the CMB we might expect that observingchanges in the actual sky map, or the ams, should be farmore promising than looking for changes in the heavilycompressed C power spectrum. Intuitively, as the shellr = rLS on the LSS grows in size, we expect the fineststructures to change first, then the larger ones. As weshall see, the difference between two sky maps measuredat different times does indeed encode much more infor-mation than the C spectra, namely the correlations be-tween the two maps, although perhaps counterintuitivelythe magnitude of a change C will dominate over the dif-ference map power spectrum for small time intervals.

1. Definitions

Consider two measurements of the ams at times and and define the difference map by

am am() am(). (53)Using Eq. (22), we can readily calculate the statisticalproperties of the difference map. We find

amam = D

mm , (54)

where we define the power spectrum of the difference

map, D

, by

D

C() + C() 2C

, (55)

and

C

4

dk

kPR(k)T(k, , rLS)T(k, , rLS). (56)

Note that the quantity amam is diagonal in andm, and that C

= C().

The quantity C

is a correlation function that relatesthe anisotropies at time with those at time , through

Re am()am() = C

mm. (57)

Since the variances C() and C() will in general differ,

the quantity C

is not the best measure of correlations,

and the spectrum D

measures not only the loss of cor-relations but also the change in variance C(). Thereforewe may consider instead the modified difference map

am

C()

C(

)

am()

am(), (58)


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13

which normalizes the modes at to have the same vari-ance as those at . Then we find

amam = 2C()

1 C mm, (59)

where we have defined the normalized correlation func-tion by

C

C

C()C()

. (60)

This normalized function is useful in that we have C

=1, 0, and 1 for perfect correlations, no correlations,and perfect anticorrelations, respectively. Similarly, the

quantity 1 C measures the loss of correlations alone.However, the spectrum D

will still be useful, since itmeasures both the loss of correlations and the change in

variance, so might be expected to be more sensitive tochanges in the CMB than the quantity 1 C .

2. Time evolutionflat sky approximation

In analogy with Eq. (34) for C, we can write thespectrum of the difference map for small increments intime = as

D

am

am

()2. (61)

Therefore the difference of the power spectrum, C,

dominates over the power spectrum of the difference map,

D

, for small enough , since C is only proportionalto the first power of . Of course for a particular we must calculate the coefficients of and ()2 beforewe decide which method is more efficient if we are inter-ested in a detection. The details of instrumental noiseare important and this will be discussed fully in [11].

Beyond the ()2 scaling, it is much more difficult to

obtain the detailed evolution ofD

than it was for C.Even when we consider only the Sachs-Wolfe plateau con-tribution, for which A1(k) = 1/5 and A2(k) = A3(k) =0, the Bessel integrals involved in Eq. (56) cannot be an-

alytically solved. In fact, this problem is related to adivergence that can be illuminated if we employ the flatsky approximation described in the Appendix.

Under that approximation, which is valid over smallpatches of sky and replaces the discrete indices and mwith the continuous two-dimensional vector , and thepolar coordinate r with a Cartesian coordinate x par-allel with the line of sight, we can readily calculate thequantity on the left-hand side of Eq. (54). Using

a() a(, xLS) a(, xLS) (62)to define the difference map, where xLS and xLS are thecomoving distances to the LSS at times and , respec-

tively, and using Eq. (A.6) for the anisotropies, we find

that the result is not diagonal in . Rather, it containsterms proportional to 2( ) and 2(1 ),where we have defined

xLS

xLS. (63)

Indeed this is not surprizing: in the flat sky approxima-tion, an anisotropy on angular scale at xLS correspondsto a physical mode with comoving wavevector component/xLS orthogonal to the line of sight. But Eq. (11) tellsus that such a mode should share correlations with thesame physical scale at xLS, which corresponds to the an-gular scale xLS/xLS. Such off-diagonal correlations arecompletely suppressed in the full spherical expansion, aswe found.

The relevant quantity to calculate in the flat sky ap-proximation is instead the power in the difference mapdefined by a() a(, xLS) a(, xLS). (64)Again applying Eq. (A.6) for a(, xLS), we finda()a( ) = D() 2( ), (65)which is diagonal in , with the power spectrum of thedifference map given by

D

() C(, ) + 2C(, ) 2C(). (66)Here C(, ) is the flat sky approximation to theanisotropy power spectrum, given by Eq. (A.9), and

C

() is the correlation function given by

C

() x2LS

dkxPR(k)|T(k, kx)|2 cos(kxxLS)

k3,

(67)where T(k, kx) is the flat sky transfer function, xLS xLS xLS, and kx is the component of the comovingwavevector parallel to the line of sight. Eqs. (64) to (67)are the flat sky analogues of Eq. (53) to (56), respec-tively. (The continuous argument () will always distin-guish quantities in the flat sky approximation from thecorresponding exact quantities, which are labelled withthe discrete indices m.)

Note that the integrand in Eq. (67), which is exactapart from the flat sky approximation, is bounded in

magnitude by the integrand in Eq. (A.9) for the powerspectrum C(, ), as we vary xLS. In place of Eq. (60),the normalized correlation function becomes in the flatsky approximation

C

() C()

C(, )2C(, ), (68)

which is bounded by |C()| 1. In the limit of shorttime interval, xLS 0, we have C() 1, corre-sponding to perfect correlation. We also have C

() 0 as xLS (recall, however, that in a CDM uni-verse, only finite conformal time is available into the fu-

ture).


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14

3. Special cases

Armed with the above flat sky approximation, we cannow calculate the difference map power spectrum and

correlation function in some special cases. First, con-sider the short time interval case, xLS 0. ExpandingEq. (66) in powers ofkxxLS xLS/xLS, where ||,we find

D

() =

xLSxLS

2

dkxPR(k)|T(k, kx)|2k2x

k3(69)

for the power spectrum of the difference map at low-est order in xLS/xLS. This expression exhibits pre-cisely the time interval dependence that we predictedin Eq. (61). (Note that in defining the difference mapthrough Eq. (64), we have fixed the observed transversewavevectors at both observation times, so the integrand

in Eq. (69) is independent of .)The integrand in Eq. (69) resembles closely that for

the anisotropy power spectrum in Eq. (A.9), but with anextra factor of k2x in the numerator. In fact, using therelation

k2 = k2x +

xLS

2(70)

we can easily rewrite Eq. (69) as

D

() =

xLSxLS

220C

(nS+2)(, ) 2C(, )

. (71)

Here C(nS+2)(, ) is the anisotropy power spectrum cal-culated using a modified primordial power spectrum de-fined by

P(nS+2)R (k)

k

k0

2PR(k), (72)

where k0 0/xLS is the pivot scale used to definethe primordial spectrum. (The result for D

() is, ofcourse, independent of the pivot scale chosen.) For thespecial case of a power law primordial spectrum PR(k),with scalar spectral index nS , the modified spectrum

P(nS+2)R (k) has spectral index nS + 2; hence our choice

of notation. Eq. (71) says that, for small time incre-ments, the shape of the power spectrum of the differ-ence map is determined entirely by the actual anisotropyspectrum blue tilted, i.e. 2C(, ), together with thespectrum C(nS+2)(, ) calculated from a blue-tilted pri-mordial spectrum, both evaluated at the same time .Therefore we expect that generically the shape of thedifference map spectrum D

() will be roughly that ofa strongly blue-tilted version of the anisotropy spectrumC(, ). The height of the spectrum of the difference mapis determined by the ratio xLS/xLS.

Next, we can specialize to the case of the pure scale-invariant (nS = 1) Sachs-Wolfe plateau, which is charac-

terized by T(k, kx) = A1(k) = const and PR(k) = const.

Eq. (A.9) gives in this case

C(, ) =2PRA21

2, (73)

in agreement with the standard Sachs-Wolfe result, toorder 1/. The normalized correlation function is then

C

() =2

2x2LS

dkxcos(kxxLS)

k3. (74)

In the short time interval limit, xLS/xLS 1,Eq. (69) becomes for the Sachs-Wolfe plateau

D

() = PRA21

xLSxLS

2

dkxk2xk3

. (75)

Note that this last integral is logarithmically divergent,

but this is just an artifact of our assumption of a scaleinvariant spectrum to arbitrarily small scales [33]. Equiv-alently, Eq. (71) cannot be applied in this case, becausethe Sachs-Wolfe integral diverges for nS 3. In real-ity, damping within the LSS imposes an effective cut-off,with essentially no structure at wavenumbers above somevalue kmax [34]. Replacing the infinite limits with kmax,we can evaluate the integral in Eq. (75) with the result(valid for /xLS kmax)

D

() 2PRA21

xLSxLS

2ln

2kmaxxLS

1

. (76)

This means that the contribution to the difference mappower from the Sachs-Wolfe plateau is independent of, apart from a logarithmic correction. This is the -dependence we expect for the Sachs-Wolfe plateau for theanisotropy power spectrum C from a strongly blue tiltedprimordial spectrum, with scalar index nS = 3, as we pre-dicted above based on Eq. (71). Comparing Eqs. (A.9)and (69) for the power spectra of the anisotropies and ofthe difference map, and recalling the expression Eq. (A.7)for the transfer function, we see that the monopole con-tribution to the spectrum D

() (the part proportionalto A1) is proportional to the dipole contribution to thespectrum C(, ) (the part proportional to A2).

Finally, we note that we can evaluate Eq. (67) for thecorrelation function analytically for all xLS for the caseof a delta-source in k-space, PR(k) = PR(k k). Such asource will be very helpful in understanding the temporalbehaviour of the normalized correlation function C

()at late times. The result for such a source is

C

() =

cos

kx()xLS

if kxLS,0 if > kxLS,

(77)

where

kx()

k2 2

x2

LS

(78)


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15

is the line-of-sight component of the source mode k cor-responding to the observed scale . This result tells usthat the normalized correlation function is initially (at = 0) unity, as expected, and subsequently oscillatoryin , with positive correlations alternating with anticor-relations, and each scale oscillating at a different rate.The largest angular scales (smallest ) reach anticorre-lation first, followed by smaller scales. The peak scale, = kxLS, never becomes anticorrelated. This behaviourcan be easily understood with the assistance of Fig. 7,by noting that at the peak scale we have kx() = 0, sothat the modes k which contribute to the peak scale areparallel to the LSS and hence cannot produce anticorre-lations. As decreases, kx() increases, i.e. k contains anincreasing component parallel to the line of sight, so the

first anticorrelations occur earlier and earlier. If we con-sider sources at different scales k, Eq. (77) tells us thatthe first anticorrelations occur earlier for smaller scales(larger k), as expected.

4. Origin of the difference map power

As we mentioned above, the power spectrum of the dif-

ference map, D

, contains two distinct contributions:the loss of correlations and the change in variance be-tween the two times of observation. To make this explicit,and to determine which contribution is more important,we can use Eqs. (55) and (60) to write

D

=

C()

C()2

+ 2

C()C()

1 C

(79)

= C()

1

4

C

C()

2+ 2

1 C

+ O

LS

3. (80)

The first line above is exact, while in the second we havedropped higher order terms in

CC()

LS

(81)

[recall Eq. (35)]. With a calculation similar to that lead-ing to Eq. (69), it is straightforward to show that, for

short time intervals (/LS 1), we have 1 C

(/LS)2, so that the two terms in square brackets inEq. (80) are of the same order in /LS.

The first term in square brackets in the expression ( 80)is due entirely to the change in variance C, while thesecond term is due solely to the loss of correlations be-tween am() and am() [recall Eq. (59)]. But fromEq. (71) we have

D

LS2

2C(). (82)

Therefore, for all but the very largest angular scales(smallest ), the second term in the brackets in (80) must

dominate over the first, and so the power spectrum D

is dominated by the loss in correlations. This can beconfirmed by a direct computation in the flat sky ap-proximation, which gives

2C(, )

1 C()

= D

(), (83)

at lowest order in /LS. This means that the flat sky

approximation to D

captures only the (dominant) con-tribution due to loss of correlations. This is not surpriz-

ing: because of the scaling relation (30), the flat sky

difference map defined in Eq. (64) is closely related tothe normalized difference map defined in Eq. (58).

One further contribution to the difference map arisesif we consider the absolute temperature anisotropies Trather than the relative quantity T/T, where T is

the mean temperature. Recall from Section IIIB that,if we consider the absolute spectrum T2C instead ofthe relative quantity C, then the difference (T

2C) T2()C() T2()C() receives an extra contributiondue to the expansion redshift. In that case we showedthat the extra contribution is of the same order as thegeometrical scaling part [recall Eq. (36)].

We can now repeat this calculation for the differencemap power spectrum. The difference map in absolutetemperature units is

(T am) = T

(aH)1am + am

(84)

at lowest order in /(aH)1, where we have used T =H T. Therefore the corresponding power spectrum be-comes

(T am)(T am) = T2

D

+

(aH)1

2C

+

(aH)1

D

C

, (85)

where we have used the expressions (54), (55), and

(57). Next, retaining only terms to lowest order in


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16

k1k2

x1x2

FIG. 7: Two source modes, k1 and k2, with the same wavenumber k. Diagonal lines indicate troughs (solid) and crests(dashed). The observer is towards the bottom. The solid horizontal line indicates the position of the LSS at initial time .The dotted horizontal lines indicate the position of the LSS at the first later time that produces perfect anticorrelations withthe initial time, so that hot spots (solid) line up with cold spots (hollow) and vice versa. Mode k1, which corresponds to a

smaller observed , reaches anticorrelation before mode k2 (since x1 < x2). A mode k parallel to the LSS would never reachanticorrelation, while a mode parallel to the line of sight would correspond to = 0 (in the flat sky approximation).

/(aH)1 /LS, and using Eq. (81), we have

(T am)(T am) = T2

D

+ O

LS

2C

.

(86)But then Eq. (82) tells us that the first term on theright-hand side of Eq. (86) dominates for all but thevery largest angular scales [just as we argued above forEq. (80)], and so

(T am)(T am) T2D

. (87)

In other words, the part of the power spectrum for the ab-solute difference map (T am) which is due to the expan-sion redshift is subdominant. Thus, contrary to the casewith C, it is irrelevant for the difference map whetherwe consider absolute or relative temperature differences(apart from the very largest scales).

In hindsight this result could have been anticipateddirectly from Eq. (84), since we expect that the changeam corresponding to the time interval should be

am

LS am, (88)

so that the first term on the right-hand side of Eq. (84),which is due to the expansion redshift, is subdominanton all but the largest scales. Intuitively, the change inam due to a change in observation time grows as thewavelength of the source modes decreases (for constantmode amplitude), since the corresponding increase in ra-dius of the LSS is a larger fraction of a shorter wavelengthmode. On the other hand, the change in T am due to theexpansion redshift is independent of scale .

Similarly, the contribution to the difference map dueto loss of correlations, which is described crudely by

Eq. (88), is expected to dominate over the contribution

due to changing variance C, which is roughly indepen-dent of , as we showed rigorously above.

B. Time evolution from CAMB

1. Power spectrum and correlation function

We have computed the correlation function C

from

Eq. (56) and the difference map power spectrum D

from Eq. (55) numerically using our modified version ofcamb to extract T(k, , rLS) at different rLS (as outlinedin Section IIIF), using the cosmological parameters of

our fiducial CDM model. In Fig. 8 we display D

forthe times and corresponding to today, aobs = 1, andfuture times when aobs = 1 + a, for the cases a = 10

4,0.001, 0.01, and 0.1. For small increments a these curves

exhibit precisely the quadratic scaling D

()2 thatwe predicted in Eq. (61), and the slope ofD

for small matches our analytical prediction for the Sachs-Wolfeplateau, Eq. (76). For large increments a the difference

map power spectrum approaches the sum of the individ-ual power spectra as the correlation function decays tozero, as we expect according to Eq. (55). Generally, thesecurves exhibit the heavily blue-tilted form we predictedin the previous subsection, due to the more rapid loss ofcorrelations on smaller angular scales.

Also shown in Fig. 8 is the curve calculated fromthe flat-sky analytical expression, Eq. (71), for the casea = 104. This curve coincides extremely well withthe numerical result for > 20. The departures at largescales are due to two factors. First, the flat sky approx-imation is poor at those scales. Second, Eq. (71) wasderived under the assumption that all anisotropies were

primary, which is not the case for the ISW contribution.


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17

1 10 100 1000

10-18

10-17

10-16

10-15

10-14

10-13

10 -12

10-11

10-10

10-9

10-8

(

+1)

//(2

)

FIG. 8: Difference map angular power spectrum D

calcu-lated from our modified version of camb (solid lines) for the

two times corresponding to aobs

= 1 and a

obs

= 1 + a, forthe cases (top to bottom) a = 0.1, 0.01, 0.001, and 104.Also shown is the analytical result (dashed curve) calculatedusing Eq. (71) for a = 104.

In Fig. 9 we plot the normalized correlation function

C

, calculated using our modified version of camb forour fiducial CDM model, between the set of ams ob-served at and . Here, corresponds to an observationof the CMB sky today at aobs = 1 and to an obser-vation at aobs = 1 + a, where we illustrate the casesa = 0.001, 0.01, 0.03 and 0.1. For the smallest inter-val a, we find very strong correlation between the twosky maps, as expected. The correlations fall off as aincreases, with the sky maps becoming somewhat anti-

correlated for intermediate intervals before C

decaysto zero at the largest intervals.

The general features of the correlation function can beunderstood by considering the detailed arguments pre-sented in the previous subsection. For a 1 the in-crease in the LSS radius corresponding to the intervala is rLS = H

10 a. For the case of a = 0.01, using

H0 = 73 kms1Mpc1, we find rLS = 40.9 Mpc, cor-

responding to a comoving wavenumber k = 0.15 Mpc1.This wavenumber is much larger than the wavenumberof the first acoustic peak, given by k = /sLS = 0.021Mpc1, where sLS 150 Mpc is the sound horizon atlast scattering. Hence, for this a, we are essentiallysampling the same set of inhomogeneities which give riseto the first acoustic peak at both times, and so we expectfluctuations to be correlated on these scales. Indeed, we

see from Fig. 9 for a = 0.01 that C

0.9 for thefirst acoustic peak scale, 220. Extending this argu-ment, we expect that C

1 as 0 for fixed a,as the largest scale (smallest k) features should be most

correlated, and of course we similarly expect C

1as a 0 for fixed .

The presence of anticorrelations was discussed in Sec-

tion IV A, where we derived the behaviour of the cor-

0 500 1000 1500 2000

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

/

0 500 1000 1500 2000

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

/

FIG. 9: Normalized correlation function C

between theCMB sky observed today (a = 1) and at a = 1 + a, calcu-lated with our modified version of camb, for a = 0.001 (toppanel, solid curve), 0.01 (top, dashed), 0.03 (bottom panel,solid), and 0.1 (bottom, dashed). Anticorrelations are seen todevelop as a increases, before the correlation function decaysto zero for large a.

relation function in the flat sky approximation for thecase of a delta-source at wavenumber k = k. The re-sult, Eq. (77), exhibited oscillating positive and negativecorrelations, with the first anticorrelations occuring ear-

lier for smaller scales (larger k), as we have confirmedhere for a realistic spectrum using camb. Eq. (77) alsodescribed anticorrelations occuring earlier for smaller ,with k fixed. This feature is not visible in the actualcorrelation function plotted in Fig. 9 since the real pri-mordial power spectrum is far from a delta-source. If weconsider a small subset ofk modes, e.g. those correspond-ing to the fourth acoustic peak scale, then some of thosemodes will be aligned nearly parallel to our line of sightand hence produce early anticorrelations at small forsome a (recall Fig. 7). However, there are many moremodes due to power at smaller k that are still tightly

correlated at the same a and hence result in C

1for small .


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18

2. Sky maps

Assuming Gaussianity, generating a single realizationof a set ofams usually involves drawing each am() in-

dependently from a Gaussian distribution with varianceC(). With the correlation function C

, we have ameasure of the degree of correlation between ams at twodifferent times. Hence, given a set of ams at the firsttime, the variance of the distribution at each time, andthe correlation between them, one can generate a real-ization of a second set of ams at some later time.

Formally, we draw the new set of ams from the likeli-hood function

P(Xi) =1

2|C|1/2 exp

12

XTi C1Xi

, (89)

with

C =

C C

C

C

, (90)

where Xi is a random 2-vector containing each am coef-ficient at and , and we have relabeled the variance of

the am() distribution at each time by C and C

.We illustrate the likelihood function in Fig. 10 for the

a2m and a5m modes, where corresponds to an observa-tion at aobs = 1 and to aobs = 2.0. The a2m coefficientsare more tightly correlated than their a5m counterparts,since for such a large a the correlation rapidly falls off as increases. It is also noticeable that the contours of the

likelihood are slightly elongated vertically, due to the in-creased variance ofam(

) on large scales resulting fromthe increasing ISW effect.

Therefore, our method of generating CMB sky mapsinvolves firstly generating a random realization at someinitial time, and then generating all subsequent realiza-tions by mapping the ams using the correlation function.For large a, where the ams are uncorrelated, we are es-sentially selecting a completely new set of coefficients.

For small a, C

approaches unity and the ams maptrivially according to am() am(). At some in-termediate intervals, anticorrelation favours a reversal ofsign of the ams, i.e. hot spots are mapped to cold spots

and vice versa.We generate maps using thehealpix code with nside =

512, corresponding to a pixel resolution of 6.87 arcmin.We present a series of these maps in Fig. 11, plotting thefractional temperature fluctuation Ti/T at each pixel i.For presentational clarity we show a patch of sky covering 1000 square degrees, and use modes up to max =1000. We generate the first map at aobs = 1, and showsubsequent maps at aobs = 1+a, where a = 0.001, 0.01,0.1, and 1.0. We also show the difference map for eachobservation relative to aobs = 1. We have checked thatthe power spectra reconstructed from our simulated skymaps agree with the intended spectra to within sample

variance.

alm

()/106

alm(

)/10

6

15 10 5 0 5 10 15

15

10

5

0

5

10

15

FIG. 10: Distribution from which the ams are drawn. Here,

corresponds to aobs = 1 and to aobs = 2.0, and contoursshow the 2 error ellipse. The distribution of a2m is shownby the solid contour, which has a correlated set of variancesC2() and C2(

). The distribution of a5m, shown by thedotted contour, has a smaller variance at both times and theseare much less correlated.

Visually, the a = 0.001 map is extremely similar tothe initial map. The variance of the map, given by

TT 2

map=

1

Npix

Npixi TiT

2

, (91)

is over four orders of magnitude higher than the differ-ence map variance. For a = 0.01, the primary temper-ature fluctuations have a variance around two orders ofmagnitude more than the difference map, and changes insmall scale structure (from the initial map) are clearlyapparent.

For a = 0.1 and 1.0, the variance of the differenceis actually larger then the temperature fluctuations atthat time, and acoustic scale structures are visible in the

difference. This is understandable from our discussion ofthe correlation functionat these times the correlationon all but the very largest scales has dropped to zero, sothat the variance of the difference approaches the sum ofthe initial and final map variance [recall Eq. (55)].

Finally, in Fig. 12 we present a simulated sky mapfor the asymptotic future. This map clearly differs fromtodays map, with the dramatic increase in large scalepower due to the ISW effect readily apparent.

High resolution versions of these skymaps, together with animations illustratingthe evolution of the CMB, are available at

http://www.astro.ubc.ca/people/scott/future.html.
http://www.astro.ubc.ca/people/scott/future.htmlhttp://www.astro.ubc.ca/people/scott/future.htmlhttp://www.astro.ubc.ca/people/scott/future.html


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19

a = 1.01 Difference

0.00020

Difference

a = 1

a = 1.001

0.00020

0.00020

0.00020

0.00020

0.00020 0.00011

1.3e05

0.00011

1.3e05

FIG. 11: Simulated map realizations (top and left panels) and difference map (relative to aobs = 1) (right panels) for a = 103

(middle panels, corresponding to a 13 Myr interval) and 102 (bottom panels, 130 Myr). Note the vastly different power scalesbetween the sky and difference maps. The maps presented here are for a patch of sky covering 1000 square degrees. Highresolution version available at http://www.astro.ubc.ca/people/scott/future.html.
http://www.astro.ubc.ca/people/scott/future.htmlhttp://www.astro.ubc.ca/people/scott/future.htmlhttp://www.astro.ubc.ca/people/scott/future.html


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20

Difference

0.000200.00020 0.000240.00019

Difference

0.00020 0.00020 0.00021 0.00021

a = 2

a = 1.1

FIG. 11: (Continued.) Simulated map realizations (left panels) and difference (right) for a = 0.1 (top) and 1.0 (bottom). Highresolution version available at http://www.astro.ubc.ca/people/scott/future.html.

V. DISCUSSION

We have systematically described the temporal evolu-tion of the CMB, beginning with the mean temperatureand dipole, and then moving to the anisotropy powerspectrum. We found that the evolution of the spectrumis described at all but the largest angular scales by asimple scaling relation. At large scales the ISW contri-bution grows to dominate even the first acoustic peak atlate times. The extra optical depth du

james p. zibin, adam moss and douglas scott- the evolution of the cosmic microwave background

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