Efficient cosmological parameter estimation from microwave background anisotropies

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PHYSICAL REVIEW D 66, 063007 ~2002!

Efficient cosmological parameter estimation from microwave background anisotropies

Arthur Kosowsky,* Milos Milosavljevic,† and Raul Jimenez‡

Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, New Jersey 08854-801~Received 2 June 2002; published 26 September 2002!

We reexamine the issue of cosmological parameter estimation in light of current and upcoming high-precision measurements of the cosmic microwave background power spectrum. Physical quantities whichdetermine the power spectrum are reviewed, and their connection to familiar cosmological parameters isexplicated. We present a set of physical parameters, analytic functions of the usual cosmological parameters,upon which the microwave background power spectrum depends linearly~or with some other simple depen-dence! over a wide range of parameter values. With such a set of parameters, microwave background powerspectra can be estimated with a high accuracy and negligible computational effort, vastly increasing theefficiency of the cosmological parameter error determination. The techniques presented here allow calculationof microwave background power spectra 105 times faster than comparably accurate direct codes~after pre-computing a handful of power spectra!. We discuss various issues of parameter estimation, including parameterdegeneracies, numerical precision, mapping between physical and cosmological parameters, systematic errors,and illustrate these considerations with an idealized model of the Microwave Anisotropy Probe experiment.

DOI: 10.1103/PhysRevD.66.063007 PACS number~s!: 98.70.Vc

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I. INTRODUCTION

By January 2003, microwave maps of the full sky at 0resolution will be available to the world, the harvest of tremarkable Microwave Anisotropy Probe~MAP! satellitecurrently taking data@1#. The angular power spectrum of thtemperature fluctuations in these maps will be determinea high precision on angular scales from the resolution liup to a dipole variation; this corresponds to about 800 statically independent power spectrum measurements. Remeasurements have given a taste of the data to comethough covering much smaller patches of the sky and wpotentially more serious systematic errors@2–4#.

The main science driving these spectacular technical fis the determination of basic cosmological parametersscribing our Universe, and resulting insights into fundamtal physics; see@5,6# for recent reviews and@7# for a peda-gogical introduction. The expected series of acoustic pein the microwave background power spectrum encoenough information to make possible the determinationnumerous cosmological parameterssimultaneously @8#.These parameters include the long-sought Hubble paramH0, the large-scale geometry of the UniverseV, the meandensity of baryons in the UniverseVb , and the value of themysterious but now widely accepted cosmological consL, along with parameters describing the tiny primordial pturbations which grew into present structures, and the rshift at which the Universe reionized due to the formationthe first stars or other compact objects. While most of thparameters and other information will be determined to

*Also at School of Natural Sciences, Institute for AdvancStudy, Einstein Drive, Princeton, NJ 08540; electronic [email protected]

†Electronic address: [email protected]‡Electronic address: [email protected]

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high precision, one near-exact degeneracy and other appmate degeneracies exist between these parameters@9,10#.

The exciting prospects of providing definitive answerssome of cosmology’s oldest questions raise a potentiallyficult technical issue, namely finding constraints on a lardimensional parameter space. Given a set of data, what rof points in parameter space gives models with an acceptgood fit to the data? The answer requires evaluating a likhood function at many points in parameter space; in partlar, finding confidence regions in multidimensional paraeter space requires looking around in the space. Thisstraightforward process, in principle. But a parameter spwith as many as ten dimensions requires evaluating a lomodels: a grid with a crude 10 values per parameter contten billion models. A direct calculation of this many modeis prohibitive, even with very fast computers. While somethe parameters are independent of the others~i.e., tensormodes!, reducing the effective dimensionality, many of thparameters will be constrained quite tightly, requiring a finsampling in that direction of parameter space. An additioproblem with brute-force grid-based methods is a lackflexibility: if additional parameters are required to correcdescribe the Universe, vast amounts of recalculation musdone. Grid-based techniques have been used for analyscosmological parameters from microwave background~e.g.,@11,12#! but clearly this method is not fast or flexible enougto deal with upcoming data sets adequately.

A more sophisticated approach is to perform a searchparameter space in the region of interest. Relevant teniques are well known and have been applied to the micwave background@13–15#. Reliable estimates of the erroregion in cosmological parameter space can be obtainedrandom sets of around 105 models, reducing the computational burden by a factor of 1000 or more compared tocruder grid search methods. On fast parallel computerspossible to compute the power spectrum for a given modea computation time on the order of a second@16#, making

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KOSOWSKY, MILOSAVLJEVIC, AND JIMENEZ PHYSICAL REVIEW D66, 063007 ~2002!

Monte Carlo error determinations feasible. For improvedficiency, useful implementations do not recalculate the enspectrum at each point in parameter space, but ratherapproximate power spectra based on smaller numbers ofculated models@17#.

Here we expand and refine this idea by presenting a separameters, functions of the usual cosmological paramewhich reflect the underlying physical effects determiningmicrowave background power spectrum and thus resulparticularly simple and intuitive parameter dependences.vious crude implementations of this idea applied to loresolution measurements of the power spectrum@18#; thecurrent and upcoming power spectrum measurements reqa far more refined implementation. Our set of parametersbe used to construct computationally trivial but highly accrate approximate power spectra, and large Monte Carlo cputations can then be performed with great efficiency. Adtional advantages of a physically based parameter set arethe degeneracy structure of the parameter space can bemuch more clearly, and the Monte Carlo computation itstakes significantly fewer models to converge.

Earlier Fisher-matrix approximations of parameter errare a rough implementation of this general idea: if the powspectrum varies exactly linearly with each parameter in soparameter set, and the measurement errors are Gaussiadom distributed and uncorrelated, then the likelihood fution as a function of the parameters can be computedexactlyfrom the partial derivatives of the model with respect to tparameters@8,19,20#. Even if the linear parameter depedence does not hold throughout the parameter space, italmost always be valid in some small enough region, bethe lowest-order term in a Taylor expansion. If this regionat least as large as the resulting error region, then the Fimatrix provides a self-consistent approximation for errortermination. The difficulty is in finding a suitable parametset. Our aim is not necessarily to find a set of paramewhich all have a perfectly linear effect on the power sptrum. Rather, more generally, we desire a set of paramefor which the power spectrum can be approximated verycurately with a minimum of computational effort. Then wcan dispense with approximations of the likelihood functioas in the Fisher matrix approach, and directly implemenMonte Carlo parameter space search with a minimumcomputational effort. This technique allows simple incorpration of prior probabilities determined from other sourcesdata, and if fast enough allows detailed exploration of pottial systematic biases in parameter values arising frommerous experimental and analysis issues such as treatmeforegrounds, noise correlations, mapmaking techniques,model accuracy.

We emphasize that with the accuracy of upcoming micwave background power spectrum measurements, themological parameters will be determined precisely enouthat their values will be significantly affected by systemaerrors from assumptions made in the analysis pipeline apotentially, numerical errors in evaluating theoretical powspectra. The measurements themselves may also be dnated by systematic errors. Extensive modelling of thepact of various systematic errors on cosmological param

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determination will beessentialbefore any parameter detemination can be considered reliable. This is the primary mtivation for increasing the efficiency of parameter erranalysis. We discuss this point in more detail in the last stion of the paper.

The following section reviews physical processes affeing the cosmic microwave background and definesphysical parameter set. The key parameter, which setsangular scale of the acoustic oscillations, has been discupreviously in similar contexts@10,21,22# but has not beenexplicitly used as a parameter. The other parameters desing the background cosmology can then be chosen to mother specific physical effects. Section III displays how tpower spectrum varies as each parameter changes withothers held fixed. The mapping between our physical pareters and the conventional cosmological parameters immately reveals the structure of degeneracies in the cosmocal parameter space. Simple approximations for the effeceach physical parameter are shown to be highly accurateall but the largest scales in the power spectrum. We tdetermine the error region in parameter space for an idized model of the MAP experiment in Sec. IV, comparinwith previous calculations, and test the accuracy of ourproximate power spectrum calculation. Finally, Sec. V dcusses the potential speed of our method, likelihood estition, mapping between the sets of parameters, accuracy,systematic errors. Appendix A summarizes an analysis pline going from a measured power spectrum to cosmologparameter error estimates; Appendix B details severalmerical difficulties with using theCMBFAST code for the cal-culations in this paper, along with suggested fixes.

Other recent attempts to speed up power spectrum cputation through approximations include theDASH numericalpackage@23# and interpolation schemes@12#. Our methodhas the advantage of conceptual simplicity and ease ofcombined with great speed and high accuracy.

II. COSMOLOGICAL PARAMETERSAND PHYSICAL QUANTITIES

We consider the standard class of inflation-like cosmlogical models, specified by five parameters determiningbackground homogeneous spacetime~matter densityVmat,radiation densityV rad, vacuum energy densityVL , baryondensityVb , and Hubble parameterh), four parameters determining the spectrum of primordial perturbations~scalarand tensor amplitudesAS and AT and power law indicesnandnT), and a single parametert describing the total opticadepth since reionization. We neglect additional complicatiosuch as massive neutrinos or a varying vacuum equatiostate. This generic class of simple cosmological modelspears to be a very good description of the Universe on lascales, and has the virtue of arising naturally in inflationacosmology. Of course, any analysis of new data shouldtest whether this class of models actually provides an accably good fit to the data, and only then commence withterminations of cosmological parameters.

What constraints do a measurement of the microwbackground power spectrum place on this parameter sp

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EFFICIENT COSMOLOGICAL PARAMETER ESTIMATION . . . PHYSICAL REVIEW D66, 063007 ~2002!

A rough estimate assumes the likelihood of a particular separameters is a quadratic function of the parameters~theusual Fisher matrix technique!; better is a Monte Carlo exploration of parameter space in the region of a best-fit moFor either technique to give reliable results, it is imperatto isolate the physical quantities which affect the microwabackground spectrum and understand their relation tocosmological parameters. If a set of physical quantitwhich have essentially orthogonal effects on the power sptrum can be isolated, then extracting parameter spacestraints becomes much more reliable and efficient.

The five parameters describing the background cosmogy induce complex dependences in the microwave baground power spectrum through multiple physical effecUsing the physical energy densitiesVmath

2, Vbh2, andVLh2, as has been done in numerous prior analyses,proves the situation but is still not ideal. The characterisphysical scale in the power spectrum is the angular scalthe first acoustic peak, so it is advantageous to use this sas a parameter which can be varied independently ofother parameters. This angular scale is in turn determinethe ratio of the comoving sound horizon at last scatter~which determines the physical wavelength of the acouwaves! to the angular diameter distance to the surface ofscattering~which determines the apparent angular size of tyardstick!. We first determine this quantity in terms of thcosmological parameters, and then choose three ocomplementary quantities. The analytic theory underlyany such choice of physical parameters has been workedin detail ~see@24–27#, and also@28,29#!.

The angular diameter distance to an object at a redshzis defined to beDA(z)5R0Sk(r )/(11z) where, followingthe notation of Peacock@30#, R0 is the ~dimensionful! scalefactor today and

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the Friedmann equation provides the connection betwredshift z and coordinate distancer. With a dimensionlessscale factora and a Hubble parameterH5a/a, the differen-tial relation is

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using the above expression forr. Here we have used threlationH0R05uV21u21/2 which follows directly from theFriedmann equation.

The sound horizon is defined asr s5r p a* , and the par-ticle horizon (r p) is

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wherecs(t) is the sound speed of the baryon-photon fluidtime t; to a very good approximation, before decoupling tsound speed is given by@24#

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The physical quantity relevant to the microwave bacground power spectrum is

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where the two functions are given by Eqs.~7! and ~4!, anda* is the scale factor at decoupling. An accurate analyticfor a* in terms of the cosmological parameters has bgiven by Hu and Sugiyama@26#:

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This fit applies to standard thermodynamic recombinationa wide range of cosmological models.~Note that while the

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formula was constructed only for models with the standvalue for V rad, the effect of a change of radiation densiwill come only through the redshift of matter-radiatioequality, and thusVmath

2 can be replaced by a factor propotional toVmat/V rad to account for a variation in the radiatiodensity. The changes inz* are small anyway and have littlimpact on our analysis.! Sincea* !1, the term proportionato VL in Eq. ~7! may be dropped and the integral performegiving @25#

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whereR53rb/4rg is proportional to the scale factor.If two cosmological models are considered which bo

have adiabatic perturbations and the same value forA, tohigh accuracy their acoustic peaks will differ only in heighnot position. This is an obvious advantage for constrainmodels. The peak positions, and thusA, will be extremelywell constrained by power spectrum measurements. Ocombinations which vary the peak heights only will be lewell determined.

Choosing four other parameters describing the baground cosmology must balance several considerations~i!the new set of parameters must cover a sufficiently laregion of parameter space;~ii ! the power spectrum shoulvary linearly or in some other simple way with the new prameters;~iii ! the new parameters should be nearly orthonal in the cosmological parameter space;~iv! the new param-eters should correspond to the most important indepenphysical effects determining the power spectrum;~v! com-mon theoretical prior constraints, like flatness or standradiation, should be simple to implement by fixing a singparameter. No parameter set can satisfy all of these cstraints perfectly. We use the following parameters, whprovide a good balance between these criteria:

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B is proportional to the baryon-photon density and thustermines the baryon driving effect on the acoustic oscitions @24,27#. R is the matter-radiation density ratio at rcombination, which determines the amount of eaintegrated Sachs-Wolfe effect.V determines the late-time integrated Sachs-Wolfe effect arising from a late vacuudominated phase, but otherwise represents a nearly exacgeneracy~sometimes called the ‘‘geometrical degeneracy!.M couples only to other small physical effects and isapproximate degeneracy direction. This choice of parameis not unique, but this set largely satisfies the above crite

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Given values forA, B, V, R, and M, they can be in-verted to the corresponding cosmological parameters bywriting the definition ofA in terms ofB, V, R, M, andh,then searching inh until the desired value forA is obtained.Vb and VL then follow immediately, whileVmat and V radcan be obtained with a few iterations to determine a precvalue for a* . A less efficient but more straightforwarmethod we have implemented is simply to search5-dimensional cosmological parameter space for the corvalues.

The other cosmological parameters which affect thecrowave background power spectrum, aside from tensorturbations, are the reionization redshift and the amplituand power law index of the scalar perturbation power sptrum. For reionization, we use the physical parameter

Z[e22t, ~12!

the factor by which the microwave background anisotropat small scales are damped due to Compton scattering byelectrons after the Universe is reionized. At large scales,temperature fluctuations are suppressed by a smaller am

The primordial perturbation amplitude cannot be mesured directly, but only the amplitude of the microwabackground fluctuations. Some care must be taken indefinition of the amplitude of the microwave fluctuationsthat the other physical parameters are not significantlygenerate with a simple change in normalization. Forample, one common normalization is some weighted averof theCl ’s over the smallestl values, say 2, l ,20 @31#. Thisis known as ‘‘COBE-normalization’’ because this is thrange of scales probed by the Cosmic Background Explo~COBE!, which tightly constrains the normalization of thtemperature fluctuations at these scales@11#. But althoughsuch a normalization is useful for interpreting the COBresults, it is a bad normalization to adopt when probingmuch larger range inl-space. The reason is that the lowemultipoles have a significant contribution from the integratSachs-Wolfe effect: if the total matter density changes snificantly between two models, the resulting lowl multipolemoments will also change. If the normalization is fixedlow l, then the result is that the two models will be offsethigh l values. As a concrete example, consider two modwhich differ only inV, with A, B, M andR held fixed. Theonly physical difference between these two models is aference in the structure growth rate at low redshifts, andparticular the fluctuations at the last scattering surfaceidentical. But if these two models are COBE normalizethen everyCl for l 520 is offset between the two modelDefining the normalization as an average band power ovwider range ofl values@10# is better but not ideal. We advocate defining a normalization parameterS by the amplitudeof the perturbations on the scale of the horizon at last stering, corrected by the amount of small-scale suppressiothe Cl ’s due to reionization. In practice, for models with thsame values ofn, this normalization essentially corresponto the amplitude of the acoustic oscillations atl values higherthan the first several peaks. Note that increasingZ whileholding S and the other physical parameters fixed keeps

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microwave background power spectrum the same at hilwhile enhancing the signal at lowl. The first few peaks cannot be used for normalization because they have beenjected to driving by the gravitational potential as they crothe horizon@27#, so their amplitudes vary significantly witB andR, as shown in the following section.

The primordial power spectrum of density perturbationsgenerally taken to be a power law,P(k)}kn, with n51 cor-responding to the scale-invariant Harrison-Zeldovich sptrum. Making the approximation thatk and l have a directcorrespondence, the effect ofn on the microwave background power spectrum can be modelled as

Cl~n!5Cl~n0!S l

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which is a good approximation for power law power spectA departure from a power law, characterized bya @32#, canbe represented in a similar way. Note that since the depdence is exponential, a linear extrapolation is never a gapproximation over the entire interesting range 2, l,3000. The choice ofl 0 is arbitrary: changingl 0 simplygives a different overall normalization.

Finally, note that the amplitude of the tensor mode powspectrum should be used as a separate parameter, nocommon choice of the tensor-scalar ratio. The tensor mocontribute at comparatively large angular scales; they arenificant only for l ,100 given current rough limits on thtensor amplitude. Varying the tensor amplitude indepdently automatically leaves the normalization parameteSfixed.

In the rest of the paper, we refer to the parametersA, B,V, R, M, S, andZ as ‘‘physical parameters,’’ as opposedthe usual ‘‘cosmological parameters’’Vb , Vmat, V rad, Vvac,h, Q2 ~a quadrupole-based normalization!, andzr .

III. POWER SPECTRA

We now present the dependence of the temperaturetuation power spectrum on these physical parameters. Afiducial model, we choose a standard cosmology withV50.99, L50.7, Vmat50.29, Vb50.04, h50.7, n51, andfull reionization at redshiftzr57, with standard neutrinosWe compute power spectra for the fiducial model andevaluating the various numerical derivatives using SeljakZaldarriaga’sCMBFAST code@33–35#. Note that the model isslightly open rather than flat to compensate for numerinstabilities in theCMBFAST code; see the discussion in thAppendix. The corresponding physical parameters areA050.0106, B050.0196, V050.338, M050.154, andR053.252. Since the dependence of the spectrum onS istrivial, its definition is arbitrary; it can be taken, for examplas the amplitude of a particular high peak, or as the equlent scalar perturbation amplitudeAS at the scale which thepower lawn is defined for a model with no reionization. Wdo not consider tensor perturbations in this paper, whichbe analyzed separately by the same techniques; the teperturbations are simpler because they depend on fewe

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rameters than the scalar perturbations. None of the followresults depend qualitatively on the fiducial model chosen

The top panel of Fig. 1 displays power spectra asparameterA varies while the other physical parameters aheld fixed. It is clear that the effect ofA is only to determinethe overall angular scale, except for the small integraSachs-Wolfe~ISW! effect for l ,200. For the bottom panelthe l-axis of each power spectrum is rescaled to giveCl as afunction of lA/A0, and then the percent difference betwethe rescaled model and the fiducial model is plotted. Tsmall variation of the scaled power spectrum between arol 550 andl 5200 is due to the early ISW effect and canaccurately modelled as a linear variation, while betweel52 andl 550 the variation withA is more complicated dueto the late-time ISW contribution. Forl .200, the scaledpower spectra match to within 1%; the residual discrepais likely due to numerical inaccuracy. SinceA will be tightlyconstrained by the peak positions, this numerical error wnot affect the determination ofA, but may contribute a systematic error to the determination of other parameters at1% level.

Figure 2 displays the power spectrum variation withB,keeping the other parameters fixed. This plot clearly shothe well-known baryon signature of alternating peak ehancement and suppression. The variation of a particmultipole Cl with B is highly linear forl .50 for variationsof up to 20%; for all multipoles, the dependence is linearvariations of up to 10%. A figure of merit for a linear approximation to the power spectrum is whether the errorsthe approximation are negligible compared to the cosmictistical error at a given multipole. The bottom panel of Figshows just how good the linear approximation is for seveselectedl values. The horizontal axis plots the fraction

FIG. 1. Temperature power spectra for the fiducial cosmologmodel~top panel, heavy line!, plus models varying the parameterAupwards by 10%~curve shifts to lowerl values! and downwards by10% ~curve shifts to higherl values! while keepingB, M, V, R, Sandn fixed. The bottom panel displays the fractional error betwethe fiducial model and the other two models withl-axes rescaled bythe factorA /A

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change in the parameterB. The vertical axis plots the ratio othe change inCl , Cl(B)2Cl(B0), to the statistical errorfrom the cosmic variance at that multipole,s l

5Cl(B0)A@2/(2l 11)#. ~Any measurement of the multipolCl is subject to a statistical error at least as large ascosmic variance error, due to the finite number of modsampled on the sky.! The linear dependence remains vaeven near the ‘‘pivot’’ points between a peak and a trouThus a linear extrapolation using computed numericalrivatives is highly accurate at reproducing theB dependencethe error in such an approximation appears to be dominby systematic numerical errors in computing the power sptrum, and funny behavior of some of the multipoles for smvariations inB is surely due to systematic errors.

Figure 3 shows the variation of the power spectrum wrespect to the parameterR, keeping the others fixed, with thcorresponding dependence of specific multipoles.smaller values ofR, matter-radiation equality occurs lateso the Universe is less accurately described as madominated at the time of last scattering, leading to ancreased amplitude of the first few peaks from the integraSachs-Wolfe effect at early times. The dependence onR isnot quite as linear as forB, but still the spectrum varieshighly linearly with R for parameter variations of 10% fol .30. Figure 4 shows the same for the parameterM, whichis an approximate degeneracy direction: the power spectchanges noticeably only in the neighborhood of the fipeak. If the standard three massless neutrino species isposed as a prior,M is fixed.

FIG. 2. Temperature power spectra for the fiducial cosmologmodel~top panel, heavy line!, plus models varying the parameterBupwards by 25%~higher first peak! and downwards by 25%~lowerfirst peak! while keepingA, M, V, R, S, andn fixed. The bottompanel displaysCl as a function ofB, for l 550, 100, 200, 500, and1000. The horizontal axis shows the fractional change inB, whilethe vertical axis gives the change inCl as a fraction of the cosmicvariance at that multipole.

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Variation of V holding the others fixed is nearly an exaphysical degeneracy@9,10#, broken only by the late-timeISW effect at low multipole moments displayed in Fig.This variation at smalll values is not well approximated bsimple linear extrapolation; accurate approximation mrely on more sophisticated schemes, but the total contrtion to constraining cosmological models will have compatively little weight due to the large cosmic variance. A

FIG. 4. Same as Fig. 2, except varying the parameterM whilekeeping the others fixed. LargerM increases the first peak heighAgain, the glitch in the bottom panel is due to systematic errorsCMBFAST.

l FIG. 3. Same as Fig. 2, except varying the parameterR whilekeeping the others fixed. LargerR increases the peak heights. Thsubstantial glitch in the bottom panel is due to systematic errorCMBFAST.

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higherl values, the power spectra should be precisely degerate, and the extent to which they are not is a measurnumerical inaccuracies. Note that the inaccuracies are inform of systematic offsets at levels smaller than a percWhen analyzing data, the variation inV can simply be set tozero for all l values higher than the first acoustic peak. Iflat universe is imposed as a prior condition, the parameteVis then fixed.

Aside from initial conditions, reionization is the othephysical effect which can have a significant impact onpower spectrum. Note that the effective normalizationuse depends onZ, so that asZ varies, the power spectrumretains the same amplitude at highl. As discussed in theprevious section, for largel the effect of changing the totaoptical depth is effectively just a change in the overall aplitude. Thus for our set of parameters, variations inZ whileholding the other parameters fixed change the power strum only at the largest scales. For this reason, we doconsider reionization further here. Note that for polarizatireionization will generate a new peak in the polarizatipower spectrum at lowl, and this effect may require morcomplicated analytic modelling than a simple linear extralation @36#.

Finally, we show the effect of changing the spectral indn of the primordial power spectrum in Fig. 6. Equation~13!

FIG. 5. Varying the parameterV, keeping the others fixed, folow l ~top! and higherl ~center! values; note the scales of the vetical axes. Only the lowest multipoles have any significant vation, arising from the integrated Sachs-Wolfe effect at late timthe bottom panel shows the dependence of thel 55, 10, and 20multipoles onV. A linear approximation is rough but reasonabhigher-order approximations will model this dependence betterhigher l, the variation between the curves is a measure of themerical accuracy of the code generating theCl curves.

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gives a very good approximation to the effect of changingn;the largest errors are at smalll and in the immediate regionof the first few peaks and troughs, amounting to less th10% of the cosmic variance for any individual multipole.

We have now displayed simple numerical or analytic aproximations for the effect of all parameters with a sizeaimpact on the cosmic microwave background power sptrum, for l .50 and for parameter variations which are in trange to which the parameters will be restricted by upcommicrowave background maps. For a model which diffefrom a fiducial model in several different parameters, it isprinciple ambiguous in what order to do the various extralations and approximations. For example, should a reiontion correction be applied before or after theA parameterextrapolation? In practice, however, most of the parameranges involved will be quite small, particularly for theAparameter, and the order in which all of the approximatioare applied to the fiducial spectrum is essentially irrelevaWe test the validity of the various approximations for arbtrary models below.

We also note that while we have demonstrated explicthe validity of linear extrapolations and other simple aproximations in computing deviations from a particular fidcial power spectrum, we have not shown explicitly that this true for any fiducial cosmological model. But current mesurements point towards a Universe well-described bmodel close to the fiducial one considered here, and inunlikely event that the actual cosmology is significantly dferent, the various approximations are simple to checkplicitly for any underlying model.

Some of the linear extrapolations lose accuracy forsmallestl values. It is not clear whether any simple appromation is sufficient for reproducing the power spectrumsmall l values, because a variety of different effects contrute in varying proportions. While these large scales hasignificant cosmic variance and do not have very muchtistical weight compared to the rest of the spectrum, sev

FIG. 6. A comparison of computed~solid! and approximated~dashed! power spectra for different values of the scalar indexn, forn50.8 andn51.2 ~dashed!, where the approximated values havbeen calculated by applying Eq.~13! to the n51 model ~dotted!.Note the displayed variation inn is roughly three times larger thathe constraint MAP will impose~see Fig. 7!.

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FIG. 7. Parameter error contours anticipated for the MAP satellite in the physical parametespace. In each panel, the entilikelihood region has been projected onto a particular plane. Thaxes refer to the fractional variation for each parameter with respect to the fiducial one, e.g.,dA5(A2Afid)/Afid . The contoursshow 68%, 95%, and 99% confidence levels.

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parameters (V, Z, T, nT) have significant effectsonly atsmall l, and accurate limits on these parameters will requapproximating these moments. One possible approachuse quadratic or higher-order extrapolations. We haveexplored the efficacy of such approximations, and they mrequire a more accurate numerical code, but promise togood solution. A more complicated approach would bedevelop analytic fitting formulas. Steps in this direction habeen taken by Durrer and collaborators@37#, but their solu-tions rapidly lose accuracy forl .10, and a variety of physi-cal effects are neglected which contribute at the few perclevel or more. Another possibility is an expanded numeriapproximation, where the effects of our parameters are filinear or quadratic, but instead of fitting the totalCl depen-dence, the various separate terms~i.e., quadratic combinations of Sachs-Wolfe, integrated Sachs-Wolfe, Doppler, aacoustic effects! contributing to eachCl are modelled inde-pendently. In the estimates below, we will not consider tsor perturbations and fix the parameterZ, while V is a de-generate direction; we thus sidestep the issueapproximating the power spectrum for lowl, and do notdiscuss the issue further here.

IV. ERROR REGIONS FOR COSMOLOGICALPARAMETERS

Given some data, we want to determine the error regioparameter space corresponding to some certain confid

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level in fitting the data. This must be done by looking arouin parameter space in the vicinity of the best-fitting modWith a fast computation of the theoretical power spectrfor different points in parameter space in hand, a straightward Metropolis algorithm can be used to construct a MoCarlo exploration of the parameter space. Refined techniqwith the label of Markov chains have recently been applto the microwave background@15,38#. Our approximatepower spectrum evaluation renders these techniques hiefficient.

As a demonstration, we consider estimated power sptrum measurement errors drawn from thex2l 11

2 probabilitydistribution derived in Ref.@39#. We assume that the Universe is actually described by a particular fiducial cosmolocal model. With the simple assumption that the measuremerrors for eachCl are uncorrelated, the likelihood of a comological model being consistent with a measurement offiducial model becomes trivial to compute. Then we maplikelihood around the fiducial model via a Markov chainpoints in parameter space, starting with the fiducial moand moving to a succession of random points in the spusing a simple Metropolis-Hastings algorithm. Sophisticaconvergence tests can be applied to the chain to determwhen the distribution of models in the Monte Carlo samplihas converged to a representation of the underlying likhood function. The chain of models should be constructedthe physical parameter space: the roughly orthogonal eff

7-8

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FIG. 8. The same as Fig. 7, except using the Fisher matrix approximation to the actual likeli-hood. The contours in the twocases are very similar.

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of the physical parameters on the power spectrum lead tefficient Monte Carlo sampling of the space. For each moin the chain, the cosmological parameters of the modelcomputed and stored along with the physical parameters

The commonly used Fisher matrix approximation@8# ex-pands the likelihood function in a Taylor series aroundmost likely model and retains only the quadratic term. WGaussian uncorrelated statistical errors, the Fisher matrixproximation becomes exact whenever the parameter dedence of theCl ’s is exactly linear in all parameters. Since thpower spectrum is nearly linear in most of our physical prameter set, we can check our Monte Carlo results by cparing with the Fisher matrix likelihood computed with thphysical variables.

As a simple illustration, we determine the error regicorresponding to a simplified model of the power spectrwhich will be obtained by the MAP satellite, currently colecting data. MAP’s highest frequency channel has a Gaian beam with a full-width-half-max size of about 0.21 dgrees. MAP will produce a full-sky map of the microwavbackground at this resolution with a sensitivity of arou35 mK per 0.3 square-degree pixel. We neglect eventualcuts and assume allCl estimates are uncorrelated. We cosider models with only scalar perturbations and fixed reiization. The physical parameter space is thusA, B, M, R, Vandn. The other parameters we have fixed~tensor perturba-tions,Z) affect the power spectrum only at smalll, and will

06300

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have little effect on the physical parameters considered.assume the underlying cosmology is described by the ficial model of the previous section. Fundamental physiconstraints have been enforced:Vb,Vmat and all densitiesmust be positive. Additionally, we only consider models wVvac,2 and discard any models which are not invertibfrom the physical to the cosmological parameters~theseamount to a handful of models with extreme parameter vues!.

Figure 7 shows the anticipated MAP error contours inphysical parameter space, extracted from a Markov chai33104 models. The full 7-dimensional likelihood regionprojected onto all pairs of parameters. The parameterV isessentially unconstrained by MAP, whileM has a 1-s errorof around 30%. Measurements extended to smaller scwill not significantly improve constraints on these two prameters: they represent true physical degeneracies inmodel space. The other parameters are well constrained.proximate 1-s errors are 0.5% forA, 5% for R, 3% for B,2% for S, and 3% forn. The error regions are ellipticalbecause the likelihood region in these parameters is wapproximated by a quadratic form. The residual correlatioinvolving S and n will be lifted by measurements out thigher l values; the parameters are largely uncorrelatedtheir power spectrum effects. Figure 8 shows a Fisher maestimate of the same likelihood. The two sets of likeliho

7-9


FIG. 9. The likelihood distri-bution in Fig. 7 mapped into thecosmological parameter space.

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contours are very close, as expected since we have shexplicitly that the power spectrum varies nearly linearlythe physical variables over a parameter region consiswith MAP-quality data. The most notable exception is tvariableA, which does not give a linear variation, but it is swell constrained that it is effectively fixed when consideriits impact on determining the other parameters. The contowill also differ slightly due to the nonlinear behavior ofn,but the general excellent agreement demonstrates thaMonte Carlo technique works as expected.

In Fig. 9, we replot Fig. 7 in the cosmological paramespace. This procedure is trivial, since we calculate the cmological parameter set for each model. The resulting ctours are deformed, reflecting the nonlinear relation betwthe physical and cosmological parameter sets. ClearlFisher matrix approximation in the cosmological variablas has been done in numerous analyses, will give sigcantly incorrect error regions@10#. Note, however, that forMAP-quality data, the Fisher matrix errors in the physicvariables projected into the cosmological variables givehighly accurate result, Fig. 10, in contrast to the claimsRef. @10#. The Fisher matrix approximation in the physicparameter space can be used to further increase the efficof error region evaluation, which may be useful in the cawhere the computation of the error regions is dominatedevaluation of the likelihood for a given model from actudata. If the likelihood evaluation is at least as efficient as

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power spectrum evaluation, then the increased efficiefrom a Fisher matrix approximation may not result in drmatic increases in computational speed, since in both cthe physical parameter space must be sampled and paramconversion performed at each point to construct the likhood contours in the space of the cosmological paramet

It is useful to plot error regions in both sets of parametePhysical parameters represent the actual physical quanwhich are being directly constrained by the measuremeand the error regions are nearly elliptical. The cosmologiparameters are useful theoretically and for comparisons wother data sources, i.e., large-scale structure or superstandard candles. One advantage of determining the eregion via a Monte Carlo technique is that prior constraion cosmological parameters can be implemented simply

When the Fisher matrix approximation is computed in tphysical parameter space, it is highly accurate. This doesactually help much, however: if information about cosmlogical parameters is desired, transforming from the physto the cosmological parameter space requires some kinsampling of the likelihood region. The most efficient waydo this is via a Monte Carlo technique, so the additioncomputational overhead in performing a Monte Carlo likehood evaluation of the physical variable likelihood insteada Fisher matrix approximation is negligible.

To test the validity of our numerical power spectrum aproximations, we choose 1000 models at random from

7-10

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FIG. 10. The likelihood distri-bution in Fig. 8 mapped into thecosmological parameter spacThese contours, derived fromFisher matrix approximation, arevery close to those in Fig. 9.

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where (s lMAP)2 is the variance ofCl in our model MAP data.

For MAP, the measured power spectrum will consist of aproximately 800 independent multipole moments, so anproximate power spectrum which differs from the actumodel power spectrum by the MAP error at each multipwill have a value for the statisticx2 of around 800. Figure 11shows the error distribution for our subset of models. Tstatisticx2 for each model is plotted against the ‘‘distancin physical parameter space of the model from the fidumodel, where the distance measure is the Fisher matrix:

d5@~P2P0!F~P2P0!#1/2, ~15!

whereP is a vector of physical parameters correspondingthe model considered,P0 is the vector of fiducial-model parameters, and

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is the Fisher matrix. This is roughly equivalent to measurthe distance along each axis in the physical parameter sin units of the 1-s error interval in that parameter directionThe solid-circle points are models with 2,Nn,4, for whichthe worst-approximated models disappear. The differencetwo possible origins: either linear extrapolation is not vaover a large enough range in the variablesR and M tohandle these models, orCMBFAST does not provide accuratcomputations in this range. The second possibility seemore likely, since the code contains an explicit trap preveing computations for these unusual numbers of neutrino scies.

For essentially all other models, the error averaged oall multipoles is small compared to the MAP statistical erors, and the error distribution is roughly independent ofdistance in parameter space, showing that our numericalproximations are valid. Furthermore, most of the total ertends to come from a small number of multipoles, indicatithat systematic errors inCMBFAST dominate the total differ-ence between our approximations and the direct numercalculations. Given this fact, we forego a more thoroucharacterization of the errors in our approximate spectrcomputations until a more accurate code is available, but

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approximations are likely accurate enough for reliablerameter analysis. It is unclear at this point whether the erin our approximations or the systematic errors inCMBFAST

will have a greater impact on the inferred error region.~Sev-eral other public CMB Boltzmann codes are in fact availabbut none are as general asCMBFAST, allowing for arbitrarycurvature, vacuum energy, and neutrino species, all of whare required for the analysis presented here.!

V. DISCUSSION AND CONCLUSIONS

Error regions similar to Fig. 9 have been previously costructed for model microwave background experime

@10,38#. The remarkable point about our calculation is ththe entire error region, constructed from power spectra33104 points in parameter space, has been computedfew seconds of time on a desktop computer. In fact, sincepower spectra can be computed with a few arithmetic options per multipole moment, the calculation time might neven be dominated by computing model power spectra,rather by computing the likelihood function or by convertinbetween the cosmological and physical parameters.

For actual nondiagonal covariance matrices describreal experiments, the computation of the likelihood wdominate the total computation time. Gupta and Heavhave recently formulated a method for computing anproximate likelihood with great efficiency@40#, based onfinding uncorrelated linear combinations of the power sptrum estimates@41#. Each parameter corresponds to a uniqcombination, so the likelihood is calculated with a handfuloperations. Moreover, the method is optimized so the pareter estimation can be done with virtually no loss of acracy. Using this technique, likelihood estimates for realiscovariance matrices will be roughly as efficient as our powspectrum estimates.

The conversion from the physical parameters to cosm

FIG. 11. A plot of the difference between approximate andmerical power spectra versus distance from the fiducial modeparameter space, for 1000 cosmological models drawn randofrom the distribution in Fig. 7. The solid circles are for models w2,Nn,4.

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logical ones requires evaluation of several numerical ingrals. The integrands will be smooth functions, and this scan likely be made nearly as efficient as the power spectevaluation, although we have not implemented an optimiroutine for parameter conversion. We make the followirough timing estimate: each multipole of the power spectrcan be approximated by a handful of floating point opetions, and an approximate likelihood and the parameter cversion both will plausibly require similar computationwork. So we anticipate on the order of 10 floating pooperations per multipole for each of 1000 multipoles. Ongigaflop machine, this results in evaluation of around 15

models per second, so a Markov Chain with tens of thsands of models can be computed in under a secondcourse, such a computation is trivially parallelizable, and ceasily be sped up by a factor of tens on a medium-sizeallel machine. In contrast, other state-of-the-art parametertimation techniques@16,17,23#, dominated by the powespectrum calculation, compute roughly one model per sond, requiring several hours for a 104 point computation.

This great increase in computational speed is highly uful. All upcoming microwave background measurements aresulting parameter estimates will be dominated by systatic errors, both from measurement errors when observthe sky and processing errors in the data analysis pipeThe only way to discern the effect of these errors is throumodelling them, requiring a determination of parametersmerous times. The microwave background has the potento constrain parameters at the few percent level. But a vation of this size in any given parameter will result in changin Cl ’s of a few percent, which is significantly smaller thathe errors with which eachCl will be measured. This meanthat small systematic effects distributed over manyCl ’s canbias derived parameter values by amounts larger thanformal statistical errors on the parameter. Exhaustive simlation of a wide range of potential systematics will be rquired before the accuracy of highly precise parameter deminations can be believed, and our techniques make sinvestigations far faster and easier.

Along with increased speed, our approximation methoalso promise high accuracy. A careful error evaluation is hdered by systematic errors inCMBFAST; our numerical ap-proximations give better accuracy than the predominantmerical code for parameter ranges generally larger thanregion which will be allowed by the MAP data. The physicset of variables presented here clarify the degeneracy sture of the power spectrum, clearly displaying degenerand near-degenerate directions in the parameter spaceexample, with the physical parameters presented here,clear that any tensor mode contribution, which affects olow-l multipoles, will have a negligible impact on the errocontours in Fig. 7.

Bond and Efstathiou used a principle component analyto claim that uncertainties in the tensor mode contributiowould be the dominant source of error for all of the cosmlogical parameters, and further claimed that as a result,Fisher matrix approximation will overestimate the errorsother apparently well-determined parameters such as ouB@10#. These results are clearly valid only for data which asignificantly less constraining than the MAP data will b

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Essentially, their approximate degeneracy trades off vations in R andB, which change the heights of the first fepeaks, with variations inns , which also impact the first fewpeaks, while holdingA fixed. From our analysis, it is cleathat this degeneracy only holds approximately, sinceR andB have a significant impact only on the first three peawhile ns affects all of the peak heights. Indeed, a careexamination of Fig. 3 in@42# shows that the power spectrare only approximately degenerate, with peak height difences which are significant for MAP. The addition of tensmodes allows a rough match between the two power speat low l values, but for highl values the discrepancy in peaheights is already evident at the third peak and will becolarger for higher peaks, since the values ofns in the twomodels are so different.

Gravitational lensing makes a negligible difference in tlinearity of the parameter dependences. We have notcluded polarization in this analysis, but the three other poized CMB power spectra can be handled in the same wathe temperature case using standard techniques@43,44#.

We have shown that a proper choice of physical paraeters enables the microwave background power spectrube simply approximated with high accuracy over a signcant region of parameter space. This region will be laenough for analyses of data from the MAP satellite, althoularger regions could be stitched together using multipleerence models. Our approximation scheme requirescomputing numerical derivatives of the power spectrum mtipoles with respect to the various parameters, but this mbe done only once and then computing further spectraextremely fast~in the neighborhood of 105 models per sec-ond or more on common computers!. The error determina-tion techniques in this paper require the numerical evaluaof only tens of power spectra rather than thousands orlions, so speed of the power spectrum code will not beparamount concern. On the other hand, we emphasizefor any high-precision parameter estimate, even small stematic errors in computingCl ’s can lead to biased parameter estimates comparable to the size of the statistical erThese two considerations argue for a significant revisionCMBFAST ~CMBSLOW, perhaps?! or construction of other in-dependent codes which focus on overall accuracy and stity of derivatives with respect to parameters rather thancomputational speed. Such a code, combined with themation techniques in this paper and efficient likelihoevaluation methods@40# will provide a highly efficient andreliable way to constrain the space of fundamental coslogical parameters.

ACKNOWLEDGMENTS

We thank David Spergel, who has independently impmented some of the ideas in this paper, for helpful discsions, and Lloyd Knox, Manoj Kaplinghat, Alan HeavenAntony Lewis, and Hiranya Peris for useful commenBruce Winstein suggested numerous clarifications andrections. We also thank Uros Seljak and Matias Zaldarrifor making public theirCMBFAST code, which has proven

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invaluable for this and so much other cosmological analyA.K. has been supported in part by NASA through graNAG5-10110.

APPENDIX A: A MODEL PIPELINE FORPARAMETERS ANALYSIS

In the interests of completeness, in this appendix we oline an analysis pipeline for cosmological parameters atheir errors. Our basic data set will be a high-resolution mcrowave background map, from which has been extractedangular power spectrum~i.e., theCl ’s! and a covariance matrix for the Cl ’s. Given this data set, plus potentially relevadata from other sources, we need to find the best-fit coslogical model and the error contours in the parameter sp

In the case of an ideal, full-sky map with only statisticerrors and uniform sky coverage, the covariance matrixduces to a multiple of the identity matrix, but in general, fmost experiments the covariance matrix may contain signcant off-diagonal terms. Even for the MAP satellite, whiwill come close to a diagonal covariance matrix, the needmask out the galactic plane induces some correlationstween nearby multipoles. The first step in the analysis isconstruct a likelihood calculator: what is the likelihood ththe measured data represent some particular theorepower spectrum? This is in general a computationally hproblem, since it involves inverting anN3N matrix, whereN is the number of multipoles in the power spectrum. Hoever, the problem becomes easier if the covariance matrclose to diagonal. Heavens and Gupta have recently forlated a method for computing an approximate likelihowith great efficiency@40#, based on an eigenvector decomposition of the covariance matrix@41#. They have demon-strated a likelihood calculator which takes an input theorcal power spectrum and outputs a likelihood basedsimulated measurements and covariances in much less thsecond of computation time. Further efficiency improvments are desirable, as this is likely to be the piece ofanalysis pipeline which dominates the total time.

The likelihood calculator can also incorporate prior proabilities obtained from other data sources, or from physiconsiderations likeL.0. Since none of the following stepdepend on the form of the likelihood function, the inclusioof other kinds of data can be handled in a completely molar way.

The second element needed is a pair of routines to conbetween the physical parameter space (A,B,M,V) and thecosmological parameter space (Vmat,Vb ,VL ,h). To gofrom the cosmological parameters to the physical paramesimply involves evaluating the integral needed in Eq.~8!.The reverse direction can be done straightforwardly bywriting A in terms ofB, M, V, R, andh and then numeri-cally finding the value ofh which gives the needed value oA. The total variation ofA with h is a smooth function, andthe root-finding step can be done with a minimal numberintegral evaluations. This procedure does not guarantee alution, however, in extreme regions of parameter spacemore general~but slower! method is to search the cosmologcal parameter space for the model which best fits the gi

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set of physical parameters, and then project this model binto the physical parameter space to check the quality offit.

Once a likelihood calculator and parameter conversroutines are in hand, we advocate the following procedfor parameter determination. First, choose a fiducial mowhich gives a good fit to the measured power spectrum. Tcan largely be done by eye, using the parameter dependedisplayed in the figures in Sec. III. Note that the fiducmodel does not need to be the formal best-fit model, but oneeds to be close enough to the best-fit model so that mothroughout the error region will be accurately approximaby the extrapolations and approximations in Sec. III.

Next, check the validity of the fiducial model. Is the fit othe model with the data adequate? It is certainly possiblethe simple space of cosmological models considered hmay not contain the actual universe. For example, somemixture of isocurvature initial conditions might be importanor the initial power spectrum might not be well describeda power law. If the best-fit model is not a good fit to the dait makes no sense to proceed with any further paramanalysis. The model space must be enlarged or modifieinclude models which provide a good fit to the data.

If the fiducial model is adequate, then compute numerpartial derivatives of eachCl with respect toB, M, andR.Also, write numerical routines to compute the power sptrum approximately forl ,50. Even thoughCMBFAST orsimilar codes run very efficiently when computing only, l ,50, this time will still greatly dominate the total computation time for parameter error determination, so an ecient analytic estimate is essential. Note that an alternpossibility is to simply ignore the lowest multipoles, whicwill be a small portion of the data for a high-resolution maat the expense of losing any leverage on the parameterV,Z, and the tensor power spectrum. Still, the other paramewill be determined much more precisely, so it may be psible to fit the rest of the parameters usingl .50 data only,and then constrain just theV-Z-tensor parameter space frothe l ,50 data independently. Now it is possible to estimthe Cl ’s highly efficiently for any model within a sizeablparameter space region around the fiducial model.

Now perform a Markov-chain Monte Carlo approximtion @15# in the physical parameter space. Each point inrameter space requires evaluation of one approximate pospectrum, one likelihood, and one conversion to cosmolocal parameters. The Monte Carlo approximation will coverge efficiently since the physical parameters are nearlythogonal in their effects on the power spectruSophisticated convergence diagnostics are available fortermining when the Markov chain of models has sufficiensampled the error region@45#.

Finally, once the chain of models in parameter spacebeen computed, extract the best-fit model and error contfrom the distribution of models. By displaying twodimensional likelihood plots for all parameter combinationa reliable picture of the shape of the likelihood region in twhole multidimensional parameter space can be visualiz

Upcoming data will be good enough that the statistierrors, given by these likelihood contours, will be smal

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than the systematic errors, at least in some parameter sdirections. Sources of systematic error include instrumeeffects, modelling of foreground emission, approximationthe covariance matrix, inaccuracies in the theoretical modand inaccuracies in the approximations in this paper. Befwe can confidently claim to have determined the cosmolocal parameters and their errors, these systematics musmodelled and investigated. Using the techniques outlinhere, where a complete Monte Carlo approximation canaccomplished in seconds, investigation of systematics isily practical, instead of a computational challenge.

APPENDIX B: SOME NUMERICAL CONSIDERATIONSWITH THE CMBFAST CODE

Seljak and Zaldarriaga’sCMBFAST code @33,35# has be-come the standard tool for computing the microwave baground temperature and polarization power spectra forinflation-type cosmological models discussed in this papHundreds of researchers have employed it for tasks ranfrom predictions of highly speculative cosmological modeto analysis of every microwave background measurementhe past few years. Its comprehensive treatment of a varof physical effects and its public availability have made it tgold standard for microwave background analysis.

The original aim of theCMBFAST code was fast evaluationof the power spectra. Previous codes had simply evolvedcomplete Boltzmann hierarchy of equations up to the mamum desired value ofl. This resulted in a coupled set oaround 3l linear differential equations, and thel-values ofinterest might be 1000 or greater. The task of evolving thsands of coupled equations with oscillatory solutions ledcodes which required many hours to run on fast computTheCMBFAST algorithm evolves the source terms first, whicinvolve angular moments only up tol 54; the same Boltz-mann hierarchy can be used, but with a cutoff ofl .10. Therest of theCl ’s can then be obtained by integrating thsource term against various Bessel functions. The Befunctions can be precomputed or approximated@46# indepen-dently for eachl. Great time savings result since~i! it is notnecessary to compute the power spectrum for everyl, butrather only, e.g., every 50thl with the rest obtained frominterpolation;~ii ! the time steps for the Boltzmann integrtion are not controlled by the need to resolve all of thecillations in the Bessel functions, and thus the Boltzmaintegrator can be significantly more efficient. It also maypossible to employ approximations to the integrals ofsource times the Bessel functions with an additional gainspeed, althoughCMBFAST does not implement this. The codoffers a speed improvement of a factor of 50 over Boltzmahierarchy codes for flat universes, although the advantagsmaller for models which are not flat or have a cosmologiconstant.

CMBFAST is nominally accurate at around the one perclevel, although no systematic error analysis has ever bpublished. While the numerical error in any given multipoCl is relatively small, the derivatives ofCl with respect tothe various cosmological parameters are significantly l

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accurate. At the time the code was written, no microwabackground experiments were of sufficient precision to wrant worrying about inaccuracies at this level, and the grutility of the code has been its combination of speed aaccuracy. But we have now entered the era of high-precismeasurements which will place significant constraintsmany parameters simultaneously, and as this paper hasonstrated, efficiently prosecuting this analysis requirescode of high precision and stable behavior with respecparameter variations. We have found that the unmodifiedsion ofCMBFAST ‘‘out of the box’’ is not sufficiently accuratefor the calculations presented here, but these shortcomare partially compensated by the following procedures.

First, CMBFAST uses slightly different algorithms for flaand nonflat background spacetime geometries. In thecase, the necessary spherical Bessel functions are preputed and called from a file; in the nonflat case, the hypspherical Bessel functions are computed via an integralresentation as they are needed. As a result, the line-of-sintegrals over the product of source term and Bessel funchave different partitions in the two cases, and the numervalues differ at the percent level. Also, the code has awhich forces evaluation of the flat case wheneveruV21u,0.001. If the fiducial model is flat, then as parametersvaried which change the geometry, many individual mupoles Cl exhibit discontinuities at the point in parametspace where the code switches from the flat to the nonevaluation scheme. These discontinuities can result innificant errors when evaluatingCl derivatives with respect toparameters. Also, even if the derivative is correctly evaated, the offset between the two cases can result in a sligbiased error estimate.

A careful fix of this problem would involve insuring thathe integral partitions used in the two cases are determconsistently. Alternately, the flat-space code could be rewten to numerically integrate the necessary Bessel functionin the nonflat case, since the two methods do not havsignificant difference in computational speed. A simppractical solution which we have employed in this paper isforce the code always to use the nonflat integration routThis can be accomplished by narrowing the tolerancewhich the code uses the nonflat integration~say to uV21u,1025), combined with always using a nonflat fiducimodel ~shifting Vmat by 1025 will never result in a statisti-cally significant difference when fitting a given data set, dto cosmic variance!.

Second, in varying theB parameter, adjacent acoustpeaks move in the opposite direction. Between the petherefore, is a pivot pointl for which Cl remains fixed asBis varied, and the nearbyCl ’s will vary only slightly. AsCMBFAST is written, it computesCl at l values separated b50; the complete spectrum is then obtained by spliningtween the calculated values. Splines are rather stiff,small changes in regions where the power spectrum hasnificant curvature ~i.e. around the acoustic peaks atroughs! can noticeably change the overall fit to the powspectrum in the regions between. In particular, the positionthe ‘‘pivot’’ l-value in the spectrum sometimes shifts a b

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which gives a spurious irregular dependence onB of the fewCl ’s in this region.

This can be addressed by a simple modificationCMBFAST forcing the explicit evaluation ofCl for more lvalues. We have used ad l of 10, which has eliminated thespurious behavior. Of course, this results in a significantcrease in the computational time needed for an individmodel. A more sophisticated approach would be to uscomparable number ofl values as the original code, but distribute them preferentially in thel-ranges corresponding tthe peaks and troughs where the curvature of the power strum is largest. This is easily implemented using the vaables defined in this paper, since the peak positions are cpletely determined byA alone. Using a finer grid inl alsoreproduces the peak positions with higher accuracy.

Third, as discussed in the body of the paper, the pospectra must be normalized according to a physical critewhich does not depend on nonlinear behavior at smalll val-ues. This is accomplished by simply compilingCMBFAST

with the UNNORM option. In this case, no normalizationCOBE is done; instead, the amplitude of the potentialC istaken to be unity at the scale of the horizon at the time ofscattering. This normalization is fixed by the primordial amplitude of scalar perturbations, often denoted asAS . We de-fine a physical variableS corresponding to the microwavbackground amplitude by also including the effect of diffeing amounts of reionization so that the power spectrumlarge l remains fixed.

Some residual numerical problems remain withCMBFAST.We have found, for example, that in the region aroundl5350 ~between the first and second acoustic peaks forfiducial model!, the power spectrum exhibits spurious osclatory behavior with an amplitude of a few percent, if computed directly for everyl instead of every 10th or 50thl ~seeFig. 12!. This likely arises from either thek or h integralover an oscillatory integrand not being evaluated on a fienough grid. As a result, directly calculatedCl values in thisregion of the parameter space effectively have non-neglig

FIG. 12. Derivatives ofCl with respect toB, calculated byCMBFAST with D l 510, from 2%, 5%, and 10% finite differences iB. The smoothest curve is for the 10% variation.

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random errors. This problem is normally hidden by the splsmoothing with every 50thl computed. When the number oCl evaluations is increased to every 10thl, the power spec-trum shows a distinct glitch in this region, and the parame

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dependence is completely unreliable here. The spline fitsforced into more oscillations to include the computed poinMore accurate numerical integrations will probably solthis problem, at the expense of a significant speed reduc

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Efficient cosmological parameter estimation from microwave background anisotropies

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