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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 08854-8019

(Dated: May, 2002)

We revisit the issue of cosmological parameter estimation in light of current and upcoming high-precision measurements of the cosmic microwave background power spectrum. Physical quantitieswhich determine the power spectrum are reviewed, and their connection to familiar cosmologicalparameters is explicated. We present a set of physical parameters, analytic functions of the usualcosmological parameters, upon which the microwave background power spectrum depends linearly(or with some other simple dependence) over a wide range of parameter values. With such a set of pa-rameters, microwave background power spectra can be estimated with high accuracy and negligiblecomputational effort, vastly increasing the efficiency of cosmological parameter error determination.The techniques presented here allow calculation of microwave background power spectra 105 timesfaster than comparably accurate direct codes (after precomputing a handful of power spectra). Wediscuss various issues of parameter estimation, including parameter degeneracies, numerical preci-sion, mapping between physical and cosmological parameters, and systematic errors, and illustratethese considerations with an idealized model of the MAP experiment.

PACS numbers: 98.70.V, 98.80.C

I. INTRODUCTION

By January 2003, microwave maps of the full sky at0.2 resolution will be available to the world, the har-vest of the remarkable MAP satellite currently takingdata [1]. The angular power spectrum of the tempera-ture fluctuations in these maps will be determined to highprecision on angular scales from the resolution limit up toa dipole variation; this corresponds to about 800 statis-tically independent power spectrum measurements. Re-cent measurements have given a taste of the data to come,although covering much smaller patches of the sky andwith potentially more serious systematic errors [2, 3, 4].

The main science driving these spectacular technicalfeats is the determination of basic cosmological parame-ters describing our Universe, and resulting insights intofundamental physics; see [5, 6] for recent reviews and[7] for a pedagogical introduction. The expected seriesof acoustic peaks in the microwave background powerspectrum encode enough information to make possiblethe determination of numerous cosmological parameterssimultaneously [8]. These parameters include the long-sought Hubble parameter H0, the large-scale geometry ofthe Universe Ω, the mean density of baryons in the Uni-verse Ωb, and the value of the mysterious but now widelyaccepted cosmological constant Λ, along with parametersdescribing the tiny primordial perturbations which grewinto present structures, and the redshift at which the Uni-

∗Also at School of Natural Sciences, Institute for Advanced Study,

Einstein Drive, Princeton, New Jersey 08540 ; Electronic address:

kosowsky@physics.rutgers.edu†Electronic address: milos@physics.rutgers.edu‡Electronic address: raulj@physics.rutgers.edu

verse reionized due to the formation of the first stars orother compact objects. While most of these parametersand other information will be determined to high preci-sion, one near-exact degeneracy and other approximatedegeneracies exist between these parameters [9, 10].

The exciting prospects of providing definitive answersto some of cosmology’s oldest questions raise a poten-tially difficult technical issue, namely finding constraintson a large-dimensional parameter space. Given a set ofdata, what range of points in parameter space give mod-els with an acceptably good fit to the data? The answerrequires evaluating a likelihood function at many pointsin parameter space; in particular, finding confidence re-gions in multidimensional parameter space requires look-ing around in the space. This is a straightforward pro-cess, in principle. But a parameter space with as many asten dimensions requires evaluating a lot of models: a gridwith a crude 10 values per parameter contains ten billionmodels. A direct calculation of this many models is pro-hibitive, even with very fast computers. While some ofthe parameters are independent of the others (i.e. tensormodes), reducing the effective dimensionality, many ofthe parameters will be constrained quite tightly, requir-ing a finer sampling in that direction of parameter space.An additional problem with brute-force grid-based meth-ods is a lack of flexibility: if additional parameters arerequired to correctly describe the Universe, vast amountsof recalculation must be done. Grid-based techniqueshave been used for analysis of cosmological parametersfrom microwave background (e.g., [11, 12]) but clearlythis method is not fast or flexible enough to deal withupcoming data sets adequately.

A more sophisticated approach is to perform a searchof parameter space in the region of interest. Relevanttechniques are well known and have been applied to the

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microwave background [13, 14, 15]. Reliable estimates ofthe error region in cosmological parameter space can beobtained with random sets of around 105 models, reduc-ing the computational burden by a factor of 1000 or morecompared to the cruder grid search methods. On fastparallel computers it is possible to compute the powerspectrum for a given model in a computation time onthe order of a second [16], making Monte Carlo errordeterminations feasible. For improved efficiency, usefulimplementations do not recalculate the entire spectrumat each point in parameter space, but rather use approx-imate power spectra based on smaller numbers of calcu-lated models [17].

Here we expand and refine this idea by presenting a setof parameters, functions of the usual cosmological param-eters, which reflect the underlying physical effects deter-mining the microwave background power spectrum andthus result in particularly simple and intuitive parame-ter dependences. Previous crude implementations of thisidea applied to low-resolution measurements of the powerspectrum [18]; the current and upcoming power spectrummeasurements requires a far more refined implementa-tion. Our set of parameters can be used to constructcomputationally trivial but highly accurate approximatepower spectra, and large Monte Carlo computations canthen be performed with great efficiency. Additional ad-vantages of a physically-based parameter set are that thedegeneracy structure of the parameter space can be seenmuch more clearly, and the Monte Carlo itself takes sig-nificantly fewer models to converge.

Earlier Fisher-matrix approximations of parameter er-rors are a rough implementation of this general idea: ifthe the power spectrum varies exactly linearly with eachparameter in some parameter set, and the measurementerrors are gaussian random distributed and uncorrelated,then the likelihood function as a function of the parame-ters can be computed exactly from the partial derivativesof the model with respect to the parameters [8, 19, 20].Even if the linear parameter dependence does not holdthroughout the parameter space, it will almost always bevalid in some small enough region, being the lowest-orderterm in a Taylor expansion. If this region is at least aslarge as the resulting error region, then the Fisher matrixprovides a self-consistent approximation for error deter-mination. The difficulty is in finding a suitable parameterset. Our aim is not necessarily to find a set of parame-ters which all have a perfectly linear effect on the powerspectrum. Rather, more generally, we desire a set of pa-rameters for which the power spectrum can be approxi-mated very accurately with a minimum of computationaleffort. Then we can dispense with approximations of thelikelihood function, as in the Fisher matrix approach,and directly implement a Monte Carlo parameter spacesearch with a minimum of computational effort. Thistechnique allows simple incorporation of prior probabil-ities determined from other sources of data, and if fastenough allows detailed exploration of potential system-atic biases in parameter values arising from numerous

experimental and analysis issues such as treatment offoregrounds, noise correlations, mapmaking techniques,and model accuracy.

We emphasize that with the accuracy of upcomingmicrowave background power spectrum measurements,the cosmological parameters will be determined preciselyenough that their values will be significantly affected bysystematic errors from assumptions made in the analysispipeline and, potentially, numerical errors in evaluatingtheoretical power spectra. The measurements themselvesmay also be dominated by systematic errors. Extensivemodelling of the impact of various systematic errors oncosmological parameter determination will be essential

before any parameter determination can be consideredreliable. This is the primary motivation for increasingthe efficiency of parameter error analysis. We discussthis point in more detail in the last Section of the paper.

The following Section reviews physical processes affect-ing the cosmic microwave background and defines ourphysical parameter set. The key parameter, which setsthe angular scale of the acoustic oscillations, has been dis-cussed previously in similar contexts [10, 21, 22] but hasnot been explicitly used as a parameter. The other pa-rameters describing the background cosmology can thenbe chosen to model other specific physical effects. Sec-tion III displays how the power spectrum varies as eachparameter changes with the others held fixed. The map-ping between our physical parameters and the conven-tional cosmological parameters immediately reveals thestructure of degeneracies in the cosmological parameterspace. Simple approximations for the effect of each phys-ical parameter are shown to be highly accurate for allbut the largest scales in the power spectrum. We thendetermine the error region in parameter space for an ide-alized model of the MAP experiment in Sec. IV, com-paring with previous calculations, and test the accuracyof our approximate power spectrum calculation. Finally,Sec. V discusses the potential speed of our method, like-lihood estimation, mapping between the sets of parame-ters, accuracy, and systematic errors. The first appendixsummarizes an analysis pipeline going from a measuredpower spectrum to cosmological parameter error esti-mates; the second appendix details several numerical dif-ficulties with using the CMBFAST code for the calcula-tions in this paper, along with suggested fixes.

Other recent attempts to speed up power spectrumcomputation through approximations include the dASHnumerical package [23] and interpolation schemes [12].Our method has the advantage of conceptual simplicityand ease of use combined with great speed and high ac-curacy.

II. COSMOLOGICAL PARAMETERS AND

PHYSICAL QUANTITIES

We consider the standard class of inflation-like cosmo-logical models, specified by five parameters determining

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the background homogeneous spacetime (matter densityΩmat, radiation density Ωrad, vacuum energy density ΩΛ,baryon density Ωb, and Hubble parameter h), four pa-rameters determining the spectrum of primordial pertur-bations (scalar and tensor amplitudes AS and AT andpower law indices n and nT ), and a single parameter τ de-scribing the total optical depth since reionization. We ne-glect additional complications such as massive neutrinosor a varying vacuum equation of state. This generic classof simple cosmological models appears to be a very gooddescription of the Universe on large scales, and has thevirtue of arising naturally in inflationary cosmology. Ofcourse, any analysis of new data should first test whetherthis class of models actually provides an acceptably goodfit to the data, and only then commence with determi-nations of cosmological parameters.

What constraints do a measurement of the microwavebackground power spectrum place on this parameterspace? A rough estimate assumes the likelihood of a par-ticular set of parameters is a quadratic function of theparameters (the usual Fisher matrix technique); betteris a Monte-Carlo exploration of parameter space in theregion of a best-fit model. For either technique to givereliable results, it is imperative to isolate the physicalquantities which affect the microwave background spec-trum and understand their relation to the cosmologicalparameters. If a set of physical quantities which haveessentially orthogonal effects on the power spectrum canbe isolated, then extracting parameter space constraintsbecomes much more reliable and efficient.

The five parameters describing the background cosmol-ogy induce complex dependences in the microwave back-ground power spectrum through multiple physical effects.Using the physical energy densities Ωmath

2, Ωbh2, and

ΩΛh2, as has been done in numerous prior analyses, im-proves the situation but is still not ideal. The character-istic physical scale in the power spectrum is the angularscale of the first acoustic peak, so it is advantageous touse this scale as a parameter which can be varied inde-pendently of any other parameters. This angular scale isin turn determined by the ratio of the comoving soundhorizon at last scattering (which determines the physi-cal wavelength of the acoustic waves) to the angular di-ameter distance to the surface of last scattering (whichdetermines the apparent angular size of this yardstick).We first determine this quantity in terms of the cosmo-logical parameters, and then choose three other comple-

mentary quantities. The analytic theory underlying anysuch choice of physical parameters has been worked outin detail (see [24, 25, 26, 27], and also [28, 29]).

The angular diameter distance to an object at a red-shift z is defined to be DA(z) = R0Sk(r)/(1 + z) where,following the notation of Peacock [30], R0 is the (dimen-sionful) scale factor today and

Sk(r) =

sin r, Ω > 1;r, Ω = 1;sinh r, Ω < 1;

(1)

the Friedmann equation provides the connection betweenredshift z and coordinate distance r. With a dimension-less scale factor a and a Hubble parameter H = a/a, thedifferential relation is

H0R0dr =−da

[(1 − Ω)a2 + ΩΛa4 + Ωmata + Ωrad]1/2. (2)

Thus the relation between scale factor and coordinatedistance becomes

r = |Ω − 1|1/2

×∫ 1

a

dx

[(1 − Ω)x2 + ΩΛx4 + Ωmatx + Ωrad]1/2(3)

and the angular diameter distance is just

DA(a) = aH−10 |Ω − 1|−1/2Sk(r), (4)

using the above expression for r. Here we have used therelation H0R0 = |Ω − 1|−1/2 which follows directly fromthe Friedmann equation.

The sound horizon is defined, in analogy to the particlehorizon, as

rs(t) =

∫ t

0

cs(t′)

a(t′)dt′ (5)

where cs(t) is the sound speed of the baryon-photon fluidat time t; to a very good approximation, before decou-pling the sound speed is given by [24]

c2s =

1

3(1 + 3ρb/4ργ)

−1. (6)

Using this expression plus the equivalent expression to Eq. (2) connecting dt and da gives

rs(a) =1

H0

√3

∫ a

0

dx[(

1 + 3Ωb

4Ωγx)

((1 − Ω)x2 + ΩΛx4 + Ωmatx + Ωrad)]1/2

. (7)

The physical quantity relevant to the microwave back- ground power spectrum is

A ≡ rs(a∗)

DA(a∗)(8)

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where the two functions are given by Eqs. (7) and (4), anda∗ is the scale factor at decoupling. An accurate analyticfit for a∗ in terms of the cosmological parameters hasbeen given by Hu and Sugiyama [26]:

z∗ =1

a∗

− 1 = 1048[

1 + 0.00124(Ωbh2)−0.738

]

×[

1 + g1(Ωmath2)g2

]

, (9)

g1 ≡ 0.0783(Ωbh2)−0.238

[

1 + 39.5(Ωbh2)0.763

]−1,

g2 ≡ 0.560[

1 + 21.1(Ωbh2)1.81

]−1.

This fit applies to standard thermodynamic recombina-tion for a wide range of cosmological models. (Note thatwhile the formula was constructed only for models withthe standard value for Ωrad, the effect of a change of ra-diation density will come only through the redshift ofmatter-radiation equality, and thus Ωmath

2 can be re-placed by a factor proportional to Ωmat/Ωrad to accountfor a variation in the radiation density. The changes in z∗are small anyway and have little impact on our analysis.)Since a∗ ≪ 1, the term proportional to ΩΛ in Eq. (7) maybe dropped and the integral performed, giving [25]

rs(a) =2√

3

3(Ω0H

20 )−1/2

(

aeq

Req

)1/2

× ln

√1 + R +

√

R + Req

1 +√

Req

. (10)

where R = 3ρb/4ργ is proportional to the scale factor.If two cosmological models are considered which both

have adiabatic perturbations and the same value for A,to high accuracy their acoustic peaks will differ only inheight, not position. This is an obvious advantage forconstraining models. The peak positions, and thus A,will be extremely well constrained by power spectrummeasurements. Other combinations which vary the peakheights only will be less well determined.

Choosing four other parameters describing the back-ground cosmology must balance several considerations:(1) the new set of parameters must cover a sufficientlylarge region of parameter space; (2) the power spectrumshould vary linearly or in some other simple way with thenew parameters; (3) the new parameters should be nearlyorthogonal in the cosmological parameter space; (4) thenew parameters should correspond to the most impor-tant independent physical effects determining the powerspectrum; (5) common theoretical prior constraints, likeflatness or standard radiation, should be simple to im-plement by fixing a single parameter. No parameter setcan satisfy all of these constraints perfectly. We use thefollowing parameters, which provide a good balance be-tween these criteria:

B ≡ Ωbh2,

V ≡ ΩΛh2,

R ≡ a∗Ωmat

Ωrad

,

M ≡(

Ω2mat + a−2

∗ Ω2rad

)1/2h2. (11)

B is proportional to the baryon-photon density and thusdetermines the baryon driving effect on the acoustic os-cillations [24, 27]. R is the matter-radiation density ratioat recombination, which determines the amount of earlyIntegrated Sachs-Wolfe effect. V determines the late-timeIntegrated Sachs-Wolfe effect arising from a late vacuum-dominated phase, but otherwise represents a nearly exactdegeneracy (sometimes called the “geometrical degener-acy”). M couples only to other small physical effects andis an approximate degeneracy direction. This choice ofparameters is not unique, but this set largely satisfies theabove criteria.

Given values for A, B, V , R, and M, they can beinverted to the corresponding cosmological parametersby rewriting the definition of A in terms of B, V , R, M,and h, then searching in h until the desired value for Ais obtained. Ωb and ΩΛ then follow immediately, whileΩmat and Ωrad can be obtained with a few iterations todetermine a precise value for a∗. A less efficient but morestraightforward method we have implemented is simplyto search the 5-dimensional cosmological parameter spacefor the correct values.

The other cosmological parameters which affect the mi-crowave background power spectrum, aside from tensorperturbations, are the reionization redshift and the am-plitude and power law index of the scalar perturbationpower spectrum. For reionization, we use the physicalparameter

Z ≡ e−2τ , (12)

the factor by which the microwave backgroundanisotropies at small scales are damped due to Comptonscattering by free electrons after the Universe is reion-ized. At large scales, the temperature fluctuations aresuppressed by a smaller amount.

The primordial perturbation amplitude cannot bemeasured directly, but only the amplitude of the mi-crowave background fluctuations. Some care must betaken in the definition of the amplitude of the microwavefluctuations so that the other physical parameters arenot significantly degenerate with a simple change in nor-malization. For example, one common normalization issome weighted average of the Cl’s over the smallest lvalues, say 2 < l < 20 [31]. This is known as “COBE-normalization” because this is the range of scales probedby COBE, which tightly constrains the normalization ofthe temperature fluctuations at these scales [11]. But al-though such a normalization is useful for interpreting theCOBE results, it is a bad normalization to adopt whenprobing a much larger range in l-space. The reason isthat the lowest multipoles have a significant contribu-tion from the Integrated Sachs-Wolfe effect: if the totalmatter density changes significantly between two models,the resulting low l multipole moments will also change.If the normalization is fixed at low l, then the result isthat the two models will be offset at high l values. As a

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concrete example, consider two models which differ onlyin V , with A, B, M and R held fixed. The only phys-ical difference between these two models is a differencein the structure growth rate at low redshifts, and in par-ticular the fluctuations at the last scattering surface areidentical. But if these two models are COBE normal-ized, then every Cl for l = 20 is offset between the twomodels. Defining the normalization as an average bandpower over a wider range of l values [10] is better but notideal. We advocate defining a normalization parameterS by the amplitude of the perturbations on the scale ofthe horizon at last scattering, corrected by the amount ofsmall-scale suppression in the Cl’s due to reionization. Inpractice, for models with the same values of n, this nor-malization essentially corresponds to the amplitude of theacoustic oscillations at l values higher than the first sev-eral peaks. Note that increasing Z while holding S andthe other physical parameters fixed keeps the microwavebackground power spectrum the same at high l while en-hancing the signal at low l. The first few peaks cannot beused for normalization because they have been subjectedto driving by the gravitational potential as they cross thehorizon [27], so their amplitudes vary significantly withB and R, as shown in the following Section.

The primordial power spectrum of density perturba-tions is generally taken to be a power law, P (k) ∝ kn,with n = 1 corresponding to the scale-invariant Harrison-Zeldovich spectrum. Making the approximation that kand l have a direct correspondence, the effect of n on themicrowave background power spectrum can be modelledas

Cl(n) = Cl(n0)

(

l

l0

)n−n0

, (13)

which is a good approximation for power law power spec-tra. A departure from a power law, characterized by α[32], can be represented in a similar way. Note that sincethe dependence is exponential, a linear extrapolation isnever a good approximation over the entire interestingrange 2 < l < 3000. The choice of l0 is arbitrary: chang-ing l0 simply gives a different overall normalization.

Finally, note that the amplitude of the tensor modepower spectrum should be used as a separate parameter,not the common choice of the tensor-scalar ratio. Thetensor modes contribute at comparatively large angularscales; they are significant only for l < 100 given cur-rent rough limits on the tensor amplitude. Varying thetensor amplitude independently automatically leaves thenormalization parameter S fixed.

In the rest of the paper, we refer to the parameters A,B, V , R, M, S, and Z as “physical parameters”, as op-posed to the usual “cosmological parameters” Ωb, Ωmat,Ωrad, Ωvac, h, Q2 (a quadrupole-based normalization),and zr.

III. POWER SPECTRA

We now present the dependence of the temperaturefluctuation power spectrum on these physical parame-ters. As a fiducial model, we choose a standard cosmol-ogy with Ω = 0.99, Λ = 0.7, Ωmat = 0.29, Ωb = 0.04,h = 0.7, n = 1, and full reionization at redshift zr = 7,with standard neutrinos. We compute power spectra forthe fiducial model and for evaluating the various numeri-cal derivatives using Seljak and Zaldarriaga’s CMBFASTcode [33, 34, 35]. Note that the model is slightly openrather than flat to compensate for numerical instabili-ties in the CMBFAST code; see the discussion in theAppendix. The corresponding physical parameters areA0 = 0.0106, B0 = 0.0196, V0 = 0.338, M0 = 0.154, andR0 = 3.252. Since the dependence of the spectrum onS is trivial, its definition is arbitrary; it can be taken,for example, as the amplitude of a particular high peak,or as the equivalent scalar perturbation amplitude AS atthe scale which the power law n is defined for a modelwith no reionization. We do not consider tensor pertur-bations in this paper, which can be analyzed separatelyby the same techniques; the tensor perturbations are sim-pler because they depend on fewer parameters than thescalar perturbations. None of the following results de-pend qualitatively on the fiducial model chosen.

The top panel of Fig. 1 displays power spectra as theparameter A varies while the other physical parametersare held fixed. It is clear that the effect of A is only todetermine the overall angular scale, except for the smallIntegrated Sachs-Wolfe effect for l < 200. For the bot-tom panel, the l-axis of each power spectrum is rescaledto give Cl as a function of lA/A0, and then the percentdifference between the rescaled model and the fiducialmodel is plotted. The small variation of the scaled powerspectrum between around l = 50 and l = 200 is due tothe early ISW effect and can be accurately modelled as alinear variation, while between l = 2 and l = 50 the vari-ation with A is more complicated due to the late-timeISW contribution. For l > 200, the scaled power spectramatch to within 1%; the residual discrepancy is likelydue to numerical inaccuracy. Since A will be tightly con-strained by the peak positions, this numerical error willnot affect the determination of A, but may contribute asystematic error to the determination of other parame-ters at the 1% level.

Figure 2 displays the power spectrum variation withB, keeping the other parameters fixed. This plot clearlyshows the well-known baryon signature of alternatingpeak enhancement and suppression. The variation of aparticular multipole Cl with B is highly linear for l > 50for variations of up to 20%; for all multipoles, the depen-dence is linear for variations of up to 10%. A figure ofmerit for a linear approximation to the power spectrumis whether the errors in the approximation are negligiblecompared to the cosmic statistical error at a given multi-pole. The bottom panel of Fig. 2 shows just how good thelinear approximation is for several selected l values. The

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FIG. 1: Temperature power spectra for the fiducial cosmological model (top panel, heavy line), plus models varying theparameter A upwards by 10% (curve shifts to lower l values) and downwards by 10% (curve shifts to higher l values) whilekeeping B, M, V, R, S and n fixed. The bottom panel displays the fractional error between the fiducial model and the othertwo models with l-axes rescaled by the factor A/A′.

horizontal axis plots the fractional change in the parame-ter B. The vertical axis plots the ratio of the change in Cl,Cl(B) − Cl(B0), to the statistical error from the cosmic

variance at that multipole, σl = Cl(B0)√

(2/(2l + 1)).(Any measurement of the multipole Cl is subject to astatistical error at least as large as this cosmic varianceerror, due to the finite number of modes sampled on the

sky.) The linear dependence remains valid even near the“pivot” points between a peak and a trough. Thus a lin-ear extrapolation using computed numerical derivativesis highly accurate at reproducing the B dependence; theerror in such an approximation appears to be dominatedby systematic numerical errors in computing the powerspectrum, and funny behavior of some of the multipoles

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FIG. 2: Temperature power spectra for the fiducial cosmological model (top panel, heavy line), plus models varying theparameter B upwards by 25% (higher first peak) and downwards by 25% (lower first peak) while keeping A, M, V, R, S , andn fixed. The bottom panel displays Cl as a function of B, for l = 50, 100, 200, 500, and 1000. The horizontal axis shows thefractional change in B, while the vertical axis gives the change in Cl as a fraction of the cosmic variance at that multipole.

for small variations in B is surely due to systematic er-rors.

Figure 3 shows the variation of the power spectrumwith respect to the parameter R, keeping the othersfixed, with the corresponding dependence of specific mul-tipoles. For smaller values of R, matter-radiation equal-ity occurs later, so the Universe is less accurately de-

scribed as matter-dominated at the time of last scatter-ing, leading to an increased amplitude of the first fewpeaks from the Integrated Sachs-Wolfe effect at earlytimes. The dependence on R is not quite as linear asfor B, but still the spectrum varies highly linearly withR for parameter variations of 10% for l > 30. Figure 4shows the same for the parameter M, which is an approx-

8

FIG. 3: Same as Fig. 2, except varying the parameter R while keeping the others fixed. Larger R increases the peak heights.The substantial glitch in the bottom panel is due to systematic errors in CMBFAST.

imate degeneracy direction: the power spectrum changesnoticably only in the neighborhood of the first peak. Ifthe standard three massless neutrino species is imposedas a prior, M is fixed.

Variation of V holding the others fixed is nearly an ex-act physical degeneracy [9, 10], broken only by the late-time ISW effect at low multipole moments displayed inFig. 5. This variation at small l values is not well approx-imated by simple linear extrapolation; accurate approx-

imation must rely on more sophisticated schemes, butthe total contribution to constraining cosmological mod-els will have comparatively little weight due to the largecosmic variance. At higher l values, the power spectrashould be precisely degenerate, and the extent to whichthey are not is a measure of numerical inaccuracies. Notethat the inaccuracies are in the form of systematic offsetsat levels smaller than a percent. When analyzing data,the variation in V can simply be set to zero for all l val-

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FIG. 4: Same as Fig. 2, except varying the parameter M while keeping the others fixed. Larger M increases the first peakheight. Again, the glitch in the bottom panel is due to systematic errors in CMBFAST.

ues higher than the first acoustic peak. If a flat universeis imposed as a prior condition, the parameter V is thenfixed.

Aside from initial conditions, reionization is the otherphysical effect which can have a significant impact on thepower spectrum. Note that the effective normalizationwe use depends on Z, so that as Z varies, the power spec-trum retains the same amplitude at high l. As discussedin the previous section, for large l the effect of chang-

ing the total optical depth is effectively just a change inthe overall amplitude. Thus for our set of parameters,variations in Z while holding the other parameters fixedchange the power spectrum only at the largest scales.For this reason, we do not consider reionization furtherhere. Note that for polarization, reionization will gener-ate a new peak in the polarization power spectrum at lowl, and this effect may require more complicated analyticmodelling than a simple linear extrapolation [36].

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FIG. 5: Varying the the parameter V, keeping the others fixed, for low l (top) and higher l (center) values; note the scales ofthe vertical axes. Only the lowest multipoles have any significant variation, arising from the Integrated Sachs-Wolfe effect atlate times; the bottom panel shows the dependence of the l = 5, 10, and 20 multipoles on V. A linear approximation is roughbut reasonable; higher-order approximations will model this dependence better. At higher l, the variation between the curvesis a measure of the numerical accuracy of the code generating the Cl curves.

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FIG. 6: A comparison of computed (solid) and approximated (dashed) power spectra for different values of the scalar index n,for n = 0.8 and n = 1.2 (dashed), where the approximated values have been calulated by applying Eq. (13) to the n = 1 model(dotted). Note the displayed variation in n is roughly three times larger than the constraint MAP will impose (see Fig. 7).

Finally, we show the effect of changing the spectral in-dex n of the primordial power spectrum in Fig. 6. Equa-tion (13) gives a very good approximation to the effectof changing n; the largest errors are at small l and inthe immediate region of the first few peaks and troughs,amounting to less than 10% of the cosmic variance forany individual multipole.

We have now displayed simple numerical or analyticapproximations for the effect of all parameters with asizeable impact on the cosmic microwave backgroundpower spectrum, for l > 50 and for parameter variationswhich are in the range to which the parameters will be re-stricted by upcoming microwave background maps. Fora model which differs from a fiducial model in severaldifferent parameters, it is in principle ambiguous in whatorder to do the various extrapolations and approxima-tions. For example, should a reionization correction beapplied before or after the A parameter extrapolation? Inpractice, however, most of the parameter ranges involvedwill be quite small, particularly for the A parameter, andthe order in which all of the approximations are appliedto the fiducial spectrum is essentially irrelevant. We testthe validity of the various approximations for arbitrary

models below.

We also note that while we have demonstrated explic-itly the validity of linear extrapolations and other simpleapproximations in computing deviations from a particu-lar fiducial power spectrum, we have not shown explic-itly that this is true for any fiducial cosmological model.But current measurements point towards a Universe well-described by a model close to the fiducial one consideredhere, and in the unlikely event that the actual cosmologyis significantly different, the various approximations aresimple to check explicitly for any underlying model.

Some of the linear extrapolations lose accuracy for thesmallest l values. It is not clear whether any simple ap-proximation is sufficient for reproducing the power spec-trum for small l values, because a variety of differenteffects contribute in varying proportions. While theselarge scales have significant cosmic variance and do nothave very much statistical weight compared to the restof the spectrum, several parameters (V , Z, T , nT ) havesignificant effects only at small l, and accurate limits onthese parameters will require approximating these mo-ments. One possible approach is to use quadratic orhigher-order extrapolations. We have not explored the

12

efficacy of such approximations, and they may require amore accurate numerical code, but promise to be a goodsolution. A more complicated approach would be to de-velop analytic fitting formulas. Steps in this directionhave been taken by Durrer and collaborators [37], buttheir solutions rapidly lose accuracy for l > 10, and a va-riety of physical effects are neglected which contribute atthe few percent level or more. Another possibility is anexpanded numerical approximation, where the effects ofour parameters are fit as linear or quadratic, but insteadof fitting the total Cl dependence, the various separateterms (i.e., quadratic combinations of Sachs-Wolfe, Inte-grated Sachs-Wolfe, doppler, and acoustic effects) con-tributing to each Cl are modelled independently. In theestimates below, we will not consider tensor perturba-tions and fix the parameter Z, while V is a degeneratedirection; we thus sidestep the issue of approximating thepower spectrum for low l, and do not discuss the issuefurther here.

IV. ERROR REGIONS FOR COSMOLOGICAL

PARAMETERS

Given some data, we want to determine the error re-gion in parameter space corresponding to some certainconfidence level in fitting the data. This must be done bylooking around in parameter space in the vicinity of thebest-fitting model. With a fast computation of the theo-retical power spectrum for different points in parameterspace in hand, a straightforward Metropolis algorithmcan be used to construct a Monte-Carlo exploration ofthe parameter space. Refined techniques with the labelof Markov Chains have recently been applied to the mi-crowave background [15, 38]. Our approximate powerspectrum evaluation renders these techniques highly effi-cient.

As a demonstration, we consider estimated power spec-trum measurement errors drawn from the χ2

2l+1 proba-bility distribution derived in Ref. [39]. We assume thatthe Universe is actually described by a particular fiducialcosmological model. With the simple assumption thatthe measurement errors for each Cl are uncorrelated, thelikelihood of a cosmological model being consistent witha measurement of the fiducial model becomes trivial tocompute. Then we map the likelihood around the fidu-cial model via a Markov Chain of points in parameterspace, starting with the fiducial model and moving to asuccession of random points in the space using a sim-ple Metropolis-Hastings algorithm. Sophisticated con-vergence tests can be applied to the chain to determinewhen the distribution of models in the Monte Carlo hasconverged to a representation of the underlying likelihoodfunction. The chain of models should be constructed inthe physical parameter space: the roughly orthogonal ef-fects of the physical parameters on the power spectrumleads to an efficient Monte Carlo sampling of the space.For each model in the chain, the cosmological parame-

ters of the model are computed and stored along withthe physical parameters.

The commonly-used Fisher matrix approximation [8]expands the likelihood function in a Taylor series aroundthe most likely model and retains only the quadraticterm. With Gaussian uncorrelated statistical errors, theFisher matrix approximation becomes exact wheneverthe parameter dependence of the Cl’s is exactly linear inall parameters. Since the power spectrum is nearly linearin most of our physical parameter set, we can check ourMonte Carlo results by comparing with the Fisher matrixlikelihood computed with the physical variables.

As a simple illustration, we determine the error regioncorresponding to a simplified model of the power spec-trum which will be obtained by the MAP satellite, cur-rently collecting data. MAP’s highest frequency channelhas a gaussian beam with a full-width-half-max size ofabout 0.21 degrees. MAP will produce a full-sky map ofthe microwave background at this resolution with a sen-sitivity of around 35 µK per 0.3 square-degree pixel. Weneglect eventual sky cuts and assume all Cl estimatesare uncorrelated. We consider models with only scalarperturbations and fixed reionization. The physical pa-rameter space is thus A, B, M, R, V and n. The otherparameters we have fixed (tensor perturbations, Z) af-fect the power spectrum only at small l, and will havelittle effect on the physical parameters considered. We as-sume the underlying cosmology is described by the fidu-cial model of the previous section. Fundamental physicalconstraints have been enforced: Ωb < Ωmat and all den-sities must be positive. Additionally, we only considermodels with Ωvac < 2 and discard any models whichare not invertible from the physical to the cosmologicalparameters (these amount to a handful of models withextreme parameter values).

Figure 7 shows the anticipated MAP error contours inthe physical parameter space, extracted from a MarkovChain of 3 × 104 models. The full 7-dimensional like-lihood region is projected onto all pairs of parameters.The parameter V is essentially unconstrained by MAP,while M has a 1-σ error of around 30%. Measurementsextended to smaller scales will not significantly improveconstraints on these two parameters: they represent truephysical degeneracies in the model space. The other pa-rameters are well constrained. Approximate 1-σ errorsare: 0.5% for A, 5% for R, 3% for B, 2% for S, and 3%for n. The error regions are elliptical, because the like-lihood region in these parameters is well-approximatedby a quadratic form. The residual correlations involvingS and n will be lifted by measurements out to higher lvalues; the parameters are largely uncorrelated in theirpower spectrum effects. Figure 8 shows a Fisher matrixestimate of the same likelihood. The two sets of likeli-hood contours are very close, as expected since we haveshown explicitly that the power spectrum varies nearlylinearly in the physical variables over a parameter re-gion consistent with MAP-quality data. The most no-table exception is the variable A, which does not give

13

FIG. 7: Parameter error contours anticipated for the MAP satellite in the physical parameter space. In each panel, the entirelikelihood region has been projected onto a particular plane. The axis refer to the fractional variation for each parameter withrespect to the fiducial one, e.g. δA = (A − Afid)/Afid. The contours show 68%, 95%, and 99% confidence levels.

a linear variation, but it is so well constrained that it iseffectively fixed when considering its impact on determin-ing the other parameters. The contours will also differslightly due to the nonlinear behavior of n, but the gen-eral excellent agreement demonstrates that our MonteCarlo technique works as expected.

In Fig. 9, we replot Fig. 7 in the cosmological parame-ter space. This procedure is trivial, since we calculate the

cosmological parameter set for each model. The resultingcontours are deformed, reflecting the non-linear relationbetween the physical and cosmological parameter sets.Clearly, a Fisher matrix approximation in the cosmolog-ical variables, as has been done in numerous analyses,will give significantly incorrect error regions [10]. Note,however, that for MAP-quality data, the Fisher matrixerrors in the physical variables projected into the cosmo-

14

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

logical variables gives a highly accurate result, Fig. 10, incontrast to the claims in Ref. [10]. The Fisher matrix ap-proximation in the physical parameter space can be usedto further increase the efficiency of error region evalua-tion, which may be useful in the cases where the compu-ation of the error regions is dominated by evaluation ofthe likelihood for a given model from actual data. If thelikelihood evaluation is at least as efficient as the powerspectrum evaluation, then the increased efficiency from a

Fisher matrix approximation may not result in dramaticincreases in computational speed, since in both cases thephysical parameter space must be sampled and parame-ter conversion performed at each point to construct thelikelihood contours in the space of the cosmological pa-rameters.

It is useful to plot error regions in both sets of param-eters. Physical parameters represent the actual physicalquantities which are being directly constrained by the

15

FIG. 9: The likelihood distribution in Fig. 7 mapped into the cosmological parameter space.

measurements, and the error regions are nearly elliptical.The cosmological parameters are useful theoretically andfor comparisons with other data sources, i.e. large-scalestructure or supernova standard candles. One advantageof determining the error region via a Monte Carlo tech-nique is that prior constraints on cosmological parame-ters can be implemented simply.

When the Fisher matrix approximation is computed inthe physical parameter space, it is highly accurate. Thisdoes not actually help much, however: if information

about cosmological parameters is desired, transformingfrom the physical to the cosmological parameter spacerequires some kind of sampling of the likelihood region.The most efficient way to do this is via Monte Carlo, sothe additional computational overhead in performing aMonte Carlo likelihood evaluation of the physical vari-able likelihood instead of a Fisher matrix approximationis negligible.

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

16

FIG. 10: The likelihood distribution in Fig. 8 mapped into the cosmological parameter space. These contours, derived from aFisher matrix approximation, are very close to those in Fig. 9.

the above Monte Carlo set of cosmological models. Wecompute each numerically using CMBFAST, and thencompare the computed and approximated models. Thetwo are compared using the statistic

χ2 =∑

l

(Capproxl − Cnum

l )2(σMAPl )−2 (14)

where (σMAPl )2 is the variance of Cl in our model MAP

data. For MAP, the measured power spectrum will

consist of approximately 800 independent multipole mo-ments, so an approximate power spectrum which differsfrom the actual model power spectrum by the MAP er-ror at each multipole will have a value for the statisticχ2 of around 800. Figure 11 shows the error distributionfor our subset of models. The statistic χ2 for each modelis plotted against the “distance” in physical parameterspace of the model from the fiducial model, where the

17

distance measure is the Fisher matrix:

d = [(P − P0)F (P − P0)]1/2

, (15)

where P is a vector of physical parameters correspond-ing to the model considered, P0 is the vector of fiducial-model parameters, and

Fij =∑

l

1(

σMAPl

)2

∂Cl

∂Pi

∂Cl

∂Pj(16)

is the Fisher matrix. This is roughly equivalent to mea-suring the distance along each axis in the physical pa-rameter space in units of the 1-σ error interval in thatparameter direction. The solid-circle points are modelswith 2 < Nν < 4, for which the worst-approximatedmodels disappear. The difference has two possible ori-gins: either linear extrapolation is not valid over a largeenough range in the variables R and M to handle thesemodels, or CMBFAST does not provide accurate compu-tations in this range. The second possibility seems morelikely, since the code contains an explicit trap prevent-ing computations for these unusual numbers of neutrinospecies.

For essentially all other models, the error averaged overall multipoles is small compared to the MAP statisticalerrors, and the error distribution is roughly independentof the distance in parameter space, showing that our nu-merical approximations are valid. Furthermore, most ofthe total error tends to come from a small number of mul-tipoles, indicating that systematic errors in CMBFASTdominate the total difference between our approxima-tions and the direct numerical calculations. Given thisfact, we forego a more thorough characterization of theerrors in our approximate spectrum computations until amore accurate code is available, but the approximationsare likely accurate enough for reliable parameter analy-sis. It is unclear at this point whether the errors in ourapproximations or the systematic errors in CMBFASTwill have a greater impact on the inferred error region.(Several other public CMB Boltzmann codes are in factavailable, but none are as general as CMBFAST, allow-ing for arbitrary curvature, vacuum energy, and neutrinospecies, all of which are required for the analysis pre-sented here.)

V. DISCUSSION AND CONCLUSIONS

Error regions similar to Fig. 9 have been previouslyconstructed for model microwave background experi-ments [10, 38]. The remarkable point about our calcu-lation is that the entire error region, constructed frompower spectra for 3× 104 points in parameter space, hasbeen computed in a few seconds of time on a desktopcomputer. In fact, since our power spectra can be com-puted with a few arithmetic operations per multipole mo-ment, the calculation time might not even be dominated

FIG. 11: A plot of the difference between approximate and nu-merical power spectra versus distance from the fiducial modelin parameter space, for 1000 cosmological models drawn ran-domly from the distribution in Fig. 7. The solid circles arefor models with 2 < Nν < 4.

by computing model power spectra, but rather by com-puting the likelihood function or by converting betweenthe cosmological and physical parameters.

For actual non-diagonal covariance matrices describingreal experiments, the computation of the likelihood willdominate the total computation time. Gupta and Heav-ens have recently formulated a method for computing anapproximate likelihood with great efficiency [40], basedon finding uncorrelated linear combinations of the powerspectrum estimates [41]. Each parameter correspondsto a unique combination, so the likelihood is calculatedwith a handful of operations. Moreover, the method isoptimized so the parameter estimation can be done withvirtually no loss of accuracy. Using this technique, like-lihood estimates for realistic covariance matrices will beroughly as efficient as our power spectrum estimates.

The conversion from the physical parameters to cos-mological ones requires evaluation of several numericalintegrals. The integrands will be smooth functions, andthis step can likely be made nearly as efficient as thepower spectrum evaluation, although we have not imple-mented an optimized routine for parameter conversion.We make the following rough timing estimate: each mul-tipole of the power spectrum can be approximated by ahandful of floating point operations, and an approximatelikelihood and the parameter conversion both will plausi-bly require similar computational work. So we anticipateon the order of 10 floating point operations per multipolefor each of 1000 multipoles. On a 1 gigaflop machine, thisresults in evaluation of around 105 models per second, soa Markov Chain with tens of thousands of models can be

18

computed in under a second. Of course, such a compu-tation is trivially parallelizable, and can easily be spedup by a factor of tens on a medium-size parallel machine.In contrast, other state-of-the-art parameter estimationtechniques [16, 17, 23], dominated by the power spectrumcalculation, compute roughly one model per second, re-quiring several hours for a 104 point computation.

This great increase in computational speed is highlyuseful. All upcoming microwave background measure-ments and resulting parameter estimates will be domi-nated by systematic errors, both from measurement er-rors when observing the sky and processing errors in thedata analysis pipeline. The only way to discern the ef-fect of these errors is through modelling them, requiringa determination of parameters numerous times. The mi-crowave background has the potential to constrain pa-rameters at the few percent level. But a variation of thissize in any given parameter will result in changes in Cl’sof a few percent, which is significantly smaller than theerrors with which each Cl will be measured. This meansthat small systematic effects distributed over many Cl’scan bias derived parameter values by amounts larger thanthe formal statistical errors on the parameter. Exhaus-tive simulation of a wide range of potential systematicswill be required before the accuracy of highly precise pa-rameter determinations can be believed, and our tech-niques make such investigations far faster and easier.

Along with increased speed, our approximation meth-ods also promise high accuracy. A careful error evalua-tion is hindered by systematic errors in CMBFAST; ournumerical approximations give better accuracy than thepredominant numerical code for parameter ranges gen-erally larger than the region which will be allowed bythe MAP data. The physical set of variables presentedhere clarify the degeneracy structure of the power spec-trum, clearly displaying degenerate and near-degeneratedirections in the parameter space. For example, with thephysical parameters presented here, it is clear that anytensor mode contribution, which affects only low-l mul-tipoles, will have negligible impact on the error contoursin Fig. 7.

Bond and Efstathiou used a principle component anal-ysis to claim that uncertainties in the tensor mode con-tributions would be the dominant source of error for allof the cosmological parmeters, and further claimed thatas a result, the Fisher matrix approximation will over-estimate the errors in other apparently well-determinedparameters such as our B [10]. These results are clearlyvalid only for data which is significantly less constrainingthan the MAP data will be. Essentially, their approxi-mate degeneracy trades off variations in R and B, whichchange the heights of the first few peaks, with variationsin ns, which also impact the first few peaks, while holdingA fixed. From our analysis, it is clear that this degen-eracy only holds approximately, since R and B have asignificant impact only on the first three peaks, while ns

affects all of the peak heights. Indeed, a careful exami-nation of Fig. 3 in [42] shows that the power spectra are

only approximately degenerate, with peak height differ-ences which are significant for MAP. The addition of ten-sor modes allows a rough match between the two powerspectra at low l values, but for high l values the discrep-ancy in peak heights is already evident at the third peakand will become larger for higher peaks, since the valuesof ns in the two models are so different.

Gravitational lensing makes a negligible difference inthe linearity of the parameter dependences. We have notincluded polarization in this analysis, but the three otherpolarized CMB power spectra can be handled in the sameway as the temperature case using standard techniques[43, 44].

We have shown that a proper choice of physical param-eters enables the microwave background power spectrumto be simply approximated with high accuracy over a sig-nificant region of parameter space. This region will belarge enough for analyses of data from the MAP satellite,although larger regions could be stitched together usingmultiple reference models. Our approximation scheme re-quires first computing numerical derivatives of the powerspectrum multipoles with respect to the various parame-ters, but this must be done only once and then computingfurther spectra is extremely fast (in the neighborhood of105 models per second or more on common computers).The error determination techniques in this paper requirethe numerical evaluation of only tens of power spectrarather than thousands or millions, so speed of the powerspectrum code will not be of paramount concern. Onthe other hand, we emphasize that for any high-precisionparameter estimate, even small systematic errors in com-puting Cl’s can lead to biased parameter estimates com-parable to the size of the statistical errors. These twoconsiderations argue for a significant revision of CMB-FAST (CMBSLOW, perhaps?) or construction of otherindependent codes which focus on overall accuracy andstability of derivatives with respect to parameters ratherthan on computational speed. Such a code, combinedwith the estimation techniques in this paper and efficientlikelihood evaluation methods [40] will provide a highlyefficient and reliable way to constrain the space of fun-damental cosmological parameters.

Acknowledgments

We thank David Spergel, who has independently im-plemented some of the ideas in this paper, for helpful dis-cussions, and Lloyd Knox, Manoj Kaplinghat, and AlanHeavens for useful comments. We also thank Uros Seljakand Matias Zaldarriaga for making public their CMB-FAST code, which has proven invaluable for this and somuch other cosmological analysis. A.K. has been sup-ported in part by NASA through grant NAG5-10110, andis a Cotrell Scholar of the Research Corporation.

19

APPENDIX A: A MODEL PIPELINE FOR

PARAMETERS ANALYSIS

In the interests of completeness, in this Appendix weoutline an analysis pipeline for cosmological parametersand their errors. Our basic data set will be a high-resolution microwave background map, from which hasbeen extracted an angular power spectrum (i.e., the Cl’s)and a covariance matrix for the Cl’s. Given this dataset, plus potentially relevant data from other sources, weneed to find the best-fit cosmological model and the errorcontours in the parameter space.

In the case of an ideal, full-sky map with only sta-tistical errors and uniform sky coverage, the covariancematrix reduces to a multiple of the identity matrix, butin general, for most experiments the coviance matrix maycontain significant off-diagonal terms. Even for the MAPsatellite, which will come close to a diagonal covariancematrix, the need to mask out the galactic plane inducessome correlations between nearby multipoles. The firststep in the analysis is to construct a likelihood calculator:what is the likelihood that the measured data representssome particular theoretical power spectrum? This is ingeneral a computationally hard problem, since it involvesinverting an N × N matrix, where N is the number ofmultipoles in the power spectrum. However, the prob-lem becomes easier if the covariance matrix is close todiagonal. Heavens and Gupta have recently formulateda method for computing an approximate likelihood withgreat efficiency [40], based on an eigenvector decompo-sition of the covariance matrix [41]. They have demon-strated a likelihood calculator which takes an input the-oretical power spectrum and outputs a likelihood basedon simulated measurements and covariances in much lessthan a second of computation time. Further efficiencyimprovements are desirable, as this is likely to be thepiece of the analysis pipeline which dominates the totaltime.

The likelihood calculator can also incorporate priorprobabilities obtained from other data sources, or fromphysical considerations like Λ > 0. Since none of the fol-lowing steps depend on the form of the likelihood func-tion, the inclusion of other kinds of data can be handledin a completely modular way.

The second element needed is a pair of routines toconvert between the physical parameter space (A, B,M, V) and the cosmological parameter space (Ωmat, Ωb,ΩΛ, h). To go from the cosmological parameters to thephysical parameters simply involves evaluating the inte-gral needed in Eq. 8. The reverse direction can be donestraightforwardly by rewriting A in terms of B, M, V ,R, and h and then numerically finding the value of hwhich gives the needed value of A. The total variationof A with h is a smooth function, and the root-findingstep can be done with a minimal number of integral eval-uations. This procedure does not guarantee a solution,however, in extreme regions of parameter space. A moregeneral (but slower) method is to search the cosmolog-

ical parameter space for the model which best fits thegiven set of physical parameters, and then project thismodel back into the physical parameter space to checkthe quality of the fit.

Once a likelihood calculator and parameter conversionroutines are in hand, we advocate the following proce-dure for parameter determination. First, choose a fidu-cial model which gives a good fit to the measured powerspectrum. This can largely be done by eye, using the pa-rameter dependences displayed in the Figures in Sec. III.Note that the fiducial model does not need to be the for-mal best-fit model, but only needs to be close enough tothe best-fit model so that models throughout the errorregion will be accurately approximated by the extrapo-lations and approximations in Sec. III.

Next, check the validity of the fiducial model. Is the fitof the model with the data adequate? It is certainly pos-sible that the simple space of cosmological models con-sidered here may not contain the actual universe. Forexample, some admixture of isocurvature initial condi-tions might be important, or the initial power spectrummight not be well described by a power law. If the best-fitmodel is not a good fit to the data, it makes no sense toproceed with any further parameter analysis! The modelspace must be enlarged or modified to include modelswhich provide a good fit to the data.

If the fiducial model is adequate, then compute nu-merical partial derivatives of each Cl with respect to B,M, and R. Also, write numerical routines to computethe power spectrum approximately for l < 50. Eventhough CMBFAST or similar codes run very efficientlywhen computing only 2 < l < 50, this time will stillgreatly dominate the total computation time for param-eter error determination, so an efficient analytic estimateis essential. Note that an alternate possibility is to sim-ply ignore the lowest multipoles, which will be a smallportion of the data for a high-resolution map, at the ex-pense of losing any leverage on the parameters V , Z, andthe tensor power spectrum. Still, the other parameterswill be determined much more precisely, so it may bepossible to fit the rest of the parameters using l > 50data only, and then constrain just the V-Z-tensor pa-rameter space from the l < 50 data independently. Nowit is possible to estimate the Cl’s highly efficiently for anymodel within a sizeable parameter space region aroundthe fiducial model.

Now perform a Markov-chain Monte Carlo [15] in thephysical parameter space. Each point in parameter spacerequires evaluation of one approximate power spectrum,one likelihood, and one conversion to cosmological pa-rameters. The Monte Carlo will converge efficiently sincethe physical parameters are nearly orthogonal in theireffects on the power spectrum. Sophisticated conver-gence diagnostics are available for determining when theMarkov chain of models has sufficiently sampled the errorregion [45].

Finally, once the chain of models in parameter spacehas been computed, extract the best-fit model and error

20

contours from the distribution of models. By displayingtwo-dimensional likelihood plots for all parameter combi-nations, a reliable picture of the shape of the likelihoodregion in the whole multidimensional parameter spacecan be visualized.

Upcoming data will be good enough that the statisticalerrors, given by these likelihood contours, will be smallerthan the systematic errors, at least in some parameterspace directions. Sources of systematic error include in-strumental effects, modelling of foreground emission, ap-proximation of the covariance matrix, inaccuracies in thetheoretical models, and inaccuracies in the approxima-tions in this paper. Before we can confidently claim tohave determined the cosmological parameters and theirerrors, these systematics must be modelled and investi-gated. Using the techniques outlined here, where a com-plete Monte Carlo can be accomplished in seconds, in-vestigation of systematics is easily practical, instead of acomputational challenge.

APPENDIX B: SOME NUMERICAL

CONSIDERATIONS WITH THE CMBFAST

CODE

Seljak and Zaldarriaga’s CMBFAST code [33, 35] hasbecome the standard tool for computing the microwavebackground temperature and polarization power spectrafor the inflation-type cosmological models discussed inthis paper. Hundreds of researchers have employed itfor tasks ranging from predictions of highly speculativecosmological models to analysis of every microwave back-ground measurement of the past few years. Its compre-hensive treatment of a variety of physical effects and itspublic availability have made it the gold standard for mi-crowave background analysis.

The original aim of the CMBFAST code was fast eval-uation of the power spectra. Previous codes had simplyevolved the complete Boltzmann hierarchy of equationsup to the maximum desired value of l. This resulted in acoupled set of around 3l linear differential equations, andthe l-values of interest might be 1000 or greater. The taskof evolving thousands of coupled equations with oscilla-tory solutions led to codes which required many hoursto run on fast computers. The CMBFAST algorithmevolves the source terms first, which involve angular mo-ments only up to l = 4; the same Boltzmann hierarchycan be used, but with a cutoff of l ≃ 10. The rest ofthe Cl’s can then be obtained by integrating this sourceterm against various Bessel functions. The Bessel func-tions can be precomputed or approximated [46] indepen-dently for each l. Great time savings result since (1) it isnot necessary to compute the power spectrum for every l,but rather only, e.g., every 50th l with the rest obtainedfrom interpolation; (2) the time steps for the Boltzmannintegration are not controlled by the need to resolve allof the oscillations in the Bessel functions, and thus theBoltzmann integrator can be significantly more efficient.

It also may be possible to employ approximations to theintegrals of the source times the Bessel functions with anadditional gain in speed, although CMBFAST does notimplement this. The code offers a speed improvementof a factor of 50 over Boltzmann hierarchy codes for flatuniverses, although the advantage is smaller for modelswhich are not flat or have a cosmological constant.

CMBFAST is nominally accurate at around the onepercent level, although no systematic error analysis hasever been published. While the numerical error in anygiven multipole Cl is relatively small, the derivatives ofCl with respect to the various cosmological parametersare significantly less accurate. At the time the code waswritten, no microwave background experiments were ofsufficient precision to warrant worrying about inaccura-cies at this level, and the great utility of the code has beenits combination of speed and accuracy. But we have nowentered the era of high-precision measurements which willplace significant constraints on many parameters simul-taneously, and as this paper has demonstrated, efficientlyprosecuting this analysis requires a code of high precisionand stable behavior with respect to parameter variations.We have found that the unmodified version of CMBFAST“out of the box” is not sufficiently accurate for the calcu-lations presented here, but these shortcomings are par-tially compensated by the following procedures.

First, CMBFAST uses slightly different algorithms forflat and non-flat background spacetime geometries. Inthe flat case, the necessary spherical Bessel functionsare precomputed and called from a file; in the nonflatcase, the hyperspherical Bessel functions are computedvia an integral representation as they are needed. Asa result, the line-of-sight integrals over the product ofsource term and Bessel function have different partitionsin the two cases, and the numerical values differ at thepercent level. Also, the code has a flag which forcesevaluation of the flat case whenever |Ω − 1| < 0.001. Ifthe fiducial model is flat, then as parameters are variedwhich change the geometry, many individual multipolesCl exhibit discontinuities at the point in parameter spacewhere the code switches from the flat to nonflat evalua-tion scheme. These discontinuities can result in signifi-cant errors when evaluating Cl derivatives with respectto parameters. Also, even if the derivative is correctlyevaluated, the offset between the two cases can result ina slightly biased error estimate.

A careful fix of this problem would involve insuringthat the integral partitions used in the two cases are de-termined consistently. Alternately, the flat-space codecould be rewritten to numerically integrate the neces-sary Bessel functions as in the non-flat case, since thetwo methods do not have a significant difference in com-putational speed. A simpler practical solution which wehave employed in this paper is to force the code always touse the non-flat integration routine. This can be accom-plished by narrowing the tolerance at which the code usesthe non-flat integration (say to |Ω−1| < 10−5), combinedwith always using a non-flat fiducial model (shifting Ωmat

21

FIG. 12: Derivatives of Cl with respect to B, calculated byCMBFAST with ∆l = 10, from 2%, 5%, and 10% finite dif-ferences in B. The smoothest curve is for the 10% variation.

by 10−5 will never result in a statistically significant dif-ference when fitting a given data set, due to cosmic vari-ance).

Second, in varying the B parameter, adjacent acousticpeaks move in the opposite direction. Between the peaks,therefore, is a pivot point l for which Cl remains fixed asB is varied, and the nearby Cl’s will vary only slightly.As CMBFAST is written, it computes Cl at l values sep-arated by 50; the complete spectrum is then obtained bysplining between the calculated values. Splines are ratherstiff, and small changes in regions where the power spec-trum has significant curvature (i.e. around the acousticpeaks and troughs) can noticeably change the overall fitto the power spectrum in the regions between. In partic-ular, the position of the “pivot” l-value in the spectrumsometimes shifts a bit, which gives a spurious irregulardependence on B of the few Cl’s in this region.

This can be addressed by a simple modification toCMBFAST forcing the explicit evaluation of Cl for morel values. We have used a δl of 10, which has eliminated

the spurious behavior. Of course, this results in a signif-icant increase in the computational time needed for anindividual model. A more sophisticated approach wouldbe to use a comparable number of l values as the originalcode, but distribute them preferentially in the l-rangescorresponding to the peaks and troughs where the cur-vature of the power spectrum is largest. This is easilyimplemented using the variables defined in this paper,since the peak positions are completely determined by Aalone. Using a finer grid in l also reproduces the peakpositions with higher accuracy.

Third, as discussed in the body of the paper, the powerspectra must be normalized according to a physical cri-terion which does not depend on non-linear behavior atsmall l values. This is accomplished by simply compilingCMBFAST with the UNNORM option. In this case, nonormalization to COBE is done; instead, the amplitudeof the potential Ψ is taken to be unity at the scale of thehorizon at the time of last scattering. This normalizationis fixed by the primordial amplitude of scalar perturba-tions, often denoted as AS . We define a physical variableS corresponding to the microwave background amplitudeby also including the effect of differing amounts of reion-ization so that the power spectrum at large l remainsfixed.

Some residual numerical problems remain with CMB-FAST. We have found, for example, that in the regionaround l = 350 (between the first and second acousticpeaks for our fiducial model), the power spectrum ex-hibits spurious oscillatory behavior with an amplitude ofa few percent, if computed directly for every l instead ofevery 10th or 50th l (see Fig. 12). This likely arises fromeither the k or η integral over an oscillatory integrand notbeing evaluated on a fine enough grid. As a result, di-rectly calculated Cl values in this region of the parameterspace effectively have non-negligible random errors. Thisproblem is normally hidden by the spline smoothing withevery 50th l computed. When the number of Cl evalu-ations is increased to every 10th l, the power spectrumshows a distinct glitch in this region, and the parameterdependence is completely unreliable here. The spline fitsare forced into more oscillations to include the computedpoints. More accurate numerical integrations will prob-ably solve this problem, at the expense of a significantspeed reduction.

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