PHYSICAL REVIEW D 73, 063528 (2006)

Reexamining the constraint on the helium abundance from the CMB

Kazuhide Ichikawa and Tomo TakahashiInstitute for Cosmic Ray Research, University of Tokyo Kashiwa 277-8582, Japan

(Received 10 January 2006; published 29 March 2006)

1550-7998=20

We revisit the constraint on the primordial helium mass fraction Yp from observations of cosmicmicrowave background (CMB) alone. By minimizing �2 of recent CMB experiments over 6 othercosmological parameters, we obtained rather weak constraints as 0:17 � Yp � 0:52 at 1� C.L. for aparticular data set. We also study the future constraint on cosmological parameters when we take accountof the prediction of the standard big bang nucleosynthesis (BBN) theory as a prior on the helium massfraction where Yp can be fixed for a given energy density of baryon. We discuss the implications of theprediction of the standard BBN on the analysis of CMB.

DOI: 10.1103/PhysRevD.73.063528 PACS numbers: 98.80.Ft, 26.35.+c, 98.70.Vc

I. INTRODUCTION

Recent precise cosmological observations such asWMAP [1] push us toward the era of so-called precisioncosmology. In particular, the combination of the data fromcosmic microwave background (CMB), large scale struc-ture, type Ia supernovae and so on can severely constrainthe cosmological parameters such as the energy density ofbaryon, cold dark matter and dark energy, the equation ofstate for dark energy, the Hubble parameter, the amplitudeand the scale dependence of primordial fluctuation.

Among the various cosmological parameters, the pri-mordial helium mass fraction Yp is the one which has beenmainly discussed in the context of big bang nucleosynthe-sis (BBN) but not that of CMB so far. One of the reason isthat the primordial helium abundance has not been consid-ered to be well constrained by observations of CMB sinceits effects on the CMB power spectrum is expected to betoo small to be measured. However, since now we havevery precise measurements of CMB, we may have a chanceto constrain the primordial helium mass fraction fromCMB observations. Since the primordial helium massfraction can affect the number density of free electron inthe course of the recombination history, the effects of Ypcan be imprinted on the power spectrum of CMB.Recently, some works along this line have been done bytwo different groups [2,3], which have discussed the con-straints on Yp from current observations of CMB. In factthey claim different bounds on the primordial helium massfraction, especially in terms of its uncertainty: the author ofRef. [2] obtained 0:160 � Yp � 0:501, on the other handthe authors of Ref. [3] got Yp � 0:250�0:010

�0:014 at 1� con-fidence level. It should be noticed that the latter bound ismuch more severe than that of the former. If the heliummass fraction is severely constrained by CMB data, itmeans that the CMB power spectrum is sensitive to thevalues of Yp. In such a case, the prior on Yp should beimportant to constrain other cosmological parameters tooand the usual fixing of Yp � 0:24 in CMB power spectrum

06=73(6)=063528(7)$23.00 063528

calculations might not be a good assumption. Especially,analyses like Refs. [4–8] predict light element abundancesincluding 4He from the baryon density which is obtainedfrom the CMB data sets with the analysis fixing the valueof Yp. Such procedure is only valid when Yp is not severelyconstrained by CMB. Thus it is very important to check theCMB bound on Yp.

One of the main purpose of the present paper is that werevisit the constraint on Yp from observations of CMBalone with a different analysis method from Markov chainMonte Carlo (MCMC) technique which is widely used forthe determination of cosmological parameters and adoptedin Refs. [2,3]. In this paper, we calculate �2 minimum as afunction of Yp and derive constraints on Yp. We adopt theBrent method of the successive parabolic interpolation tominimize �2 varying 6 other cosmological parameters ofthe �CDM model with the power-law adiabatic primordialfluctuation. We obtain the constraint on Yp by this methodand compare it with previously obtained results.

We also study the constraint on Yp from future CMBexperiment. A particular emphasis is placed on investigat-ing the role of the standard BBN theory. Since the primor-dial helium is synthesized in BBN, once the baryon-to-photon ratio is given, the value of Yp is fixed theoretically.Thus, using this relation between the baryon density andhelium abundance, we do not have to regard Yp as anindependent free parameter when we analyze CMB data.We study how the standard BBN assumption on Yp affectsthe determination of other cosmological parameters in thefuture Planck experiment using the Fisher matrix analysis.

The structure of this paper is as follows. In the nextsection, we briefly discuss the effects of the heliummass fraction on the CMB power spectrum, in particular,its effects on the change of the structure of the acousticpeaks. Then we study the constraint on the primordialhelium mass fraction from current observations ofCMB using the data from WMAP, CBI, ACBAR andBOOMERANG. In Sec. IV, we discuss the expected con-straint on Yp from future CMB observation of Planck and

-1 © 2006 The American Physical Society

KAZUHIDE ICHIKAWA AND TOMO TAKAHASHI PHYSICAL REVIEW D 73, 063528 (2006)

also study how the standard BBN assumption on Yp canaffect the constraints on cosmological parameters. Thefinal section is devoted to the summary of this paper.

II. EFFECTS OF THE CHANGE OF Yp ON CMB

In this section, we briefly discuss the effects of thechange of the helium abundance on the CMB power spec-trum. More detailed description of this issue can be foundin Ref. [2].

The main effect of Yp on the CMB power spectrumcomes from the diffusion damping at small scales. WhenYp is large, since it is easier for electrons to recombine with4He than with H, the number of free electron becomessmall. Thus the Compton mean free path becomes largerfor larger Yp, which means that the diffusion length ofphoton becomes also larger. Since the photon-baryon tightcoupling breaks down at the photon diffusion scales, thefluctuation of photon is exponentially damped due to thediffusive mixing and rescattering. Hence the CMB powerspectrum is more damped for larger values of Yp. To seethis tendency, we plot the CMB power spectra for severalvalues of Yp in Fig. 1. We clearly see that the damping atthe small scales is more significant for the cases with largervalues of Yp. The effect of diffusion damping causes thechange in the power spectrum at a percent level for 10%change of Yp [2].

To see this more quantitatively, we consider the ratio ofthe second peak height to the first which is defined as [9]

H2 �

��T�l � l2��T�l � l1�

�2; (2.1)

and the third peak height to the first

0

1000

2000

3000

4000

5000

6000

0 200 400 600 800 1000 1200 1400

l(l+

1)C

l / 2

π

multipole l

FIG. 1 (color online). The CMB power spectra for the caseswith Yp � 0:1 (green dashed line), 0.24 (red solid line) and 0.5(blue dotted line). Other cosmological parameters are taken to bethe WMAP mean values for the power-law �CDM model.

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H3 �

��T�l � l3��T�l � l1�

�2; (2.2)

where ��T�l��2 � l�l� 1�Cl=2�. We do not discuss thefirst peak position and height because they are almostunaffected by the change in Yp. We calculate the responsesof these quantities with respect to the change in the cos-mological parameters around the fiducial values, !m �0:14, !b � 0:024, �� � 0:73, � � 0:166, ns � 0:99 andYp � 0:24 where !i � �ih

2 with �i being the energydensity of component i normalized by the critical energydensity. The subscript b denotes baryon and m stands formatter which is the sum of baryon and CDM. h is theHubble parameter, � is the reionization optical depth andns is the scalar spectral index of primordial power spec-trum. These values (except for Yp) are the mean values ofWMAP for the power-law �CDM model [10]. Also, wekeep the flatness of the universe when we change parame-ters. We found

�H2 � �0:30�!b

!b� 0:015

�!m

!m� 0:41

�nsns

� 0:0125�YpYp

; (2.3)

�H3 � �0:18�!b

!b� 0:21

�!m

!m� 0:56

�nsns

� 0:029�YpYp

; (2.4)

where we neglected the dependence on �� and � sincetheir coefficients are very tiny even if compared to that ofYp. From these expressions, we can see that the response ofCl to the change in Yp is very sluggish. This is one of thereasons why we do not expect to obtain a meaningfulconstraint on Yp from CMB until recently. Moreover, thechange of Cl caused by varying Yp is readily canceled byshifting other parameters. However, since observations ofCMB now have become precise and cover wider multipolerange, we may have a chance to constrain Yp from currentobservations of CMB, which will be discussed in the nextsection.

III. CONSTRAINT ON Yp FROM CURRENT CMBOBSERVATIONS

Now we study the constraint on the helium abundancefrom current observations of CMB. For this purpose, weuse the data from WMAP [1], CBI [11] and ACBAR [12].We also include the recent data from BOOMERANGexperiment [13–15]. To calculate �2 from WMAP data,we used the code provided by WMAP [16–18]. For CBI,ACBAR and BOOMERANG, we made use of modules inCOSMOMC [19]. As mentioned in the introduction, twogroups have reported different bound on Yp using CMB

-2

0

1000

2000

3000

4000

5000

6000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

l(l+

1)C

l / 2

π [µ

K2 ]

multipole l

WMAPBOOMERANG03

ACBARCBI

FIG. 3 (color online). The CMB power spectra for the casewith Yp � 0:24, !m � 0:135, !b � 0:023, h � 0:72, � �0:117, ns � 0:96 (red solid line), Yp � 0:1, !m � 0:130, !b �

0:023, h � 0:72, � � 0:101, ns � 0:95 (green dashed line) andYp � 0:5, !m � 0:148, !b � 0:023, h � 0:72, � � 0:117, ns �0:99 (blue dotted line). Notice that these power spectra arealmost indistinguishable up to multipole region l� 1000.

REEXAMINING THE CONSTRAINT ON THE HELIUM . . . PHYSICAL REVIEW D 73, 063528 (2006)

data alone, especially in terms of its uncertainties. Onegroup has obtained 0:160 � Yp � 0:501 [2] at 1� C.L., onthe other hand the authors of Ref. [3] give the bound asYp � 0:250�0:010

�0:014. The authors of Refs. [2,3] use the CMBdata which cover the similar multipole region as that ofours. For details of their analysis, we refer the reader toRefs. [2,3]. If the severe bound on Yp is obtained fromcurrent CMB data, it means that the CMB power spectrumis sensitive to the value of Yp and the prior on Yp wouldaffect the constraints on other cosmological parameters.Thus it is important to check the bound independently.

For the analysis in this paper, we adopted a �2 minimi-zation by nested grid search instead of Markov chainMonte Carlo (MCMC) method which was used in theiranalysis. In our analysis, we apply the Brent method [20] ofthe successive parabolic interpolation to find a minimumwith respect to one specific parameter with other parame-ters at a given grid, then we iteratively repeat the procedureto find the global minimum. For the detailed description ofthis method, we refer the readers to Ref. [21]. Here weassume a flat universe and the cosmological constant fordark energy. We also assume no contribution from gravitywave. In Fig. 2, we show the values of �2 minimum as afunction of Yp. As seen from the figure, we do not have asevere constraint from current observations of CMB, whichsupports the result of Ref. [2]. Reading Yp values whichgive ��2 � 1, we obtained the constraint at 1� C.L. as0:17 � Yp � 0:52 for the case where the data fromWMAP, CBI and ACBAR are used. When the data fromBOOMERANG is added, we got 0:25 � Yp � 0:54. Wehave also made the analysis for different data sets forcomparison. In fact, we cannot obtain a significant con-straint in the region 0:1< Yp < 0:6 using WMAP data

FIG. 2 (color online). The values of ��2 are shown as afunction of Yp. Other cosmological parameters are taken tominimize �2. The red solid line is for the data of WMAP, CBIand ACBAR. The green dashed line includes BOOMERANGdata in addition.

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alone. Even if we add the data from BOOMERANG, wecannot constrain the value of Yp. Thus the data from CBIand ACBAR which cover high multipole regions are im-portant to constrain Yp although the constraint is ratherweak.

As discussed in the previous section, the CMB powerspectrum can be affected by changing the value of Yp.However this change can be canceled by tuning othercosmological parameters to give almost the same CMBpower spectra. To see this clearly, in Fig. 3, we show theCMB power spectra for several values of Yp with othercosmological parameters being chosen to give almost in-distinguishable angular power spectra. As seen from thefigure, even if we take much larger or smaller values of Ypthan usually assumed, we can fit such values of Yp to thedata by tuning other cosmological parameters.

When we include the data from BOOMERANG, thefavored values of Yp are shifted to larger Yp. Notice thathigher multipoles are more suppressed by increasing Yp,which is almost the same effect as decreasing ns.

1 SinceBOOMERANG data favors red-tilted initial power spec-trum compared to other data such as WMAP [22], it isreasonable that larger values of Yp are favored byBOOMERANG. Particularly, Yp � 0:24 which is used inusual analysis is just out of the 1� bound. However, it is

1Equations (2.3) and (2.4) show these properties quantitatively.We remark that they are also useful to understand the tendencythat the values of ns which give the minimum �2 for fixed Ypbecome larger as we increase Yp. This is because the suppressionof higher multipoles caused by increasing Yp can be compen-sated by increasing ns.

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KAZUHIDE ICHIKAWA AND TOMO TAKAHASHI PHYSICAL REVIEW D 73, 063528 (2006)

allowed at 2� C.L. so we do not take this as a seriousdiscrepancy from the standard assumption.

2The negative values of �N can also arise in a scenario withlow reheating temperature Tref �O �MeV�. However, in thiskind of scenario, the neutrino distribution functions are deviatedfrom the thermal ones so the primordial helium abundance ismodified in a way that the fitting formula Eq. (4.5) does notapply [26–29]. The most recent Yp calculation in the lowreheating scenario including the effects of neutrino oscillationsare given in Ref. [29].

IV. CONSTRAINT ON Yp FROM FUTURE CMBOBSERVATIONS AND THE ROLE OF STANDARD

BBN THEORY

In this section, we discuss the future constraint on theprimordial helium mass fraction and other cosmologicalparameters. We especially want to investigate how theconstraints are modified when we take account of therelation between !b and Yp fixed by the standard BBNtheory. As mentioned in the introduction, the CMB anisot-ropies can be measured more precisely in the future, thusthe primordial helium mass fraction may well be deter-mined from CMB observations alone. The future con-straints on Yp has already been investigated in Ref. [2]using the expected WMAP 4 yr data, Planck and cosmicvariance limited experiments. Here we study this issuesupplementing the consideration regarding the predictionof the BBN theory.

When we only consider observations of CMB alone, theprimordial helium mass fraction Yp can be viewed as oneof free independent parameters. However, when we takeaccount of the BBN theory, Yp is not an independentparameter any more but is related to the value of thebaryon-to-photon ratio or the energy density of baryon.Below, we discuss how such relation derived from the BBNcalculation affects the determination of cosmological pa-rameters in the future Planck experiment.

First we give the relation between Yp and the baryon-to-photon ratio � from the calculation of the standard BBN.�and the baryon density !b are related as 1010� �273:49!b. Some groups have reported the fitting formulafor Yp as a function of � [8,23–25]. Here we adopt thefitting formula given in Ref. [8]

10Yp ��X8

n�1

anxn�1 �

X8

n�1

bnxn�1�N

�X8

n�1

cnxn�1��N�2 �

X8

n�1

dnxn�1��N�3

�

exp�X6

n�1

enxn�; (4.5)

where x � log10�1010��, the coefficients an, bn, cn, dn anden are given in Ref. [8] and �N represents the number ofeffective degrees of freedom of extra relativistic particlespecies. The standard BBN case is obtained with �N � 0.According to Ref. [8], the accuracy of this formula is betterthan 0.05% for the range of 5:48 10�10 <�< 7:1210�10 (0:02<!b < 0:026) which corresponds to the 3�range obtained from WMAP and �3< �N < 3. Sincescenarios with �N � 0 have been discussed in the litera-ture including the possibility of negative �N such as dark

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radiation in brane world scenario, varying gravitationalconstant and so on,2 we consider two cases when wediscuss the future constraints. The first one is the case ofthe standard BBN, in other words, we assume that theenergy density of extra radiation component as a fixedparameter with �N � 0. For the other case, we treat �Nas a usual cosmological parameter which we vary, namely,we assume �N � 0. In this case, we use Eq. (4.5) to obtainYp for given !b and �N.

Now we discuss the expected constraint from futureCMB observations. For this purpose, we adopt the Fishermatrix method. Thus first we briefly review the Fishermatrix analysis which is widely used in the literature tostudy the future constraints on cosmological parameters.Detailed descriptions of this analysis method can be foundin Refs. [30–33]. For the CMB data, the Fisher matrix canbe written as

Fij �Xl

XX;Y

@CXl@xi

Cov�1�CXl ; CYl �@CYl@xj

; (4.6)

where X; Y � TT; TE; EE, xi represents a cosmologicalparameter and Cov�CXl ; C

Yl � is the covariance matrix of

the estimator of the corresponding CMB power spectrumwhich is given explicitly in Ref. [31]. The 1� uncertainty

can be estimated as����������������F�1�ii

pfor a cosmological parameter

xi. For the fiducial model, we assumed the cosmologicalparameters as A � 0:86, !m � 0:14, !b � 0:024, �� �0:73, � � 0:166 and ns � 0:99. Here, A represents theamplitude of scalar perturbation whose normalization istaken to be same as that of WMAP team [10]. The fiducialvalue for Yp is fixed using the BBN relation Eq. (4.5) for agiven!b and �N unless otherwise stated. A flat universe isassumed and we do not consider the contribution from thetensor mode. For dark energy, we assumed the cosmologi-cal constant. To forecast uncertainties, we use the futuredata from Planck [34] whose expected instrumental speci-fications can be found in Ref. [33].

Now we show our results. First we present the case withYp being treated as a free parameter, namely, we do notconsider the BBN relation. In Fig. 4, we show the expected1� contour in the !b vs Yp plane. Other cosmologicalparameters are marginalized. We also draw the band for Ypas a function of !b from theoretical calculation of thestandard BBN with 1� error due to the uncertainties inthe reaction rates. The uncertainty is dominated by that ofthe neutron lifetime, which is very small. As is clearly seen

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FIG. 4 (color online). Expected contour of the 1� constraintfrom Planck experiment. We take Yp as an independent freeparameter. Other cosmological parameters are marginalized overin this figure. The thin and nearly horizontal band is the theo-retical BBN calculation of Yp as a function of !b with its widthrepresenting 1� error from reaction rates. Here we assumed�N � 0.

REEXAMINING THE CONSTRAINT ON THE HELIUM . . . PHYSICAL REVIEW D 73, 063528 (2006)

from the figure, the constraint on Yp from Planck is notsignificant compared to the uncertainties of the standardBBN calculation. Hence as far as we take account of thestandard BBN, the helium mass fraction can be fixed usingthe BBN relation of Eq. (4.5) even for the precision mea-surements of CMB such as Planck.

Next we discuss how the theoretical BBN relation canaffect the determinations of cosmological parameters. Firstwe consider the case with �N � 0. When we determinethe value of Yp for a given !b using Eq. (4.5) (to be morespecific, when we calculate numerical derivatives withrespect to !b, we simultaneously varied Yp followingEq. (4.5)) we can expect that the uncertainties of othercosmological parameters are reduced to some extent. InTables I and II, we show the uncertainties of cosmologicalparameters from the future Planck experiment using theinformation of TT spectrum alone and that including po-larization spectrum, respectively. The first and second rowsin the tables correspond to the case without and with the

TABLE I. Expected 1� uncertainties from Planck experiment usinvalues used in the analysis.

A !b !m

w/o BBN rel. ��N � 0� 0.132 0.00068 0.0033w/ BBN rel. ��N � 0� 0.071 0.00022 0.0018w/o BBN rel. ��N � 0� 0.164 0.00098 0.0041w/ BBN rel. ��N � 0� 0.120 0.00074 0.0031

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BBN relation. For these cases, we assumed �N � 0.Furthermore, we also show the cases with �N � 0 in thethird and forth row in the tables.

Now we discuss the cases with �N � 0. As seen fromthe tables, when we assume the BBN relation, the uncer-tainties become smaller by a factor of O�1� compared tothat for the case with Yp being an independent free pa-rameter. The parameter which receives the benefit most isns. This is consistent with the fact that this parameter is themost degenerate parameter with Yp. Meanwhile, we notethat the value of Yp fixed by the BBN relation for !b �

0:024 is slightly different from Yp � 0:24 which is usuallyused in the literature. We also evaluated the uncertaintiesfixing the helium mass fraction as Yp � 0:24 independentof !b and checked that the 1� errors are quite similar tothose for the case with Yp being related to !b by Eq. (4.5).Since the change of Cl with respect to that of Yp is verysmall, in other words the derivative of Cl with respect to Ypis very small, we can have almost the same result even ifwe use a slightly different value for Yp. Thus we can just fixthe value of Yp instead of using Eq. (4.5) even for the futureCMB experiments such as Planck.

Here the discussion for the case with �N � 0 is in order.Here we treat �N as one of the cosmological parameterswhich should be varied. Notice that the addition of an extraradiation component can affect the CMB power spectrumthrough the speed up of the Hubble expansion and the earlyISW effect because of the change of the radiation-matterequality epoch. In fact, some authors have discussed thefuture constraints on cosmological parameters paying at-tention to �N [35,36] but without considering the effect ofYp. Here we study this issue allowing Yp to vary and alsoinvestigate the implications of the BBN relation on thefuture constraints on them. For the purpose of the Fishermatrix analysis, we assumed �N � 0 as the fiducial value.As already discussed, Yp, !b and �N are related byEq. (4.5) from the BBN calculation. In Fig. 5, we showthe expected 1� contour in the !b vs �N plane fromPlanck experiment for the cases with and without assumingthe BBN relation. Naturally, when we assume the BBNrelation, the uncertainties become smaller because itreduces the number of independent parameters. Also noticethat, since the BBN theory relates !b with �N, thecontour shrinks to the direction of correlation between!b and �N

g the temperature fluctuation alone. See the text for the fiducial

�� � ns Yp �N

0.025 0.067 0.025 0.030

0.010 0.036 0.005

0.041 0.086 0.038 0.031 0.410.035 0.065 0.029 0.41

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TABLE II. Expected 1� uncertainties from Planck experiment using both the temperature and polarization data. See the text for thefiducial values used in the analysis.

A !b !m �� � ns Yp �N

w/o BBN rel. ��N � 0� 0.0106 0.00024 0.0012 0.0081 0.0054 0.0075 0.011

w/ BBN rel. ��N � 0� 0.0097 0.00015 0.0012 0.0067 0.0052 0.0035

w/o BBN rel. ��N � 0� 0.0107 0.00025 0.0029 0.0090 0.0055 0.0082 0.014 0.19w/ BBN rel. ��N � 0� 0.0099 0.00023 0.0020 0.0091 0.0054 0.0078 0.14

KAZUHIDE ICHIKAWA AND TOMO TAKAHASHI PHYSICAL REVIEW D 73, 063528 (2006)

In the third and forth rows of Tables I and II, theuncertainties of cosmological parameters are also shownfor the cases with and without the BBN relation beingimposed among Yp, !b and �N respectively. As expected,the uncertainties for cosmological parameters for the casewith the BBN relation are smaller, however the differencesare not so large.

Here we comment on the implications of the BBNrelation on the constraints from current observations ofCMB. We have also made the analysis adopting the BBNrelation to fix the value of Yp for given !b. The constrainton !b for this case is almost unchanged compared to thecase with Yp � 0:24 being fixed. This can be more or lessexpected from the result we have shown in the previoussection. This again shows that current observations ofCMB are not so sensitive to the values of Yp. Thus predict-ing the helium abundance by the BBN theory using theCMB value of !b, namely, the procedure adopted inRefs. [4–8], is valid at least with the current quality ofthe CMB data.

FIG. 5 (color online). Expected contours of 1� constraintsfrom Planck experiment for the cases without the BBN relation(solid red line) and with it (dashed green line). We marginalizedover other cosmological parameters in this figure. Yp is deter-mined by the BBN relation.

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V. SUMMARY

We revisited the constraint on the primordial heliummass fraction Yp from current observations of CMB.Some authors have already studied the constraint [2,3],however their results were different especially in terms ofthe uncertainty. One of the main purpose of the presentpaper is to study the constraints on Yp from current obser-vations adopting a different analysis method. Instead ofMCMC method which was adopted by the authors ofRefs. [2,3] to obtain the constraint, here we adopted a �2

minimization by a nested grid search. We did not obtain asevere constraint in agreement with Ref. [2]. Using thedata from WMAP, CBI and ACBAR as well as recentBOOMERANG data, we get 1� constraint as 0:25 � Yp �0:54 and 0:17 � Yp � 0:52 for the cases with and withoutthe data from BOOMERANG. It might be interesting tonote that usual assumption of Yp � 0:24 is not in the 1�error range of BOOMERANG combined analysis but it isnot of high significance at this stage so we can safelyassume Yp � 0:24 for current CMB data analysis.

We also studied the future constraint from CMB on Yptaking account of the standard BBN prediction as a prior onYp. Although we cannot obtain a severe constraint atpresent, observations of CMB can be much more precisein the future. Thus we may have a chance to obtain aprecise measurement of Yp from upcoming CMB experi-ments. On the other hand, since the primordial helium hasbeen formed during the time of BBN, once the baryon-to-photon ratio is given, the value of Yp can be evaluatedtheoretically assuming the standard BBN. Thus, in thiscase, we do not have to assume Yp as an independentfree parameter when we analyze CMB data. We studiedhow such BBN theory prior on Yp affects the determinationof other cosmological parameters in the future Planckexperiment. We evaluated the uncertainties for the casewith Yp being an independent free parameter and Yp beingfixed for a given !b using the BBN relation. We showedthat the BBN prior improves the constraints on othercosmological parameters by a factor of O�1� and also itinduces some correlations among the parameters whichappear in the BBN relation. As shown in Fig. 4, as far aswe consider the standard scenario of cosmology, the he-lium mass fraction can be fixed for CMB analysis even inthe future experiments since we can expect that the con-

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REEXAMINING THE CONSTRAINT ON THE HELIUM . . . PHYSICAL REVIEW D 73, 063528 (2006)

straint from Planck is much weaker than the uncertainty ofthe theoretical calculation of the standard BBN. However,it is worthwhile to do CMB analysis treating Yp as a freeparameter and measure the helium mass fraction indepen-dently from the baryon density since it provides a consis-tency test for the standard BBN theory (of course,measurements of primordial light element abundances byastrophysical means provide further consistency tests). Bychecking the robustness of the consistency from various

063528

observations, the golden age of precision cosmology canpush us toward the accurate understanding of the universe.

ACKNOWLEDGMENTS

We acknowledge the use of CMBFAST [37] package forour numerical calculations. The work of T. T. is supportedby Japan Society for the Promotion of Science.

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