Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory
Title Multiple Peaks in the Angular Power Spectrum of the Cosmic Microwave Background: Significance and Consequences for Cosmology
Permalink https://escholarship.org/uc/item/8c5997k0
Authors de Bernardis, P. Ade, P.A.R. Bock, J.J. et al.
Publication Date 2001-05-17
eScholarship.org Powered by the California Digital Library University of California arXiv:astro-ph/0105296 v1 17 May 2001 .d Bernardis de P. ..Netterfield C.B. utpePasi h nua oe pcrmo h Cosmic the of Spectrum Power Angular the in Peaks Multiple .Martinis L. 13 ..Contaldi C.R. ftesrn orlto ihteotcldepth. optical the with correlation strong the of ujc headings: Subject akrudfo h OMRN xeiet at experiment, BOOMERANG the from background and .Hivon E. ftebscprdg scret ets h outeso u eut ycmaigBayesian comparing by pipeline, results computational Ω independent our agreement: an of excellent sec- using find a robustness found space, and to the parameter be applied techniques cosmological test should maximization 7-dimensional We likelihood dips ond with the and space correct. with this peaks agreement on is techniques few good marginalization paradigm next gives basic the which the where models, amplitudes if and such forecast locations of inflation- and dip and space adiabatic estimates, peak parameter the model-independent the 7-dimensional with of variance large consistent and a are mean the These in calculate also significant amplitude. We statistically and are model. location ary features five their all measure that we find and we analyses, model-independent Using tively. irwv akrud infiac n osqecsfor Consequences and Significance Background: Microwave ula n srpyisLbrtr,Uiest fOfr,KbeRa,Ofr,O R,UK 3RH, OX Oxford, Road, Keble Oxford, of University Laboratory, Astrophysics and Nuclear he ek n w ishv endtce ntepwrsetu ftecsi microwave cosmic the of spectrum power the in detected been have dips two and peaks Three 10 n 5 s eateto srnm,SaeSine a n etrfrPril Astrophysics, Particle for Center and Lab Sciences Space Astronomy, of Department ainlEeg eerhSinicCmuigCne,LN,Bree,C,USA CA, Berkeley, LBNL, Center, Computing Scientific Research Energy National 4 =0 aainIsiuefrTertclAtohsc,Uiest fTrno Canada Toronto, of University Astrophysics, Theoretical for Institute Canadian 9 . 96 ,V.V.Hristov 11 1 ...Ade P.A.R. , 14 − +0 ,S.Masi 14 7 0 9 . . 4 1 et.o hsc n srnm,Uiest fTrno Canada Toronto, of University Astronomy, and Physics of Depts. 10 09 omcMcoaeBcgon nstoy Cosmology Anisotropy, Background Microwave Cosmic ,E.Pascale et fPyis nv fClfri,SnaBraa A USA CA, Barbara, Santa California, of Univ. Physics, of Dept. .Prunet S. ..Crill B.P. , PC aionaIsiueo ehooy aaea A USA CA, Pasadena, Technology, of Institute California IPAC, iatmnod iia nvria aSpez,Rm,Italy Roma, Sapienza, La Universita’ Fisica, di Dipartimento vs. 8 12 aionaIsiueo ehooy aaea A USA CA, Pasadena, Technology, of Institute California 2 0 et fPyisadAtooy adffUniversity, Cardiff Astronomy, and Physics of Dept. 3 . 90 ue ayadWsfil olg,Lno,UK London, College, Westfield and Mary Queen 1 15 e rplinLbrtr,Psdn,C,USA CA, Pasadena, Laboratory, Propulsion Jet ,P.Mason nvriyo A ekly A970USA 94720 CA Berkeley, CA, of University ± siuoNzoaed efiia oa Italy Roma, Geofisica, di Nazionale Istituto 4 8 2 0 ,G.Romeo .Iacoangeli A. , ,J.J.Bock 8 . 6 8 h eito npioda pcrlindex spectral primordial in deviation The 08. ,G.DeTroia tot .Piacentini F. , adffC2 Y,Wls UK Wales, 3YB, CF24 Cardiff 6 =1 RECR iez,Italy Firenze, IROE-CNR, 11 . 8 NA rsai Italy Frascati, ENEA, 02 Cosmology ABSTRACT ..Mauskopf P.D. , − +0 3 0 15 ,J.R.Bond . . 06 05 ,J.E.Ruhl 1 vs. 1 ,A.H.Jaffe 1 ,P.Farese 1 .Pogosyan D. , 1 . 04 ± ∼ 0 4 . 7 ,J.Borrill 5 Ω 05, 210 12 .Scaramuzzi F. , 7 .Melchiorri A. , 10 ,K.Ganga , 540 ,W.C.Jones b h 2 4 , =0 ,G.Polenta 4 and 840 5 . ,A.Boscaleri 022 9 − +0 ,M.Giacometti 0 . . 11 n ∼ 004 003 13 8 s ,A.E.Lange 420 ,T.Montroy saconsequence a is vs. 1 .Pongetti F. , , 5,respec- 750, 0 . 019 6 ,K.Coble − +0 0 . . 005 004 8 1 7 , , , , 15 , 7 , 1. Introduction Furthermore, the new BOOMERANG spec- 2 trum gives a value for the baryon fraction Ωbh BOOMERANG has recently produced an im- that is in excellent agreement with independent C proved power spectrum of the cosmic microwave constraints from standard big bang nucleosynthe- background (CMB) temperature anisotropy rang- sis (BBN), eliminating any hint of a conflict be- ∼ ∼ ing from 100 to 1000 (Netterfield et al. tween the BBN and the CMB-derived values for 2001, hereafter B01). Three peaks are evident the baryon density (see e.g. Peebles et al. 2000). in the data, the first, at ∼ 210 confirms the re- In this paper we support the conclusions of sults of previous analysis of a small subset of the B01 by presenting two complementary analyses BOOMERANG data set (de Bernardis et al 2000, of the measured power spectrum. In Section 2 hereafter B00) as well as results from other exper- we briefly compare these latest BOOMERANG iments (see e.g. Miller et al. 1999; Mauskopf et al. results with those of B00. In Section 3, us- 2000; Hanany et al. 2000). Analysis of the bulk ing “model-independent” methods similar to that of the remaining data has improved the precision used in Knox & Page (2000), we measure the posi- of the power spectrum and extended the coverage tions and amplitudes of the peaks in the spectrum. to ∼ 1000, revealing for the first time a second We also compute the probability distribution of peak at ∼ 540, and a third at ∼ 840. the theoretical power spectra used in B01 given The results of two other experiments, released the measured C ’s to estimate the averages and simultaneously with B01, are in good agreement variances of the peaks and dips, as in Bond et al. with the BOOMERANG data. In addition to the (2000). first peak, the results from DASI (Halverson et al. In Section 4, we extract the distribution of cos- 2001) show a peak coincident with that seen in mological parameters using a different grid of the- our data near ∼ 540, and a rise in the spectrum oretical power spectra and a different method for toward high that is consistent with the leading projecting onto one and two variable likelihoods edge of the peak seen in our data near ∼ 840. than that used in B01, and show good agreement The results from MAXIMA (Lee et al. 2001) are of between the two methods. The B01 C -database lower precision, but are consistent with both the used the cosmological parameter set {Ωtot,ΩΛ, DASI and the BOOMERANG results. 2 2 Ωbh ,Ωch , ns, τC , C10},wherens is the spec- These detections are the first unambiguous con- tral index of primordial density fluctuations, Ωc firmation of the presence of acoustic oscillations and ΩΛ are the cold dark matter and vacuum en- in the primeval plasma before recombination (Pee- ergy densities in units of the critical density, and bles et al. 1970; Sunyaev & Zeldovich 1970), as ex- C10 is an overall normalization of the power spec- pected in the standard inflationary scenario (Bond trum. The alternate grid described here uses Ωb, & Efstathiou 1987). Ωm ≡ Ωc +Ωb and the Hubble parameter h in- 2 2 If the adiabatic cold dark matter (CDM) model stead of Ωbh ,Ωch and Ωtot =Ωm +ΩΛ,which with power-law initial perturbations describes our are now derived parameters. cosmogony, then the angular power spectrum of The determination of cosmological parameters the CMB temperature anisotropy is a powerful is affected by the presence of near-degeneracies tool to constrain cosmological parameters (see e.g. among them (Efstathiou & Bond 1999). In B01 Kamionkowski & Kosowsky 1999, and references and Lange et al. (2001), this is improved through therein). the use of parameter combinations which minimize In B01 a rigorous parameter extraction has the effects of these degeneracies, except for the been carried out with the methods of Lange et al. important one between Ωtot and ΩΛ. However, (2001), significantly improving the constraints ob- as long as the database is sufficiently extensive tained from previous CMB analyses (see e.g. Do- and finely gridded, one should get nearly the same delson & Knox 2000; Melchiorri et al. 1999; Lange answer. In addition to using different parameter et al. 2001; Balbi et al. 2000; Tegmark & Zaldar- choices to generate one and two dimensional like- riaga 2000; Jaffe et al. 2001; Kinney et al. 2000; lihood functions, we use likelihood maximization Bond et al. 2000; Bridle et al. 2000) on key cos- rather than marginalization (integration) over the mological parameters. other variables. Maximization could have a dif-
2 ferent (and often more conservative) response to in the same paper, and to derive a full set of cos- the presence of degeneracies in the model space. mological parameters in Lange et al. (2001). The We obtain excellent agreement between the two cosmological result of B00 - that the geometry of treatments, and also show that when we apply our the universe is nearly flat - is not altered by the methods to the DASI data, we get the same results correction, since the correction does not move sig- as Pryke et al. (2001). In Section 5 we report our nificantly the location of the first peak. That anal- conclusions. ysis was done assuming a ”medium” prior on h and 2 Ωbh . When the full parameter analysis is done, assuming weaker priors as in Lange et al. (2001), 2. The Power Spectrum: Comparison the effect of the correction is to drive Ωtot even with Previous Results closer to unity. With the new data set of B01, . +0.08 < The first results from the 1998/99 flight of Ωtot =104−0.07 using the 600 part of the new +0.06 BOOMERANG were reported one year ago by data, Ωtot =1.02−0.05 with all of it. B00. These included the first resolved images of The largest effect of the correction is on the the CMB over several percent of the sky, and a baryon density derived in Lange et al. (2001). preliminary power spectrum based on analysis of The uncorrected spectrum gives a baryon density 2 +0.006 a small fraction of the data. The power spectrum Ωbh =0.036−0.005 for the ”weak” prior case. The obtained by B01 incorporates a 14-fold increase in value derived from the B01 spectrum if only the +0.005 effective integration time, and an ∼ 1.8-fold in- data at ≤ 600 are used is 0.027−0.005. Adding crease in sky coverage. Thus, the B01 result ob- the data at higher removes degeneracies and fur- +0.004 tains higher precision both at low , where the pre- ther reduces the value to 0.022−0.003, as described cision is limited by sample variance, and at high below. , where it is limited by detector noise. Moreover, the B01 results make use of an improved pointing 3. Significance and Location of the Peaks solution that has led to qualitative improvements and Dips in our knowledge of the physical beam and our understanding of the pointing jitter, which adds In this section we investigate the significance in quadrature with the physical beam to give the of the detection of peaks and dips in our data, effective angular resolution of the measurement. and estimate their locations using both model- We now understand that the pointing solution independent, frequentist methods and model- used in B00 produced an effective beam size of dependent, Bayesian methods. (12.7±2.0) arcmin, rather than the (10±1)arcmin 3.1. Model-Independent Analyses of Peaks assumed in that analysis, due to an underestimate and Dips of the pointing jitter. The effect of underesti- mating the effective beamwidth is to suppress the 3.1.1. Does a Flat Spectrum Fit the Data? power spectrum at small angular scales. However, As a first step, we answer the question, ”How this is not the reason preventing the detection of likely is it that the measured bandpowers C are a second peak from that dataset. In fact, when CT C C the effects of the pointing jitter are corrected for, just fit by a first order polynomial = A+ B ”. Since the first peak is evident, we limit this anal- the signal-to-noise ratio of the B00 spectrum at ysis to the data bins centered at 450 ∼< ∼< 1000. >350 is still insufficient to detect the second 2 T −1 T χ C −C M C −C peak, and the data are still compatible with flat We compute =( b b ) bb ( b b ), where Mbb is the covariance matrix of the measured bandpower. In Fig. 1 we plot the spectrum of B00 T bandpowers Cb and C is an appropriate band av- corrected for the jitter underestimate and for an b erage of C . For this exercise, we used a Gaussian overall gain adjustment (+20% in C , within the distribution in the CT , but have also checked a ±20% uncertainty assigned to the absolute cali- b CT M bration in both B00 and B01). The corrected B00 lognormal in b ,with suitably transformed. We find similar answers using these two limits of spectrum is in excellent agreement with the B01 the offset-lognormal distribution for the bandpow- spectrum. ers recommended by Bond, Jaffe & Knox (2000). The data of B00 have been used to derive Ωtot
3 Table 1 Peaks and dips: location and amplitude.
2 p Cp (μK )
Features model independent no priors weak model independent no priors weak +10 +1200 +1300 +1280 Peak 1 213−13 220 ± 7 221 ± 5 5450−1100 5440−1050 5140−1030 +22 +440 +390 +390 Dip 1 416−12 413 ± 12 412 ± 7 1850−410 1640−320 1590−320 +20 +560 +620 +630 Peak 2 541−32 539 ± 14 539 ± 8 2220−520 2480−500 2420−500 +20 +550 +470 +500 Dip 2 750−750 688 ± 22 683 ± 23 1550−610 1530−360 1590−380 +12 +790 +720 +760 Peak 3 845−25 825 ± 21 822 ± 21 2090−850 2170−540 2270−570 +290 +310 Dip 3 - 1025 ± 24 1024 ± 26 - 880−220 910−230 +380 +400 Peak 4 - 1139 ± 24 1138 ± 24 - 1100−280 1130−290 +220 +240 Dip 4 - 1328 ± 31 1324 ± 33 - 570−160 610−170 +290 +310 Peak 5 - 1442 ± 30 1439 ± 32 - 680−200 730−220 +130 +130 Dip 5 - 1661 ± 37 1660 ± 36 - 300−90 320−90
Note.—Location (columns 2-4) and amplitude (columns 5-7) of peaks and dips in the power spectrum of the CMB measured by BOOMERANG. In column 2 we list the values measured by means of a parabolic fit to the bandpowers (Δ = 50). In columns 3 and 4 we list the values estimated by integrating the peak and dip properties of the theoretical spectra used in Lange et al. (2001) and B01 over the probability distribution for the database, assuming either the no prior or weak prior restrictions of those papers. In column 5,6,7 we do the same for the amplitude of the peaks. Note that Peaks 4 and 5, and Dips 3, 4, 5 are all outside the range directly measured, so they are forecasts of what is likely to emerge if the database has components that continue to describe the data well, as they do now. All the errors are at 1-σ and include the effect of gain and beam calibration uncertainties. 2-σ confidence intervals can be significantly broader than twice the 1-σ intervals reported here, as is clear from Fig. 2.
4 7000 B01 B01-1σ[beam] 6000 B01+1σ[beam] )
2 B00+1σ[gain]+jitter correction K 5000 B00 original μ (
π 4000 /2 l 3000
l(l+1)c 2000
1000
0 0 200 400 600 800 1000 multipole
Fig. 1.— The angular power spectrum of the CMB measured by BOOMERANG. The filled circles are the spectrum reported by B01; the up and down triangles indicate the systematic error in B01 resulting from the error in the beam. Three peaks are visible at multipoles ∼ 210, ∼ 540, ∼ 840. The open squares are the spectrumreportedbyB00,basedonananalysisof∼ 7% of the data analyzed in B01. The filled squares present the B00 spectrum after correction for pointing jitter and scaled by 10% (1σ) in overall calibration (see text).
We vary CA and CB to find the minimum of the 3.1.2. Peak and Dip Location and Amplitude 2 2 χ , χmin, and compute the probability of having a Likelihood Maps χ2 <χ2 . A first order polynomial is rejected in min We now describe some methods we have used to all cases. P (χ2 <χ2 ) is 95.2%, 96.5%, 94.8% for min locate the peaks and dips of the power spectrum. the multipole ranges 401-1000, 401-750, 726-1025 Results shown in Table 1 used a parabolic fit for C . respectively (using a binning of Δ =75forthe (As described below, we have also used parabolas bandpowers). These conclusions are robust with in ln C . The use of more complex functions, see respect to variations in the location and width of e.g. Knox & Page (2000), is not required by the the -ranges, as well as for variations of the beam data.) The criterion for robust peak/dip detec- FWHM allowed by the measurement error. We tion is that the mean curvature of the parabola be conclude that the features measured in the spec- above some threshold (e.g., some multiple of the trum are statistically significant at approximately rms deviation in the curvature). For the model- 2σ. independent entries of Table 1, for each amplitude Cp and location of the peak p, we found the cur- 2 vature which gave minimum χ (Cp, p). We re-
5 peat this procedure for different data ranges. The contours corresponding to Δχ2 of (2.3,6.17 and 11.8) are plotted in Fig. 2 for the ranges where this procedure provides a detection of both Cp and p. Three peaks and two dips are clearly detected by this procedure, at the multipoles and with the amplitudes reported in the table. The curvature is required to be negative for peak detection, posi- tive for dip detection. Note that zero curvature is equivalent to the flat bandpower case. Approach- ing this zero curvature limit is why the contours sometimes open up in p at ∼ 2σ and become hor- izontal at ∼ 3σ. This is a visual reiteration of the point made above, that the flat C is rejected at about 2σ. We have found that the location of the peaks is robust against variations of the gain and beam calibration inside the reported error inter- vals, 10% for the gain calibration error, producing a 20% error in the Cp,and1.4 (1σ) corresponding to 13% in the beam. The different best fit parabo- las for the peaks and dips of the BOOMERANG power spectrum are shown in Fig. 3.
3.1.3. Fisher Matrix Approach with Sliding Bands We have tried a number of other model- independent peak/dip finding algorithms on the data. For the parabolic model in for ln C ,we analyzed the quadratic form in blocks of 3, 4 or 5 bins, each bin being of width Δ = 50, de- termining the best fit and the variance about it as defined by the inverse of the likelihood curva- ture matrix (Fisher matrix). For this exercise, we adopted a lognormal distribution of the bandpow- ers (but checked for robustness by also assuming a Gaussian distribution) and marginalized over Fig. 2.— Δχ2 contours for the position and ampli- the beam uncertainty. We slid across the -space tude of the peaks and dips in the BOOMERANG CMB with our fixed bin group and estimated the sig- temperature power spectrum. The crosses in the two nificance of the peak or dip detection by the ratio χ2 upper panels correspond to the minima for the ; of the best-fit curvature to the rms deviation in χ2 . , . , . the three contours are plotted at Δ =23 6 17 11 8 it. For BOOMERANG, using three bins of width (corresponding to 68.3%, 95.4%, 99.7% confidence in- Δ = 50 gives stronger detection, but results in tervals for a gaussian likelihood in two parameters). The two upper panels refer to a Δ = 50 binning of larger error bars (as estimated from the inverse the power spectrum. The multipole ranges used for Fisher matrix). Here, we quote our 5-bin results, the parabolic fits on peak 1, dip 1, peak 2, dip 2 and and among these we use for each peak/dip the peak 3 are (76 - 375), (276 - 475), (401 - 750), (676 5-bin template which gives the largest ratio of - 875) and (726 -1025) respectively. The two lower the mean curvature to standard deviation of the panels refer to binnings with Δ = 75. While the sta- curvature. tistical significance of the detections is improved, the We find interleaved peaks and dips at = location of the features is less robust against the cen- 215±11, 431±10, 522±27, 736±21 and 837±15. tering of the bins: continuous and dashed lines refer The error bars quoted correspond to the variance to bin centers shifted by 35.
6 in peak position p given the curvature of the To test whether this works using just the data parabola fixed at the best-fit value ( if one at- at hand, we restrict ourselves to using only the tempts to marginalize over the curvature, p loses <600 data and see how well the peaks be- localization due to contribution from fits with yond are found. As expected, the features with vanishing curvature, i.e. straight lines). The am- <600 are quite compatible with those in Ta- +344 +196 plitudes of the features are 5760−324, 1890−178, ble 1 derived using all of the data: for the weak +330 +500 +900 2 2290−290, 1640−380 and 2210−640 μK , corre- prior case, the first peak, the first dip and the spondingly. When the 10% calibration uncertainty second peak for <600 are found to be at = is included, the errors are similar to those in Table 219 ± 5, 403 ± 10 and 530 ± 23, with amplitudes +1450 +440 +750 2 1, but of course the estimates move up and down 6000−1160, 1750−350 and 2700−590 μK . The sec- together in a coherent way, so here we have chosen ond dip and third peak are out of range, predicted to indicate the purely statistical error bars. The to be found at = 640 ± 49 and 789 ± 40, with +790 +1410 2 significance of the detection as estimated by the amplitudes 2080−570 and 3160−980 μK , in good distinction of the best-fit curvature from zero is agreement with the = 683 ± 23 and 822 ± 21, +500 +760 2 1.7 σ for the second peak and dip, and 2.2 σ for with amplitudes 1590−380 and 2270−570 μK ac- the third peak. tually found. If we further restrict the prior prob- abilities, the forecasted positions can move a bit: 3.2. Model-Dependent Analysis of Peaks e.g., we find the <600 cut with the weak prior and Dips gives Ωtot =1.04±0.07, suggesting a constraint to the theoretically-motivated flat universe is reason- 3.2.1. Ensemble-Averaging Method able. If we adopt as well the “large scale structure” We have also estimated peak/dip positions and prior of Lange et al. (2001), we get = 653 ± 45 +580 +990 amplitudes on a large set of prescribed C shapes and 798±41 at amplitudes 1950−450 and 2900−740 weighted by the probability that each shape has μK2; and the fourth peak for <600 is forecasted when confronted with the data. For the cases to be at = 1107±46, similar to the = 1138±24 shown in Table 1, the prescribed shapes are the forecast of the table. Our conclusion is that even elements of the C database used in Lange et al. if we restrict ourselves to <600, the forecasts are (2001) and B01. Explicitly, we compute for the very good. We expect that using all of the data to BOOMERANG+DMR data the ensemble aver- = 1000, with multiple peaks and dips, the fore- ages < ln p > and < ln Cp > and their variances, casts of the Table should be even more accurate. 2 2 < (Δ ln p) >,and< (Δ ln Cp) >, with respect to the product of the likelihood function for the 3.3. Peak and Dip Finding in the DASI parameters and their prior probability. Two cases data are shown, for the no prior and weak prior cases We have applied the sliding band procedure of of Lange et al. (2001). Note the good agreement Section 3.1.3 to the DASI data using the 3-band using this method with the model-independent re- and 4-band sliders. For the 3-band, we find the sults. (The results are nearly the same if we use first two peaks and interleaving dip at = 202±15, < p > instead of < ln p >.) When this method +530 407 ± 14 and 548 ± 10, with amplitudes 5200−480, was applied to all of the CMB data published be- +163 +270 1590 and 2760 μK2. Again, we have not fore 2001, including that of BOOMERANG and −148 −250 included the coherent 8% calibration uncertainty Maxima, a first peak location of 212 ± 7 was ob- (16% in C )intheseCp numbers. It may seems tained (Bond et al. 2000). quite incongruous that the error bars on the peak and dip positions can be so small relative to the 3.2.2. Forecasting Peaks and Dips bin size, but we emphasize that these give the er- One virtue of this procedure is that, because we ror contours in the immediate neighborhood of the know the spectral shape for all , we can forecast maximum, as described by the Fisher matrix. Just where peaks and dips will lie beyond the -region as in Fig. 2, the contours open up at levels lower we observe. Thus, Table 1 gives predictions for the than 1σ, resulting in imprecise localization at the locations and amplitudes of the subsequent (4,5) 2σ level. DASI’s detection of the second dip is 1.6σ peaks and (3,4,5) dips. in curvature by the Fisher error accounting, but it
7 6000 5000
) 4000 2 K
μ 3000 (
π 2000 /2 l 1000 0 l(l+1)c 4000 3000
2000 1000 0 200 400 600 800 1000 multipole
Fig. 3.— Best fit parabolas for the different peaks and dips of the BOOMERANG spectrum.
+260 2 is found, at = 656 ± 19 and Cd = 1650−220μK . al. 1996). Although the main effect giving rise We have also applied the prescribed shape to them is regular sound compression and rarefac- method of Section 3.2.1 to the DASI data (Halver- tion of the photon-baryon plasma at photon de- son et al. 2001). We find that the location and coupling, there are a number of influences that forecasts are quite compatible with those given in make the regularity only roughly true. Nonethe- Table 1 for BOOMERANG. For example, for the less, the “catalogue” of peak and dip positions weak prior, the first three peaks and interleaved used to construct the table could be searched to two dips are at = 216 ± 6, 401 ± 10, 523 ± 12, find best-fitting sequences and the associated cos- +1310 655 ± 30 and 794 ± 30, with amplitudes 5240−1050, mological parameters giving rise to them. Indeed +370 +590 +410 +660 2 we know that peaks and dips and only a few points 1570−300, 2410−470, 1690−330 and 2480−520 μK . These Cp now include the 8% calibration uncer- in between are enough to characterize the mor- C tainty. The forecasted third dip is at = 988 ± 34 phology of the spectra (e.g. Sigurdson & Scott +260 2 2000). However, it is clear that it is better to work with amplitude 980−200μK , compatible with the BOOMERANG result. with the full shapes to test the theories. In this section, we show that the extracted cosmological 4. Robustness of Cosmological Parame- parameters using the full shapes are robustly de- C ters termined, by comparing results for the database and Bayesian marginalization techniques used in The multiple peaks and dips are a strong pre- Lange et al. (2001) and B01 with those obtained diction of the simplest class of adiabatic inflation- using the variables and likelihood maximization ary models, and more generally of models with techniques described below and in de Bernardis et passive, coherent perturbations (e.g. Albrecht et al. (1997), Dodelson & Knox (2000), Melchiorri et
8 al. (1999) and Balbi et al. (2000). We also test our to 0.85. This is very similar to the weak prior of methods on the DASI data set. Lange et al. (2001), Jaffe et al. (2001) and B01, except the upper limit was 0.95: this results in 4.1. Extraction of Cosmological Parame- only tiny differences in the extracted parameters. ters 4.2. Cosmological Parameter Results The {Ωm, Ωb, ΩΛ,h,ns,τC ,C10} space has pa- rameters sampled as follows: Ωm =0.11, ..., 1.085, Our basic results, on method testing and cos- in steps of 0.025; Ωb =0.015, ..., 0.20, in steps mological implications, are shown in Fig. 4 and of 0.015; ΩΛ =0.0, ..., 0.975, in steps of 0.025; the associated contour plots Figs. 5 and 6. Fig. 4 h =0.25, ..., 0.95, in steps of 0.05; spectral in- clearly shows that it does not make that much dex of the primordial density perturbations ns = difference in constructing the likelihood function 0.50, ..., 1.50, in steps of 0.02, τC =0., .., 0.5, in for a target cosmological variable if we marginal- steps of 0.1. The overall amplitude C10, expressed ize (integrate) over the other cosmological param- COBE in units of C10 , is allowed to vary continu- eters or find the maximum likelihood value. Nor ously. The theoretical models are computed us- does it matter which database is used. Similar suc- ing the cmbfast program (Seljak & Zaldarriaga cess in agreement is found with other prior choices, 1996), as were those used in B01. Here, we have but for this paper we will restrict ourselves to the ignored the role gravity waves may play. We used 0.45
9 Fig. 4.— Likelihood curves derived from the BOOMERANG and COBE/DMR datasets as a function of Ωtot, 2 ns and Ωbh show the relative insensitivity to whether marginalization over the other variables is done (solid) or the maximum likelihood point in the other variables is chosen (dashed). These curves were constructed using a prior on the C -database used in B01 designed to mimic the ranges in the alternate C -database that uses {Ωb, Ωm,h} (open circles). A further weak prior, with 0.45
10 primordial fluctuations gives information about the shape of the primordial potential of the in- flaton field which drove inflation. While there is no fundamental constraint on this parameter, the simplest and least baroque models of inflation do give values that are just below unity. We obtain ns =0.90 ± 0.09 using the maximization proce- dure and the ns =0.96 ± 0.09 using the preferred marginalization method.
Fig. 5.— Constraints in the Ωm vs. ΩΛ plane from the combined BOOMERANG and COBE/DMR datasets, assuming the weak prior 0.45
11 τC = 0, the maximization procedure effectively that takes millions of C shapes with very differ- ignores τC > 0, whereas marginalization averages ent peak/dip amplitudes and locations, weighted over the allowed values of τC , accounting for the their peak and dip locations with the probability of increase from 0.90 to 0.96. (If we restrict ourselves the (normalized) shapes, and finds well-localized +.09 to τC =0,weget0.89−.08 with maximization, p and Cp for each of the peaks and dips. The +.08 0.92−.07 with marginalization.) agreement between BOOMERANG and DASI us- The values of the spectral index, the curvature ing this method is very good, and completely con- and the cosmological constant affect the shape sistent with the model-independent results. It will and the amplitude of the power spectrum of the of course be of great interest whether the forecasts 2 allowed by this model-dependent technique for the matter distribution. The bandpower σ8, giving the variance in (linear) matter fluctuations aver- positions and amplitudes of the next peaks and aged in 8h−1Mpc spheres, is often used to char- dips will be borne out by even higher resolution acterize the results of many large scale struc- observations. ture observations. In Lange et al. (2001), Jaffe The natural interpretation of this much-antici- 0.56 +.02,+.11 C et al. (2001) and B01, σ8Ωm =0.55−.02,−.08 was pated sequence of peaks and dips in is that adopted, where the combination of Gaussian and we are seeing phase-coherent pressure waves in top hat error bars were used to generate a wide the photon-baryon fluid at photon decoupling distribution “weak” large scale structure prior to at redshift ∼ 1100, expected in adiabatic mod- be imposed upon the CMB data. This compares els of structure formation, and which therefore 0.5 with σ8Ωm ∼ 0.50±0.05 estimated by Pen (1998) the simplest inflation models give. This is cer- 0.47 and a best fit value σ8Ωm ∼ 0.56 from Viana tainly the most economical interpretation, espe- & Liddle (1999). We have computed σ8 for each cially since the locations within this paradigm model in our database using the matter fluctua- correspond to widely-anticipated cosmological pa- tions power spectrum. From the likelihood analy- rameter choices, namely Ωtot ≈ 1, Ωm ∼ 1/3, 2 sis in the space (σ8, Ωm), we find the 1-sigma range ΩBh ≈ 0.02, with a nearly scale invariant spec- < < < < 0.5 ∼ σ8 ∼ 0.8for0.3 ∼ Ωm ∼ 0.7. For the de- trum, ns ∼ 1. We have shown how well deter- tailed application of the weak large scale structure mined and robust these parameter values are to prior of Lange et al. (2001), which also included changes in the C database, in using likelihood a constraint on the shape of the power spectrum, maximization or Bayesian marginalization, and to see B01. using either the BOOMERANG or DASI C sin conjunction with the DMR C s. 5. Discussion and Conclusions Of course this does not clinch the case: the de- rived cosmological parameters could be quite dif- We have analyzed the most recent results from ferent if we allowed ourselves much further free- BOOMERANG, and shown that there are a series dom beyond a single slope to characterize the pri- of 5 features, 3 peaks with 2 interleaved dips, each mordial spectrum. It could even be that the peaks of which is detected using a cosmological model in- and dips reflect early universe structures rather ∼ dependent method at 95% or better confidence. than sound wave structures. It is also possible Because BOOMERANG is able to achieve resolu- that isocurvature modes with artful enough initial ∼ ∼ tion in -space of Δ 50, with only 10% (anti)- condition choices could mimic the simple adiabatic correlation between bins, the data determine the case (e.g. Turok 1996). The detection of the asso- positions as well as the amplitudes of each of the ciated polarization peaks and dips would rule out features with reasonable precision. these more exotic possibilities. Although there are The positions and amplitudes of the features no glaring anomalies between theory and data at are consistent with the results of the DASI exper- this stage requiring a revisit of basic CMB assump- iment. A direct, cosmological model-independent tions, the improved precision in this -range that comparison of the two experiments is made diffi- more sky coverage will give, and the extension to cult by the lower -space resolution of the DASI higher that other CMB experiments will give, experiment. However, the two results can be ac- could well reveal that our forecasts are wrong. curately compared via a model-dependent analysis Even with the current data, there is certainly room
12 for other cosmological parameters not treated in Jaffe A., et al. 2001, Phys. Rev. Lett., 86, 3475 our minimalist inflation databases, and such ex- Kamionkowski M. & Kosowsky A. 1999, Ann. Rev. plorations are underway. Nucl. Part. Sci., 49, 77
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