A Measurement of Omega from the North American Test Flight Of

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arXiv:astro-ph/9911445v2 27 Nov 1999 rpittpstuigL using typeset Preprint odfnto,psil airto netite n the and uncertainties likeli- calibration the possible of function, non-Gaussianity the 68% hood performed the incorporating was at analysis DK99, thorough data in that more CMB A found with level. consistent author confidence was the universe where flat con- (1999) a been is Tegmark has universe parameters by the of sidered range that extensive CMB most the The from flat. evidence strong now et is Ganga DK99, the on experi- and now 1997). from the 1999), Knox, al. in & uncertainties (Dodelson al. ments systematic (Lineweaver et of Melchiorri parameters incorporation at 1999, possible attempts Tegmark of 1999), 1998, range al et the Knox Bartlett enlarging & BJK98, Jaffe methods hereafter (Bond, improved cosmologi- 1998, Estimation to the lead Likelihood and has Maximum universe activity of the This of constant. density cal quantities energy sophistica- fundamental the to such increasing measure as to lead with attempt an has datasets in tion parameters existing Back- of cosmological Microwave analyses to Cosmic (CMB) the the of of ground prop- sensitivity (e.g. spectrum The scale power precision angular unprecedented 1999). large Kosowsky with the & universe Kamionkowski measuring our cosmologists of of presented has erties possibility years few the past with the in data ical ihntecaso daai nainr oesthere models inflationary adiabatic of class the Within astronom- of quality the in improvement dramatic The h ls fCl akMte oes efidta h vrl fract 0 overall the be that to find we constrained models, is Matter Dark Ω, Cold of class the the of flight test American esrmn n h ihrdhf uenvedt eoti new headings: obtain Subject constant. we cosmological data the supernovae and redshift density high the and measurement eueteaglrpwrsetu fteCsi irwv Backgr Microwave Cosmic the of spectrum power angular the use We ESRMN FΩFO H OT MRCNTS LGTOF FLIGHT TEST AMERICAN NORTH THE FROM Ω OF MEASUREMENT A 13 Boscaleri .d Gasperis de G. hsqeCrucliee omlge olg eFac,1 France, de College Cosmologie, et Corpusculaire Physique ..Mauskopf P.D. .Melchiorri A. 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Physics, of Dept. 2 . iatmnod iia nvria aSpez,Rm,Ita Roma, Sapienza, La Universita’ Fisica, di Dipartimento iatmnod iia nvria o egt,Rm,Ita Roma, Vergata, Tor Universita’ Fisica, di Dipartimento , 85 Boomerang 4 9 ...Ade P.A.R. , aionaIsiueo ehooy aaea A USA CA, Pasadena, Technology, of Institute California 4 3 5 ≤ 12 .D Troia De G. , Romeo ue ayadWsfil olg,Lno,UK London, College, Westfield and Mary Queen 16 e rplinLbrtr,Psdn,C,USA CA, Pasadena, Laboratory, Propulsion Jet hoyDvso,CR,Gnv,Switzerland Geneva, CERN, Division, Theory Ω siuoNzoaed efiia oa Italy Roma, Geofisica, di Nazionale Istituto 11 ≤ 1 , 16 15 ETA S,Lso,Portugal Lisbon, IST, CENTRA, 1 8 ..Ruhl J.E. , ..Netterfield C.B. , 1 . xeiet ocntantegoer fteuies.Within universe. the of geometry the constrain to experiment, 5a h 8 ofiec ee.Cmie ihteCOBE the with Combined level. confidence 68% the at 25 ..Hristov V.V. , RECR iez,Italy Firenze, IROE-CNR, 3 .d Bernardis de P. , ABSTRACT 1 .Farese P. , 5 France 05, 1 10 n .Vittorio N. and twt te omlgcldt ooti e constraint combine new constant. and a cosmological obtain findings to the our data on cosmological discuss other we with 4 it section in Finally rcinleeg est fteuies,Ω=Ω the = on Ω obtain universe, we the constraints of explored, the density have and energy we use space fractional sec- we parameter In the method out the obtained. spell spectrum we power 3 charac- tion angular the the and of undertaken teristics analysis the data describe briefly the we experiment, 2 models section In inflationary adiabatic universe. of family set the constraints within primary data our this obtain of We flight from 1999). test al. 1997 et Mauskopf the from spectrum the power angular the fvle o h rcinldniyo atr Ω matter, of range density allowed fractional the the parame- search for for a for space values perform methods parameter of we the BJK98, cosmological in Using in described CMB. estimation the ter from verse hs rvosaaye eersrce otecaso open All of class models. the flat to Universe. and restricted Sitter were analyses Einsten-de previous com- these contours the likelihood encompass 68% the fortably again, data: recent most omlgclcntn,Ω constant, cosmological 4 nthis In .H Jaffe H. A. , 10 lc aclnBrhlt 53 ai Cedex Paris 75231 Berthelot, Marcelin place 1 Boomerang .G Ferreira G. P. , 15 1 .Pascale E. , ..Bock J.J. , letter ,Bree,C,USA CA, Berkeley, a, oa nrydniyo h universe, the of density energy ional osrit ntefatoa matter fractional the on constraints ,Ahrt A USA MA, Amherst, s, epeetfrhreiec o a uni- flat a for evidence further present we ekly A USA CA, Berkeley, , ud esrddrn h North the during measured ound, 6 2 ..Lange A.E. , 4 xeiet(e h opno paper companion the (see experiment , 8 5 alone .Piacentini F. , .Borrill J. , 9 land ly ly , ada 11 A , 12 Λ n n opligevidence, compelling find and ie h eetetmt of estimate recent the given , .Ganga K. , ns power ents, 4 .Masi S. , 6 Boomerang , 7 A. , 1 G. , 4 , 13 1 , , Boomerang M o flat a for M Ω + and , Λ . 2

2. THE DATA pling. The amplitude and shape of the power spectrum is primarily sensitive to the overall curvature of the uni- The data we use here are from a North American test flight of Boomerang (Boomerang/NA), a balloon- verse, Ω (Doroskevich, Zeldovich, & Sunyaev 1978); other n borne telescope designed to map CMB anisotropies from parameters such as the scalar spectral index, S, the fractional energy density in baryons, Ω , the cosmological a long-duration, balloon-borne (LDB) flight above the B constant, ΩΛ, and the Hubble constant, H0 ≡ 100h km Antarctic. A detailed description of the instrument can −1 be found in Masi et al. 1999. A description of the data sec , will also affect the height of the peak and there- fore some “cosmic confusion” will arise if we attempt in- and observations, with a discussion of calibrations, systematic effects and signal reconstruction can be found in dividual constraints on each of the parameters (Bond et al 1994). In our analysis we shall restrict ourselves to the Mauskopf et al. 1999. This test flight produced maps of family of adiabatic, CDM models. This involves consid- the CMB with more than 200 square degrees of sky coverage at frequencies of 90 and 150 GHz with resolutions of erable theoretical predjudice in the set of parameters we choose to vary although, as the presence of an acoustic 26 arcmins FWHM and 16.6 arcmins FWHM respectively. peak at ℓ ∼ 200 becomes more certain, the assumption The size of the Boomerang/NA 150 GHz map (23,561, 6′ pixels) required new methods of analysis able to incor- that structure was seeded by primordial adiabatic pertur- bations becomes more compelling (Liddle 1995; however, porate the effects of correlated noise and new implementa- counterexamples exist, Turok 1997, Durrer & Sakellari- tions capable of processing large data sets. The pixelized map and angular power spectrum were produced using the adou 1997, Hu 1999). We should, in principle, consider an 11-dimensional MADCAP software package of Borrill (1999a, 1999b) (see space of parameters; sensible priors due to previous con- http://cfpa.berkeley.edu/∼borrill/cmb/madcap.html) on Boomerang the Cray T3E-900 at NERSC and the Cray T3E-1200 at straints and the spectral coverage of the /NA angular power spectrum reduce the space to 6 dimensions. CINECA. In particular, we assume τc = 0 (lacking convincing evidence for high redshift reionization), we assume a neglige- able contribution of gravitational waves (as predicted in the standard scenario), and we discard the weak effect due to massive neutrinos. The remaining parameters to vary are ΩCDM , ΩΛ, ΩB, h, nS and the amplitude of fluctu- COBE 2 ations, C10, in units of C10 . The combination ΩBh is constrained by primordial nucleosynthesis arguments: 2 0.013 ≤ ΩBh ≤ 0.025, while we set 0.5 ≤ h ≤ 0.8. For the spectral index of the primordial scalar fluctuations we make the choice 0.8 ≤ nS ≤ 1.3 and we let a 20% variation in C10. As our main goal is to obtain constraints in the (ΩM = ΩCDM +ΩB, ΩΛ) plane, we let these parameters vary in the range [0.05, 2] × [0, 1]. Proceeding as in DK99, we attribute a likelihood to a point on this plane by find- ing the remaining four, “nuisance”, parameters that maxi- mize it. The reasons for applying this method are twofold. First, if the likelihood were a multivariate Gaussian in all the parameters, maximizing with regards to the nuisance parameters corresponds to marginalizing over them. Sec- ond, if we define our 68%, 95% and 99% contours where the likelihood falls to 0.32, 0.05 and 0.01 of its peak value FIG. 1.— The Likelihood function of each of the eight band (as would be the case for a two dimensional Multivariate powers, ℓeff = (58, 102, 153, 204, 255, 305, 403, 729), reported in Gaussian), then the constraints we obtain are conserva- Mauskopf et al. 1999 computed using the offset lognormal tive relative to any other hypersurface we may choose in ansatz of BJK98. parameter space in the sense that they rule out a smaller The angular power spectrum, Cℓ, resulting from the range of parameter space than other usual choices. analysis of the 150 GHz map was estimated in eight bins The likelihood function for the estimated band powers spanning ℓ with seven bin’s centered between ℓ = 50 to is non-Gaussian but one can apply the “radical compres- ℓ = 400 and one bin at ℓ = 800. sion” method proposed by BJK98; the likelihood function The bin correlation matrix is diagonalized as in BJK98 is well approximated by an offset lognormal distribution resulting in eight orthogonalized (independent) bins. We whose parameters can be easily calculated from the output present the likelihood for each orthogonalized band power of MADCAP. The theory Cℓs are generated using CMB- in Figure 1, using the offset lognormal ansatz proposed in FAST (Seljak & Zaldarriaga 1996) and the recent imple- BJK98. As described in Mauskopf et al. 1999 the data mentation for closed models CAMB (Lewis, Challinor & show strong evidence for an acoustic peak with an ampli- Lasenby 1999). We search for the maximum along a 4 di- tude of ∼ 70µKCMB centered at ℓ ∼ 200. mensional grid of models, using the fact that variations in C10 and ns are less CPU time consuming. We searched 3. MEASURING CURVATURE also for the multidimensional maxima of the likelihood The Boomerang/NA angular power spectrum covers adopting a Downhill Simplex Method (Press et al. 1989), a range of ℓ corresponding to the horizon size at decou- obtaining consistent results. 3

FIG. 2.— The Likelihood function of Ω = ΩM + ΩΛ normalized to unity at the peak after marginalizing along the ΩM −ΩΛ direction. The dashed line is the cumulative likelihood.

In Figure 2, we plot the likelihood of Ω normalized to 1 at the peak where, again, we have maximized along the ΩM − ΩΛ direction. The likelihood shows a sharp peak near Ω = 1 and this result is insensitive to the tradeoff between ΩM and ΩΛ. (see Figure 3 and explanation in following paragraphs). This is an extreme manifestation of the “cosmic degeneracy” problem (because we are fo- cusing on just the first peak): we are able to obtain robust constraints on Ω without strong constraints on ΩM and ΩΛ individually. Within the range of models we are considering, we find that 68% of integrated likelihood corresponds to 0.85 ≤ Ω ≤ 1.25 (0.65 ≤ Ω ≤ 1.45 at 95%). The best fit is a FIG. 3.— The likelihood contours in the ΩM ,ΩΛ plane, eval- marginally closed model with ΩCDM = 0.26, ΩB = 0.05, Ω = 0.75, n = 0.95, h = 0.70, C = 0.9. An almost uated at the maxima of the remaining four “nuisance” param- Λ S 10 etes. The top panel is from the Boomerang/NA data, the equivalent good fit is given by ΩCDM = 0.39, ΩB =0.07, bottom panel is from Boomerang/NA+COBE. The contours ΩΛ =0.65, nS =0.90, h =0.55, C10 =1.0. In Figure 3 (top panel) we estimate the likelihood of correspond to 0.32, 0.05 and 0.01 of the peak value of the likelihood. The small triangle indicates the best fit. The dashed the data for a 20 × 20 grid in (ΩM ,ΩΛ) by applying the maximization/marginalization algorithm described above. line corresponds to the flat models. The effect of marginalizing is, as expected, to expand the contours along the Ω =constant lines but has little effect in the perpendicular direction and we are able to rule out 4. DISCUSSION a substantial region of parameter space. In the previous section we have obtained a constraint ∼ For ΩM 1 models, the position of the peak is solely on Ω using only the Boomerang/NA data. These new dependent on the angular-diameter distance, with a good − 1 results are consistent with Lineweaver (1998), Tegmark approximation being ℓpeak ∝ Ω 2 ; this approximation (1999) and DK99. However, the Boomerang/NA data breaks down when ΩM → 0 where the early time inte- on its own does not constrain the shape and amplitude grated Sachs-Wolfe effect becomes important and ℓpeak is of the power spectrum at ℓ ≤ 25 and limits our ability far more sensitive to Ω (White & Scott 1996). This effect to independently determine the parameters nS,ΩB, h,Ω → Λ leads to a convergence of contour levels as ΩM 0 in and C10. We combine the Boomerang/NA data with the Figure 3. 4-year COBE/DMR angular power spectrum to attempt to break this degeneracy. In Figure 3 (bottom panel) we 4 plot the likelihood contours, again maximized over the nui- of the Boomerang/NA test flight to constrain the curva- sance parameters for the combined Boomerang/NA and ture of the universe. Given that we have based our results COBE data. The inclusion of the COBE data does not on this data set alone, our results are completeley inde- greatly affect the constraints at high ΩM or the confidence pendent from previous analysis of the CMB. At the time levels on Ω, but, as expected, it helps to close of the con- of submission, this letter is also the first analysis of this tours at low values of ΩM . The best fit model changes kind to include closed models in the computation. to have ΩCDM = 0.46, ΩB = 0.05, ΩΛ = 0.50, nS = 1.0, We find strong evidence against an open universe: we h =0.70, C10 =0.94. We find that for the likelihood to be find that 0.65 ≤ Ω ≤ 1.45 at the 95% confidence level, sig- greater than 0.32 of its peak value then ΩM > 0.2, again nificantly ruling out the current favourite open inflationary similar to the results of DK99. models for structure formation (Lyth & Stewart 1990, Ra- One can combine our constraints with those obtained tra & Peebles 1995, Bucher, Goldhaber & Turok 1995). from the luminosity-distance measurements of high-z su- Much tighter constraints will soon be placed on these and pernovae (Perlmutter et al. 1998, Schmidt et al 1998): others cosmological parameters from future data sets, in- using the 1-σ constraint from Perlmutter et al (1998), cluding data obtained during by the Antarctic LDB flight ΩM − 0.6ΩΛ = −0.2 ± 0.1, we find 0.2 ≤ ΩM ≤ 0.45 of Boomerang, which mapped over 1200 square degrees ′ and 0.6 ≤ ΩΛ ≤ 0.85. of the sky with 12 angular resolution and higher sensitiv- A few comments are in order about the robustness of our ity per pixel than the Boomerang/NA. analysis. Firstly we have not truly marginalized over the nuisance parameters. However the constraints we obtain We acknowledge useful conversations with Dick Bond, in this way are, if anything, more conservative. Secondly, Ruth Durrer, Eric Hivon, Tom Montroy, Dmitry Pogosyan although we are limiting ourselves to standard adiabatic and Simon Prunet. The Boomerangprogram has been models, a strong case can be made against the rival the- supported by Programma Nazionale Ricerche in Antar- ory of topological defects: the presence of a fairly local- tide, Agenzia Spaziale Italiana and University of Rome La ized rise and fall in the data around ℓ of 200 indicates that Sapienza in Italy, by NASA grant numbers NAG5-4081 the characteristic broadening due to decoherence of the & NAG5-4455 in the USA, and by the NSF Science & either cosmic strings (Contaldi, Hindmarsh & Magueijo Technology Center for Particle Astrophysics grant num- 1999) or textures (Pen, Seljak & Turok 1997) is strongly ber SA1477-22311NM under AST-9120005, by the NSF disfavoured. Office of Polar Programs grant number OPP-9729121 and Finally we have restricted ourselves to only four ex- by PPARC in UK. This research also used resources of tra nuisance parameters. Again we believe this does not the National Energy Research Scientific Computing Cen- affect our main result (our constraints on Ω) although ter, which is supported by the Office of Science of the it may affect the low ΩM constraints when we combine U.S. Department of Energy under Contract No. DE- the Boomerang/NA data with COBE; the results from AC03-76SF00098. Additional computational support for Tegmark (1998) and DK99 lead us to believe that the ef- the data analysis has been provided by CINECA/Bologna. fect will not greatly change our results. We also acknowledge using the CMBFAST, CAMB and To summarize we have used the angular power spectrum RADPack packages.

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