View metadata, Newcitation and CMB similar constraintspapers at core.ac.uk on the cosmic matter budget: trouble for nucleosynthesis?brought to you by CORE provided by CERN Document Server Max Tegmark Dept. of Physics, Univ. of Pennsylvania, Philadelphia, PA 19104; [email protected] Matias Zaldarriaga Institute for Advanced Study, Princeton, NJ 08540; [email protected] (Submitted 30 April 2000) We compute the joint constraints on ten cosmological parameters from the latest CMB measure- ments. The lack of a significant second acoustic peak in the latest Boomerang data favors models with more baryons than Big Bang nucleosynthesis predicts, almost independently of what prior information is included. The simplest flat inflation models with purely scalar scale-invariant fluctu- 2 ations prefer a baryon density 0:024 <hΩb < 0:042 and a total (hot + cold) dark matter density 2 0:16 <h Ωb < 0:51 at 95% confidence, and allow reionization no earlier than z ∼ 30. One of the main challenges in modern cosmology is to have used the measured cosmic microwave background refine and test the standard model of structure formation (CMB) fluctuations to constrain subsets of these param- by precision measurements of its free parameters. The eters. CMB data has improved dramatically since fluctu- cosmic matter budget involves at least the four parame- ations were first detected [9]. The measurement of a first ters Ωb,Ωcdm,Ων and ΩΛ, which give the percentages of acoustic peak at the degree scale [10], suggesting that the critical density corresponding to baryons, cold dark mat- Universe is flat (Ωk = 0), has now been beautifully con- ter, massive neutrinos and vacuum energy. A “budget firmed and improved by using the ground-breaking high deficit” Ωk ≡ 1 − Ωb − Ωcdm − Ων − ΩΛ manifests itself fidelity maps of the Boomerang experiment [11]. As can as spatial curvature. The description of the initial seed be seen in Figure 1, perhaps the most important new in- fluctuations predicted by inflation require at least four formation from Boomerang is its accurate measurements parameters, the amplitudes As & At and slopes ns & nt of the angular power spectrum C` on even smaller scales, of scalar and tensor fluctuations, respectively. Finally, out to multipole ` ∼ 600. The striking lack of a sig- the optical depth parameter τ quantifies when the first nificant second acoustic peak places strong constraints stars or quasars reionized the Universe and the Hubble on the cosmological parameters [8,11,12], making a new parameter h gives its current expansion rate. full-fledged analysis of all the CMB data very timely. In this Letter, we jointly constrain the following 10 cosmological parameters: τ,Ωk,ΩΛ, ns, nt, As,the tensor-to scalar ratio r ≡ At=As, and the physical matter 2 2 2 densities !b ≡ h Ωpb, !cdm ≡ h Ωcdm and !ν ≡ h Ων . The identity h = !cdm + !b + !ν =1 − Ωk − ΩΛ fixes the Hubble parameter. We use the 10-dimensional grid method described in [5]. In essence, this utilizes a tech- nique for accelerating the the CMBfast package [13] by a factor around 103 to compute theoretical power spectra on a grid in the 10-dimensional parameter space, fitting these models to the data and then using cubic interpo- lation of the resulting 10-dimensional likelihood function to marginalize it down to constraints on individual or pairs of parameters. We use the 77 data points shown in Figure 1, combining the 65 tabulated in [5] with the 12 new Boomerang points [11]. Our 95% confidence limits on the best constrained pa- rameters are summarized in Table 1. Figure 2 shows that CMB alone suggests that the Universe is either flat FIG. 1. The 77 band power measurements used. Circles ≡ show the new Boomerang points. The curve shows the simple (near the diagonal line Ωm +ΩΛ =1,whereΩm inflationary model with τ =Ωk = !ν = r =0,ΩΛ =0:19, Ωb +Ωcdm +Ων ) or closed (upper right). These con- !cdm =0:19, !b =0:03, ns =1. straints come largely from the location of the first peak, which is well-known to move to the right if the curva- During the past year or so, a number of papers [1–8] ture Ωk is increased [14]. Very closed models work only 1 because the first acoustic peak can also be moved to the τ 2Ω 2Ω right by increasing the tilt ns and bringing the large-scale h b h dm Ωk = 0 Ω = 0 Ω = 0 COBE signal back up with tensor fluctuations (gravity n = 1 k k s ns= 1 ns= 1 waves) [15,18]. Galaxy clustering constraints disfavor r= 0 r= 0 r= 0 such strong blue-tilting, and Figures 2 and 3 show that closing this loophole by barring gravity waves (r =0) favors curvature near zero and ns near unity. This is a striking success for the oldest and simplest inflation mod- Ω = 0 Ω Ω r k Λ k ns els, which make the three predictions =0,Ωk =0and ns= 1 = 0 r= 0 r= 0 ns = 1 [16,17]. Another important success for inflation is r that the first peak is so narrow — if the data had revealed the type of broad peak expected in many topological de- fect scenarios, none of the models in our grid would have provided an acceptable fit. Because of these tantalizing hints that “back to basics” inflation is correct, Table 1 FIG. 3. Marginalized : <h< : and Figure 3 include results assuming this inflation prior likelihoods assuming that 0 5 0 8 (95%) and the in- flationary priors shown above. 2 − σ limits are roughly where r =Ωk =0;ns =1. the curves drop beneath the dashed lines. The constraints in Table 1 are seen to be much more interesting than those before Boomerang [5], thanks to Λ 1.0 CMB alone new information on the scale of the second peak. CDM and neutrinos have indistinguishable effects on the CMB Ω CMB + h except for very light neutrinos (small !ν ), and the cur- CMB + h + r rent data still lacks the precision to detect this sub- tle difference. The predicted height ratio of the first two peaks therefore depends essentially on only three parameters [8,12]: ns, !b and !dm, where the total 0.5 ! ≡ ! ! CMB +SN h 1a dark matter density dm cdm + ν . Let us focus CMB alone on the constraints on these parameters. Increasing !b tends to boost the odd-numbered peaks (1, 3, etc.) at the expense of even ones (2, 4, etc.) [14], whereas in- creasing !dm suppresses all peaks (see the CMB movies at www:hep:upenn:edu=∼max or www:ias:edu=∼whu). 0.0 The low second peak can therefore be fit by either de- Vacuum density creasing the tilt ns or by increasing the baryon density 0.0 0.5 1.0 !b [8,12] compared to the usually assumed values ns ≈ 1, !b ≈ 0:02. As illustrated in Figure 4, this conclusion Matter density Ω is essentially independent of what priors are assumed. m However, reducing ns below 0.9 is seen to make things FIG. 2. The regions in the (Ωm; ΩΛ)-plane that are ruled worse again, as the first peak becomes too low relative to out at 95% are shown using (starting from the outside) no COBE-normalization. priors the prior that 0:5 <h<0:8 (95%) and the additional In short, there are two very simple ways of explain- constraint r = 0 The SN 1a constraints are from White98. ing the lack of a prominent second acoustic peak: more baryons or an red-tilted spectrum [8,12]. However, as we will now discuss, both of these solutions have problems Table 1 – Maximum-likelihood values and 95% confidence limits. of their own. The “inflation prior” for each parameter is indicated in Figure 3 Figure 5 shows that when more baryons are added, more dark matter is needed to keep the first peak height 10 free parameters Inflation prior constant. When the tilt ns is fixed by the inflation prior, Quantity Min Best Max Min Best Max the constraints on the remaining two parameters !dm τ 0.0 0.0 0.35 0.0 0.0 0.31 2 and !b are seen to become quite tight. Intriguingly, h Ωb .015 .05 − .024 .03 .045 2 the preferred baryon fraction is of the same order as h Ωdm 0 −−0.16 0.30 0.51 preferred by Big Bang nucleosynthesis, but nonetheless ΩΛ − 0.72 0.85 −0.44 .19 0.73 higher than the tight nucleosynthesis error bars [20,21] Ωk −−0.39 0.22 −0.24 0 0.15 !b =0:019 0:0024 allow. Even if the nucleosynthesis ns 0.71 1.53 − 0.84 1.0 1.16 2 error bars have somehow been underestimated so that permitted to be low enough to agree with nucleosynthesis > !b ∼ 0:03 as required by the CMB data plus simple in- for any value of the tilt ns, so the tilt solution may have flation is allowed, this solution may conflict with other worked using all the data merely because of the above- astrophysical constraints. For instance, X-ray observa- mentioned dilution effect. tions of clusters of galaxies can be used to determine the ratio of baryons to dark matter [22], and !b =0:03 can only be reconciled with these observations by having CMB + h + r Ωdm ∼ 0:6 which would conflict with the supernova 1a results and other estimates of the dark matter density CMB only [23]. CMB + h On the other hand, the tilt solution is no panacea ei- ther. In a wide range of inflationary models, the am- plitude of the tensor component is related to the tilt of the scalar spectrum, r =7(1− ns) [24].