4 Cosmological parameter constraints

  
4.1 Archeops

We first find constraints on the cosmological parameters using the Archeops data alone. The cosmological model that presents the best fit to the data has a $\chi^2_{\rm gen}= 6/9$. Figure 3 gives confidence intervals on different pairs of parameters. The Archeops data constrain the total mass and energy density of the Universe ( $\Omega_{\rm tot}$) to be greater than 0.90, but it does not provide strong limits on closed Universe models. Figure 3 also shows that $\Omega_{\rm tot}$ and h are highly correlated (Douspis et al. 2001b). Adding the HST constraint for the Hubble constant, $H_0=72\pm8 \rm ~km~s^{-1}~Mpc^{-1}$ (68% CL, Freedman et al. 2001), leads to the tight constraint $\Omega_{\rm tot}= 0.96^{+0.09}_{-0.04}$ (full line in Fig. 3), indicating that the Universe is flat.

Using Archeops data alone we can set significant constraints neither on the spectral index n nor on the baryon content $\Omega_{\rm b}h^2$because of lack of information on fluctuations at small angular scales.

4.2 COBE, Archeops, CBI

We first combine only COBE/DMR, CBI and Archeops so as to include information over a broad range of angular scales, $2 \le \ell \le 1500$, with a minimal number of experiments. The results are shown in Fig. 4, with a best model $\chi^2_{\rm gen}= 9/20$. The constraint on open models is stronger than previously, with a total density $\Omega_{\rm tot}= 1.16 ^{+0.24}_{-0.20}$ at 68% CL and $\Omega_{\rm tot}> 0.90$ at 95% CL. The inclusion of information about small scale fluctuations provides a constraint on the baryon content, $\Omega_{\rm b}h^2= 0.019^{+0.006}_{-0.007}$ in good agreement with the results from BBN (O'Meara et al. 2001: $\Omega_{\rm b}h^2= 0.0205\pm0.0018$). The spectral index n=1.06^+0.11_-0.14 is compatible with a scale invariant Harrison-Zel'dovich power spectrum.

Figure 4: Likelihood contours for (COBE + Archeops + CBI) in the $(\Omega _{\Lambda }, \Omega _{\rm tot})$, $(H_0, \Omega _{\rm tot})$, $(\Omega _{\rm tot}, n)$ and ( $\Omega _{\rm b}h^2,n$) planes.

Figure 5: Likelihood contours in the $(\tau , n)$ and $(\tau , \Omega _{\rm b}h^2)$ planes using Archeops + CBDMVC datasets.

Figure 6: Likelihood contours in the $(\Omega _{\rm tot}, \Omega _{\Lambda })$ and $(\Omega _{\rm tot}, \Omega _{\rm b}h^2)$ planes. Left: constraints using Archeops+CBDMVC datasets. Right: adding HST prior for H0.

Figure 7: Best model obtained from the Archeops + CBDMVC + HST analysis with recalibrated actual datasets. The fitting allowed the gain of each experiment to vary within their quoted absolute uncertainties. Recalibration factors, in temperature, which are applied in this figure, are 1.00, 0.96, 0.99, 1.00, 0.99, 1.00, and 1.01, for COBE, Boomerang, Dasi, Maxima, VSA, CBI and Archeops respectively, well within 1 $\sigma$ of the quoted absolute uncertainties (<1, 10, 4, 4, 3.5, 5 and 7%).

4.3 Archeops and other CMB experiments

 

 

Table 2: Cosmological parameter constraints from combined datasets. Upper and lower limits are given for 68% CL. See text for details on priors. The central values are given by the mean of the likelihood. The quoted error bars are at times smaller than the parameter grid spacing, and are thus in fact determined by an interpolation of the likelihood function between adjacent grid points.

Data	$\Omega_{\rm tot}$	$n_{\rm s}$	$\Omega_{\rm b}h^2$	h	$\Omega_{\Lambda}$	$\tau$	$\chi^2_{\rm gen}/{\rm d.o.f.}$
Archeops	>0.90	1.15+0.30-0.40	-	-	<0.9	<0.45	6/9
Archeops + COBE + CBI	1.16+0.24-0.20	1.06+0.11-0.14	0.019+0.006-0.007	>0.25	<0.85	<0.45	9/20
CMB	1.18+0.22-0.20	1.06+0.14-0.20	0.024+0.003-0.005	0.51+0.30-0.30	<0.85	<0.55	37/52
Archeops + CMB	1.15+0.12-0.17	1.04+0.10-0.12	0.022+0.003-0.004	0.53+0.25-0.13	<0.85	<0.4	41/67
Archeops + CMB + $\tau=0$	1.13+0.12-0.15	0.96+0.03-0.04	0.021+0.002-0.003	0.52+0.20-0.12	<0.80	0.0	41/68
Archeops + CMB + $\Omega_{\rm tot}=1$	1.00	1.04+0.10-0.12	0.021+0.004-0.003	0.70+0.08-0.08	0.70+0.10-0.10	<0.40	41/68
Archeops + CMB + HST	1.00+0.03-0.02	1.04+0.10-0.08	0.022+0.003-0.002	0.69+0.08-0.06	0.73+0.09-0.07	<0.42	41/68
Archeops + CMB + HST + $\tau=0$	1.00+0.03-0.02	0.96+0.02-0.04	0.021+0.001-0.003	0.69+0.06-0.06	0.72+0.08-0.06	0.0	41/69
Archeops + CMB + SN1a	1.04+0.02-0.04	1.04+0.10-0.12	0.022+0.003-0.004	0.60+0.10-0.07	0.67+0.11-0.03	<0.40	41/69
Archeops + CMB + BBN	1.12+0.13-0.14	1.04+0.10-0.12	0.020+0.002-0.002	0.50+0.15-0.10	<0.80	<0.25	41/68
Archeops + CMB + BF(H)	1.11+0.12-0.11	1.03+0.12-0.14	0.022+0.004-0.004	0.46+0.09-0.11	0.45+0.10-0.10	<0.40	43/69
Archeops + CMB + BF(L)	1.22+0.18-0.12	1.03+0.07-0.13	0.021+0.003-0.004	<0.40	<0.3	<0.40	45/69

By adding the experiments listed in Fig. 1 we now provide the best current estimate of the cosmological parameters using CMB data only. The constraints are shown in Figs. 5 and 6 (left). The combination of all CMB experiments provides $\sim$10% errors on the total density, the spectral index and the baryon content respectively: $\Omega_{\rm tot}=1.15^{+0.12}_{-0.17}$, n=1.04^+0.10_-0.12 and $\Omega_{\rm b}h^2= 0.022^{+0.003}_{-0.004}$. These results are in good agreement with recent analyses performed by other teams (Netterfield et al. 2002; Pryke et al. 2002; Rubino-Martin et al. 2002; Sievers et al. 2002; Wang et al. 2002). One can also note that the parameters of the $\Lambda$CDM model shown in Fig. 1 are included in the 68% CL contours of Fig. 6 (right).

As shown in Fig. 5 the spectral index and the optical depth are degenerate. Fixing the latter to its best fit value, $\tau=0$, leads to stronger constraints on both n and $\Omega_{\rm b}h^2$. With this constraint, the prefered value of n becomes slightly lower than 1, n=0.96^+0.03_-0.04, and the constraint on $\Omega_{\rm b}h^2$ from CMB alone is not only in perfect agreement with BBN determination but also has similar error bars, $\Omega_{\rm b}h^2_{\rm (CMB)}=0.021^{+0.002}_{-0.003}$. It is important to note that many inflationary models (and most of the simplest of them) predict a value for n that is slightly less than unity (see, e.g., Linde 1990; Lyth & Riotto 1999 for a recent review).

4.4 Adding non-CMB priors

In order to break some degeneracies in the determination of cosmological parameters with CMB data alone, priors coming from other cosmological observations are now added. First we consider priors based on stellar candles like HST determination of the Hubble constant (Freedman et al. 2001) and supernovæ determination of $\Omega_{\rm m}$ and $\Lambda$ (Perlmutter et al. 1999). We also consider non stellar cosmological priors like BBN determination of the baryon content, (O'Meara et al. 2001), and baryon fraction determination from X-ray clusters (Roussel et al. 2000; Sadat & Blanchard 2001). For the baryon fraction we use a low value, BF(L), $f_{\rm b} = 0.031h^{-3/2} +0.012\; (\pm 10\%)$, and a high value, BF(H), $f_{\rm b} = 0.048h^{-3/2} + 0.014\; (\pm 10\%)$(Douspis et al. 2001b and references therein). The results with the HST prior are shown in Fig. 6 (right). Considering the particular combination Archeops + CBDMVC + HST, the best fit model, within the Table 1 gridding, is $(\Omega_{\rm tot}, \Omega_{\Lambda}, \Omega_{\rm b}h^2, h, n, Q, \tau)=(1.00,0.7,0.02,0.665,0.945,19.2{\rm\mu K},0.)$ with a $\chi^2_{\rm gen}=41/68$. The model is shown in Fig. 7 with the data scaled by their best-fit calibration factors which were simultaneously computed in the likelihood fitting process. The constraints on h break the degeneracy between the total matter content of the Universe and the amount of dark energy as discussed in Sect. 4.1. The constraints are then tighter as shown in Fig. 6 (right), leading to a value of $\Omega_{\Lambda}= 0.73^{+0.09}_{-0.07}$ for the dark energy content, in agreement with supernovæ measurements if a flat Universe is assumed. Table 2 also shows that Archeops + CBDMVC cosmological parameter determinations assuming either $\Omega_{\rm tot}=1$ or the HST prior on h are equivalent at the 68% CL.