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Astron. Astrophys. 329, 799-808 (1998)

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4. Non-CMB constraints in the h [FORMULA] plane

Four independent non-CMB cosmological measurements constrain the acceptable regions of the h [FORMULA] plane.

4.1. Nucleosynthesis

Primordial nucleosynthesis gives us limits on the baryonic density of the Universe. Although deuterium measurements seem to be currently the most accurate baryometer, there is an active debate about whether they yield high (Tytler et al. 1996, Tytler & Burles 1997) or low (Songaila et al. 1994, Carswell et al. 1994, Rugers & Hogan 1996) baryonic densities. We have adopted the range [FORMULA] because it encompasses most published results. These limits are plotted in Figs. 1-4 and are labelled "BBN" in Fig. 4. We use [FORMULA] as a central value. The BBN constraints are independent of [FORMULA], [FORMULA], n and Q and thus do not depend on our [FORMULA], [FORMULA] assumptions. Lyman limit systems yield a somewhat model-dependent lower limit for the baryonic density, lending support to the higher values of [FORMULA] (Weinberg et al. 1997, Bi & Davidsen 1997).

[FIGURE] Fig. 4. This plot has no CMB information in it. The bands are the constraints from 4 non-CMB measurements discussed in Sect. 4: Big Bang nucleosynthesis ("BBN"), cluster baryonic fraction ("Clusters"), the galaxy and cluster scale density fluctuation shape parameter (" [FORMULA] ") and the age of the oldest stars in globular clusters ("Age": region between the vertical lines). To quantify the combination of these 4 constraints, we show contours obtained by assuming for each constraint a two-tailed Gaussian probability around the central values. The 1 and [FORMULA] contours are thus the result of an approximate joint likelihood of 4 non-CMB constraints for [FORMULA], [FORMULA] CDM models. A low h is preferred. This was pointed out in Bartlett et al. (1995). The combination of these four, independent, non-CMB measurements yields [FORMULA] and [FORMULA]. This region of the [FORMULA] plane is very similar to the region preferred by the CMB data (compare Fig. 3). The point of this diagram is not to show that h is low since we have ignored the numerous, more direct, [FORMULA] measurements of h which find [FORMULA]. The point of this diagram is to show that an important set of independent constraints also favors low values of h in [FORMULA], [FORMULA] CDM models.

4.2. X-ray cluster baryonic mass fraction

Observations of the X-ray luminosity and the angular size of galaxy clusters can be combined to constrain the quantity [FORMULA]. We adopt the range [FORMULA] (White et al. 1993) with a central value of 0.06 (Evrard 1997). These limits seem to be inconsistent with BBN if [FORMULA] and [FORMULA]. This is known as the baryon catastrophe and has led some to believe that [FORMULA]. The severity of this catastrophy in [FORMULA] models can be examined in Fig. 4 by comparing the "BBN" region with the "Clusters" region. For [FORMULA], [FORMULA] models allow consistency between the nucleosynthetic and cluster data for low values of h.

4.3. Matter power spectrum shape parameter [FORMULA]

Peacock & Dodds (1994) made an empirical fit to the matter power spectrum using a shape parameter [FORMULA]. For [FORMULA] models, [FORMULA] can be written as (Sugiyama 1995)


We adopt the [FORMULA] limits of the empirical fit of Peacock & Dodds (1994)(see also Liddle et al. 1996a) and include the n dependence


Under the assumption that [FORMULA], Eq. 2becomes [FORMULA]. This is the [FORMULA] constraint used in Fig. 4. We use [FORMULA] as a central value.

4.4. Limits on the age of the Universe from the oldest stars in globular clusters

There is general agreement that the Universe should be older than the oldest stars in our Galaxy. Thus a lower bound on the age of the Universe comes from an age determination of the oldest stars in the most metal-poor (= oldest) globular clusters. A reasonably representative sample of globular cluster ages, [FORMULA], in the literature is,


Allowing [FORMULA] Gyr for the formation of globular clusters, we adopt the range [FORMULA] Gyr with a central value of 14 Gyr. We use this relatively large interval to avoid overconstraining the models and to encompass most published results. Age determinations are [FORMULA] and [FORMULA] independent but converting them to limits on Hubble's parameter depends on our [FORMULA] and [FORMULA] assumptions. In the models we are considering here, our age limits are converted directly into limits on Hubble's constant using [FORMULA] which yields [FORMULA] (with the central value 14 Gyr corresponding to [FORMULA]). This region is marked "Age" in Fig. 4.

4.5. Summary of non-CMB constraints and comparison to CMB constraints

The constraints we adopt from BBN, cluster baryonic fraction, [FORMULA], and the ages of the oldest stars in globular clusters (as described above) are,


Bands illustrating these constraints are plotted in Fig. 4. Since these constraints are independent, it is not obvious that they should be consistent with each other. They are consistent in the sense that there is a region of overlap. This consistency is improved if [FORMULA] turns out to be high as indicated by Tytler & Burles (1997). To visualize more quantitatively the combination of these four constraints, for each constraint we assume a two-tailed Gaussian distribution around the central values. This allows the flexibility to account for asymmetric error bars. We then calculate a joint likelihood,




The upper and lower limits of the four constraints determine the [FORMULA] 's for the two-tailed Gaussians. For example [FORMULA] and [FORMULA]. The joint likelihood of the four terms [FORMULA] is shown in Fig. 4. The contour levels are [FORMULA]. The combined non-CMB constraints from BBN, cluster baryonic fraction, [FORMULA] and stellar ages yield [FORMULA] and [FORMULA] for [FORMULA], [FORMULA].

The point of Fig. 4 is not to show that h is low since we have of course ignored the numerous, more direct, [FORMULA] measurements of h which find [FORMULA] (see for example Freedman 1997). The point of this diagram is to show that an important set of independent constraints do overlap and are consistent with each other for low values of h in the [FORMULA], [FORMULA] models considered here.

Fig. 4 should be compared with Fig. 3 which has contours of [FORMULA]. There is an interesting consistency between the non-CMB constraints and the CMB constraints of Fig. 3. Although they extend over relatively small regions in the [FORMULA] plane, the [FORMULA] regions of the non-CMB joint likelihood and the CMB overlap.

In Fig. 4 we see that the combination of four independent cosmological measurements indicate that a low value of h could make the CDM theory viable, as Bartlett et al. (1995) argued. The amplitude of small scale matter fluctuations is an additional consistency check on this model. The value of [FORMULA] at the minimum in Fig. 3 is [FORMULA]. This agrees quite well with values inferred from X-ray cluster data (Viana & Liddle 1996, Oukbir et al. 1996).

Liddle et al. (1996b) studied critical density CDM models. Based on the COBE normalization, peculiar velocity flows, the galaxy correlation function, abundances of galaxy clusters, quasars and damped Lyman alpha systems, they found that [FORMULA] and [FORMULA] is preferred. Adams et al. (1997) come to similar conclusions.

Bartlett et al. (1995) listed the advantages of a low h in critical density universes, the main point being that there exists a region of parameter space in which this simplest of models remains consistent with observations of the large-scale structure of the Universe. For example, there is the question of the age crisis. Recent h measurements point to values in the interval [FORMULA]. In a critical density universe [FORMULA] implies an age of 9.3 Gyr, younger than the estimated age of many globular clusters. [FORMULA] yields an age for the universe of 21.7 Gyr comfortably in accord with globular cluster ages.

What is perhaps surprising is the fact that the CMB data do not rule out such a low value of [FORMULA] but seem to favor it within the context of this type of scenario. Of course, these low [FORMULA] values are in disagreement with current measurements of the Hubble constant.

4.6. Other projections

Figs. 5 through 9show our 4-D [FORMULA] ellipsoid projected on to 2-D planes orthogonal to the [FORMULA] plane. The limits obtained on h from Fig. 5 are the same as we obtained from Fig. 3 since we are projecting the same 4-D [FORMULA] surface. h and n are positively correlated for [FORMULA] and possibly negatively correlated for [FORMULA]. In Fig. 6 we see that with n fixed at 1, a high precision determination of Q is possible, [FORMULA] K. In Fig. 7 the [FORMULA] constraint is dropped.

[FIGURE] Fig. 5. Likelihood contours in the [FORMULA] plane for [FORMULA] free and Q free. [FORMULA] and [FORMULA]. The [FORMULA] region has [FORMULA] K. At [FORMULA], [FORMULA], and the largest n is for [FORMULA]. In the [FORMULA] contour there is a strong h and n correlation: the lower values of h go with the lower values of n. When different values of n are allowed, the amplitude of the Doppler peak varies up and down but the location of the peak does not.

[FIGURE] Fig. 6. Likelihood contours in the [FORMULA] plane for [FORMULA] and [FORMULA] free. [FORMULA]. [FORMULA] K. Notice that since n has been fixed at 1, the value of Q does not rise in the high h region as it does in the next figure. The [FORMULA] contours are round: there is no correlation between h and Q.

[FIGURE] Fig. 7. Likelihood contours in the [FORMULA] plane for n free and [FORMULA] free. Notice that in the high h region, [FORMULA] K is preferred. A comparison with Fig. 5 shows that n and Q are anti-correlated but this is seen more easily in the next two figures.

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Online publication: December 16, 1997