4. Non-CMB constraints in the h plane
Four independent non-CMB cosmological measurements constrain the acceptable regions of the h plane.
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 because it encompasses most published results. These limits are plotted in Figs. 1-4 and are labelled "BBN" in Fig. 4. We use as a central value. The BBN constraints are independent of , , n and Q and thus do not depend on our , assumptions. Lyman limit systems yield a somewhat model-dependent lower limit for the baryonic density, lending support to the higher values of (Weinberg et al. 1997, Bi & Davidsen 1997).
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 . We adopt the range (White et al. 1993) with a central value of 0.06 (Evrard 1997). These limits seem to be inconsistent with BBN if and . This is known as the baryon catastrophe and has led some to believe that . The severity of this catastrophy in models can be examined in Fig. 4 by comparing the "BBN" region with the "Clusters" region. For , models allow consistency between the nucleosynthetic and cluster data for low values of h.
4.3. Matter power spectrum shape parameter
Under the assumption that , Eq. 2becomes . This is the constraint used in Fig. 4. We use 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,
, in the literature is,
4.5. Summary of non-CMB constraints and comparison to CMB constraints
The constraints we adopt from BBN, cluster baryonic fraction,
, and the ages of the oldest stars in globular
clusters (as described above) are,
The upper and lower limits of the four constraints determine the 's for the two-tailed Gaussians. For example and . The joint likelihood of the four terms is shown in Fig. 4. The contour levels are . The combined non-CMB constraints from BBN, cluster baryonic fraction, and stellar ages yield and for , .
The point of Fig. 4 is not to show that h is low since we have of course ignored the numerous, more direct, measurements of h which find (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 , models considered here.
Fig. 4 should be compared with Fig. 3 which has contours of . 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 plane, the 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 at the minimum in Fig. 3 is . 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 and 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 . In a critical density universe implies an age of 9.3 Gyr, younger than the estimated age of many globular clusters. 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 but seem to favor it within the context of this type of scenario. Of course, these low values are in disagreement with current measurements of the Hubble constant.
4.6. Other projections
Figs. 5 through 9show our 4-D ellipsoid projected on to 2-D planes orthogonal to the 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 surface. h and n are positively correlated for and possibly negatively correlated for . In Fig. 6 we see that with n fixed at 1, a high precision determination of Q is possible, K. In Fig. 7 the constraint is dropped.
© European Southern Observatory (ESO) 1998
Online publication: December 16, 1997