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Astron. Astrophys. 329, L17-L19 (1998)

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2. Cosmological conditions

The Hubble constant [FORMULA] used to be uncertain by a factor of two, but it now is converging into the (one standard deviation) range


This value is the result by Nevalainen & Roos (1997) of combining [FORMULA] determinations from four HST-observed galaxies, and applying the correction due to the metallicity dependence of the Cepheids, as determined by Beaulieu & al. (1997) and Sasselov & al. (1997), to Cepheids in M96 (Tanvir & al. 1995) and M100 (Freedman & al. 1994) and to the supernovæNGC 5253 (Sandage & al. 1994) and IC 4182 (Saha & al. 1994, Sandage & al. 1994). If one takes into account the correction to the Cepheid period-luminosity relation measured by Hipparcos (Feast & Catchpole 1997) the above [FORMULA] value may still go down by 10%, but we feel that it is still not quite settled whether this is justified at the distances in question.

An age limit of the Universe can be taken from the age of the oldest globular clusters. Making use of the new distance measurements by Hipparcos, Chaboyer & al. (1997) estimate their mean age to be


To which the unknown age of the Universe at the time of their formation must be added. An even more stringent limit is posed by the discovery (Dunlop & al. 1996, Kashlinsky & Jimenez 1996) of a weak and extremely red radio galaxy 53W091 at [FORMULA] whose spectral data indicate that its stellar population is at least 3.0 Gyr old. Although Bruzual & Magris (1997) have claimed that this age determination is an artifact, we shall explore the consequences of such an old object.

In the Friedman-Lemaître model the age [FORMULA] of the Universe at redshift z can be expressed in the form


Here [FORMULA] is the ratio of pressure to energy density, thus it defines the equation of state. For ordinary or dark non-relativistic pressureless matter [FORMULA]. For a flat, static Universe with a cosmological constant [FORMULA].

Using the Dunlop & al. (1996) age of the radio galaxy 53W091,


Eq. (4) constrains the space of [FORMULA] and [FORMULA]. In Fig. 1 we plot the solution to Eq. (4) for [FORMULA] and the limit (4) for several values of [FORMULA]. It is obvious that the standard corner [FORMULA] is then strongly disfavoured by the [FORMULA] value (2).

[FIGURE] Fig. 1. Regions allowed in the [FORMULA] -space. The set of solid lines are imposed by the observation of the radio galaxy 53W091 assuming [FORMULA]. The numbers indicate the value of the Hubble constant in units of km s-1 Mpc-1. The limits [FORMULA], [FORMULA] are indicated.

The only way to reduce the age below 3.0 Gyr is to reduce the product [FORMULA], but this then reduces the small scale power in the primordial density field beyond allowed limits. One degree of freedom which can be used to improve this situation has been pointed out by Steinhardt (1996). Although [FORMULA] is traditionally taken to be zero, in a universe with interacting fields and topological defects [FORMULA] can vary between -1 and 0. This implies the presence of strings causing more small scale power than in simple inflation. In Fig. 2 we plot the situation with [FORMULA] for different [FORMULA] -values. This moves the allowed region towards the standard corner. However, the price paid for this improvement may seem too high, also in view of the next conditions below.

[FIGURE] Fig. 2. As Fig. 1, but for [FORMULA] = -0.1

Let us alternatively look at the consequences of taking [FORMULA] to be given by the age of the globular clusters. The Chaboyer & al. (1997) determination carries a [FORMULA] error of 1.3 Gyr. Let us assume that the age of the Universe at the time of their formation is short enough to be included in this error, then we may take


Requiring Eq. (4) to yield this value when [FORMULA] has the value in Eq. (2), one finds that


in a flat Universe. Thus a considerable [FORMULA] -component is required. In an open Universe with [FORMULA], [FORMULA] would be very small, 0.3 at most.

The second condition ruling out the standard model is the observational value of [FORMULA] as determined by X-ray studies of gas in clusters, e.g. the Coma cluster (White & al. 1993) and the rich cluster A85 (David & al. 1995, Nevalainen & al. 1997). Assuming that the gravitating matter seen in these galaxies out to a radius of about 3 Mpc extrapolate well to the average matter density of the Universe, one can obtain a value for the quantity [FORMULA]. To evaluate [FORMULA] one needs to know the baryonic density parameter [FORMULA] which is highly controversial, due to the conflicting deuterium observations (Rugers & Hogan 1996, Tytler & al. 1996, 1997, Webb & al. 1997). Pushing the gas density profile parameters to their [FORMULA] confidence limit, and the Hubble parameter at its lower (two-sided) [FORMULA] limit, [FORMULA], and taking the D/H ratio to be maximal (Webb & al. 1997), on obtains


Large-scale structures (galaxy correlation functions, the abundance of rich clusters, cluster-cluster correlations etc.) do not constrain [FORMULA] very tightly, but in the [FORMULA] range (2) they prefer values larger than Eq. (8), [FORMULA] or so. Thus there is some conflict but in no way driving the solution into the standard corner. Perhaps the solution to this conflict is anti-biasing, or perhaps the X-ray studies of gas in clusters reflect local conditions which are different from cosmological values.

Plotting the limit (8) in Figs. 1 and 2 it is obvious that it strongly rules out the standard corner, but it can be made to agree with a flat [FORMULA] universe if the vacuum energy density contribution is large.

The third condition ruling out the standard model is due to strong and weak gravitational lensing. From observations of the clusters Cl [FORMULA] and A370, Mellier & al.(1997) conclude that


Note that previous lensing studies only gave upper limits, e.g. Kochanek (1996) found


for a flat Universe.

As can be seen in the Figures, these limits are in agreement with the limit (9) and the [FORMULA] range (2) when taking the age of the Universe from the 53W091. It also agrees roughly with the upper limit to [FORMULA] from the observations of Perlmutter & al (1997) of the light-curves of seven high redshift supernovae. When [FORMULA] is taken from the globular clusters, these conclusions are somewhat softened, but qualitatively the same.

We have not made any use of CMB data, because the height of the Doppler peak has not been measured well enough yet to yield information of precision comparable to what was used here. Note that both large-scale structures and the CMB Doppler peak depend on further adjustable parameters which we have not referred to, such as bias b and the spectral index n.

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© European Southern Observatory (ESO) 1998

Online publication: November 24, 1997