3 Discussion

The age of the universe predicted in such any theory with non-zero $\Lambda$ is given as (e.g. Weinberg 1972; Carrol et al. 1992)

$$t_0= \frac{1}{H_0}\!\! \int_0^1\!\!{\rm d}x [(1-\Omega_{\rm ... ...m m} x^{2-3(1+ \alpha)} + \Omega_\Lambda x^2]^{- \frac{1}{2}}.$$ (11)

Here $\alpha$ defines the equation of state of the present matter, being effectively the ratio of pressure to energy density. For $\Omega=1$case, this reduces to the well-known relation (e.g. Singh 1995)

$$t_0 = \frac{2}{3 H_0} \frac{1}{\sqrt{\Omega_\Lambda}} \ln \l... ... \sqrt{\Omega_\Lambda}}{\sqrt{1- \Omega_\Lambda}} \right]\cdot$$ (12)

In Fig. 2 we present the age estimates for the same two cases as in Fig. 1 in the $\Lambda$ + DDM theory. This is to be compared with the best estimate of the current age of the universe for $\Omega_\Lambda$ given by Eq. (1) is (Perlmutter et al. 1999)

$$t_0 = 14.9^{+1.4}_{-1.1} \times \frac{0.63}{h} \;{\rm Gyr}.$$ (13)

In addition, it should be compared to the age of globular clusters recently carefully measured with accounting for the revised Hipparcos distance scale (Chaboyer et al. 1996, 1998). We perceive that the estimates presented in Fig. 2 are rather significantly smaller from the result in Eq. (13), although they have a correct correspondence limit $\Lambda=0$of $t_0 \approx 12$ Gyr as in Sciama (1997). So low ages are inconvenient from the point of view of globular cluster ages, as well as our understanding of the most distant galaxies observed.

Figure 1: The value of the Hubble parameter, h, as a function of the cosmological constant $\Lambda$, i.e. its contribution to the total cosmological density (assumed to be unity). For the sake of clarity, we plotted $1-\Omega _\Lambda$which represents contribution of the neutrinos and baryonic matter to the total cosmological density. Vertical dotted line, plotted at $1 - \Omega _\Lambda \approx 0.28$ corresponds to the realistic contribution of matter in the universe. It is obvious that in this case h tends to the unrealistically high value of $h\sim 1$. The different possible contributions of the baryonic matter to the total cosmological density are represented by two curves: solid (realistic higher $\Omega _{\rm b}$, see text) and dashed (used by Sciama 1997).

Considering the current trend in observational estimates of cosmological parameters, the impact of cosmological constant on parameter values in DDM theory is largely negative. Resulting values of corrected parameters for $\Lambda \neq 0$ version of Sciama's theory make the entire scheme less plausible. In that respect, recent results of the EURD mission are highly indicative of the observational verdict. This mission failed to observe the emission of the dark sky at wavelengths slightly lower than 912 Å, with the limit (95% confidence) of only a third of the predicted intensity.

The negative EURD result is not the only indication of problems of the DDM theory. Recently, Maloney & Bland-Hawthorn (1999) calculated the flux from a full-neutrino halo and obtained: $\phi\approx 2-3\times 10^5$ photons cm^-2 s^-1. They find that the observed emission is much fainter: $\phi\sim 10^4$ photons cm^-2 s^-1. The detrimental consequences of this result for DDM cannot be remedied by introduction of $\Lambda$, since the latter does not impact galactic dynamics, and therefore the estimates of necessary amount of dark matter in galactic haloes. Of course, one could always assert that DDM is only a small ($\sim$10%) part of the dark halo, most of it being in the form of baryons (i.e. MACHOs and/or molecular clouds; see Fields et al. 1998; Samurovic et al. 1999, also Sciama 2000). In that version, DDM is dominant only on larger than galactic scales. However, the entire rationale of the theory is undermined in this way, since there is no more any direct connection between cosmology and the ISM physics, and the properties of the decaying neutrino can not be constrained with remarkable precision any more. The same criticism applies to the open DDM models (i.e. $\Omega_\nu \approx \Omega < 1$) which requires even more fine-tuning, in particular in view of the consequent properties of matter in galaxy clusters.

However, even the fine-tuned version of the theory fails if confronted with negative results in particle experiments on neutrino masses. Although recent results on the oscillations of atmospheric neutrinos (Fukuda et al. 1998) are sensitive only to the mixing angles and mass difference  $\Delta m^2$ between the two neutrino flavors, the results are somewhat indicative in suggesting rather low, probably sub-eV neutrino masses. Although the DDM theory was correct in assuming neutrino masses - the first empirical result in particle physics outside of the Standard Model - only experiments currently in progress will show whether the required neutrino masses are compatible with empirical limits.

Figure 2: The ages predicted by $\Lambda$ + DDM flat cosmological model. The same notation as in Fig. 1 is used.

Obviously, the simplicity and elegance of the original DDM theory is lost with any complication such as discussed in the present paper. Any attempt of bringing it in accordance with the observational data must result, it seems, in more and more contrived versions of the original beautiful idea. In this sense, we may compare it with the classical steady state cosmological model of Bondi & Gold (1948), as well as Hoyle (1948), which has been discredited in the course of progress of observations, but which has had an epochal impact on the very formation of modern cosmology (Kragh 1996). In the same manner, Sciama's DDM theory, although it may be regarded as disproved by now, has inspired and provoked an immense theoretical and observational activity in astrophysics and cosmology. The results of these efforts will certainly present its lasting legacy.

Acknowledgements

The authors wholeheartedly thank Vesna Milosevic-Zdjelar for help in finding several important references. S.S. acknowledges the financial support of the Abdus Salam International Centre for Theoretical Physics, Trieste. This research has made use of NASA's Astrophysics Data System Abstract Service.