1 Introduction

Decaying dark matter (DDM) theory is an attempt to simultaneously solve two important problems of contemporary astrophysics: the dark matter problem in spiral galaxies, like the Milky Way, and the problem of ionization of the interstellar and intergalactic medium (Sciama 1993). To achieve these goals, theory introduces massive decaying neutrino with the mass $m_\nu \sim$ 30 eV. This neutrino has a decay lifetime of $2\pm 1\times 10^{23}$ s, that produces a decay photon of energy of $13.7\pm 0.1$ eV (Sciama 1998). This theory is heavily constrained, i.e. its parameters are very well defined, with extremely small uncertainties. An experiment, EURD, has been proposed in order to test the theory (Sciama 1993). Results recently published suggest that this theory is no longer viable, because the emission predicted by the DDM theory was not registered (Bowyer et al. 1999). In this Paper we wish to investigate the values of the Hubble constant and predicted age of the universe in the DDM theory, in the light of two recent important empirical discoveries: first that neutrinos do have mass (Fukuda et al. 1998), and the second one according to which there exists a large positive cosmological constant (Perlmutter et al. 1998, 1999; Reiss et al. 1998).

In this respect, it seems that we are in the middle of a major change of cosmological paradigm (not unexpected, however, as even the cursory look at the relevant literature could show). Recent results of the surveys of the type I supernovae at cosmological distances indicate the possible presence of a large cosmological constant (Perlmutter et al. 1998, 1999; Reiss et al. 1998). If the total cosmological density parameter corresponds to the flat ($\Omega=1$) universe, the contribution due to matter density is (total 1$\sigma$ statistical + systematic errors quoted)

$$\Omega_{\rm m} = 0.28^{+0.14}_{-0.12}\cdot$$ (1)

This result suggests not only that the universe will expand indefinitely, but that it will expand in an (asymptotically) exponential manner, similar to the early inflationary phase in its history. In addition to these observations, we use results from the primordial nucleosynthesis which are entering the high-precision phase (Schramm & Turner 1998), and limit the combination of baryonic density fraction and the Hubble parameter. We shall use the following (conservative) limits:

$$\Omega_{\rm b} h^2= 0.025 \pm 0.005.$$ (2)

These are larger values than those used by Sciama (1997), but this can be justified on several counts. First of all, later measurements of deuterium abundance at high redshift unambiguously indicate lower abundances than previous controversial values (Burles & Tytler 1998). In addition, measurements of HeII Gunn-Peterson effect at high redshift (Jakobsen 1998) gave very high values for $\Omega_{\rm b} h^2$, even higher than those in Eq. (2). For the sake of completeness, we have used both this realistic, and the lower value of Sciama (1997) in further calculations.

One should add the following epistemological consideration. Being the property of the quantum vacuum itself, addition of the non-zero cosmological constant does not prima facie increase the conceptual complexity of the theory for dark matter. However, if we believe in classical prediction of the inflationary scenario $\Omega = 1 \pm \epsilon$with the precision $\epsilon \simeq 10^{-5}$, we have to take into account this additional constraint on the distribution of total energy density in the universe. We shall use this assumption in the further considerations.

We shall use the following notation: symbol $\Omega$ without any subscripts will be reserved for the total density parameter of the universe, which, according to our present understanding can be written as the sum of densities of matter and vacuum density (which is manifested in the form of the cosmological constant $\Lambda$), i.e.

$$\Omega \equiv \Omega_{\rm m} + \Omega_\Lambda.$$ (3)

The contribution of matter can be written as

$$\Omega_{\rm m} \equiv \frac{\rho_{\rm m}}{\rho_{\rm crit}} = \frac{8 \pi G \rho_{\rm m}}{3 H_0^2},$$ (4)

and the one of the cosmological constant as

$$\Omega_\Lambda = \frac{c^2 \Lambda}{3H_0^2} = 2.8513 \times 10^{55} \, h^{-2} \Lambda,$$ (5)

$\Lambda$ being in units of cm^-2. The present day Hubble constant is parametrized in a standard way as $H_0 \equiv 100 \, h \; {\rm km \; s}^{-1} \; {\rm Mpc}^{-1}$. $\Lambda$ enters the Einstein field equations as

$$R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R - \Lambda g_{\mu \nu} = - \frac{8\pi G}{c^4} T_{\mu \nu},$$ (6)

and $\Lambda$-universes are the homogeneous and isotropic solutions of these tensor equations (for other notation see any of the standard General Relativity textbooks, e.g. Weinberg 1972; for history and phenomenology of the cosmological constant, see the detailed review of Carroll et al. 1992, and references therein). We now wish to investigate whether an inflationary DDM universe can be reconciled with non-zero $\Lambda$ and still perform its explanatory tasks.