2 Simple parameter estimates

It is not possible to proceed in as simple way as in Sciama (1997), since the combination $\Omega_\Lambda h^2$ does not have an obvious physical meaning. However, we shall use this circumstance in order to establish plausible values for h first. For establishing connection between ionizing flux F and $m_\nu$ we follow the same simple procedure outlined by Sciama (1997), except that it is not possible any more to simply plug in the "final'' value for h as it has been done in that study. This situation gives rise to the term linear in h, which is the main source of difficulties here. Therefore, we obtain for the decaying neutrino mass

$$m_\nu = (27.2 + 0.39 h) \pm 0.39 h \; {\rm eV}.$$ (7)

Hence, the contribution to the cosmological density fraction is

$$\Omega_\nu h^2= \frac{27.2 + 0.39 h}{93.6} = 0.2906 + 0.0042 h.$$ (8)

Assuming in the spirit of DDM theory that $\Omega_{\rm m} = \Omega_\nu + \Omega_{\rm b}$, we can write

$$\Omega h^2 = \Omega_\Lambda h^2+ \Omega_\nu h^2 + \Omega_{\rm b} h^2,$$ (9)

or, equivalently, taking into account Eqs. (2) and (8), we have

 

$$(\Omega - \Omega_\Lambda) h^2 0.025 + 0.2906 + 0.0042\, h$$ =

$$0.3156 + 0.0042\, h.$$ = (10)

Now we may use the theoretical prejudice for $\Omega=1$that is in agreement with the recent Boomerang result (de Bernardis et al. 2000), and therefore $\Omega_\Lambda$ is determined by Eq. (1). Later we shall discuss the consequences of variations in $\Omega$in the observationally allowed range (approximately 0.3-1.1). The physically acceptable solution of this quadratic equation in h is h = 1.07.

The physical picture here is highly intuitive: for the fixed total density parameter, introduction of a term with negative effective pressure in the Friedmann equation results in a faster expansion rate. However, all recent observational measurements have suggested lower values for the Hubble parameter, in the 0.5-0.8 range, even tending toward the lower limit (e.g. Paturel et al. 1998; Schaefer 1998). We notice that we recover the particular value h = 0.55for $\Lambda=0$ obtained by Sciama (1997), as expected.