2. The decay lifetime
2.1. The ionisation of the interstellar medium
An upper limit on will now be derived from recent estimates of the maximum permitted density of dark matter near the sun and of the free electron density which is here being attributed to the ionisation of hydrogen by decay photons.
In Sect. 2.2 a lower limit on will be determined from recent observational upper limits on the extragalactic diffuse background at , which impose an upper limit on the flux of red shifted decay photons emitted by the cosmological distribution of neutrinos.
Our derived upper and lower limits on are only just consistent with one another, and so lead to a highly constrained value for this quantity.
To derive the upper limit on consider a region near the sun whose atomic hydrogen density makes it opaque to ionising decay photons. In ionisation equilibrium one would have
where is the local number density of neutrinos and is the recombination coefficient (excluding recombinations directly to the ground state). The value of which the theory attributes to ionisation by decay photons has been recently rediscussed by Sciama (1997a). Using observations of dispersion measures for nearby pulsars with known radio parallaxes (Gwinn et al 1986, Bailes et al 1990), and HST observations of the absorption spectrum of the halo star HD93521 (Spitzer & Fitzpatrick 1993) (which are somewhat less straightforward to interpret), the result obtained was (Reynolds 1990, Sciama 1990b, 1997a).
Finally, to derive an upper limit on an upper limit on the number density must be established. This can be obtained from an upper limit on the local mass density of neutrinos , since earlier discussions of the decaying neutrino theory have already provided a sufficiently accurate value for , namely (Sciama 1990 a, 1995). An upper limit on near the sun was recently derived by Sciama (1997 b) in connexion with values for the largest permitted flattening of the neutrino halo and for the rotational velocity of the Galaxy at the sun's position. By taking into account various estimates for column densities of material near the sun an upper limit for of 0.03 pc-3 was obtained. This upper limit would be associated with the largest permitted flattening of the dark halo (Dehnen & Binney 1997), corresponding to an axial ratio of 0.2 (shape E8). It is reassuring that some other galaxies do seem to exhibit a similar flattening (Sackett et al 1994, Olling 1996, Becquaert & Combes 1997).
The local mass density can also be determined by comparing dynamical estimates of the total density near the sun (the Oort limit) with the densities of known stars and gas. The value of has been controversial; an estimate will be used here which is based on recent Hipparcos observations of F stars, and so may lead to a more reliable result. This result is pc-3 (Pham 1996, 1997). However, a lower value for has recently been obtained from Hipparcos data by Crézé et al (1998), namely pc-3. This result seems rather low when compared to the density of known matter (see below) and the disagreement remains to be clarified. Here we provisionally adopt Pham's value.
The contribution of known stars and gas to is itself somewhat uncertain. Often quoted values for each are pc-3 (e.g. Bienaymé, Robin & Crézé 1987, Crézé et al 1998). The stellar contribution may have to be increased to allow for the existence of faint stars, but the HST deep survey suggests that this additional contribution may be small (e.g. Gould et al 1996). If this is correct, one again obtains an upper limit for of pc-3. This upper limit was previously derived by Bienaymé et al (1987). If now , as derived in Sect. 2.2, there follows an upper limit for near the sun of .
Our limits on the values of , and now lead, in conjunction with (1), to the conclusion that
2.2. The extragalactic background at 1500 Å
Some of the earliest lower limits on (for a decaying neutrino not then related to the ionisation of the interstellar medium) were based on observational estimates of the cosmic background in the far ultra-violet, which was compared with the red shifted decay flux produced by the cosmological distribution of neutrinos (Stecker 1980, Kimble, Bowyer & Jakobsen 1981). Estimates of this background are still controversial (compare Bowyer 1991 with Henry 1991). Recent contributions to the discussion have been made by Henry & Murthy (1993), Witt & Petersohn (1994) and Witt, Friedmann & Sasseen (1997). We adopt from their discussions an upper limit of about 300 photons cm-2 sec-1 ster (continuum units or CU) at . From this value one must subtract the contribution due to galaxies, which has been evaluated by Armand, Milliard and Deharveng (1994) as about , leaving an upper limit of about available for the decay flux.
Theoretical aspects of this flux have been re-discussed by Sciama (1991), Overduin, Wesson & Bowyer (1993), Dodelson & Jubas (1994) and most recently by Overduin & Wesson (1997). These calculations show that a decay flux of at corresponds to a lifetime of sec. Hence
Our overall conclusion from this discussion is that, if the decaying neutrino theory for the ionisation of the interstellar medium is correct, the decay lifetime is determined as
The uncertainty in this estimate is difficult to pin down. A reasonable guess would be at most. Hence we adopt
We note, however, that if the Crézé et al (1998) value for is correct no solution for is possible, and the decaying neutrino theory would be ruled out.
© European Southern Observatory (ESO) 1998
Online publication: June 12, 1998