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Astron. Astrophys. 335, 12-18 (1998)

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3. The photon energy E

3.1. The [FORMULA] flux from intergalactic HI clouds

Existing attempts to observe [FORMULA] radiation from opaque intergalactic HI clouds lead to an upper limit on the hydrogen-ionising photon flux F incident on the clouds arising from the cosmological distribution of neutrinos. This upper limit on F then leads to an upper limit on [FORMULA] because of the role of the red shift in reducing the energy of a decay photon to below 13.6 eV. Write


Then, if [FORMULA], F at zero red shift [FORMULA] is given by


where [FORMULA] is the standard cosmological number density of neutrinos at [FORMULA], (namely [FORMULA] of the number density of photons in the cosmic microwave background), c is the velocity of light, and [FORMULA] is the present value of the Hubble constant. In the decaying neutrino theory [FORMULA] km sec-1 Mpc-1 (Sciama 1997c).

The [FORMULA] upper limit observations of Vogel et al (1995) and of Donahue, Aldering & Stocke (1995) imply that [FORMULA] photons cm-2 sec-1. Since [FORMULA] (Sciama 1997c) it follows that


Combining this inequality with our previous value for [FORMULA], one obtains


so that indeed [FORMULA]. It follows that


We need not be too surprised that [FORMULA] is required to be so close to the ionisation potential of hydrogen, since cosmological considerations alone had previously suggested that [FORMULA] ev, so that [FORMULA] eV, which is within [FORMULA] of 13.6 eV.

The upper limit on [FORMULA] is based on the assumption that the intergalactic clouds are so opaque that essentially every incident ionising photon actually produces a free electron which then recombines with a proton. However, the clouds concerned may contain holes in their HI distribution which would then permit a larger ionising flux to be compatible with the observed upper limits on the [FORMULA] flux (Bland-Hawthorn 1996). The covering factor of the clouds concerned is in fact unknown. It is therefore desirable to establish an independent upper limit on [FORMULA]. Such a limit can be obtained from constraints on the hydrogen-ionising flux at red shifts in the range 2 to 4.5. These constraints can be obtained in two independent ways: (i) from the proximity effect in Lyman [FORMULA] clouds at these red shifts (Bajtlik, Duncan & Ostriker 1988) and (ii) from the recently observed absorption by He II at [FORMULA] in the spectra of high red shift QSOs.

3.2. The proximity effect in Lyman [FORMULA] clouds

The proximity effect is the reduction in the number of Lyman [FORMULA] clouds near a QSO (Murdoch et al 1986). This effect can be used to derive an upper limit on the ionising decay flux F(z) at a red shift z in the range [FORMULA]. The relation [FORMULA] (Sciama 1990a) then leads to an upper limit on [FORMULA]. Absorption of decay photons by intergalactic clouds does not have to be taken into account in deriving this relation even at [FORMULA] or greater, where in principle the absorption would be much larger than at [FORMULA] (eg. Haardt & Madau 1996), because with our small value for [FORMULA] the red shift reduces the energy of a decay photon to below 13.6 eV in a shorter distance than the absorption mean free path even at large z.

The proximity effect was first used by Bajtlik, Duncan & Ostriker (1988) to derive the background ionisation rate [FORMULA] per H atom for [FORMULA] by attributing this effect to the additional and known direct ionising influence of the QSO on nearby clouds. They obtained [FORMULA], with an uncertainty of a factor 3 either way. They also found that, within their uncertainty, [FORMULA] did not vary significantly with z. Of the many later attempts to determine [FORMULA] and its possible variation with z, the most recent are due to Giallongo et al (1996), Lu et al (1996), Savaglio et al (1997) and Cooke, Epsey & Carswell (1997). Despite the considerable remaining uncertainties there is a consensus that [FORMULA], with no significant z variation out to [FORMULA].

There has been much discussion in the literature as to whether the population of QSOs alone can provide a sufficient ionising flux to account for [FORMULA]. According to Haardt & Madau (1996) [FORMULA] at [FORMULA], but only [FORMULA]. The discrepancy of a factor [FORMULA] at [FORMULA] appears to be significant, even if the one at [FORMULA] lies within the uncertainties.

All the estimates of [FORMULA] from the proximity effect which have so far been made have been based on the simple assumption that the influence of the QSO on the nearby clouds is entirely due to the additional direct ionisation which it produces. However, as Miralda-Escudé & Rees (1994) (MR) pointed out, one should also allow for the additional heat input due to the radiation from the QSO. The resulting expansion of the clouds reduces their electron density, and the increase of temperature reduces the recombination coefficient, so that the recombination rate of the clouds is reduced. The net additional ionisation produced by the QSO is thus increased, and so the implied value of the background ionisation rate is also increased. MR further pointed out that, since the main contribution to the additional heating comes from the ionisation of He II in the clouds, the effect is larger where the QSO (but not the general UV background) is capable of ionising He II. This remark has become particularly pertinent now, because it seems that ionisation breakthrough for He II in the intergalactic medium may not have occurred until [FORMULA] (Songaila & Cowie 1996, Hogan et al 1997, Reimers et al 1997, Boksenberg 1998, Songaila 1998), a situation which was theoretically anticipated by MR and by Madau & Meiksin (1994).

While the numerical value of the MR effect is model dependent (preliminary attempts to estimate it having been made by MR themselves) it is clear that it has two consequences for our discussion. It leads to an increase in the (small) discrepancy between [FORMULA] and [FORMULA] at [FORMULA], and to an increase of [FORMULA] with z, in contrast to the rapid decrease in [FORMULA].

It has often been suggested that any discrepancy between [FORMULA] and [FORMULA] could be resolved by appealing to ionising radiation emitted by hot stars in galaxies (eg. Miralda-Escudé and Ostriker 1990). Recent discussions of this possibility have been given by Giroux & Shapiro (1996) and by Madau & Shull (1996). Metal-enrichment arguments imply that if [FORMULA] of the Lyman continuum photons emitted by hot stars escape from their parent galaxies, then these photons would be responsible for an ionisation rate [FORMULA] per H atom at [FORMULA]. The actual escape fraction is not known at high z, but at [FORMULA] it is less than [FORMULA] (Deharveng et al 1997). This strong constraint was deduced by relating the [FORMULA] luminosity density of star-forming galaxies in the local universe (Gallego et al 1995) to the [FORMULA] observations of Vogel et al (1995) and Donahue et al (1995) which, as already mentioned, lead to an upper limit on the hydrogen-ionising flux at [FORMULA]. Although the escape fraction is not known at high z, there exists evidence for considerable dust extinction in galaxies at these redshifts (Meurer 1997, Cimatti et al 1997). Absorption by atomic hydrogen in these galaxies may also be important.

This argument has recently been much strengthened by the observations of Spinrad et al (1998) which failed to detect any Lyman continuum radiation escaping from [FORMULA] Lyman-limit galaxies. According to these authors it is implausible that ionising radiation from young galaxies can replace QSO ionisation at [FORMULA].

Another much-discussed possibility is that radiation from stars at red shifts much greater than 5 (the so-called Population III stars) might make an appreciable contribution to [FORMULA] and be responsible for the reionisation of the universe at [FORMULA]. Recent discussions of this possibility have been given by Haiman & Loeb (1997), Gnedin & Ostriker (1997) and Gnedin (1998). An important aspect of this proposal concerns the level of metallicity which would result from the required stellar activity. It is not clear whether this level agrees with observation, especially in view of the low metallicity recently derived by Songaila (1997) for the Lyman [FORMULA] clouds. We cannot go into this intricate question here,and since we are seeking an upper limit on the intergalactic flux of decay photons, it will now be assumed that most of the missing ionising photons at [FORMULA] are produced by the cosmological distribution of decaying neutrinos and not by hot stars.

This assumption would enable us to understand why the usual interpretation of the proximity effect leads to a value of [FORMULA] which is approximately independent of z. If, at [FORMULA], [FORMULA] is of the same general order as [FORMULA] then, as z increases, [FORMULA] would increase as [FORMULA] while [FORMULA] decreases, leaving [FORMULA] approximately constant.

It would seem that a rough upper limit for [FORMULA] can be derived by setting [FORMULA] at [FORMULA]. Then, since at this red shift [FORMULA] according to Haardt & Madau (1996), [FORMULA] would [FORMULA] and [FORMULA]. This excess of [FORMULA] over the proximity effect value of [FORMULA] could be attributed to the MR effect. Indeed, since in this picture [FORMULA] is the dominant contributor to [FORMULA], the spectrum of the QSO would differ appreciably from that of the background, and the MR effect would then be greater than in the pure QSO case (Rees 1990, Sciama 1995). Finally, it should be noted that, with our adopted values, [FORMULA] would increase by a factor 1.7 between [FORMULA] and [FORMULA]. This increase could be consistent with the dependence of the MR effect on z.

Our upper limit on [FORMULA] at [FORMULA] implies an upper limit on [FORMULA] at [FORMULA] of [FORMULA]. Since the decay photons have an energy close to the Lyman limit this ionisation rate converts to a photon flux by using the photoionisation cross-section at this limit, which is [FORMULA]. Hence one obtains [FORMULA], which is not essentially different from the upper limit [FORMULA] derived by Vogel et al (1995) and by Donahue, Aldering & Stocke (1995), the precise value of which in fact depends on the uncertain shapes of the intergalactic clouds which they observed.

3.3. HeII absorption at [FORMULA]

We now show that the recently derived Gunn-Peterson optical depth in HeII at [FORMULA], [FORMULA] (Zheng et al 1998), in conjunction with the known upper limit on [FORMULA], leads to a lower limit on [FORMULA] at [FORMULA]. It turns out that this lower limit is close to the upper limit estimated in Sec. 3.2.

There now exist a number of observations of HeII absorption at [FORMULA] to 3 in the spectra of QSOs (Jakobsen et al 1994, Tytler et al 1995, Jakobsen 1996, Davidsen, Kriss & Zheng 1996, Hogan, Anderson & Rogers 1997, Reimers et al 1997). There has been considerable discussion in these and other papers (Madau & Meiksin 1994, Fardal, Giroux & Shull 1998) as to whether this absorption is entirely due to the HeII in Lyman [FORMULA] clouds,or whether part of it must be attributed to the Gunn-Peterson effect arising in an essentially diffuse intergalactic medium. We here follow the calculations of Zheng, Davidsen & Kriss (1998), which lead to a definite value of [FORMULA] for this effect at [FORMULA]. This would imply that [FORMULA] cm-3. To see whether this result is reasonable we derive from it the implied value of the total diffuse intergalactic gas density [FORMULA] at [FORMULA], using estimates for the HeII-ionising flux due to QSO radiation filtered through the absorbing medium of Lyman [FORMULA] clouds and Lyman limit systems (Haardt & Madau 1996, Fardal, Giroux & Shull 1998), and the value 0.08 for the He/H number ratio. Using [FORMULA] sec-1 one obtains [FORMULA] cm-3. Comparing this with the higher of the two competing values for the total baryon density [FORMULA], based on measurements of the deuterium abundance and the theory of big bang nucleosynthesis (Schramm & Turner 1998), one finds that [FORMULA].

This result is compatible with the somewhat model-dependent estimate of [FORMULA] made by Giallongo, Fontana & Madau (1997). These authors found that at [FORMULA] about half of [FORMULA] could be attributed to gas in Ly [FORMULA] clouds, leaving about half for the IGM (since the contribution from galaxies can here be neglected (Persic & Salucci 1992)). Given the uncertainties, this fraction of 1/2 is compatible with our derived value of 0.36.

The next step is to use the [FORMULA] upper limit of 0.04 for [FORMULA] (Giallongo, Cristiani & Trevese 1992). Since


it follows that


so that [FORMULA]. This lower limit is compatible with the upper limit for [FORMULA] ([FORMULA] sec-1) proposed in Sec. 3.2.

This comparison is not strictly self-consistent because, by choosing a value of [FORMULA] greater than that due to QSOs, a disturbance has been introduced into the calculation of the opacity of the universe,since the ionisation state of the absorbers would be affected. This disturbance would be reduced if in fact [FORMULA] from QSOs had to be increased by a factor of the same order as the ratio [FORMULA], since then [FORMULA] would not be much altered.

An increase in the HeII - ionising power of QSOs over that arising from the usual power-law spectrum has already been proposed by Sciama (1994), who needed to ensure that an increase in [FORMULA] due to decay photons would not drive the universe to become completely opaque at the HeII edge. This proposal was based on existing observational hints that many QSOs possess a soft x-ray excess in their spectra. This excess was attributed to a Guilbert-Rees (1988) thermal bump with [FORMULA] eV, resulting from the reprocessing of harder x-rays from the central regions of QSOs by optically thick cold material. Since 1994 further observational evidence has accumulated for the prevalence of a soft x-ray excess in the spectra of QSOs. This evidence has been reviewed by Gondhalekar, Rouillon-Foley & Kellett (1996). It is also noteworthy that a 50 eV bump would fit nicely in the gap (due to galactic absorption) in the composite spectrum shown in fig.6 of Laor et al (1997). On the theoretical side it has been found recently that a slim accretion disk around a black hole at the centre of a QSO would produce a soft x-ray excess without any Guilbert-Rees reprocessing (Shimura & Takahara 1995, Szuszkiewicz 1996, Szuszkiewicz, Malkan & Abramowicz 1996).

These arguments have recently been strengthened by the considerations of Korista, Ferland & Baldwin (1997) who pointed out that a QSO emission spectrum without a bump at 50 ev would not account (via excitation effects) for the observed strengths of the HeII emission lines in QSO spectra. These authors suggested that either the QSOs have a suitably complicated geometry, or that their emission spectrum contains a significant bump in the vicinity of the HeII ionisation edge at 54.4 ev.

A rough estimate for the resulting increase in [FORMULA] led to a factor [FORMULA] (Sciama 1994). If the actual factor were closer to 2 the absorption analysis would still not be much changed, while the lower limit on [FORMULA] would be increased to [FORMULA] sec-1 which is the same as our proposed approximate upper limit. A self-consistent solution is thus possible. For this solution one would have [FORMULA], which is in good agreement with the estimate implied by the calculations of Giallongo, Fontana & Madau (1997).

Our proposed introduction of an appreciable flux of decay photons at high z also has implications for the abundance of HeI at these red shifts. It was argued by Miralda-Escudé & Ostriker (1992) and by Reimers et al (1993) that the low values of [FORMULA] observed in Lyman limit systems and Lyman [FORMULA] clouds are incompatible with the decaying neutrino theory. However, Sciama (1994) showed that if QSOs possess a soft x-ray bump in their spectra, the resulting high ionisation of HeII in the various cloud systems would sufficiently lower their abundance of HeI. A further reduction in [FORMULA] could arise from hot stars in galaxies, since the escape fraction of HeI-ionising photons would be expected to exceed that of HI-ionising photons. Accordingly the existence of an appreciable flux of decay photons at high z is not incompatible with the observed values and upper limits on [FORMULA].

Our conclusion from all these considerations is that it is unlikely that F(0) can be increased by a substantial factor, say [FORMULA], over the upper limit derived from the [FORMULA] measurements of intergalactic clouds, by appealing to a small HI covering factor in these clouds. Accordingly the constraint [FORMULA] eV still holds good, so that



and, from Sect. 2


These are our updated parameters for the decaying neutrino theory. It should be noted that these parameters lead to precise values for the Hubble constant [FORMULA] and the age of the universe [FORMULA] Gyr) (Sciama 1997c), if the cosmological constant is zero.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998