3. Dust opacity model
We adopt a recent model for intergalactic extinction in which dust is clumped into damped Ly absorption systems whose numbers and density profiles are sufficient to obscure a portion of the light reaching us from , but not to account fully for the turnoff in quasar population (Fall & Pei 1993; hereafter "FP"). The mean opacity at observed wavelength due to these dusty clumps (out to redshift z) is:
Here is the comoving dust density in units of optical depth per Hubble radius and is the ratio of extinction at wavelength to that in the B-band (4400 Å). Earlier treatments (Ostriker & Cowie 1981 , Ostriker & Heisler 1984) took constant and , so that or , depending on cosmology. We follow the more sophisticated approach of FP, in which dust opacity depends on redshift:
where and are fixed by observational data on extinction due to hydrogen at and (for damped Ly systems) at , together with estimates of dust-to-gas ratios (see FP for discussion). The data are consistent with lower limits of , (model A), best-fit values of , (model B), or upper limits of , (model C), assuming a critical density .
To calculate the extinction in the 300-2000 Å range, we use numerical Mie scattering routines in conjunction with various interstellar dust models. In performing these calculations, we have tacitly assumed that intergalactic and interstellar dust are similar in nature, which is a reasonable assumption that is, of course, very difficult to test. Many people have constructed dust grain models that reproduce the average interstellar extinction curve for Å (as given by Mathis 1990, for example), but there have been far fewer studies of the optical properties of the diffuse interstellar medium (DISM) at shorter wavelengths. One such study is that of Martin & Rouleau (1991; hereafter "MR"). These authors extended the silicate/graphite grain synthetic extinction curves produced by Draine & Lee (1984) to photon energies in excess of 100 eV. Like Draine & Lee, they assumed: (1) two populations of homogeneous spherical dust grains composed of graphite and silicates respectively; (2) a power law size distribution of the form where a is the grain radius; (3) the range of grain radii is 50-2500 Å; and (4) an interstellar abundance of carbon and silicon relative to hydrogen that was the same as that for the Sun (as given by Meyer 1979). It has recently been discovered that the heavy element abundances in the DISM are significantly lower than the Solar values. For example, Snow & Witt (1996) report interstellar abundances of carbon and silicon to be H and H respectively. This represents a 50% reduction in the elemental abundances used by Draine & Lee and MR, which imposes a severe handicap on the simple silicate/graphite model's ability to reproduce the observed DISM extinction curve.
These difficulties have prompted us to perform fresh extinction calculations based on new grain models. Since we are primarily interested in obtaining the most conservative bounds possible on the neutrino decay hypothesis, we have attempted to find dust grain populations with optimal extinction efficiency in the neighborhood of 861 Å. In so doing, we do not aim to create a comprehensive new dust model, or to reproduce the average interstellar extinction curve to a high degree of accuracy over all wavelengths. However, the average characteristics of interstellar dust do provide reasonable constraints on intergalactic dust which we try to adhere to as closely as possible.
In the interests of simplicity, the materials used to comprise our grain populations are characterized by the graphite and "astronomical silicates" dielectric functions provided by Draine (1995). In calculating the extinction profiles of spherical graphite particles, we employ the 1/3-2/3 approximation discussed in Draine & Malhotra (1993). Due to the lack of standard dielectric functions in the FUV we do not consider contributions due to amorphous carbon (AC), hydrogenated amorphous carbon (HAC), glassy carbons or other materials. We first considered a grain model identical to that of MR as a test of our code. We successfully reproduced the shape of their extinction curves, although the absolute intensity of our results somewhat weaker (presumably due to differences in the new dielectric functions). Having satisfied ourselves with the functionality the Mie scattering algorithm, we then calculated the extinction due to the same grain model, but now with the DISM abundances given by Snow & Witt (1996) and the maximum possible depletions of C and Si from the gas phase. The results of this calculation are plotted in Fig. 1 as the continuous solid line (population 1). We have also plotted the average interstellar extinction curve (open circles; Mathis 1990) for reference, along with the position of the neutrino decay peak (the vertical dotted line). The inadequacies of the old dust model are apparent from this figure; while the shape of the average curve is well approximated, the magnitude of the synthetic extinction is far too low. The amount of extinction in the vicinity of the neutrino decay peak is also relatively weak (the largest FUV extinction is ), making this grain model a rather poor mechanism for hiding excess EBL intensity.
It has recently been suggested that the heavy element abundance crisis in the DISM might be resolved by the presence of "fluffy" grains (Mathis 1996). In these models, the bare silicate grains are replaced with composite grains consisting of silicates and carbonaceous materials like AC or HAC with a high porousity or void fraction. By suitable variation of the model parameters, very good fits to the average interstellar curve are possible. To study the extinction characteristics of such model in the FUV, we modify the previous population by replacing the ordinary silicate grains by silicate grains with a 45% void fraction as in Mathis (1996). The dielectric function of fluffy silicates is obtained using the Bruggmann effective medium approximation (Bohren & Huffmann 1983, Eq. 8.51). We have also modified the size distribution of the graphite grains to include only small particles (a = 50-250 Å) and decreased the carbon depletion to 60% to better match the average curve. Following Dwek (1997), we assume a Si abundance of /H for this population alone. The result is the dotted (population 2) curve of Fig. 1. This grain population has much better success in matching the shape of the average curve and shows significantly more FUV extinction than population 1 (). The fit to the average curve would be improved by the inclusion of AC in the fluffy grains.
It has also been recently suggested that the carrier of the 2175 Å absorption bump is not a population of spherical graphite particles with radii 50 Å, but rather polycyclic aromatic hydrocarbon (PAH) nanoparticles. These structures are thought to be consist of stacks of PAH-like molecules resembling coronene, circumcoronene and larger species in various states of edge hydrogenation. There are several attractive properties of this model, including a very good fit to the 2175 Å feature shape and position (Duley & Seahra 1998) and an explanation of the unidentified infrared emission (UIR) bands detected in the DISM at 3.4-25 µm (Dwek 1997, Seahra & Duley, in prep.). Also, PAH nanoparticles could be responsible for the extended red emission (ERE) observed in nebular environments (Seahra & Duley 1999) and 3.4 µm infrared absorption in in the DISM (Duley & Seahra, in prep.). Now, these particles have sizes of 7-30 Å, much smaller than the canonical graphite grain populations. However, as discussed in Duley & Seahra (1998), the dielectric function of PAHs should be similar to that of graphite with the major difference coming from the free -electron characteristics of each material. In the limit, the two dielectric functions merge (see Fig. 1 of Duley & Seahra 1998). So as a first approximation to the extinction curves produced by nanoparticle populations, we use spherical graphite particle populations with very small radii. A truly rigorous calculation would use the discrete dipole array (DDA) formalism of Draine (1988), but this is computationally expensive at FUV wavelengths.
We treat the bounds on the power-law size distribution of the PAH nanostructures as variable parameters in order to maximize the FUV extinction. We find the optimal extinction in the neighborhood of the neutrino decay peak is achieved for a PAH population with radii ranging from 3-150 Å. On Fig. 1, the dashed line (population 3) reflects the extinction produced by such a PAH nanoparticle population with a silicate population identical to that of population 1. The dash-dot-dot-dashed line on Fig. 1 (population 4) represents the same nanoparticle population with a "fluffy" silicate population identical to that of population 3 except for a 45% porousity. Both populations 3 and 4 give poor fits to the average interstellar curve, as expected since the Mie scattering formalism is not expected to reproduce the behaviour of the nanoparticles near the 2175 Å resonance. Also, the lack of opaque materials such as AC reduces the synthetic extinction in the visible and near UV part of the spectrum. However, in the FUV region, we see a doubling in the magnitude of the extinction peak at 14 µm-1 compared to that of population 2 (). Evidently, the small particles are much more efficient at extinguishing light at shorter wavelengths. This is partly the reason that there is little difference between populations 3 and 4; the extinction curves are dominated by small particle contributions, so the void fraction of the silicates does not have a large effect.
Fig. 1 shows that population 1 and population 3 (or 4) dust models lead to lower and upper limits respectively on dust extinction in the FUV waveband. We therefore use these populations to obtain conservative constraints on the decaying neutrino hypothesis (regardless of the fact that they do not produce a perfect fit to the average extinction curve). We do not consider possible secondary scattering of photons back into the line of sight. This is a comparatively small effect at the wavelengths of interest here (Martin & Rouleau 1991). Moreover, neglect of this factor can only enhance the overall dust opacity along the line of sight, strengthening our conclusions further.
© European Southern Observatory (ESO) 1999
Online publication: August 25, 1999