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The deduction of past star formation rates from rest-frame UV radiation in the Hubble and other deep fields as a function of red-shift is tied to "metal" production through the Lilly-Cowie theorem (Lilly & Cowie 1987):

Equation 5   (5)
Equation 6   (6)
     Equation 7   (7)

where (1 + a) appeq 2.6 is a correction factor to allow for production of helium as well as conventional metals and beta (probably between about 1/2 and 1) allows for nucleosynthesis products falling back into black-hole remnants from the higher-mass stars. epsilon is the fraction of total energy output absorbed and re-radiated by dust and nuH is the frequency at the Lyman limit (assuming a flat spectrum at lower frequencies). The advantage of this formulation is that the relationship is fairly insensitive to details of the IMF.

Eq (7) is the same as eq (13) of Madau et al (1996), so I refer to the metal-growth rate derived in this way as rhodotZ(conventional).

Assuming a Salpeter IMF from 0.1 to 100 Msun with all stars above 10 Msun expelling their synthetic products in SN explosions, one then derives a conventional SFR density through multiplication with the magic number 42:

Equation 8   (8)

In general, we shall have

Equation 9   (9)

where gamma is some factor. E.g., for the IMF adopted by FHP, gamma = 0.67, whereas for the Kroupa-Scalo one (Kroupa et al 1993) gamma = 2.5.

Finally, the present stellar density is derived by integrating over the past SFR and allowing for stellar mass loss in the meantime, and the metal density is related to this through the yield, p:

Equation 10   (10)
Equation 11   (11)

(where alpha is the lockup fraction), whence (if a = 1.6)

Equation 12   (12)

which can be compared with Zsun appeq 1/60. It was pointed out by Madau et al (1996) that the Salpeter slope gives a better fit to the present-day stellar density than one gets from the steeper one - a result that is virtually independent of the low-mass cutoff if one assumes a power-law IMF.

Eq (8), duly corrected for absorption, forms the basis for numerous discussions of the cosmic past star-formation rate or "Madau plot". Among the more plausible ones are those given by Pettini (1999) shown in Fig 2 and by Rowan-Robinson (2000), which leads to similar results and is shown to explain the far IR data. Taking gamma = 0.62 (corresponding to a Salpeter IMF that is flat below 0.7Msun) rather than Pettini's value of 0.4 (for an IMF truncated at 1Msun), and alpha = 0.7, we get the data in the following table.

Figure 2

Figure 2. Global comoving star formation rate density vs. lookback time compiled from wide-angle ground-based surveys (Steidel et al. 1999 and references therein) assuming E-de S cosmology with h = 0.5, after Pettini (1999). Courtesy Max Pettini.

Table 2 indicates that the known stars are roughly accounted for by the history shown in Fig 2 (or by Rowan-Robinson) and the metals also if beta is close to unity, i.e. the full range of stellar masses expel their nucleosynthesis products. At the very least, beta has to be 1/2, to account for metals in stars alone. The other point arising from the table, made by Pettini, is that at a red-shift of 2.5, 1/4 of the stars and metals have already been formed, but we do not know where the resulting metals reside.

Table 2. Inventory of stars and metals at z = 0 and z = 2.5

z = 0 z = 2.5

rho* = alpha gamma integ rhodot*(conv.) dt 3.6 × 108 Msun Mpc-3 9 × 107 Msun Mpc-3
Omega* = rho* / 1.54 × 1011 h702 .0024h70-2 6 × 10-4 h70-2
Omega*(FHP 98) .0035h70-1
rhoZ = p rho* = betarho* / (42alpha gamma) 2.0 × 107 beta Msun Mpc-3 5 × 106 beta Msun Mpc-3
OmegaZ (predicted) 1.3 × 10-4 beta h70-2 3.2 × 10-5 beta h70-2
OmegaZ (stars, Z = Zsun) 7 × 10-5 h70-1
OmegaZ (hot gas, Z = 0.3Zsun) 1.0 × 10-4h70-1.5
-> 0.5 leq beta leq 1.3
OmegaZ (DLA, Z = 0.07Zsun) 2 × 10-6 h70-1
OmegaZ (Ly. forest, Z = 0.003Zsun) 1 × 10-6 h70-2
OmegaZ (Ly. break gals, Z = 0.3Zsun) ?
OmegaZ (hot gas) ?

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