tokens: ΔΩ j indicates an increase of the quantity with cosmic time. 3 EXPECTED DUST DENSITY The basic ingredient of the calculation is represented by the cosmic star formation history (CSFH), ψ( z ). Adopt the analytical fit to the available data provided by Madau Dickinson ( 2014 ) with ψ 0 = 0.015. We can then compute the stellar mass density at any given redshift, z , by integrating In the previous expression H ( z ) = H 0 [Ω m (1 + z ) 3 + Ω Λ ] 1/2 is the Hubble parameter; for consistency with data, we will adopt the following values of the cosmological parameters (Ω m , Ω Λ , h ) = (0.3, 0.7, 0.7). The return fraction of gas from stars appropriate for a Chabrier initial mass function and instantaneous recycling approximation is R = 0.41. The previous expression can be cast in a more handy form where the is the non-dimensional integral in equation ( 4 ); further define , and note that , and . For the adopted cosmology, the stellar density parameter is . The density of stars formed from z = 1 to the present is ΔΩ * = 1.43 × 10 −3 . The metal density associated with Ω * is we have assumed a metal yield y = 0.033 ± 0.01, with errors accounting for uncertainties in the nucleosynthetic yields (Peeples et al. 2014 ; Vincenzo et al. 2016 ). Finally, the dust mass can be computed by knowing the dust-to-metal ratio, i.e. the fraction of metals locked into dust, , where and Z are the dust-to-gas ratio and gas metallicity, respectively. In the Milky Way, (Zubko, Dwek Arendt 2004 ); assuming solar metallicity, Z = 0.0142 (Asplund et al. 2009 ), then f d = 0.43 (Draine 2011 ). At higher redshifts f d in measured from the gas depletion patterns in, e.g. damped Lyman α (DLA) systems (Péroux Howk 2020 ). In the redshift range 0 z 1, (see Fig. 1 and Table 1 ) in the cold, neutral medium out of which stars form. We note that combines the effects of dust production, growth, and destruction, while we aim here at isolating the first process. Hence, we write the dust density associated with the metal density as where, in general, 2 . To determine f d , we proceed as follows. The minimum value, , is obtained by neglecting growth after grains are injected in the ISM by sources [supernovae (SNe) and asymptotic giant branch (AGB) stars]. SNe with progenitor mass form on average 0.3 M ⊙ of dust, 80 per cent of which is typically destroyed by the reverse shock on site (Todini Ferrara 2001 ; Bianchi Schneider 2007 ; Leśniewska Michałowski 2019 ). The AGB stars (>2 M ⊙ ) contribution per unit stellar mass formed for Z > 0.2–0.3 Z ⊙ is ≈4 times higher than the SN one (Zhukovska, Gail Trieloff 2008 ; Dell’Agli et al. 2017 ; Valiante et al. 2017 ). Hence, the combined effective dust yield per SN is y d = 0.3 M ⊙ , which gives , where is the number of SNe produced per stellar mass formed, according to the adopted Chabrier ( 2003 , equation 17) IMF. Larger values of f d might arise as a result of grain growth, which depends on ambient conditions (Asano et al. 2013 ;
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