A Simplified Proof of Cosmic Metallicity Evolution

1. Assume a 1-zone model of galaxy of fixed size with average density ρ. As a simplifying assumption, we assume that no matter is permanently locked up into compact remnants (white dwarfs, neutron stars, black holes), brown dwarfs, planets, or smaller astro-bodies. We can easily dispense with this assumption if want to.

2. A differential equation determining the change in ρ with time t is

   (dρ/dt) = (dρ/dt)_inflow - κρ ,

   where (dρ/dt) is rate of change in ρ, (dρ/dt)_inflow is a constant inflow rate, κ = 1/τ is the rate of outflow constant, and τ is time parameter that is the inverse of κ. The solution of the differential equation via an integrating factor is

   ρ = (dρ/dt)_inflow*τ*[1-exp(-t/τ)] + ρ_0*exp(-t/τ)     in general;

   ρ = (dρ/dt)_inflow*t + ρ_0                                            for t << τ;

   ρ = ρ_∞ = (dρ/dt)_inflow*τ                                           for t → ∞,

   where ρ_0 is the initial density at time zero whatever that is for the galaxy. We see that ρ saturates at ρ_∞ = (dρ/dt)_inflow*τ. Note, as the rate of outflow constant decreases (i.e., τ increases), the saturation density increases: i.e., τ ↑ ρ_∞ ↑.

3. Now metallicity times density Zρ obeys the differential equation

   [d(ρZ)/dt] = Z_IGM(dρ/dt)_inflow - κ(Zρ) + γρ ,

   where Z_IGM is the constant metallicity of the intergalactic medium (IGM) and γ is the rate of creation of metallicity constant which has units of inverse time. For the saturation solution, we set [d(ρZ)/dt] = 0 and ρ = ρ_∞, and obtain

   Z = γτ + Z_IGM .

   If γτ = 0, then the metallicity is just that of the intergalactic medium (IGM). Note, as (γτ) ↑ Z ↑.