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tokens: by supergiant shells. Simulations of supernova-driven turbulence in a representative section of a stratified galactic disk by Joung Mac Low ( 2006 ) came to a similar conclusion. Neglecting the action of self-gravity on the gas, they found that only 10% as much gas was swept up into regions able to gravitationally collapse as would have been required to maintain the star formation rate implied by the input supernova rate. 3. Toomre Instability Therefore, we now turn to large-scale gravitational instability in disks (Safronov 1960 ; Toomre 1964 ; Goldreich Lynden-Bell 1965 ), which offers an alternative mechanism to assemble gas dense enough to form molecules. Galactic dynamics appears to be driven by such instabilities, which at the largest scales drive spiral arm formation (Agertz et al. 2009 ; Wada et al. 2011 ). When these instabilities grow strongly enough because of gas infall or galaxy mergers they lead to the formation of giant clumps (Li et al. 2005 ) in what is called violent disk instability (Agertz et al. 2009 ; Dekel et al. 2009 ). (Those clumps may well themselves collapse into clusters of small clumps as shown by Behrendt et al. 2016 .) That molecular clouds can form more quickly by gravitational instability than collisional agglomeration was shown by Elmegreen ( 1990 ), despite their apparently having the same dependences on density and velocity dispersion. The Toomre instability criterion for gaseous disks (Goldreich Lynden-Bell 1965 ) iswhere Σ g is the surface density of the gas, σ g is its velocity dispersion including both thermal and nonthermal components, G is the gravitational constant, and κ 2 = −4 ωB is the square of the epicyclic frequency, with Oort’s constantIn a galaxy with a flat rotation curve, the second term in B can be neglected. In this case, , so that . The timescale for growth of a perturbation in an isothermal gas disk can be expressed in terms of Q g as (Wang Silk 1994 )At the solar circle, with r = 8 kpc, the rotational velocity ωr = 220 km s −1 , dropping only slowly to 170 km s −1 at a Galactic radius of 60 kpc (Xue et al. 2008 ). Thus, the epicyclic frequency κ = 1.26 × 10 −15 s −1 , and the growth time τ g = f ( Q ) κ −1 , where κ −1 ≃ 25 Myr and f ( Q ) is a function of Q with a value of order unity. If the diffuse gas with mass M d ∼ 4 M m (Ferrière 2001 ) forms into molecular clouds on the Toomre timescale, then the molecular gas formation rate would beIf the molecular gas is being dispersed on a timescale of the order of 10 Myr, this formation rate can replenish it at more than the required rate given in Equation ( 3 ). 4. Variation with Redshift We now consider how the timescale for molecular gas formation by gravitational instability varies with redshift. The Toomre timescale τ g ∝ ω −1 . The disk angular velocity of a galaxy ω can be characterized by its value at the exponential scale length of the disk ω ( R d ) = V c / R d , where V c is the roughly constant circular velocity of the disk