# The Relative and Absolute LGRB Rate with Metallically and Its Lack of Evolution

There is now strong evidence that Long-duration Gamma-Ray Bursts (LGRBs) have an intrinsic preference for low-metallicity environments despite the existence of some exceptions to this trend (Graham & Fruchter 2013). Here I will present a pair of results expanding on this work. First we will use the sample from Graham and Fruchter 2013 to give a quantitative answer to the fundamental question: How much more likely is an LGRB to form at one metallicity to another? In particular, we find that per unit star-formation an LGRB occurs about 30 times more often at metallicities below 8.3 on the KK04 scale than above. Then, employing this result and the work of Lien et al. 2014, we relate the LGRB rate as a function of redshift to the cosmic star-formation rate, and show that assuming a semi-nominal LGRB beaming factor of 100, in low-metallicity environments one LGRB occurs for approximately every 40 broad-lined Type Ic SNe. However the metallicity distribution of LGRB host galaxies shows a curious lack of evolution with redshift seen in both host galaxies with measured metallicities and the expected metallicities of LGRB host given their mass and redshift. We also find that the metallicities estimated from mass and redshift are about twice as metal rich as the population with actually measured values which resolves much of the difference between the LGRB formation metallicity cutoff of about a third solar in Graham & Fruchter 2015 with the cutoff value of solar claimed in Perley et al. 2016 in favor of the former.