Gamma-Ray Burst Luminosity Functions Based On a Newly Discovered
  Correlation Between Peak Spectral Energy and V/V_max
  We have discovered a correlation between the observed peak spectral energy
E_pk,obs and the average Euclidean value of V/V_max of gamma-ray bursts (GRBs).
We present the evidence for the correlation in the GUSBAD catalog and use it to
derive the luminosity function of GRBs without using any redshifts. The
procedure involves dividing GUSBAD GRBs in five spectral classes based on their
E_pk,obs. The overall luminosity function is derived assuming that each of the
spectral classes contributes a gaussian luminosity function. Their central
luminosity is derived from the average observed Euclidean V/V_max. We explore
various forms for the GRB rate function GR(z) in predicting redshift
distributions of GRBs detected by Swift. We find that GR(z) peaks at a higher
redshift than the typical star formation history currently favored in the
literature. We consider two examples of GR(z) that successfully predict the
observed redshift distribution of Swift GRBs. With the luminosity functions in
hand, we convert the E_pk,obs-V/V_max correlation into an E_pk,obs-L_iso
correlation and a rest frame E_pk-L_iso correlation. In comparing the
E_pk-L_iso correlation with a published correlation based on GRBs with known
E_pk,obs and redshifts, we discuss the effect of Malmquist bias.
