The principal goal of this paper is to use attempts at reconciling the Swift long gamma-ray bursts (LGRBs) with the star formation history (SFH) to compare the predictions of ΛCDM with those in the R h = ct Universe. In the context of the former, we confirm that the latest Swift sample of GRBs reveals an increasing evolution in the GRB rate relative to the star formation rate (SFR) at high redshifts. The observed discrepancy between the GRB rate and the SFR may be eliminated by assuming a modest evolution parameterized as (1+z) 0.8 -perhaps indicating a cosmic evolution in metallicity. However, we find a higher metallicity cut of Z = 0.52Z ⊙ than was seen in previous studies, which suggested that LGRBs occur preferentially in metal poor environments, i.e., Z ∼ 0.1 − 0.3Z ⊙ . We use a simple power-law approximation to the high-z ( 3.8) SFH, i.e., R SF ∝ [(1 + z)/4.8] α , to examine how the high-z SFR may be impacted by a possible abundance evolution in the Swift GRB sample. For an expansion history consistent with ΛCDM, we find that the Swift redshift and luminosity distributions can be reproduced with reasonable accuracy if α = −2.41 +1.87 −2.09 . For the R h = ct Universe, the GRB rate is slightly different from that in ΛCDM, but also requires an extra evolutionary effect, with a metallicity cut of Z = 0.44Z ⊙ . Assuming that the SFR and GRB rate are related via an evolving metallicity, we find that the GRB data constrain the slope of the high-z SFR in R h = ct to be α = −3.60 +2.45 −2.45 . Both cosmologies fit the GRB/SFR data rather well. However, in a one-on-one comparison using the Aikake Information Criterion, the 2 Wei et al.best-fit R h = ct model is statistically preferred over the best-fit ΛCDM model with a relative probability of ∼ 70 % versus ∼ 30 %.

In this paper, we explore some aspects of the gravitational lens effects due to a Kerr black hole. Under the eikonal approximation of the Maxwell equations in curved space, the spin function of a photon in the degenerate metric is determined. Furthermore, we present an investigation of the phase factor that a photon acquires in Kerr spacetime. The resulting phase consists of two parts: a real and an imaginary one. The real part has been interpreted as contributing a rotational angle of the plane polarization for linearly polarized light, and the imaginary one results in the light intensity amplification along with the photon's trajectory in the gravitational field. Finally, we provide the so-called "Sagnac factor" related to the phase shift. PACS numberb): 04.40. +c, 03.50.De, 03.65.Sq

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