Puzzles in Time Delay and Fermat Principle in Gravitational Lensing

The current standard time delay formula (CSTD) in gravitational lensing and its claimed relation to the lens equation through Fermat's principle (least time principle) have been puzzling to the author for some time. We find that the so-called geometric path difference term of the CSTD is an error, and it causes a double counting of the correct time delay. We examined the deflection angle and the time delay of a photon trajectory in the Schwarzschild metric that allows exact perturbative calculations in the gravitational parameter $GM$ in two coordinate systems -- the standard Schwarzschild coordinate system and the isotropic Schwarzschild coordinate system. We identify a coordinate dependent term in the time delay which becomes irrelevant for the arrival time difference of two images. It deems necessary to sort out unambiguously what is what we measure. We calculate the second order corrections for the deflection angle and time delay. The CSTD does generate correct lens equations including multiple scattering lens equations under the variations and may be best understood as a generating function. It is presently unclear what the significance is. We call to reanalyze the existing strong lensing data with time delays.Comment: 6 p., 1 fi

COBE's Constraints on the Global Monopole and Texture Theories of Cosmic Structure Formation

We report on a calculation of large scale anisotropy in the cosmic microwave background radiation in the global monopole and texture models for cosmic structure formation. We have evolved the six component linear gravitational field along with the monopole or texture scalar fields numerically in an expanding universe and performed the Sachs-Wolfe integrals directly on the calculated gravitational fields. On scales $> 7^\circ$, we find a Gaussian distribution with an approximately scale invariant fluctuation spectrum. The $\Delta T/T$ amplitude is a factor of 4-5 larger than the prediction of the standard CDM model with the same Hubble constant and density fluctuation normalization. The recently reported COBE-DMR results imply that global monopole and texture models require high bias factors or a large Hubble constant in contrast to standard CDM which requires very low $H_0$ and bias values. For $H_0 = 70 {\rm {km\over sec} Mpc^{-1}}$, we find that normalizing to the COBE results implies $b_8 \simeq 3.2\pm 1.4$ (95\% c.l.). If we restrict ourselves to the range of bias factors thought to be reasonable before the announcement of the COBE results, 1.5 \lsim b_8 \lsim 2.5, then it is fair to conclude that global monopoles and textures are consistent with the COBE results and are a {\it better} fit than Standard CDM.Comment: 8 pages, 5 figures (not included, but available by mail), CfPA-TH-92-2