The gravitational time delay effect of primordial ‘seed’ density perturbation on the light curve of Gamma Ray Bursts

The ‘rise time’ of a gamma-ray burst, which may be defined as the radiation pulse width from a small segment of a fireball’s spherical shell wherein second order effects like shell curvature are unimportant, is conventionally attributed to photon time delay across the thickness of the shell. If the intervening space between the observer and the fireball is inhomogeneous at the minimum level given by primordial density variations, two photons propagating towards us from opposite ends of this small segment along different directions will not arrive simultaneously even when they were emitted at the same depth, because of the gravitational potential gradient transverse to the light paths. The phenomenon is first order in the ‘off-axis’ angle of the photon wavevector, and should not be ignored. We demonstrate that, for bulk fireball Lorentz factors of Γ ∼ 100 – 300 this delay is usually much longer than that due to shell thickness, and can therefore be used to scrutinize the standard cosmological model, by looking for a marked increase of GRB ‘rise times’ with redshift (a factor of ∼ 2.5 between z = 0.25 and z = 3), and by investigating whether a GRB emission code that incorporates the extra delay can still match observational data. If Γ & 10, even the decaying tail of the GRB light curve will be a function of cosmology, because curvature delay at the shell will no longer be the only influence on the arrival times of photons emitted from angles ∼ 1/Γ near the edge of the jet.