Constraining gamma-ray burst initial Lorentz factor with the afterglow onset feature and discovery of a tight Λ₀-E _γ,iso correlation

The onset of gamma-ray burst (GRB) afterglow is characterized by a smooth bump in the early afterglow light curve as the GRB fireball is decelerated by the circumburst medium. We extensively search for GRBs with such an onset feature in their optical and X-ray light curves from the literature and from the catalog established with the Swift/XRT. Twenty optically selected GRBs and 12 X-ray-selected GRBs are obtained, among which 17 optically selected and 2 X-ray-selected GRBs have redshift measurements. We fit these light curves with a smooth broken power law and measure the width (w), rising timescale (t _r), and decaying timescale (t_d) at full width at half-maximum. Strong mutual correlations among these timescales and with the peak time (t_p) are found. The ratio t_r/t_d is almost universal among bursts, but the ratio t_r/t_p varies from 0.3 to ∼1. The optical peak luminosity in the R band (L_R,p) is anti-correlated with tp and w in the burst frame, indicating a dimmer and broader bump peaking at a later time. The isotropic prompt gamma-ray energy (E_γ,iso) is also tightly correlated with L_R,p and t_p in the burst frame. Assuming that the bumps signal the deceleration of the GRB fireballs in a constant density medium, we calculate the initial Lorentz factor (Λ₀) and the deceleration radius (R_d) of the GRBs with redshift measurements. The derived Λ0 is typically a few hundreds, and the deceleration radius is R_dec ∼ 2×10¹⁷ cm. More intriguingly, a tight correlation between Λ₀ and E_γ,iso is found, namely Λ₀ ≃ 182(E_γ,iso/10⁵² erg)^0.25. This correlation also applies to the small sample of GRBs which show the signature of the afterglow onset in their X-ray afterglow, and to two bursts (GRBs 990123 and 080319B) whose early optical emission is dominated by a reverse shock. The lower limits of Λ₀ derived from a sample of optical afterglow light curves showing a decaying feature from the beginning of the observation are also generally consistent with such a correlation. The tight lower limits of Λ₀ of GRBs 080916C and 090902B derived from the opacity constraints with Fermi/LAT observations are also consistent with the correlation at the 2σ confidence level, but the short GRB 090510 is a clear outlier of this relation. This correlation may give insight to GRB physics and could serve as an indicator of Λ₀ for long GRBs without early afterglow detections. A comparison of the early X-ray and optical afterglow light curves shows that the early bright X-ray emission is usually dominated by a non-forward-shock component, but occasionally (for one case) the forward shock emission is observable, and an achromatic deceleration feature is observed. The superposition of the internal and external components in X-rays causes the diversity of the observed X-ray light curves.