Luminosity-time and luminosity-luminosity correlations for GRB prompt and afterglow plateau emissions

Sco X-1 has been the subject of many multiwavelength studies in the past, being the brightest persistent extrasolar X-ray source ever observed. Here, we revisit Sco X-1 with simultaneous short cadence Kepler optical photometry and Monitor of All-sky X-ray Image X-ray photometry over a 78 d period, as well as optical spectroscopy obtained with High Efficiency and Resolution Mercator Echelle Spectrograph (HERMES). We find Sco X-1 to be highly variable in all our data sets. The optical fluxes are clearly bimodal, implying the system can be found in two distinct optical states. These states are generally associated with the known flaring/normal branch X-ray states, although the flux distributions associated with these states overlap. Furthermore, we find that the optical power spectrum of Sco X-1 differs substantially between optical luminosity states. Additionally we find rms-flux relations in both optical states, but only find a linear relation during periods of low optical luminosity. The full optical/X-ray discrete correlation function displays a broad a parts per thousand 12.5 h optical lag. However, during the normal branch phase, the X-ray and optical fluxes are anticorrelated, whilst being correlated during the flaring branch. We also performed a Cepstrum analysis on the full Kepler light curve to determine the presence of any echoes within the optical light curve alone. We find significant echo signals, consistent with the optical lags found using the discrete cross-correlation. We speculate that whilst some of the driving X-ray emission is reflected by the disc, some is absorbed and re-processed on the thermal time-scale, giving rise to both the observed optical lags and optical echoes. We present an analysis of 123 gamma-ray bursts (GRBs) with known redshifts possessing an afterglow plateau phase.We reveal that L-a-T-a correlation between the X-ray luminosity L-a at the end of the plateau phase and the plateau duration, T-a(*) in the GRB rest frame has a power-law slope different, within more than 2 sigma, from the slope of the prompt L-f-T*(f) correlation between the isotropic pulse peak luminosity, L-f, and the pulse duration, T-f(*) from the time since the GRB ejection. Analogously, we show differences between the prompt and plateau phases in the energy duration distributions with the afterglow emitted energy being on average 10 per cent of the prompt emission. Moreover, the distribution of prompt pulse versus afterglow spectral indexes does not show any correlation. In the further analysis we demonstrate that the L-peak-L-a distribution, where L-peak is the peak luminosity from the start of the burst, is characterized with a considerably higher Spearman correlation coefficient, rho = 0.79, than the one involving the averaged prompt luminosity, L-prompt-L-a, for the same GRB sample, yielding rho = 0.60. Since some of this correlation could result from the redshift dependences of the luminosities, namely from their cosmological evolution we use the Efron-Petrosian method to reveal the intrinsic nature of this correlation. We find that a substantial part of the correlation is intrinsic. We apply a partial correlation coefficient to the new de-evolved luminosities showing that the intrinsic correlation exists.