The Hubble diagram ( HD ) is a plot of the measured distance modulus versus the measured redshift , with the slope giving the expansion history of our universe .
In the late 1990 ’ s , observations of supernova out to a redshift of near unity demonstrated that the universal expansion is now accelerating , and this was the first real evidence for the mysterious energy we now call Dark Energy .
One of the few ways to measure the properties of Dark Energy is to extend the HD to higher redshifts .
Many models have been proposed that make specific predictions as to the shape of the HD and so this offers a means of testing and eliminating models .
Taking the HD to high redshifts provides a way to test models where their differences are large .
For example , in a comparison of the now concordance model ( a flat universe with \Omega _ { M } = 0.27 or so and constant Cosmological Constant ) with a representative model of evolving Dark Energy ( for which I ’ ll take w ( z ) = -1.31 + 1.48 z from Riess ’ analysis of the gold sample of supernovae ) , the predicted distance moduli differ by 0.15 mag at z = 1.7 and 1.00 mag at z = 6.6 .
The only way to extend the HD to high redshift is to use Gamma-Ray Bursts ( GRBs ) .
GRBs have been found to be reasonably good standard candles in the usual sense that light curve and/or spectral properties are correlated to the luminosity , exactly as for Cepheids and supernovae , then simple measurements can be used to infer their luminosities and hence distances .
GRBs have at least five properties ( their spectral lag , variability , spectral peak photon energy , time of the jet break , and the minimum rise time ) which have correlations to the luminosity of varying quality .
All of these properties provide independent distance information and their derived distances should be combined as a weighted average to get the best value .
For GRBs which have an independently measured redshift from optical spectroscopy , we have enough information to plot the burst onto the HD .
In this paper , I construct a GRB HD with 69 GRBs over a redshift range of 0.17 to > 6 , with half the bursts having a redshift larger than 1.7 .
This paper uses over 3.6 times as many GRBs and 12.7 times as many luminosity indicators as any previous GRB HD work .
For constructing the GRB HD , it is important to perform the calibration of the luminosity relations for every separate cosmology considered , so that we are really performing a simultaneous fit to the luminosity relations plus the cosmological model .
I have made detailed calculations of the gravitational lensing and Malmquist biases , including the effect of lensing de/magnification , volume effects , evolution of GRB number densities , the GRB luminosity function , and the discovery efficiencies as a function of brightness .
From this , I find that the biases are small , with an average of 0.03 mag and an RMS scatter of 0.14 mag in the distance modulus .
This surprising situation arises from two causes , the first being that burst peak fluxes above threshold do not vary with redshift and the second being that the four competing effects nearly cancel out for most GRBs .
The GRB HD is well-behaved and nicely delineates the shape of the HD .
The reduced chi-square for the fit to the concordance model is 1.05 and the RMS scatter about the concordance model is 0.65 mag .
This accuracy is just a factor of 2.0 times that gotten for the same measure from all the big supernova surveys .
I claim that GRBs will not suffer from effects due to evolution in the progenitors as we look back in time , with the reason being that the luminosity indicators are results of light travel time delays , conservation of energy in the shocked material , and the degree of relativistic beaming , with these not changing with metallicity or age .
That is , even though distant bursts might be more luminous on average than nearby bursts , the luminosity indicators will still operate to return the correct luminosity .
I fit the GRB HD to a variety of models , including where the Dark Energy has its equation of state parameter varying as w ( z ) = w _ { 0 } + w _ { a } z / ( 1 + z ) .
I find that the concordance model is consistent with the data .
That is , the Dark Energy can be described well as a Cosmological Constant that does not change with time .