## Abstract

The desire to extend the Hubble Diagram to higher redshifts than the range of current Type 1a Supernovae observations has prompted investigation into spectral correlations in gamma-ray bursts (GBRs), in the hope that standard candle-like properties can be identified. In this paper, we discuss the potential of these new ‘cosmic rulers’ and highlight their limitations by investigating the constraints that current data can place on an alternative Cosmological model in the form of Conformal Gravity. By fitting current Type 1a Supernovae and GRB data to the predicted luminosity distance redshift relation of both the standard Concordance Model and the Conformal Gravity, we show that currently *neither* model is strongly favoured at high redshift. The scatter in the current GRB data testifies to the further work required if GRBs are to cement their place as effective probes of the cosmological distance scale.

## 1. The Ghirlanda relation as a cosmic ruler

Recently a number of authors have highlighted the potential of long-duration gamma-ray bursts (GRBs) as distance indicators. The most promising indicator appears to be the so-called ‘Ghirlanda Relation’ (Ghirlanda *et al*. 2004); the tight correlation between the isotropic equivalent energy and the peak energy of the GRB integrated spectrum.

However, several authors (e.g. Friedman & Bloom 2005) have pointed out potential sources of systematic error which may undermine the application of the Ghirlanda Relation as a distance indicator, for example, Dai *et al.* (2004) assume a cosmology when calibrating the relation, which presents a circularity issue when using it to fit cosmological parameter values.

Friedman & Bloom conclude that the Ghirlanda Relation provides no significant improvement in the constraints on *Ω*_{M} and *Ω*_{Λ}. In their view, this is mainly due to the currently small number of GRB calibrators, including the lack of low-redshift GRBs. Contributions to the uncertainty also arise from the sensitivity to data selection choices and to the values, and ranges assumed for the number density of the surrounding medium and the efficiency of each event.

Notwithstanding the caveats of Friedman & Bloom, we have recently considered their application to test the viability of Conformal Gravity theories (Mannheim 2003). Mannheim's theory makes a specific, and very strong, prediction; the expansion of the Universe has always been accelerating. However, the Hubble Diagram for this model does not diverge from the corresponding Friedman model until *z*>1.

In Mannheim's Conformal Gravity theory, the luminosity distance redshift relation is given by(1.1)where *z* is the redshift of the source, *H*_{0} is the Hubble parameter, *c* is the speed of light and *q*_{0} is the model deceleration parameter, related to the Concordance Model parameters by .

## 2. Results and conclusions

We have used data on 150 Gold Sample Type 1a Supernovae from Riess *et al.* (2004), 71 SN from the first year results of the Supernova Legacy Survey (Astier *et al.* 2006) and 19 GRBs compiled by Friedman & Bloom (2005), employing a cut at *cz*<5000 km s^{−1} to remove the effect of peculiar velocities from the SN data. The Hubble diagram for these datasets can be seen in figure 1. We have compared these data with distance moduli predicted for the Standard Model with (*Ω*_{M}=0.3, *Ω*_{Λ=0.7}) and the corresponding Conformal Gravity model with *q*_{0}=−0.55. Values for *σ*_{obs} for our SN were taken from the published data source, while for the GRBs they were calculated following Dai *et al.* (2004). These fit gives *Χ*^{2}_{/d.f.}=4.91 and 5.65, respectively.

From our results, we see that—with current SN+GRB data—the specific prediction of Mannheim's Conformal Gravity that the universe did not undergo a deceleration phase remains viable. However, the large *Χ*^{2} per degree of freedom for *both* the Conformal Gravity fand the Friedman models shown in figure1 should sound an important note of caution regarding the efficacy of GRBs as distance indicators. The GRB data do not yet appear good enough to discriminate reliably between models which accelerate and decelerate above *z*=1.

## Footnotes

One contribution of 35 to a Discussion Meeting Issue ‘Gamma-ray bursts’.

- © 2007 The Royal Society