SIMBAD references

In this work we describe a model-independent method of developing a plot of scale factor a(t) versus lookback time t_L from the usual Hubble diagram of modulus data against redshift. This is the first plot of this type. We follow the model-independent methodology of Daly & Djorgovski used for their radio-galaxy data. Once the a(t)data plot is completed, any model can be applied and will display as described in the standard literature. We then compile an extensive data set to z = 1.8 by combining Type Ia supernovae (SNe Ia) data from SNLS3 of Conley et al., high-z SNe data of Riess et al., and radio-galaxy data of Daly & Djorgovski to validate the new plot. We first display these data on a standard Hubble diagram to confirm the best fit for ΛCDM cosmology, and thus validate the joined data set. The scale factor plot is then developed from the data and the ΛCDM model is again displayed from a least-squares fit. The fit parameters are in agreement with the Hubble diagram fit confirming the validity of the new plot. Of special interest is the transition time of the universe, which in the scale factor plot will appear as an inflection point in the data set. Noise is more visible in this presentation, which is particularly sensitive to inflection points of any model displayed in the plot, unlike on a modulus-z diagram, where there are no inflection points and the transition-z is not at all obvious by inspection. We obtain a lower limit of z ≥ 0.6. It is evident from this presentation that there is a dearth of SNe data in the range z = 1-2, exactly the range necessary to confirm a ΛCDM transition-z around z = 0.76. We then compare a "toy model" wherein dark matter is represented as a perfect fluid with an equation of state p = -(1/3) ρ to demonstrate the plot sensitivity to model choice. Its density varies as 1/t ² and it enters the Friedmann equations as Ω_dark/t ², replacing only the Ω_dark/a ³ term. The toy model is a close match to ΛCDM, but separates from it on the scale factor plot for similar ΛCDM density parameters. It is described in the Appendix. A more complete transition time analysis will be presented in a future paper.