Model-Independent Plotting of the Cosmological Scale Factor as a   Function of Lookback Time
  In the present work we describe a model-independent method of developing a plot of scale factor versus lookback time 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 and Djorgovski (2004) used for their radio-galaxy data. Once the data plot is completed, any model can be applied and will display accordingly as described in standard literature. We then compile an extensive data set to z = 1.8 by combining SNe Ia data from SNLS3 of Conley et al. (2011), High-z SNe data of Riess et al. (2004) and radio-galaxy data of Daly & Djorgovski (2004) to be used to validate the new plot. We first display these data on a standard Hubble diagram to confirm the best fit for LCDM cosmology and thus validate the joined data set. The scale factor plot is then developed from the data and the LCDM 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 on this presentation which is particularly sensitive to inflection points of any model displayed on 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 LCDM transition-z in the neighborhood of z = 0.76. 
Harry I. Ringermacher and Lawrence R. Mead
astro-ph.CO