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.
