EVIDENCE FROM SUPERNOVA DATA

In this section a precise formula for the magnitudes of supernovae with redshift is derived, which can be tested by future observations.

The Flux F due to distant supernovae is given by

where d is the luminosity distance and L is luminosity.
Traditionally,

Here the normal relation between redshift and scale factor is assumed.
However in rescaling theory z in the integral is replaced by .  The rescaling constant H, is constant, and half of Hubbles constant.  These changes combine to give

the measured magnitude is

There is an excellent match to the supernovae data (Riess 2007), gold set.  The chi-squared fit is 183.8 for 182 degrees of freedom.  This close match is obtained with , constant, with no requirement for a dark energy component of the universe.

Figure 6 shows a comparison between the rescaling model and the dark energy model for the supernova data.  Hubbles constant is a variable parameter in these plots.  The top curve is for the rescaling model with  = 65.1kms-1Mpc-1.  The bottom curve is for the best flat dark energy model with  = 63.8kms-1Mpc-1 (Wright 2007).

The dark energy model also has another variable parameter, the matter density, for the curve shown omega(matter) = 0.27.  The match from rescaling theory uses no such extra variable parameter. 

For rescaling theory, the deceleration parameter q(z) = -1 (constant), whereas for dark energy theory q(z) varies, in a way that is not understood (Shapiro & Turner, 2006).

Supernova data against redshift, for rescaling and dark energy models.

The curves are similar.  Both curves give a close match to the data at low redshifts (where the curves are almost identical).

Rescaling theory gives a slightly closer match for intermediate redshifts  z = 0.4 – 0.6.  The dark energy model gives a closer match at higher redshifts, although the number of points is fewer at the higher redshifts.  For the data point  z = 1.755, the standard deviation is 0.36 mag, whereas the typical standard deviation for the other points is about 0.2 mag.