Using Supernovae to Study Cosmology
The RedshiftDistance Test
As a function of redshift, the apparent magnitude of distant objects changes under different cosmologies, for two reasons:
 The shape of space determines how photons spread out as they move outwards (the classic 1/d^{2} effect)
 The expansion history determines how the photons are redshifted.
This can be worked out under different cosmologies to derive a form akin to our regular magnitudedistance expression:
mM = 5logd_{L}(z) 5
where d_{L}(z) is the luminosity distance,
and depends on H_{0}, Omega_{M}, Omega_{L}, and k. We typically plot this using
the distance modulus, not the distance, though (from Carroll and
Ostlie):
If we had an object of fixed brightness  a standard candle  we
could plot its apparent magnitude as a function of distance and work
out the cosmology.
Remember Type Ia supernovae:
the explosion of a ~ 1.4 Msun white dwarf. These are pretty good
approximations to a standard candle, and they are extremely bright.
That's
exactly
what we want to use for the redshiftdistance test.
But are SN Ia's
really
standard candles?
Type Ia supernovae in galaxies w/ Cepheid distances (From Riess etal 2016):
Which gives an average peak absolute magnitude of 19.26 +/ 0.16.
This uncertainty in peak mag includes the distance
uncertainties to the galaxies, so the real dispersion in peak magnitude
is even smaller, about 0.1 mags or so. That's a pretty good standard
candle.
But there's a significant drawback to using Type Ia
SNe.
You
gotta find them...
Using supernovae to study cosmology

Take a BIG picture of the sky.

Come back next month and take the same picture.

Compare the two. Differences?

If you find a possible supernova, take a spectrum
of it and
make sure it is a Type Ia SNe.

Also take a spectrum of the galaxy it lives in, to
find its
redshift.

Watch the supernova as it fades, so we can get its
peak apparent
magnitude. This is important  you probably didn't catch it when it
was
at its peak, so we need to fit it to a standard light curve to derive
its
peak magnitude.

Keep doing this so you have a big sample of high
redshift
supernovae. Then compare those supernovae to ones at lower redshift.
Some plots, courtesy of the supernovae
cosmology project at LBL and the high
z supernova search team at CfA:
More recent data: HST discovered supernovae, extending to higher redshift (Riess et al 2007).
In this plot mu=mM, the distance modulus. Curvature in the data is inconsistent with models
that use dust or evolution to explain faintness of highz SNe; instead
it is indicative of the "jerk" in the expansion history when lambda
began to dominate and the universe went from decelerating to
accelerating.