How can BAO and SN constraints be compared for measuring dark energy?

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wolram
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How do BAO help us measure dark energy?

thank you for all the help PF members give in the cosmology forum.
 
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This is what i have found so far.

To get competitive constraints on dark energy we need to be able to see changes in H(z) at the 1% level -- this would give us statistical errors in the DE equation of state of O(10%).
  • We need to be able to calibrate the ruler accurately over most of the age of the universe.
  • We need to be able to measure the ruler over much of the volume of the universe.
  • We need to be able to make ultra-precise measurements of the ruler.
 
BAO measurements provide an accurate standard ruler that can be used to estimate H(z) more accurately than supernovae, with enough galaxy observations.
 
But how are they better than supernova for distance measurement and are they independent of
sn1a? and how do they help us measure DE.
 
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They are completely independent of SN1A measurements. They help to measure DE by measuring the rate of expansion over time.

They're better primarily because there are huge numbers of galaxies out there, but only so many supernovae. Also, each individual supernova has a lot of error.

This isn't to say that supernova measurements are worthless. SN and BAO measurements have very different error properties, so that the combination of the two is significantly better than either one by itself.