Cześć, Michał Frąckowiak has been playing around with some ideas on quintessence. He suggested to me some ideas and asked if I could test these on the data that Staszek, Gary and I have already analysed.
Well, creativity generally requires that you generate 100 ideas and try to throw away 99. If you end up throwing away 100, well, you have to go on to the next 100...
Or you can modify the ideas and gradually come up with something unexpected. In fact, there's often something interesting enough to be worth talking about. So, let's see.
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Michał, I'm not sure I understand your calculation.
You express w as a Taylor series of (a-1), but normally this only makes sense if (a-1) is much smaller than 1.
[ a is the expansion factor: coordinates can be chosen in which the Universe is static, and all the expansion is represented by a single function of cosmological time: a(t). By convention, a(t)=1 now. The Big Bang is simply the extrapolation a(t) -> 0 as t -> 0^+ , i.e. arbitrary separations between any coordinates get multiplied by nearly zero as t -> 0 from above.
So outside of the Sun, a-1 is a negative number between 0 and -1. ]
You wrote:
w(a)= w^(0) (a-1)^0 + w^(1) (a-1)^1 + w^(2) (a-1)^2 + ...
What is interesting, it seems absolutely sufficient to consider only the 2 first terms in w !!! (up to z=500). Then the evolution reads:
w(a)= w^(0) + w^(1) (a-1) (equation (*))
Well, a = 1/(1+z), so a-1 = -z/(1+z),
so at, say, z=2, a-1 = -2/3. Are you interested in redshifts where z=0.01 or where z=2? The SNeIa data are mostly at about z=0.4-0.8, and the quasar data at about z=0.5 to z=2.4 (roughly).
This makes the (a-1)^i terms decrease (in magnitude) very slowly.
I can only see a rapid convergence if you already know something about w^(1), w^(2), ..., that they go very quickly to zero.
Do you have in mind a quintessence model where these derivatives go to zero very fast? Or have I misunderstood?
In any case, it is certainly possible for me to test equation (*) on the data, though at the moment it seems to me more likely that we would get reasonable error bars by estimating
w(z=0.8) and w(z=1.4) and w(z=1.9)
independently. From those three points with error bars, it would of course be easy (apart from error propagation ;-) ) to calculate w^(0) and w^(1) under the assumption that other terms can be neglected.
If we could simply obtain significantly different values in the three redshift bands, this would be a big enough result, though of course it would be useful to have some theoretical comments of what that might mean.
Pozdrawiam Boud