Michal: quintessence

Boud Roukema boud w astro.uni.torun.pl
Czw, 13 Gru 2001, 15:48:18 CET

   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.


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

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 

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.   


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