Boud: quintessence

Michal Frackowiak michalf w
Pon, 17 Gru 2001, 13:39:59 CET


Sorry for not writing for a few days, but I had a very bad time with my
studies/personal life ;)

> You express  w  as a Taylor series of (a-1), but normally this
> only makes sense if (a-1) is much smaller than 1. 

Not that much. You can always expand a function of one variable around
certain point. Expanding w(a) has the advantage that constraining only to
the first term you get 'static' X-field, for which w is fixed for all a.
Introducing additional terms allowes you to model the behaviour of w(a) in
the vicinity of 'now'. So the expansion around a=1 seems natural to me. 
Another question is: how many terms should we take into account for
proper approximation of evolution of X-field.
My idea is (after conducting some calculations) that with the quality of
available data (e.g. SNIa) and almost linear behaviour of w(a) for some
potentials, one can safely restrict the analysis to only first 2 terms.

> [
> 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. 

Look at the picture I have attached. For interesting redshifts linear
approximations looks really very fine.

> Do you have in mind a quintessence model where these derivatives go
> to zero very fast? Or have I misunderstood?
What I am trying to do is only to approximate w(a) with respect to a=1.
It is more suitable for analysing data (such as SNIa or quasars I
hope) than including the full evolution of x-field (which behaves very
'badly' much earlier than z=2).

> 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

During weekend I have partly analysed SNIa data searching for any
constrains for w^(1) and w^(2). The results are terrible. My next step is
CMB anisotropy. SNIa gives terrible confidence areas, but thats something
at least. 

Included picture: w(a) vs. a for a "inverse power-law" potential. You can
see that linear approximation is 'quite' good here, but for other
potentials is a bit worse and the quadratic term should be included. 
I am curious if one can put some constrains from available or wait for
future projects/missions.

I hope I have answered your questions (at least partly) and convinced you
that it if worth to expand w(a) near a=1. ;)



P.S. In the picture I have chosen the evolution that fits best to linear
approximation of w(a)...
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