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.
-----
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
Hi!
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. ;)
Bye!!!!!!!!!!!!
Michal
P.S. In the picture I have chosen the evolution that fits best to linear approximation of w(a)...
Hi Micha�,
Sorry for not writing for a few days, but I had a very bad time with my studies/personal life ;)
There's no urgence. We each handle time usage differently (and each have different personal lives)... :-)
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.
OK, I think this clears things up.
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).
OK, it seems to me you have chosen one particular potential, a "power-law potential", for which the derivatives go to zero very fast. If they didn't go to zero fast, your plot would not be close to linear for redshifts of interest.
So, to put this in language which everyone on the list should understand (hopefully!), you've plotted how one particular theoretical model of quintessence (a substitute for the cosmological constant, which has some theoretical physics motivation), is close to linear in the range 0.1 < a < 1, i.e. 9 > z > 0.
I agree that this covers the entire range of interest to observers (apart from the CMB)!
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.
...
Just an aside on the mathematics, which was where I was confused (before). Suppose we take the Taylor series for:
* the exponential, around x=0, and evaluate at the point x=-0.7.
This is similar to calculating a Taylor series for:
* w(a) around a=1, and evaluating it at quasar redshifts, with z=2, i.e. a= 1/(1+z) = 0.333
Well, exp(x) = 1 + x + x^2/2! + x^3/3! + .... So exp(-0.7) = 1 + (-0.7) + ... = 0.3 if we just take the linear term
And this is 40% off from the true value 0.49658... So the linear approximation can be a poor approximation for an exponential when you're too far from the expansion point.
Look at the picture I have attached. For interesting redshifts linear approximations looks really very fine.
So there must be something in the physics of the power-law potential you're talking about which makes a linear approximation valid over sub-CMB redshift ranges.
I hope I have answered your questions (at least partly) and convinced you that it if worth to expand w(a) near a=1. ;)
For this potential, you have convinced me :-).
You've also shown that the sympa software (or maybe mhonarc, which is called by sympa) has turned your postscript attachment into an html link - nice!
Here's a check to see what precision would be required: * visual - make a plot of w(z) in the range 3 < z < 0 * calculate mean (z-weighted) values of w(z) in the ranges:
0.6 < z < 1.1 1.1 < z < 1.6 1.6 < z < 2.2
for the potential you chose. From this, we will know the precision required in the individual w(z) estimates (as I propose to do) in order to obtain a significant value of w_1 .
(Of course, simply to get a significant estimate of w_0 would be a big result!)
P.S. In the picture I have chosen the evolution that fits best to linear approximation of w(a)...
Does this mean that other potentials are very non-linear?
Cze�� Boud
-
Boud Roukema -
Boud Roukema -
Michal Frackowiak