[astroph/0501104] DarkEnergy Evolution Across the CosmologicalConstant Boundary
Authors:  Robert R. Caldwell, Michael Doran 
Abstract:  We explore the properties of dark energy models for which the equationofstate, w, defined as the ratio of pressure to energy density, crosses the cosmologicalconstant boundary w = 1. We adopt an empirical approach, treating the dark energy as an uncoupled fluid or a generalized scalar field. We describe the requirements for a viable model, in terms of the equationofstate and sound speed. A generalized scalar field cannot safely traverse w = 1, although a pair of scalars with w > 1 and w < 1 will work. A fluid description with a welldefined sound speed can also cross the boundary. Contrary to expectations, such a crossing model does not instantaneously resemble a cosmological constant at the moment w = 1 since the density and pressure perturbations do not necessarily vanish. But because a dark energy with w < 1 dominates only at very late times, and because the dark energy is not generally prone to gravitational clustering, then crossing the cosmologicalconstant boundary leaves no distinct imprint. 
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 Posts: 18
 Joined: September 25 2004
 Affiliation: Simon Fraser University
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[astroph/0501104] DarkEnergy Evolution Across the Cosmolog
This looks like a nice paper that actually implements
what Anthony suggested in his Dec 17 posting to CosmoCoffee.
Namely, there is no singularity at the w>−1 > w<−1 crossing when
equations are recast in terms of the physical momentum transfer.
The authors point out that, while the effect of shear is to dump dark
energy perturbations on small scales in the w>−1 regime, it turns
into an amplification of perturbations in the w<−1 regime.
I wonder if it is possible to observationally constrain
this sign change, e.g. with the ISW effect. The ISW is quite
sensitive to clustering properties of dark energy (see e.g.
astroph/0408456).
what Anthony suggested in his Dec 17 posting to CosmoCoffee.
Namely, there is no singularity at the w>−1 > w<−1 crossing when
equations are recast in terms of the physical momentum transfer.
The authors point out that, while the effect of shear is to dump dark
energy perturbations on small scales in the w>−1 regime, it turns
into an amplification of perturbations in the w<−1 regime.
I wonder if it is possible to observationally constrain
this sign change, e.g. with the ISW effect. The ISW is quite
sensitive to clustering properties of dark energy (see e.g.
astroph/0408456).

 Posts: 6
 Joined: January 15 2005
 Affiliation: Dartmouth College
[astroph/0501104] DarkEnergy Evolution Across the Cosmolog
Thanks for noticing the paper. Michael and I had been looking into the problems of dark energy fluctuations crossing the cosmologicalconstant boundary, when I chatted with Antony at the galaxy cluster workshop at Fermilab last month. To clarify, our results suggest that there is no striking consequence of a model that crosses w=−1, except in the influence on the cosmic expansion. We did not discuss the role of shear, and explicity turned it off. I would be surprised if there is any signature of a crossing even in models with shear, however, since a crossing necessarily occurs at very low redshift leaving little time for an imprint.

 Posts: 1353
 Joined: September 23 2004
 Affiliation: University of Sussex
 Contact:
Re: [astroph/0501104] DarkEnergy Evolution Across the Cosm
I thought about this a bit more. Although the momentum transfer is regular, it's still true that the synchornous gauge [tex]\delta'[/tex] diverges at [tex]w=1[/tex] if the rest frame sound speed remains a constant.
I think the real message of your paper is that having a rest frame sound speed constant is not very sensible. In particular in my notation the sound speed in a general frame is related to the rest frame sound speed [tex]\hat{c}_s^2[/tex] by
[tex]\delta\, c_{s}^2 = \delta\, \hat{c}_s^2 + \frac{3 H v}{k} (1+w)\left(\hat{c}_s^2  \frac{p'}{\rho'}\right)[/tex]
Since the synchronous gauge dark energy velocity [tex]v[/tex] diverges if [tex]\hat{c}_s^2[/tex] is a constant at [tex]w=1[/tex], this indicates that choosing a frame where [tex]v=0[/tex] is not very sensible. i.e. the dark energy rest frame is not well defined at [tex]w=1[/tex]. Fixing the sound speed in the dark energy frame to be constant makes the sound speed in any regular frame divergent.
This is consisent with what we know about quintessence (where [tex]\hat{c}_s^2 =1[/tex]) because we know quintessence can't reach w=−1 except with w'=0 (it can't cross). There's no reason why a model with [tex]\hat{c}_s^2 =1[/tex] everywhere (including crossing w=−1) should be sensible.
If the sound speed is chosen to be regular in a well defined frame, then I don't think there are any problems going through [tex]w=1[/tex]: both the energy flux and [tex]\delta'[/tex] will be regular. Do you agree?
I think the real message of your paper is that having a rest frame sound speed constant is not very sensible. In particular in my notation the sound speed in a general frame is related to the rest frame sound speed [tex]\hat{c}_s^2[/tex] by
[tex]\delta\, c_{s}^2 = \delta\, \hat{c}_s^2 + \frac{3 H v}{k} (1+w)\left(\hat{c}_s^2  \frac{p'}{\rho'}\right)[/tex]
Since the synchronous gauge dark energy velocity [tex]v[/tex] diverges if [tex]\hat{c}_s^2[/tex] is a constant at [tex]w=1[/tex], this indicates that choosing a frame where [tex]v=0[/tex] is not very sensible. i.e. the dark energy rest frame is not well defined at [tex]w=1[/tex]. Fixing the sound speed in the dark energy frame to be constant makes the sound speed in any regular frame divergent.
This is consisent with what we know about quintessence (where [tex]\hat{c}_s^2 =1[/tex]) because we know quintessence can't reach w=−1 except with w'=0 (it can't cross). There's no reason why a model with [tex]\hat{c}_s^2 =1[/tex] everywhere (including crossing w=−1) should be sensible.
If the sound speed is chosen to be regular in a well defined frame, then I don't think there are any problems going through [tex]w=1[/tex]: both the energy flux and [tex]\delta'[/tex] will be regular. Do you agree?
[astroph/0501104] DarkEnergy Evolution Across the Cosmolog
Hello Antony,
Concerning your post about the dark energy soundspeed as w crosses −1:
In astroph/0411102 I deal with exactly this problem  v is not well defined
as w crosses −1. So, how does one pick a "good" value for c_{s}? For w away
from −1 it makes sense to set the dark energy rest frame c_{s} to be time and
scale independent, and then get the CDM rest frame value by transforming
to the CDM rest constant time hypersurfaces. For w=−1 I see the problem
as no rest frame exists. Since as one passes through w=−1 the energy density
goes from decreasing with time to increasing, it is not possible "around"
w=−1 to arbitrarily redefine the density contrast by deforming the constant
time hypersurfaces forwards & backwards. A similar argument can be used to
explain why rapidly oscillating scalar fields have c_{s} of approximately 0
instead of 1. So there is a well defined rest frame away from w=−1 
just around w=−1 there is a problem. So if you define some ε
such that for w−1 > ε the rest frame constant t hypersurface
does not intersect the w=−1 hypersurface, then for w−1 > ε
you can define a soundspeed in the rest frame normally.
For w−1 < ε you have to treat the soundspeed in some special way.
In astroph/0411102. I deal with this problem crudely  I just interpolate
1/(w+1) in the soundspeed transformation equation between the w=−1+ε
and w=−1ε. The resulting CMB and matter power spectra are, as one
might expect, insensitive to the value of ε around ε approx 0.01
for dark energy fluids that cross w=−1 at late times.
So I think using a constant restframe soundspeed for w−1 > ε
is still a good idea. You might be able to suggest a better method for
handling the soundspeed for w−1 < ε than I used though.
Greg Huey
Concerning your post about the dark energy soundspeed as w crosses −1:
In astroph/0411102 I deal with exactly this problem  v is not well defined
as w crosses −1. So, how does one pick a "good" value for c_{s}? For w away
from −1 it makes sense to set the dark energy rest frame c_{s} to be time and
scale independent, and then get the CDM rest frame value by transforming
to the CDM rest constant time hypersurfaces. For w=−1 I see the problem
as no rest frame exists. Since as one passes through w=−1 the energy density
goes from decreasing with time to increasing, it is not possible "around"
w=−1 to arbitrarily redefine the density contrast by deforming the constant
time hypersurfaces forwards & backwards. A similar argument can be used to
explain why rapidly oscillating scalar fields have c_{s} of approximately 0
instead of 1. So there is a well defined rest frame away from w=−1 
just around w=−1 there is a problem. So if you define some ε
such that for w−1 > ε the rest frame constant t hypersurface
does not intersect the w=−1 hypersurface, then for w−1 > ε
you can define a soundspeed in the rest frame normally.
For w−1 < ε you have to treat the soundspeed in some special way.
In astroph/0411102. I deal with this problem crudely  I just interpolate
1/(w+1) in the soundspeed transformation equation between the w=−1+ε
and w=−1ε. The resulting CMB and matter power spectra are, as one
might expect, insensitive to the value of ε around ε approx 0.01
for dark energy fluids that cross w=−1 at late times.
So I think using a constant restframe soundspeed for w−1 > ε
is still a good idea. You might be able to suggest a better method for
handling the soundspeed for w−1 < ε than I used though.
Greg Huey