Ah, I think I see. If you play with a(t) you're effectively changing Einstein's equations and you need to also go and change the equations for the metric perturbations. But if I'm understanding this correctly, the Boltzmann equations should be still be fine. [tex]\dot\delta_{cdm} = \frac12 \dot h[/tex] in the synchronous gauge regardless, right? But the equation for [tex]\dot h[/tex] is different if [tex](\dot a/a)^2 \ne 8\pi G \rho /3[/tex], and your point is that if you leave the [tex]\dot h[/tex] equation unchanged you'll get inconsistent and nongaugeinvariant results?Niayesh Afshordi wrote:Again, this is not about the nature dark energy, but rather a very practical question about how you calculate say CMB power spectrum.
For example, if you only modify the bcakground evolution through some \rho_{DE}(z), and don't change perturbation equations (except for the implicit change in the timedependce of background variables), CMBfast (which is synchronous) and CMBeasy (in gauge invariant mode), will not give you the same answer.
[astroph/0603449] Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results:
Authors:  D. N. Spergel, R. Bean, O. Dore', M. R. Nolta, C. L. Bennett, G. Hinshaw, N. Jarosik, E. Komatsu, L. Page, H. V. Peiris, L. Verde, C. Barnes, M. Halpern, R. S. Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weila 
Abstract:  A simple cosmological model with only six parameters (matter density, Omega_m h^2, baryon density, Omega_b h^2, Hubble Constant, H_0, amplitude of fluctuations, sigma_8, optical depth, tau, and a slope for the scalar perturbation spectrum, n_s) fits not only the three year WMAP temperature and polarization data, but also small scale CMB data, light element abundances, largescale structure observations, and the supernova luminosity/distance relationship. Using WMAP data only, the best fit values for cosmological parameters for the powerlaw flat LCDM model are (Omega_m h^2, Omega_b h^2, h, n_s, tau, sigma_8) = (0.127+0.0070.013, 0.0223+0.00070.0009, 0.73 + 0.03, 0.951+0.0150.019, 0.09 + 0.03, 0.74+0.050.06). The three year data dramatically shrink the allowed volume in this sixdimensional parameter space. Assuming that the primordial fluctuations are adiabatic with a power law spectrum, the WMAP data_alone_ require dark matter, and a spectral index that is significantly less than the HarrisonZel'dovichPeebles scaleinvariant spectrum (n_s=1,r=0). Models that suppress largescale power through a running spectral index or a largescale cutoff in the power spectrum are a slightly better fit to the WMAP and small scale CMB data than the powerlaw LCDM model (Delta chi^2 = 3) The combination of WMAP and other astronomical data yields significant constraints on the geometry of the universe, the equation of state of the dark energy, the gravitational wave energy density, and neutrino properties. Consistent with the predictions of simple inflationary theories, we detect no significant deviations from Gaussianity in the CMB maps. 
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Re: [astroph/0603449] Wilkinson Microwave Anisotropy Probe

 Posts: 49
 Joined: December 17 2004
 Affiliation: Perimeter Institute/ University of Waterloo
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Re: [astroph/0603449] Wilkinson Microwave Anisotropy Probe
Exactly!Ben Gold wrote:
Ah, I think I see. If you play with a(t) you're effectively changing Einstein's equations and you need to also go and change the equations for the metric perturbations. But if I'm understanding this correctly, the Boltzmann equations should be still be fine. [tex]\dot\delta_{cdm} = \frac12 \dot h[/tex] in the synchronous gauge regardless, right? But the equation for [tex]\dot h[/tex] is different if [tex](\dot a/a)^2 \ne 8\pi G \rho /3[/tex], and your point is that if you leave the [tex]\dot h[/tex] equation unchanged you'll get inconsistent and nongaugeinvariant results?
One way to see this explicitly is by looking at the superhorizon perturbations. For these perturbations, due to causality, the evolution within a Hubble patch can be obtained by pertubing the Friedmann equations. Therefore, if you change the Friedmann equations (through introducing dark energy or modified gravity), you will also change the equations for superhorizon perturbations. For adiabatic perturbations, this is equivalent to saying that the Bardeen parameter:
[tex]\zeta \equiv \phi + {2(H^{1}(z)\dot{\phi}+\phi)\over 3(1+w(z))}[/tex]
is constant on superhorizon scales, where \phi is the metric perturbation in the longitudinal gauge. Notice that the perturbation equation that you arrive at by taking the derivative of [tex]\zeta[/tex], contains w(z) and its derivative, and thus, does not only depende on H(z).
While this gives a unique way of modifying perturbation equations when you modify the background evolution, it cannot be uniquely extended to subhorizon scales. One may do this phenomenologically by introducing a speed of sound [tex]c^2_{s}[/tex] as the coefficient of the k^2 correction to the perturbation equation. However, things can be much more complicated, as perturbations won't remain adiabatic, and thus there will be additional degrees of freedom, and/or a more general dependence on k.
As Josh was suggesting, may be the easiest selfconsistent thing to do is to set [tex]c^2_s=0[/tex], which is equivalent to extending the superhorizon equation (which is derived uniquely) to subhorizon scales without any change. However, this is physically different from the limit of switching off DE perturbations. Instead, this may be the maximal amount of DE perturbations that you may hope to get, as DE clusters similar to dark matter. [tex]c^2_s=1[/tex], might be the closest physical limit to no perturbations.
There is also the [tex]c^2_s \rightarrow \infty[/tex] limit, but intuitively I think it should break causality.

 Posts: 54
 Joined: September 28 2004
 Affiliation: University College London
Re: [astroph/0603449] Wilkinson Microwave Anisotropy Probe
Jochen,Jochen Weller wrote: I think they just wanted to show that even if they might have messed it up last time (which I am not sure if they did or did not), that it does not make a difference as soon as you combine data sets.
Year One was only done with perturbations switched on.
Cheers
Hiranya

 Posts: 54
 Joined: September 28 2004
 Affiliation: University College London
Re: [astroph/0603449] Wilkinson Microwave Anisotropy Probe
Susana,Susana Landau wrote:what about reionization and degeneration with primordial scalar index fluctuations (n_s) and scalar amplitude (A_s) ?
In section 3.2 they say that the polarization measurements now strongly constrain tau, and that other parameters (n_s, A_s) are insesitive to the reionization history.
However from fig 10 it follows that \tau is still degenerated with A_S , and n_s.
what do you think?
Fig 10 shows exp(2\tau), not \tau. Compared to the first year results, the extent of degeneracy of \tau with n_s and A_s has significantly decreased (e.g. Fig 1). Moreover, this degeneracy is broken not just in the LCDM model but also in a variety of others.
Cheers
Hiranya

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 Joined: September 24 2004
 Affiliation: LudwigMaximiliansUniversity Munich
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 Posts: 45
 Joined: September 24 2004
 Affiliation: LudwigMaximiliansUniversity Munich
 Contact:
[astroph/0603449] Wilkinson Microwave Anisotropy Probe (WMA
Hiranya
where they also switched on for w<1 ?
where they also switched on for w<1 ?

 Posts: 11
 Joined: November 27 2005
 Affiliation: University of Alabama, Huntsville
[astroph/0603449] Wilkinson Microwave Anisotropy Probe (WMA
Jochen Weller mentioned involvement of other datasets. Well, I do not
know if this is the reason, but does anyone notice that Fig. 6a and 6b
are quite different? The redline is, as I understand it, the `WMAP only' model
power spectrum, but compare the P(k) value as given by this line, at e.g. k=0.03 (h/Mpc),
and you will see what I mean. Does anybody know how to resolve this?
I'd be grateful upon hearing.
And by the way, the yaxis has the wrong label. P(k) is in units of Mpc^3.
This is likely to be a trivial typo.
Another point of contention (nontrivial I think). Has anyone noticed that Fig.6a
and 6b are not only different, but either of them is also not the same as the WMAP
1st year spectrum, viz. Fig. 9 of Spergel et al (2003)? In that earlier plot, k
was just labelled in units of Mpc^1 rather than h/Mpc, BUT no matter how I
played around with the various combinations of multiplying/dviding k by h=0.71,
and/or P(k) by h=0.71^3, I could not get Fig. 9 to agree with Fig. 6. Not even
close.
Obviously I am the unenlightened one who missed something. Should there ever be a
scenario under which the WMAP papers/authors might wish to explain things better?
What about the (highly improbable) scenario that there is in fact something strange
going on?
know if this is the reason, but does anyone notice that Fig. 6a and 6b
are quite different? The redline is, as I understand it, the `WMAP only' model
power spectrum, but compare the P(k) value as given by this line, at e.g. k=0.03 (h/Mpc),
and you will see what I mean. Does anybody know how to resolve this?
I'd be grateful upon hearing.
And by the way, the yaxis has the wrong label. P(k) is in units of Mpc^3.
This is likely to be a trivial typo.
Another point of contention (nontrivial I think). Has anyone noticed that Fig.6a
and 6b are not only different, but either of them is also not the same as the WMAP
1st year spectrum, viz. Fig. 9 of Spergel et al (2003)? In that earlier plot, k
was just labelled in units of Mpc^1 rather than h/Mpc, BUT no matter how I
played around with the various combinations of multiplying/dviding k by h=0.71,
and/or P(k) by h=0.71^3, I could not get Fig. 9 to agree with Fig. 6. Not even
close.
Obviously I am the unenlightened one who missed something. Should there ever be a
scenario under which the WMAP papers/authors might wish to explain things better?
What about the (highly improbable) scenario that there is in fact something strange
going on?

 Posts: 27
 Joined: September 25 2004
 Affiliation: University of Barcelona
Re: [astroph/0603449] Wilkinson Microwave Anisotropy Probe
Um, that was really my point. :)Jochen Weller wrote:Fergus
Perturbations in DGP are more complicated then just changing H(z).
Look at: http://arxiv.org/abs/astroph/0407489
I was criticising the fact that so many theories assume that the growth factor and H(z) are controlled by the same physics. Why not consider the adhoc approach of modifying H(z) alone?
This is only an effective theory, trying to mimic a pure H modification, and not intending to represent the underlying physics. So the implications of setting [tex] c^2_s=\infty[/tex] don't worry me.Niayesh Afshordi wrote: There is also the [tex]c^2_s \rightarrow \infty[/tex] limit, but intuitively I think it should break causality.
Not sure how WMAP chose to evaluate the perturbationless DE approach, and as Alessandro pointed out it was probably just to show that they are currently insignificant. But an infinite sound speed is how I would envisage it. Erases all dark energy perturbations, and thus the DM perturbations are solely dictated by H. The idea is to permit DE to influence the kinematics but not the dynamics.