CosmoCoffee

Hi all,
I am studying this very interesting paper that appeared on the archive in October,
I am usually a factor [tex]\pi[/tex] months late with the respect to new posts so I apologize for being late, but I am very much puzzled and maybe someone in the forum may
light on.

In section VII (first paragraph at the bottom of the column), the authors state:

The above discussion assumes that the acceleration is due to a cosmological constant.....For [tex]w>-1[/tex], the ISW signal is enhanced, increasing the best fit value of [tex]\Omega_M[/tex] relative to the [tex]w=-1[/tex] case, while [tex]w<-1[/tex] reduces the signal, allowing for even lower values of [tex]\Omega_M[/tex].

I found this very confusing, shouldn't be exactly the opposite? At least in the case [tex]w>-1[/tex], the acceleration starts earlier for [tex]w\rightarrow -1[/tex],

hence the expansion rate starts deviating from the Matter Dominated behaviour earlier and consequently produces a larger ISW signal.

In particular if we limit to models with [tex]w>-1[/tex] the LCDM has the largest signal and I think there is pretty much consensus on this as it has been found in a number of papers. For models with [tex]w<-1[/tex] the ISW is actually enanced and not suppressed.

I tried to check their plot in Fig.12, so using their galaxy window function I have computed the cross-correlation power spectrum [tex]X_l[/tex] for different values of constant [tex]w[/tex] and keept the cosmological parameters fixed to the values of Fig.12.
However I cannot reproduce their trend, where the ratio [tex]x=\frac{X_l(w)}{X_l(-1)}[/tex] increases as function of [tex]w=-1.3 \rightarrow -0.7[/tex]. In agreement with the previous argument I found exactly the opposite.

In the case with no dark energy perturbations, such that the ISW is entirely caused by the background expansion, I found [tex]x \rightarrow 0[/tex] for [tex]w \rightarrow 0[/tex], as can been seen here
While accounting for the dark energy perturbations slightly modify this behaviour since they can affects the late time evolution of the gravitational potentials,

in particular I found that [tex]x[/tex] increases up to [tex]w=-0.8[/tex] and then decreases for [tex]w>-0.8[/tex].

Comments are welcome I wish to understand the cause of the discrepancy.

All the best,

Pier-Stefano

I guess I'm missing something here. The epoch of dark energy domintion, which is what should matter for ISW, is

[tex]1 + z_{DED} = ({\Omega_{\Lambda}\over \Omega_{m}})^{-{1\over 3w}}[/tex]

As DE dominates today, for larger w's, this is pushed to larger z's. For [tex]w\rightarrow 0[/tex], [tex]z_{DED} \rightarrow \infty[/tex]

Hi Niayesh,
the redshift of the acceleration is given by

[tex](1+z_{acc})^3=-(1+3 w)\frac{\Omega_{DE}}{1-\Omega_{DE}}[/tex]

while the redshift at which the DE starts dominating, as you quote, is given by

[tex](1+z_{DED})^3=\frac{\Omega_{DE}^{-1/w}}{1-\Omega_{DE}}[/tex].

I plotted both, solid blue line is [tex]z_{acc}[/tex] and the dash blue line is the [tex]z_{DED}[/tex], as you can see for fixed value of [tex]\Omega_{DE}=0.7[/tex]
the acceleration starts earlier than the domination for [tex]w<-0.6[/tex].
So for [tex]\Omega_{DE}=0.7[/tex] what you say is true but only for [tex]w>-0.6[/tex]. On the contrary for [tex]w<-0.7[/tex] what matters is when the acceleration starts and one can easily show that the derivative of the gravitational potential [tex]\frac{d\Phi}{dz} \approx H(z)\frac{dH}{dz}[/tex].

So to summarize for a fixed value of [tex]\Omega_{DE}[/tex], the amplitude of the ISW will be approximately determined by the redshift of the acceleration for all the values of w for which the acceleration starts earlier the epoch of dark energy domination and viceversa for those values of w for which the domination starts before the acceleration.
The best,

Pier-Stefano

Hi all,
it s me again, I checked the formula in my previous post and I noticed I made a mistake, now they should be correct, but the results are pretty much unchanged. Indeed

[tex](1+z_{DED})=\left(\frac{1-\Omega_{DE}}{\Omega_{DE}} \right)^{\frac{1}{3w}}[/tex]
while
[tex](1+z_{acc})=\left( -\frac{1}{1+3w}\right)^{\frac{1}{3w}}(1+z_{DED})[/tex]

here is their plot:


So for [tex]w\rightarrow 0[/tex] we have [tex]z_{DED}\rightarrow \infty[/tex], but for [tex]w< -0.65[/tex] we have [tex]z_{acc}>z_{DED}[/tex].

Feedback is very welcome,
cheers

Pier-Stefano

Hi everyone,

I evaluated the ratio X_l(w)/X_l(-1) using my code. I find a good agreement with Pier Stefano for both cases: with and without the perturbations. On the plots below

solid line: w=-1
dot: w=-0.9
dash: w=-0.8
long dash: w=-0.7
dash-dot: w=-0.6
long dash-dot: w=-0.5.

First one below is the plot with perturbations turned on. As you see, the
ratio increases with w until w=-0.65 or so, then it goes down,
just as Pier Stefano saw.



The next plot is with perturbations turned off. Here the correlation
changes sign for w>-0.65 or so.



I didn't use the same galaxy selection function as the on in the paper, just took a wide Gaussian window centered at z=0.5 with s.d. of 0.2, which should have been good enough to reproduce the general trend.

Neither of the two plots above are consistent with Fig 12 of the paper
under discussion and agree with Pier Stefano's. So I tend to agree with PS that Fig 12 is wrong.

Levon

Hi Pier-Stefano and Levon,

I was wondering if you fixed [tex]\Omega_{m} [/tex]when you change [tex] w[/tex] in your plots, or adjust it to fix the angular distance to the LSS. Padamnabhan et al. fix [tex]\Omega_m[/tex] in their Fig. 12.

Niayesh

Yep, I have fixed the cosmological parameters as quoted in the caption of Fig.12 and I have also used the same window function as shown in their Fig.2.

But let me explain with a simple argument why [tex]ISW \rightarrow 0[/tex] as [tex]w \rightarrow 0[/tex] at least in the presence of DE perturbations. Suppose that the flat universe is made of two clustered fluid components, standard CDM and a X fluid with [tex]w_X=0[/tex] and adiabatic perturbations with [tex]\Omega_X=0.7[/tex]. In such a case the latter always dominatse the expansion of the universe corresponding to the fact that for [tex]w_X \rightarrow 0[/tex] [tex]z_{DED}\rightarrow \infty[/tex]. This implies that throughout the expansion history of the universe the growth factor [tex]D(a)= const * a[/tex], henceforth [tex]\Phi=\frac{D}{a}=\frac{2\Omega_m}{5\Omega_X}=const[/tex] and therefore no ISW at all in the same way it happens in SCDM cosmologies (I have neglected the radiation dominated era whose transition to MDE leads to the early ISW which anyway would not be picked up by the galaxy selection function).

So to summarize for clustered DE models with [tex]w_X=0[/tex] and [tex]\Omega_X>\Omega_m[/tex] there is no ISW.

From the text of the Padmanabhan et al. paper is not clear whether they are including the DE perturbations, but either they have included or not Fig.12 does not match with physical arguments or our numerical results.

By the way thanks Levon for your runs
I can also confirm your result in the absence DE perturbations, I got that [tex]\frac{X_l(w)}{X_l(-1)}<0[/tex] for [tex]w<-0.66[/tex] . I still have to understand this flipping in the sign, but I think is correct.
The best,

Pier-Stefano

Hi guys,
To answer Niayesh's question, yes I kept Omega_M, as well as all other parameters,
fixed for those plots.
Levon

Hi everyone,

I've been following your discussion, and now I'm jumping in. Not too late I hope...

Pier Stefano's argument for why ISW -> 0 as w->0 prompted me to try the following experiment. I ran the CMBfast v4.5 Boltzman code twice,

with only matter:
----------
Omega_b = 0.05
Omega_c = 0.95

and with some matter-like dark energy:
----------
Omega_b = 0.05
Omega_c = 0.65
Omega_de = 0.30
w=0
dark energy perturbations=YES

I expected to obtain identical CMB spectra, but I did not. The acoustic peak positions were basically the same, but the dark energy model produced significantly higher peaks. I can't think of a reason why this should be true - is there one? I turned off COBE normalization, so it's not a normalizing issue.

Chaz

Hi Charles,
In CMBFAST dark energy (DE) is set to be due to scalar field quintessence, which means that the effective speed of sound is c_s^2=1. In physical terms, this means that DE is bound to be weakly clustering -- practically smooth on sub-horizon scales. CDM has c_s=0, i.e. it clusters. So, even if you set w=0 for DE it wouldn't be the same as CDM.
cheers
Levon

Yes that certainly explains it - thank you. I knew I must have been doing SOMETHING wrong...

Best,
Chaz