356. Cosmological parameters in the context of time varying w
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Rahul Biswas is a graduate student in Benjamin Wandelt’s research group working on estimation of cosmological parameters. His main interests also include models of dark energy and supernova cosmology. Dmitry.
Let me start by saying that I highly appreciate Dmitry’s idea of having this online journal club. I hope to learn something from the discussions here while talking about a recent paper (arxiv:0903.2532) with Benjamin Wandelt about parameter constraints in the context of dark energy models with a time varying equation of state 
.
More and more observations, mostly related to supernovae but supported by CMB/Galaxy survey data point to the fact that the expansion of the universe is accelerating (There have been other attempts to explain this in terms of modified gravity or backreaction of perturbations which I shall not discuss here). This calls for an otherwise unobserved currently dominant component dubbed dark energy, which is plagued by “the cosmological constant” problem, (widely discussed, including on this blog itself as a fundamental open problem in physics). Ironically, the problem is not finding a candidate that could lead to possible acceleration, but in finding a candidate for dark energy which would cause small enough observed acceleration. Therefore, a great deal of effort is being directed at quantifying the phenomenology of the dark energy.
As I said, there is no dearth of models of dark energy in terms of one or more almost homogeneous fields with different potentials… there are way too many to work out the constraints for each. A common approach is to map a bunch of dark energy models to a fluid model of dark energy, parametrized by the function 
where is the equation of state, and the speed of sound 
. The final aim for surveys would be to try to reconstruct these functions; however this is still too large a parameter space to be able to constrain to any useful level by data that is currently available, or will be available in the near future. So what people usually do is write down a specific parametrization of the equation of state and then try to constrain the parameters of the equation of state from the data. Knowing these functions, one can compute (with a bit of computer time) the power spectrum of anisotropies of the CMB, the power spectrum of matter inhomogeneities, and the luminosity redshift. The challenge for dark energy phenomenology would be to pin down the value of and see if it differs from the cosmological constant.
Let us consider two examples of parametrized Equation of State w:
- constant equation of state. This is a w which was allowed to be different from -1 (as it would be for a cosmological constant, but constrained to remain at the same value throughout). We have pretty strong constraints for these today (see here for some very recent constraints), but what if w was actually varying (which would be the case usually in say a generic scalar field model)? Could it be that we are fooled into thinking that we have a cosmological constant because
represents some sort of mean value of a time varying ? - Time Varying Equations of State: A common example that is being widely discussed today is the CPL (Chevaliar-Polarski-Linder) parametrization being used by the Dark Energy task force
. This is a simple example of an equation of state that varies between 
at early times to 
at late times. It includes the cosmological constant, and parametrizes a departure from it. Obviously, this (or any other choice for that matter) may not be how the equation of state really depends on redshift, so the equation of state constraints may be biased. However, if this does exclude the cosmological constant 
, it would imply some form of dynamics. So while you might take the precise constrained with a pinch of salt, you could use this to distinguish if dark energy evolves in a non-trivial way. Among arbitrary parametrizations of a time varying w, this is also a useful choice from a practical perspective today, since there has been a consensus (Dark Energy Task Force) to compare various observational methods using this parametrization.
You can see how the power spectrum varies as you change the parameters around (look here for example) . The point is that different characteristic change when you change the parameters. This enables one to compute the constraints on cosmological parameters by comparing the differences of these computed spectra with the observed spectra the expected statistical deviations (these come from noise in the instrument, as well sampling effects) which can be computed separately.
Computing parameter constraints this way requires repeated theoretical computation of the CMB and matter power spectrum, and that is a computation intensive process. So for dark energy parameter estimation only (other parameters can definitely be sensitive to the shape of the spectrum), sometimes people summarize the power spectrum data by quantities that are representative of the position of peaks in the angular power spectrum and the redshift of the CMB… we refer to these parameters as summary parameters (look into the paper for more deails) .These are computed from some standard Lambda CDM model, and the assumption is that the values or the expected deviations don’t change when you change the model. Think about the CMB spectrum: dark energy mainly changes the evolution of the background (acceleration of the expansion). This shows up clearly in the shifted position of peaks. However, it could also change the potentials, and thereby the growth of perturbations. This would show up in changes of magnitude (which may be different for different modes) of the CMB and matter power spectrum, resulting in a difference of shape as distinct from the peak positions. Now it turns out, that for the first case of a constant equation of state, there is not much of an effect of changes in height or shape… this can be linked to the fact that the supernovae data would require w to be close to -1, which means the dark energy density was negligible before. In contrast, dark energy with a time varying equation of state can satisfy the supernovae constraints at late times, and have a role to play earlier. This means for a general time varying w, there could be a lot more information in spectrum than the summary parameters.
We do a couple of things in this paper. First, we wanted to ask how well the constraints computed by using the summary parameters approximates the correct constraints computed by using the power spectra for the case of the gently varying CPL equation of state.
Here is what the constraints on the dark energy parameters look like when using WMAP 5 year data, SDSS, the Union compilation of the supernovae and the BBN constraints. The black dots represent samples from the posterior distribution of the dark energy parameters given the data when comparing the power spectra. The blue dots represent samples when using the summary parameters. The contours represent the ‘
and ‘
constraints, meaning that they enclose about 68 percent and 95 percent samples of the distribution. The black ones correspond to the use of comparison of power spectra, while the blue ones come from summary parameters.
While you can see the difference between the use of summary parameters and power spectra, there are some smoothing effects when you talk about contours. For example, there are hardly any sample points beyond the line 
, but this is not evident from the contours which go well beyond that straight line (check the paper for more details). Hence, when we compare the results of the likelihoods we show the differences in terms of pixelized density plots, so that there are no artifacts of smoothing effects. These quantify the level of approximation in using summary parameters in a dark energy with CPL equation of state which tell you how much you are off when you use them. These differences might be larger for other dark energy models. In particular, methods that attempt to reconstruct the equation of state from observations, can allow quite arbitrary functional forms for where such a difference might be significant, so it could be important to compare power spectra rather than use summary parameters there.
Summary Parameters
Power Spectra
What about other cosmological parameters … how well can we constrain them if we don’t know what dark energy exactly is? For example think about the curvature, or the parameters that describe the tilt of the primordial perturbation spectrum 
. Inflation is usually constructed to make the universe spatially flat, so the curvature 
would be close to zero. Similarly, one would expect that would be less than one. Remarkably, we have stringent constraints on these parameters from WMAP that would confirm these statements. But those constraints are for specific models, mostly for Lambda CDM. Are the constraints robust to changes in the dark energy model? (And recall while we said that dark energy constraints can be computed reasonably from summary parameters, this was not the case for other parameters. So here we will need to compare the full power spectrum). So, we compute the constraints on other parameters for a dark energy with the CPL Equation of state. Of course the constraints will broaden, but how much? So, here are the interesting ones.
The blue lines represent constraints for the CPL model computed by the comparing power spectrum. The solid black lines were the corresponding constraints for a constant equation of state model, while the red constraints come from the use of summary parameters (they don’t work, but this was to be expected). We can see that a Harrison-Zeldovich spectrum (
) cannot be ruled out (as can be for a standard Lambda CDM model), and a somewhat larger value of spatial curvature is allowed. The difference between the blue and black plots in these figures (as also other standard constraints for example in Lambda CDM) from suggest that it would be hard to put very stringent constraints on other cosmological parameters unless we are more specific about dark energy.
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