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356. Cosmological parameters in the context of time varying w

Posted by Rahul Biswas

at April 15, 2009

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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 w(z) where is the equation of state, and the speed of sound cs^2(k, z) . 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 w(z) 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 $w(z)=w_0 + w_1 \frac{z}{z+1}$ . 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.

356. Cosmological parameters in the context of time varying w

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 ‘ $1\sigma$ and ‘ $2\sigma$ 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 w_0 + w_1=0 , 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 w(z) where such a difference might be significant, so it could be important to compare power spectra rather than use summary parameters there.

356. Cosmological parameters in the context of time varying w

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 n_s . Inflation is usually constructed to make the universe spatially flat, so the curvature $\Omega_k$ would be close to zero. Similarly, one would expect that n_s 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.

356. Cosmological parameters in the context of time varying w

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 ( n_s=1 ) 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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355. Introduction into thermonuclear reactors

Posted by Dmitry

at April 15, 2009

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After a brief layman review of the theory of thermonuclear fusion let me get more practical and discuss a bit how thermonuclear reactors are supposed to work.

Basically, we want the energy release of the thermonuclear reactor to be larger than the energy we pump into the reactor. Depending on a particular scenario of the energy pumping, we will distinguish two types of thermonuclear reactors.

Type A reactors are such that the energy is pumped in the initial moment of time just to start the reaction, which then becomes self-sustained. On one hand, the energy of the plasma gets lost due to its finite heat conductivity and radiation – the temperature of the plasma naturally wants to drop. On the other hand, the temperature of the plasma may be supported by the energy released in thermonuclear reaction. For example, if the fuel is d-t, the reaction can become self-sustained due to $\alpha$ – particles – products of the reaction or, more accurately, due to the Coulomb interaction with particles of plasma.

We can write the following criterion for the reaction to become self-sustained:

$n_e\tau_E>\frac{T}{\langle\sigma{}v\rangle{}E_\alpha-{\rm brak.rad.}}$ .

Here n_e is the electron density in the plasma, – its temperature, $\tau_E$ – a characteristic time scale at which the energy of the plasma remains constant, $\langle\sigma{}v\rangle$ – thermonuclear reaction rate averaged over Maxwell distribution, $E_\alpha$ – energy of released $\alpha$ particles (about 3.5 MeV for d-t reaction), ${\rm brak.rad.}$ – energy loss due to the braking radiation.

Two examples of Type A reactor are a Tokamak or a stellator.

If released energy of the products of thermonuclear reaction is not enough to keep the temperature of plasma sufficiently high, we will denote such reactors Type B. Apparently, to sustain the nuclear reaction in the Type B reactor, we need to constantly pump energy into the plasma. It is still fine as long as the energy release of the thermonuclear reaction is larger than the energy cost to support the nuclear reaction.

Reactors can be also classified according to how we are going to confine the plasma. Usually, stability of the plasma is achieved by using external magnetic field, but there are also attempts to build reactors with so called inertial confinement mechanism (for example, HiPER).

HiPER scheme

In the latter case, the energy sufficient for the start of thermonuclear reaction is injected into the reactor by a short ( $10^{-8}-10^{-7}$ sec) laser impulse (or by ion/particle beam).

Miniupdate: I was corrected – for HiPER, $10^{-8}-10^{-7}$ sec is the length of pulses for lasers managing compression, the ignition energy is delivered by $10^{-11}$ sec pulses.

A reactor with inertial confinement will work in the regime of short impulses, unlike a reactor with magnetic confinement – the latter can work in a continuous regime.

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354. Vortex line representation. Clebsch variables

Posted by Dmitry

at April 14, 2009

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Let us continue our brief discussion of behavior of the vorticity field in the Eulerian flow.

(and that’s how vortex lines look like in reality… as if you wouldn’t know )

This time I would really like to derive some equations describing dynamics of vortex lines. For this, it is convenient to use so called Clebsch variables $\lambda$ and $\mu$ .

The physical meaning of Clebsch variables is the following: $\lambda={\rm Const.}$ and $\mu={\rm Const.}$ are two surfaces in space, and their intersection gives the vortex line. Vorticity $\Omega$ can be rewritten in terms of Clebsch variables $\lambda$ and $\mu$ as

$\Omega=[\nabla\lambda\times\nabla\mu]$ .

Probably, the nicest thing about them is that both variables actually remain Lagrange invariants, if the flow is uncompressible (and that’s exactly the kind of the flow we discussed in the previous post – the reason being that we would actually like to separate sound waves from vortex degrees of freedom and only discuss the latter):

$\frac{\partial\lambda}{\partial t}+(\mathbf{v}\cdot\nabla)\lambda=0,\,\,\,\frac{\partial\mu}{\partial t}+(\mathbf{v}\cdot\nabla)\mu=0$ , (1)

so Clebsch variables are actually markers for vortex lines. Namely, we can write

$\lambda=\lambda(x,y,z),\,\,\,\mu=\mu(x,y,z),\,\,\, l=l(x,y,z)$ , (2)

where is parameter along the vortex line. Then,

$\mathbf{\Omega}=\frac{1}{J}\cdot\frac{\partial\mathbf{R}}{\partial l}$ ,

where the Jacobian of the mapping $\mathbf{r}=\mathbf{R}(\lambda,\mu,l)$ is equal to

$J=\frac{\partial(x,y,z)}{\partial(\lambda,\mu,l)}$ .

Finally, we are ready to derive equations describing motion of vortex lines. Let us write an equality

$\nabla\mu\left(\frac{\partial\lambda}{\partial t}+(\mathbf{v}\cdot\nabla)\lambda\right)-\nabla\lambda\left(\frac{\partial\mu}{\partial t}+(\mathbf{v}\cdot\nabla)\mu\right)=0$ ,

which trivially follows from the conditions (1) above since $\nabla\mu$ and $\nabla\lambda$ are linearly independent vectors. Using the transformation (2) we find

$\left[\frac{\partial\mathbf{R}}{\partial l}\times\left(\frac{\partial\mathbf{R}}{\partial t}-\mathbf{v}(\mathbf{R},t)\right)\right]=0$ ,

or, in other words,

$\frac{\partial\mathbf{R}}{\partial t}=\mathbf{v}_{\perp}(\mathbf{R},t)$ ,

where $\mathbf{v}_{\perp}$ is the component of velocity perpendicular to the vorticity vector $\Omega$ . As we see, any motion along the vortex line does not change its form.

Exercise 1: try to derive equation of motion for vorticity field $\Omega$ itself. Answer: it actually has the form

$\frac{\partial\mathbf{\Omega}}{\partial t}={\rm curl}[\mathbf{v}\times\mathbf{\Omega}]$ .

Exercise 2 (funnier): check out that the flow described by Clebsch variables actually has zero helicity

$\int dV(\mathbf{v}\cdot{\rm curl}\mathbf{v})$ .

The latter is topological invariant of the flow – it describes degree of knottiness of vortex lines. What to do if the topology of the flow is non-trivial?

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353. Vortex line representation. Cauchy invariant

Posted by Dmitry

at April 13, 2009

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Several days ago I’ve promised in comments to discuss dynamics of vortex lines in turbulent flows, today is probably a good day to start. And the natural starting point of course is the Kelvin theorem and Cauchy invariant.

Let us consider an ideal (inviscid, uncompressible) fluid described by the Euler equation

$\frac{\partial\mathbf{v}}{\partial t}+(\mathbf{v}\nabla)\mathbf{v}=-\nabla p$

and incompressibility condition

${\rm div}\mathbf{v}=0$ .

A rather non-trivial fact (I bet you don’t know it ) is that these two equations have infinite number of (non-local) integrals of motion. Of course, existence of these integrals does not make the Euler equation integrable – the latter can be clearly seen from the fact that there may exist solutions of the Euler equation that feature chaotic behaviour, turbulence. Nevertheless, it is interesting to understand the nature of these integrals, since it might also shed some light on physics of viscous fluid, described by Navier-Stokes equation.

Another video explaining how turbulence looks like in Lagrangian coordinates

The simplest way to derive these integrals of motion is to use Kelvin theorem. The latter states that velocity circulation

$\Gamma=\oint(\mathbf{v}\cdot d\mathbf{l})$

is conserved along an arbitrary time-dependent contour $C(\mathbf{r}(t))$ . Let us rewrite the expression above using Lagrangian markers $\mathbf{a}$ instead of Eulerian coordinates $\mathbf{r}(t)$ . We have

$\Gamma=\oint\dot{x}_{i}\cdot\frac{\partial x_{i}}{\partial a_{k}}da_{k}$ ,

where the integration is now taken along a static, time-independent contour $C(\mathbf{a})$ . Since the contour can really be taken arbitrary, we apply the Stokes theorem and conclude that

$\mathbf{I}={\rm curl}_{a}\left(\dot{x}_{i}\cdot\frac{\partial x_{i}}{\partial a}\right)$

is conserved in time for each point $\mathbf{a}$ . The expression above is known as Cauchy invariant. Its physical meaning is actually simple – if $\mathbf{a}$ coinside with Lagrangian markers corresponding to particles of the fluid in the initial moment of time,

$\mathbf{I}=\Omega_{0}(\mathbf{a})$ ,

initial vorticity of the flow. In other words, vorticity is frozen in into the motion of the fluid – fluid particles cannot leave the vortex line they belonged to in the initial moment of time, i.e., the only relevant degree of freedom for them is along the vortex line.

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352. 48 years ago

Posted by Dmitry

at April 12, 2009

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Those were the days of our glory. By us I mean all intelligent people, all passionate, sensible people, all people who are able to discriminate objective reality from the bare wishful thinking. And, of course, people who are able to distort objective reality to match their wishful thinking.

But those days have passed. In reality, nothing really went wrong, we did what we could. It was the very objective reality who was saying since 1961 “no”, “no” and “No” to our space program. Space did not make us rich. Space did not open those multiple possibilities for us which, as seemed, were so promising. The most successful space missions that took place since 1961 were and are unmanned missions. As my favorite writer, Sir Arthur Clarke, said in “Childhood’s end”, the stars are not for Man.

But you know… I think that our very failure just signals the end of our childhood. I think we will return to space, maybe not as Men, but we eventually will.

I don’t believe in God, but I do believe in human spirit, human knowledge, human curiosity, human race. Either we or our children will reshape this Reality itself, the Reality that did not open the doors of Space for us.

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