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N-body simulation of DGP model July 1, 2009

Posted by keithkchan in Cosmology, Journal club.
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I am very happy to have Kwan Chuen Chan from Center for Particles Physics and Cosmology, New York University to talk about their new paper. He is a grad student at NYU, working with Roman Scoccimarro. His office is in Room 538 CCPP.
Keith

Thank Keith for inviting me to blog about our recent paper. In this post I will briefly talk about the paper that Roman Scoccimarro and I just uploaded to the arXiv. I will keep it brief and elementary, so for more details, please refer to the original paper arXiv:0906.4548.

Here is the abstract

Large-Scale Structure in Brane-Induced Gravity II. Numerical Simulations
Authors: K. C. Chan, Roman Scoccimarro
(Submitted on 24 Jun 2009)
Abstract: We use N-body simulations to study the nonlinear structure formation in brane-induced gravity, developing a new method that requires alternate use of Fast Fourier Transforms and relaxation. This enables us to compute the nonlinear matter power spectrum and bispectrum, the halo mass function, and the halo bias. From the simulation results, we confirm the expectations based on analytic arguments that the Vainshtein mechanism does operate as anticipated, with the density power spectrum approaching that of standard gravity within a modified background evolution in the nonlinear regime. The transition is very broad and there is no well defined Vainshtein scale, but roughly this corresponds to k_*= 2 h/Mpc at redshift z=1 and k_*=1 h/Mpc$ at z=0. We checked that while extrinsic curvature fluctuations go nonlinear, and the dynamics of the brane-bending mode $C$ receives important nonlinear corrections, this mode does get suppressed compared to density perturbations, effectively decoupling from the standard gravity sector. At the same time, there is no violation of the weak field limit for metric perturbations associated with $C$. We find good agreement between our measurements and the predictions for the nonlinear power spectrum presented in paper I, that rely on a renormalization of the linear spectrum due to nonlinearities in the modified gravity sector. A similar prediction for the mass function shows the right trends but we were unable to test this accurately due to lack of simulation volume and mass resolution. Our simulations also confirm the induced change in the bispectrum configuration dependence predicted in paper I.

DGP model is an extra-dimension model, which has one co-dimension, and ordinary matter lives on the 3-brane. The graviton propagator is modified in the infrared. One of the interesting properties of this model is that it exhibits self-accelerating solution. The hope was that the recent observed cosmic acceleration may be due to modification of gravity rather than the mysterious dark energy. However, both theoretically and observationally, this model is proved to be unfavorable. However, this model has inspired a bunch of more sophisticated models such as degravitation, galleon. One of the serious problem in modification of gravity is that it induces new degrees of freedom. The theory can usually be approximated as a scalar-tensor theory. But any scalar degree of freedom is likely to be highly constrained by current solar system experiments. There are two nice ways have been put forward to evade this kind of constraints. One of them is chameleon mechanism, which has been realized in f(R) gravity. The other mechanism is called Vainshtein effect, which is incorporated in DGP and some massive gravity models. The scalar degree of freedom becomes strongly coupling and frozen because of the derivative self-interactions. The theory effectively becomes GR.

In this paper, using numerical simulations, we study this type of brane-induced gravity in the nonlinear regime, in particular the Vainshtein effect. We compute the cosmological observables: the power spectrum, bispectrum, mass function, and bias, which give us the signatures of the DGP model, and help us to differentiate modified gravity model from dark energy. In the companion paper arXiv:0906.4545 by Scoccimarro, the model is studied by perturbative calculations. Some of the results are checked against the numerical results in this work.

The method we used is the N-body simulation, which is largely similar to the standard gravity one. However, in GR, the field equation in the subhorizon, non-relativistic regime is just the Poisson equation, now we need to solve a fully nonlinear partial differential equation. Let me write down the equations although I am not attempting to explain it in details
\bar{\nabla}^2 \phi  -  \frac{1}{\eta} \sqrt{ - \bar{\nabla}^2  } \phi +    \frac{1}{2 \eta} \bar{\nabla}^2 C   +      \frac{ 3 \eta^2 - 5 \eta + 1 }{2 \eta^2  (2 \eta -1)  }  \sqrt{ - \bar{\nabla}^2  } C      =  \frac{3}{2} \frac{\eta  -1  }{\eta} \delta
(\bar{\nabla}^2 C)^2    +    \alpha  \bar{\nabla}^2 C   - (\bar{ \nabla}_{ij} C)^2 +    \frac{ 3 \beta (\eta -1) }{2 \eta-1 }    \sqrt{ - \bar{\nabla}^2  } C     = \frac{ 3( \eta -1 ) } {\eta } ( 1- \beta \bar{\nabla}^{-1} ) \delta,
The first equation is analogous to the Poisson equation, but now we have one more field C, whose equation of motion is given by the second one. The nonlocal term like \sqrt{ - \bar{\nabla}^2  } C can be easily handled in the Fourier space. The real headache comes from the nonlinear derivative terms (\bar{\nabla}^2 C)^2 and (\bar{ \nabla}_{ij} C)^2 . One of the major achievement in this paper is that we developed a convergent method to solve this set of equations consistently. It involves alternate use of relaxation and Fast Fourier transform (so we call it FFT-relaxation method). Although that is a main result of the paper, I am not going to talk about it in details so as not to get too technical and dry. But interested readers are welcome to read the original paper.

Let me get to the results. As I have mentioned, from the simulations we have measured the power spectrum, bispectrum, mass function and bias. Here I only show the power spectrum.
PDGP_Grun_z0
PDGP_GRH_z0
In the first figure we show the power spectrum from three different models, which are the fully nonlinear DGP model (nlDGP), linearized DGP model (lDGP) and the GR with the same expansion history as the DGP model (GRH), which essentially is the GR limit. In order to see the difference more clearly, we have shown the ratios of power spectrum from various models, P_{\rm  nlDGP} / P_{\rm lDGP}  and P_{\rm GRH} / P_{\rm nl DGP} in the lower figure. In the large scales (small k), the full nonlinear DGP model reduces to the linear one. More interestingly in the nonlinear regime (large k limit), the fully nonlinear DGP model approaches the GR with the same expansion history. This demonstrates that Vainshtein effect drives the model towards GR limit in the large k regime. The transition is broad and the limit is not yet fully attained in the range shown here.

OK, let me summarize some of the main results here. We have developed a convergent algorithm, FFT-relaxation method, to solve the fully nonlinear field equations in the DGP model. This enables to compute the observables like the power spectrum in the DGP model using numerical simulations. We have demonstrated the Vainshtein effect, and the Vainshtein radius at z =0 is about 1 h/Mpc. For more details, please refer to our original paper arXiv:0906.4548.

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From spin to mechanical osicillation June 14, 2009

Posted by keithkchan in fun stuffs, Journal club.
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I haven’t updated the blog for some time, so I should say something now.
Well, there is a rather interesting report appear in Nature, Entangled mechanical oscillators. You can you find it on arXiv 0901.4779.

You probably have heard of quantum entanglement many times, which means that the state is not factorizable. A famous example is the Schr\ddot{o}dinger’s cat. Measurement causes the wave function to collapse. When you do measurement you either push the cat to the hell or drag it out of the hell. But this is still a thought experiment. Not just because physicists are kind to animals, but also it is impossible to do it on macroscopic scales because of decoherence. All the examples I heard of are limited to entanglement of spin or polarization. But in this the Nature report, a group of physicists at NIST have managed to convert spin entanglement to mechanical oscillations. The experimental details are technical, and I don’t really understand. In (and only in) simple terms, they first entangle the spin of two magnesium and two beryllium ions, and then separate them into two potential wells. In each well, there is one magnesium and one beryllium ion, which form an oscillator in the potential well. They then carry some measurements which create the motional entangle state. I don’t understand how they really do it. For those interested in, you should consult their paper.

The significance of this paper is they have for the first time created mechanical entangled states. This is one important step towards Schr\ddot{o}dinger. But, cats, no panic, there may be still 500 steps away.

Incidentally, there is an article in Science describing this paper. The article is fine. But don’t read the comments if you don’t know much about this subject. I find them dubious, if not totally ridiculous. You may want to check it against John Baez’s crackpot index.

Horava-Lifshitz Gravity April 28, 2009

Posted by keithkchan in Journal club.
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Recently, there are a lot of papers on the a new quantum gravity model, now dubbed Horava-Lifshitz gravity. It has been published in Physical Review. This model appears to be very exciting since it claims to be renormalizable. Many authors have immediately started calculating various things: inflation in this model, bouncing model, Schwarzschild solution, non-Gaussianity, blah, blah, blah.

Beside being renormalizable, another thing that is particularly exciting is that this model seems to be able to generate scale invariant spectrum without inflation. That sounds very cool!

Because of these seemingly attractive features in this model, in last week’s journal club, people in another journal club tried to read this paper. Since the model is novel and complicated, the smart people in our centre failed to understand the model. Gruzinov was very excited, and asked Kleban to talk about it in next week. He even criticized that the high energy people in our centre for their ignoring this interesting model.

So this week, Kleban presented this model. Before the start of the talk, Gruzinov asked Porrati for his comments. It turns out Porrati often makes funny comments. He said that I haven’t really read it, but if it was right, I would change my career to study swine flu. I find his comment very funny.

After all these prelude, I now briefly describe the model. Frankly speaking, I don’t really understand it, so I am not going to comment on it. The model is motivated by the Lifshitz point, which is used to model the triple point in condensed matter physics. By the way, this is another example that ideas in high energy physics originate from condensed matter physics. In this model there is critical exponent z, if it is 1, we get relativistic dispersion relation. This model is z=3 in the UV, and it becomes z=1 in the IR, and Lorentz symmetry is recovered as an accidental symmetry. The model is power counting renormalizable. In this model, space and time scales in differently, so that the the theory is rendered power counting renormalizable. Note that power counting renormalizable is just a first test of renormalizability. One really needs to check in details if it is indeed renormalizable. During the talk, some of the audience, in particular Gruzinov was increasingly impatient as the Lagrangian introduced was rather ad hoc. I don’t really understand it so I am not going to comment on that, just to say that the terms added are mostly spatial, and of high derivatives. Needly to say, it is very different from the usual GR and field theory. Gruzinov even said that life was short, why do we spend time on such a theory. Kleban reminded him that he should blame himself for asking him to present the paper. People were worried about about a parameter introduced in the theory, which start as 1/3 in UV, it has to flow to 1 in order to agree with GR in the IR regime. After browsing the whole paper, Horava did not say (or know ) how to realize this flow. Porrati said that it was almost impossible for any perturbative effect drive the parameter to flow from 1/3 to 1. At the end, nobody was happy about this paper. Gruzinov seemed to be pissed off. I personally find the talk rather amusing. The comments by various people, in particular Porrati and Gruzinov very interesting and illuminating.

So it was pretty funny that last week everybody was excited about this paper and this week everybody loses interest in it. We have to wait for another breakthrough…

Update: I would like to add a few comments on it. Nobody really knows how to come up a workable quantum theory of gravity. So anybody can take strong opinion in this regard. But anybody that asks the right question is much much more likely to succeed than others. So Porrati and Gruzinov think that Horava’s approach is not going to work. Fine.

But shouldn’t it be obvious that anything that is obvious is not going to work. Anything that is going to work is not going to be obvious. The reason that those smart people fail to come up with a sensible theory of quantum gravity maybe because they are too strongly biased. I defend this theory does not mean that I believe that this the right theory. But I certainly think that this is an interesting and novel idea. OK, Horava did not lay out the model in details. But I think he has set up a proposal that is interesting enough to pursue it further. One of the reason that so many people have jumped into this theory because they think that it is interesting and worth trying. So my philosophy to these kind of difficult open questions is that we should keep mind opened.

Up-update: Apparently, this post is controversial and stimulating. Lubos Motl has written a post about this post. And he tries to dissect each comments I said. So I should feel honored since nobody has taken my words so seriously before. But I am going to de-comment since I got much more publicity than I want. You check out his comments to see if it is justified.

I got so much publicity that I may need to put up a stupid disclaimer that the contents of this blog has nothing to do with NYU, don’t sue the director of graduate studies if you don’t like this post. OK, anyway I now shut up.

Evidence of cosmological extra dimensions? February 25, 2009

Posted by keithkchan in Journal club.
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Yesterday, we had the journal club. It was my turn to present a paper. I would like to summarize the paper I discuss here.

The paper I chose was a recent paper arXiv:0812.2244 , which claims that there might be evidences of cosmological extra dimension. The authors presented several evidences which seems to be at odd with the standard model LCDM. The evidences are:
1. the cross power spectrum between the galaxy and CMB, the power is larger than the expected value by 2 \sigma
2. existence of large scale bulk flow which is also larger than expected by 2 \sigma
3. Cosmic Background Imager measure the power in CMB larger than expected by 35% at l~3000
4. Lyman-alpha forest power spectrum, which probe the structure around z~3 find that the the power is larger by 35%
5. The CMB correlation at angular scale larger than 60 degree is much smaller than expected. This corresponds to the quadrupole in the CMB power spectrum. People have parametrized the unnaturalness of the smallness in many ways. If it is due to statistical fluctuation, the chance of having such a small value is less than 0.03%.

Evidences 1 to 4 require that the structure in the large scales are more evolved than expected from LCDM model, while evidence 5 usually mean that the evolution cannot be too large. The authors fit the data using the a phenomenological generalization of the DGP model. In their model, the number of extra dimensions is more than 1. In this kind of model, our 3-brane embeds in more than one brane. e.g. our 3-brane embeds in some 4-brane, which is again embedded in 5-brane. This kind of construction seems to avoid the ghost instability that kills DGP. In intermediate scales the extra scalar degree of freedom can mediate extra force and enhances the gravitational strength. They tune some free parameter so that the Sachs-Wolf effect cancels the Integrated Sachs-Wolf effect. It improves the fitting to CMB a little bit, so that the problem 5 is alleviated a little bit. Their model also seem to alleviate the tension between the theory and observations for problem 1 to 4.

Although I am not very familiar with most of these observations, my impression is that the potential systematics in these observations are rather large. They may be rather nonlinear, difficult to model, e.g. evidence 3. The differences from LCDM by 2 2 \sigma is not so significant. Among all the evidences, I take No. 5 most seriously. But their model can only improve it a little bit. Also their calculations are rather crude. I don’t really believe that there are already evidences for cosmological extra dimensions.

On the other hand it is interesting to note that this model seems to alleviate the tension between LCDM and data in several different observations. It is useful to keep this in mind to see what happens when the measurements improve.

An Intersting Paper about Landau January 19, 2009

Posted by keithkchan in fun stuffs, Journal club.
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Today is one day before the start of the spring semester. I try to do something light. I found an interesting papers hep-th/0204295 about Lev Landau. It is a recollection by Boris Ioffe. Apparently I only know that this author is Russian, nothing else. Maybe more, he was one of the 43 guys (I supposed all of them were male, although i have no clue from their names) who passed the theoretical minimum.

In this article, the guy talks about Landau’s theoretical minimum, which is the entrance exam set by Landau, and people who wanted to be Landau’s students must pass. Obviously the student has to be very smart in order to pass the exam. One interesting thing is that the students had to know other foreign languages, e.g. German. Landau conducted the Landau’s seminar. His requirements on the presenter was very high. The presenter had to know as much as (maye even more than) the author of the paper when they presented it. Also the topics in the seminar span from particle physics to material science. Simply because Landau knew everything.  Landau sometimes gave a very different perspective from the author of the paper and ever said that the authors did not understand what they did. Landau had very strict rules about his seminar, e.g. (almost) nobody could be late.  But apparently there were two guys that got privileged in Landau’s seminar. They were Ginzburg and Midgal. BTW, I haven’t heard of Migdal much, am I ignorant?   There were also two interesting jokes played by Migdal and Pauli respectively, in Landau’s seminar. What are they? Look it up yourself. This paper is a very good passtime.

Ab initio QCD Calculation of the Proton Mass November 23, 2008

Posted by keithkchan in Journal club.
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In Science this week, physicists report that they successfully calculated the proton mass using lattice Quantum Chromodynamics (QCD). This marks the efforts of physicists in the last 30 years to calculate the mass of nucleons from first principles. We believe that about 90% of the mass in the nucleons is due to the strong interaction between quarks and gluons, and the mass of the quarks and/or EM interaction between the quarks contribute the remaining. Because of the strong coupling the QCD, this has not been confirmed satisfactorily. The latest calculation by Durr et al is a vivid demonstration of the validity of the standard model.

In lattice QCD, the spacetime is discretized, so that the calculations are reduced to finite integrals that can be done on the computer. The difficulties in lattice QCD are that the vacuum is strongly fluctuating, so that to describe it accurately, many snapshots are required. Another obstacle is that extremely high computational power is required to include the influence of the the quark-antiquark pair on the gluon vacuum. In last couple of years, researchers develop better mathematical description and more accurate formulas to describe the quark-antiquark pair. The above simulation from Derek Leinweber is the visualization of the fluctuating vacuum due to quarks and gluons. So finally, researchers are able to calculate the mass of the proton on a computer with the needed physics included and put the error under control. The is another triumph of the standard model. Cheers =:).

May the Force be with You November 12, 2008

Posted by keithkchan in Journal club.
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Today, we don’t have the usual journal club. Instead I’ll talk about a more interesting paper (seriously). In Scientific American and else where, they report Bamford et al put the concept of magnetic shield into action. The original paper is published in Plasma and Controlled Fusion.

When I watched cartoon in my childhood, I often saw that the spaceship can put up some sort of force field to protect itself from radiation or more often enemy’s attack. Now Bamford et al have takena step toward realizing this kind of shield. Actually, the main potential application of this kind of device is to shield the spaceship in outer space from the dangerous radiation from the Sun. On the earth, we are protected from the lethal radiation by the atmosphere. Without the protection from the atmosphere, the energetic particles from the Sun can penetrate deep inside our body and change our DNA. This will cause various genetic diseases and causes mutations. We may see X-men in the street, not just in the theater.
In that paper, the researchers shot a fast flowing magnetized hydrogenic plasma on the dipole field, and they found that the dipole field was able to deflect the plasma, so that the central region of the magnetic field was devoid of plasma. They also did a computer simulation, and the simulation result agreed well with the experiment that the magnetic field region act as a diamagnetic cavity.
This paper is the still preliminary, more work needs to done, but the result is clearly exciting. However, we note the shield can only deflect the radiation from the Sun, but not the high energy cosmic rays.