The Breakthrough Prize in Life Sciences, Mathematics, and, Physics

You won’t want to miss the ceremonies later this Month. See the broadcast times at the end of the blog entry.

Life Sciences: Art Levinson, Sergey Brin, Anne Wojcicki, Mark Zuckerberg, Priscilla Chan and Yuri Milner announced the winners of the Breakthrough Prize in Life Sciences:

  • Cornelia I. Bargmann
  • David Botstein
  • Lewis C. Cantley
  • Hans Clevers
  • Napoleone Ferrara
  • Titia de Lange

Eric S. Lander

  • Charles L. Sawyers
  • Bert Vogelstein
  • Robert A. Weinberg
  • Shinya Yamanaka
    each recipient receives $3 Million as their prize.

    Mathematics: The Breakthrough Prize in Mathematics winners are:

  • Simon Donaldson, Stony Brook University and Imperial College London.
  • Maxim Kontsevich, Institut des Hautes Études Scientifiques.
  • Jacob Lurie, Harvard University.
  • Terence Tao, University of California, Los Angeles, and

Richard Taylor, Institute for Advanced Study.

The laureates will be presented with their trophies and $3 million each at the Breakthrough Prize ceremony (November 9, 2014).

Mark Zuckerberg said: “Mathematics is essential for driving human progress and innovation in this century. This year’s Breakthrough Prize winners have made huge contributions to the field and we’re excited to celebrate their efforts.”

Yuri Milner commented: “Mathematics is the most fundamental of the sciences – the language they are all written in. The best mathematical minds benefit us all by expanding the sphere of human knowledge.”

Physics: The laureates of the 2014 Physics Frontiers Prize are:

  • Joseph Polchinski, KITP/University of California, Santa Barbara.
  • Michael B. Green, University of Cambridge, and
  • Andrew Strominger and Cumrun Vafa, Harvard University.

The laureates of 2014 New Horizons in Physics Prize are:

  • Freddy Cachazo, Perimeter Institute.
  • Shiraz Naval Minwalla, Tata Institute of Fundamental Research, and
  • Vyacheslav Rychkov, CERN/Pierre-and-Marie-Curie University/École Normale Supérieure.

Ceremonies

Discovery Channel And Science Channel To Simulcast Premiere Gala On November 15 In The U.S.

BBC World News To Air Worldwide Weekend Of November 22.

Presenters to Include Kate Beckinsale, Benedict Cumberbatch, Cameron Diaz, John Hamm and Eddie Redmayne.

Honoring the world’s top scientists and mathematicians, the 2nd Annual Breakthrough Prize Ceremony will be hosted by Seth MacFarlane (of Cosmos fame.) The special will be televised in the U.S. as a simulcast on Discovery Channel and Science Channel on November 15 at 6 PM ET/PT and globally the weekend of November 22 on BBC World News.

The Breakthrough Prize’s goal is to celebrate scientists and generate excitement about the pursuit of science as a career.

The exclusive ceremony, co-hosted by the Breakthrough Prize founders and Vanity Fair editor Graydon Carter, will take place in Silicon Valley on November 9.

Award presenters include Kate Beckinsale, Benedict Cumberbatch, Cameron Diaz, Jon Hamm and Eddie Redmayne.

There was a special simulcast of the ceremonies on the UC Berkeley campus hosted by MSRI.

Interstellar!

By now you’ve probably heard of the excitement about the new movie “Interstellar.” There have been many interesting reviews and even a cover story, by Jefferey Kluger, in Time Magazine. I would recommend reading the Time article, if you have a chance.

The amazing thing about the making of this movie, to me, is that it is a collaboration between Kip Thorne and Chris Nolan, the movie maker. Actually, Kit that is an executive producer on this film. He personally did the ray tracing for the special effects views of black holes and wormholes. Finally, Kit, a professor at Caltech, is probably one of the top five General Relativists in the world. He and the special effects Director published two papers in astronomical journals that describe new ways that astronomers could look for both Black holes and wormholes. These are methods of discovery that were not known before.

I recommend reading more on the wonderful blog of Sean Carroll.

www.preposterousuniverse.com/blog/

He had a blog entry on October 29 of this year about the movie, his participation in documentary, and, Kip’s participation, viewing schedules for the documentary, a preview, and a link to a wonderful Wired story on the science.

It’s truly amazing to me, the amount of respect that seems to be coming to the scientific community from the entertainment community. Life is good.

Some ways of thinking about “black holes.”

Well let’s talk about black holes. We’ve been discussing about the Schwartschild metric, which describes the simplest kind of black hole. These black holes are not rotating and are static in time. What exactly is a black hole. There are a couple of different ways to think about this, but, the simplest description is: “A black hole is a region of space time, such that, any object within this region has an escape velocity from this region, that is greater than or equal to the speed of light.

Dark Stars

Obviously this is a concept that, with this definition, applies in Newtonian mechanics. And, in fact, there was contemporary of Newton’s, a scientist by the name of John Mitchell first developed this concept. During 1783, the geologist John Michell wrote a long letter to Henry Cavendish. In it, he outlined the expected properties of dark stars. His results were published by The Royal Society in their 1784 volume. These ideas were also developed by Pierre-Simon Laplace.

Michell calculated that when the escape velocity at the surface of a star was equal to or greater than c, the generated light would be gravitationally trapped, so that the star would not be visible to a distant astronomer. Michell’s idea for calculating the number of such “invisible” stars anticipated 20th century astronomers’ work: he suggested that a certain proportion of double-star systems might be expected to contain at least one “dark” star. Astronomers could search for and catalogue as many double-star systems as possible, and then identify cases where only a single circling star was visible. This would then provide some sort of statistical baseline for calculating the amount of other unseen stellar matter that might exist in addition to the visible stars.

Escape Velocity and The Schwarzschild Radius

In Newtonian mechanics, the escape velocity is obtained by setting the sum of the kinetic energy and the potential energy to zero. The idea is that the body will travel up the gravitational potential well and just reach “infinity.” At infinity, the kinetic energy will be zero, i.e., the speed will be zero. Similarly, the potential energy, -\frac{G\, M \, m}{r} will be zero because r is infinite. So,

-\frac{G\, M \, m}{r}=\frac{m\,v^2}{2} \implies v=\sqrt{\frac{2\,G\,M}{r}}

Please notice that if we substitute v \rightarrow c, this formula, when solved for r, the result is the formula for r_S, the Schwarzschild radius or “event horizon.” So these ideas are closely linked. However, Newtonian mechanics would say that an object leaving from a distance r_s would travel a distance up the well and then fall back. This is not true in General Relativity. In GR, space falls into the black hole.

Space Falling into a Black Hole

 

Space Waterfall

This picture is borrowed, wop, from the April 1999 article by Leonard Susskind in Scientific American. You should think of space as the water falling over the waterfall. The “light fish” (photons) can swim in space faster than it flows until they go over the waterfall. Then the water falls so fast, that no matter how fast they swim, (namely the speed of light) they cannot go back over the crest of the waterfall. That crest is known as the Schwarzschild radius.

The picture of spacing falling into a black hole has a sound mathematical basis, first discovered in 1921 by the Nobel prize-winner Alvar Gullstrand, and independently by the French mathematician and politician Paul Painlevé, who was Prime Minister of France in 1917 and then again in 1925.

It is not necessary to understand the mathematics, but I do want to emphasize that, because the concept of space falling into a black hole is mathematically correct, inferences drawn from that concept are correct.

The Gullstrand-Painlevé metric is

ds^2=dt_{ff}^{2}+(dr-v dt_{ff})^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)

which is just the Schwarzschild metric expressed in a different coordinate system. You can look up the derivation in Wikipedia:

http://tinyurl.com/Wiki-Painlev

if you are interested. I won’t bother here. The free-fall time (t_{ff}) is the proper time experienced by observers who free-fall radially from zero velocity at infinity. The velocity v in the Gullstrand-Painlevé metric equals the Newtonian escape velocity from a spherical mass M that we calculated previously. This velocity has a minus sign because space is falling inward, to smaller radius.

Physically, the Gullstrand-Painlevé metric describes space falling into the Schwarzschild black hole at the Newtonian escape velocity. Outside the horizon, the infall velocity is less than the speed of light. At the event horizon, the velocity equals the speed of light. And inside the event horizon, the velocity exceeds the speed of light. Technically, the Gullstrand-Painlevé metric encodes not only a metric, but also a complete orthonormal tetrad, a set of four locally inertial axes at each point of the spacetime. The Gullstrand-Painlevé tetrad free-falls through the coordinates at the Newtonian escape velocity.

It is an interesting historical fact that the mathematics of black holes was understood long before the physics. Einstein himself misunderstood how black holes work. He thought that the Schwarzschild geometry had a singularity at its horizon, and that the regions inside and outside the horizon constituted two separate spacetimes. They are, in fact, one spacetime describing the vacuum outside a massive object. Although they are the same spacetime, they have different geometrical properties on either side of the Schwarzschild radius (or event horizon.) A fact that we have alluded to earlier and will explore further as we go along in this series of posts.

Karl of the Schwarzschild Manifold

Karl Schwarzschild was an amazing intellect. I thought I would share with you the short biography of KS as given by Schlomo Sternberg, in his book Curvature and Mathematics and Physics [without permission].

Karl Schwarzschild’s parents were Henrietta Sabel and Moses Martin Schwarzschild. The family was Jewish, with Karl’s father being a well-off member of the business community in Frankfurt.

He attended a Jewish primary school in Frankfurt up to the age of eleven, then he entered the Gymnasium there. It was at this stage that he became interested in astronomy and saved his pocket money to buy himself materials such as a lens from which he could construct a telescope. Karl’s father was friendly with Professor J Epstein, who was professor at the Philanthropin Academy and had his own private observatory. Their friendship arose through a common interest in music. Professor Epstein has a son, Paul Epstein, who was two years older than Karl and the two boys became good friends. They shared an interest in astronomy, and Karl learnt how to use a telescope and also learnt some advanced mathematics from his friend Paul Epstein.

It was in large part what he learnt through his friendship with Epstein which led to Schwarzschild mastering celestial mechanics by the age of sixteen. Such was this mastery that he wrote his first two papers on the theory of orbits of double stars at this age (!) while still at the Frankfurt Gymnasium. The papers were published in Astronomische Nachrichten in 1890.

Schwarzschild studied at the University of Strasbourg during the two years 1891-93 where he learnt a great deal of practical astronomy, then at the University of Munich where he obtained his doctorate. His dissertation, on an application of Poincar ́e’s theory of stable configurations of rotating bodies to tidal deformation of moons and to Laplace’s origin of the solar system, was supervised by Hugo von Seeliger. Schwarzschild found great inspiration from Seeliger’s teaching which influenced him throughout his life.

After the award of his doctorate, Schwarzschild was appointed as an assistant at the Von Kuffner Observatory in Ottakring which is a suburb of Vienna. He took up his appointment in October 1896 and held it until June 1899. While at the Observatory he worked on ways to determine the apparent brightness of stars using photographic plates.

From 1901 until 1909 he was Extraordinary Professor at G ̈ottingen and also director of the Observatory there. In G ̈ottingen he collaborated with Klein, Hilbert and Minkowski. In less than a year he had been promoted to Ordinary Professor.

Schwarzschild published on electrodynamics and geometrical optics during his time at G ̈ottingen. He carried out a large survey of stellar magnitudes while at the G ̈ottingen Observatory, publishing Aktinometrie (the first part in 1910, the second in 1912). In 1906 he studied the transport of energy through a star by radiation and published an important paper on radiative equilibrium of the atmosphere of the sun.

He married Else Posenbach, the daughter of a professor of surgery at G ̈ottingen, on 22 October 1909. They had three children, Agathe, Martin who was born on 31 May 1912 and went on to became a professor of astronomy at Princeton, and Alfred.

After his marriage, near the end of 1909, Schwarzschild left G ̈ottingen to take up an appointment as director of the Astrophysical Observatory in Potsdam. This was the most prestigious post available for an astronomer in Germany and he filled the position with great success. He had the opportunity to study photographs of the return of Halley’s comet in 1910 taken by a Potsdam expedition to Tenerife. He also made major contributions to spectroscopy which became a topic of great interest to him around this time.

On the outbreak of war in August 1914 Schwarzschild volunteered for military service. He served in Belgium where he was put in charge of a weather station, France where he was assigned to an artillery unit and given the task of calculating missile trajectories, and then Russia.

He contracted an illness while in Russia called pemphigus, which is a rare autimmune blistering disease of the skin, with a more common frequency among Ashkenazic Jews. For people with this disease the immune system mistakes the cells in the skin as foreign and attacks them causing painful blisters. In Schwarzschild’s time there was no known treatment and, after being invalided home in March 1916, he died two months later.

Today, he is best known for providing the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical nonrotating mass, which he accomplished in 1915, the same year that Einstein first introduced general relativity.

The Schwarzschild solution, which makes use of Schwarzschild coordinates and the Schwarzschild metric, leads to the well-known Schwarzschild radius, which is the size of the event horizon of a non-rotating black hole. He wrote this paper while suffering from a painful disease and with full knowledge of his impending death.

End of biography.

I am inspired by the achievement of intellect that this man could achieve in such a short life.

Have fun folks. More technical aspects of the Schwarzschild metric and manifolds to come.

Schwarzschild Metric: Coordinate Singularity

Continuing with the Schwarzschild metric, we have to look at the term multiplying dr^2, namely (1-\frac{r_s}{r})^{-1}. Since the term becomes infinite at r=r_s, we might think that there is a problem with the metric. If the metric is proper, then we should be able to invert it, i.e., compute g^{\mu\nu} from g_{\mu\nu}. Written as a matrix, g_{\mu\nu} looks like:

\left( \begin{array}{cccc}    1-\frac{r_s}{r}& 0&0&0 \\    0&1/(1-\frac{r_s}{r})&0&0 \\    0&0&r^2&0 \\    0&0&0&r^2\sin^2\theta    \end{array} \right)

Since this is a diagonal matrix the determinant, Det(g_{\mu\nu})=r^4\sin^2\theta, is non-zero at r=r_s. Thus, the metric is invertible. So, the metric is not singular at this point, it is what is known as a “coordinate singularity.” I means that we need an atlas (collection) of charts (co-ordinate systems over open sets) to reach all points on the manifold. So, the Schwarzschild metric is fine outside and inside r_s, but we need another coordinate system to calculate trajectories across or in the region near r=r_s.

There are many such coordinates systems, Eddington-Finklestein, Kruskal, Penrose, turtle, rain and so forth. As the radial coordinate gets large, the Schwarzschild metric becomes the flat space spherical coordinate system. As it approaches the Schwarzschild radius, if there is enough mass, it describes the event horizon of a black hole. Inside the event horizon of a black hole, it gets even more interesting.

In the next post, we’ll address geodesics and null geodesics. In particular, we’ll look at the behavior of the light cone (causality) outside the event horizon, on the event horizon and eventually [probably the post after that] inside the event horizon. There are some very interesting effects inside the event horizon. You’ve already gotten a taste if you read the article on Einstein-OnLine.

The Schwarzschild Metric

In Chapter 9 of Moore, A General Relativity Workbook, we have been covering the Schwarschild metric. This is the first vacuum solution to Einstein’s filed equations, G_{\mu\nu}=0 that were found soon after Einstein published his general theory in 1916. The metric describes space-time outside a massive body (in the surrounding vacuum).

A few words about metrics. The metric that is most well known is the Minkowski metric.

-d\tau^2=-dt^2+dx^2+dy^2+dz^2

This metric is constant and has no cross terms, e.g. dx dy. In general the metric can contain all possible cross terms, and the short hand version written in tensor notation is d\tau^2=g_{\mu\nu}dx^\mu dx^\nu. This means that \mu and \nu each can take on values of 0,1,2,4. The coordinates are x^0=t, x^1=x, x^2=y,x^3=z.

The Minkowski metric g_{\mu\nu} can be though of as a matrix:

\left(\begin{array}{cccc}  -1 & 0 & 0 & 0 \\  0 & 1 & 0 & 0 \\  0 & 0 & 1 & 0 \\  0 & 0 & 0 & 1  \end{array}\right)

and the sign of the diagonal entries are know at the “metric signature.” So, for the Minkowski metric, the signature is -,+,+,+. There is a tremendous amount of symmetry in Einstein’s equations and there are many, different, sign conventions that are adopted by different authors. Steven Weinberg’s book on cosmology devotes and entire page to showing different sign conventions for various computed objects by various authors. Luckily, we only have to worry about one right now. The signature for the Minkowsky metric is fully equivalent to the one above multiplied by -1. With my convention, the key is to notice how we identify the time coordinate. The coordinate with a negative entry in the signature corresponds to time.

Now, lets finally look at the Schwarzschild metric. It is usually written, in spherical coordinates as opposed to cartesian coordinates, as:

-d\tau^2=-(1-\frac{r_s}{r})dt^2+(1-\frac{r_s}{r})^{-1}dr^2+r^2d\theta^2+r^2 \sin^2\theta\,d\phi^2

r_s is know as the Schwarzschild radius. This metric is also diagonal, similar to the Minkowski metric, but the entries are not constant. The Schwarzschild metric signature is also -,+,+,+. This metric allows us to calculate planetary orbits around the sun and calculate the advance of the perihelion of mercury, one of the famous first tests of general relativity.

It is also the metric that was first used to predict the existence of black holes. Now, as you know from one of my previous posts, we read an article about black holes that erroneously said you could use rockets to blast out of a black hole. There are many reasons why that is in error, as I pointed out. However, this also led to exploring what happens to space-time when you enter a black hole. I found a wonderful site “Einstein on-line” which has a lot of very excellent descriptions of different special and general relativity effects described at the level of the proverbial “man in the street.” I recommend this site highly. For black holes, I recommend the article http://www.einstein-online.info/spotlights/changing_places. I hope you read this link and come back to see a little more discussion about space-time inside a black hole.

A Solution Approach for the Cosmic Scale Factor (Friedman Equation)

{\bf Objective}

I want to describe one approach to solving the differential equation for the cosmological scale factor. In the previous post, I used numerical methods to solve the differential equation. However, in most cases analytic solutions exist. The general analytic solution is very difficult to deal with because it uses the roots of a few fourth degree polynomials, involves Elliptic functions, requires matching solution regions in the complex plane, and many other difficult issues. I will restrict my choices of analytic solutions to special cases where the solutions are ‘simple.’

{\Large {\bf Friedman\;Equation}}

Consider the ‘Friedman equation’ (see Susskind lectures or Moore) for the scale factor in the FRW metric:

a'(\eta )^2=\left(a(\eta )^2 \Omega _{\Lambda }+\frac{\Omega _{\text{Matter}}}{a(\eta )}+\frac{\Omega _{\text{Rad}}}{a(\eta )^2}\right)+\left(\Omega _k\equiv K T_{\text{H0}}^2\right)

I use the convention is that the scale factor is unitless and the time is also unit free. You may consider it to be a fraction of the current Hubble time (age of the universe). The units in the FRW metric, in my use, are attached to the x, y & z coordinates. Due to the reasons discussed previously, I only consider the case of “flat space,” i.e., \Omega _k=0. Use the normalization that the scale factor today is 1, so a(1)\equiv 1\therefore a'(1)=1. We can determine the Taylor Series exapansion of this function to as many terms as needed. The process is outlined in the next section.

For computer memory and computation time reasons, it is best to do this with numerical, in fact rational, values of \Omega _{\Lambda }\,\&\,\Omega _{\text{Rad}} \equiv \Omega _r\,\&\,\Omega _{\text{Matter}}\equiv \Omega _m .

{\bf Taylor\;Series\;Method}

The first six terms of the Taylor series of a(\eta ) about 1 is given by Mathematica as:

Series[a[\eta],{\eta, 1, 6}]


a(1) + (\eta-1) a'(1) + \frac{1}{2} (\eta-1)^2 a''(1) + \frac{1}{6} a^{(3)}(1) (\eta-1)^3 + \frac{1}{24} a^{(4)}(1) (\eta-1)^4 + \frac{1}{120} a^{(5)}(1) (\eta-1)^5 + \frac{1}{720} a^{(6)}(1) (\eta-1)^6 + O\left((\eta -1)^7\right)

Supose we want to solve for the coefficients of the Taylor series about \eta=1. We have a differential equation of the form:

a'(\eta )=f(a(\eta ))

and two initial conditions, a(1)=1\,\&\,a'(1)=1. When we differential the left hand side, we get higher derivatives of a, i.e., a^{(n)}\;\forall\,n\,=\,2,\,3,\ldots The derivatives of the rhs will always involve derivatives of a(\eta) to one order less. Hence, we can evaluate the current derivative at 1 and recurse using the new information. We can take this process to as many places as we have computer memory and CPU time.

{\bf Matter\;Dominated\;Example}

Now lets consider the case of matter dominated universe, \Omega _{\text{Matter}}=1. The analytic solution is

matterSoln = 
 DSolve[{a'[\eta]^2 == 1/a[\eta], a[1] == 1}, a[\eta], \eta]

Which has two solutions \frac{(5-3 \eta )^{2/3}}{2^{2/3}} and \frac{(3 \eta -1)^{2/3}}{2^{2/3}}. The scale factor goes to zero in the first solution at \eta=5/3, i.e. the future (today is \eta=1) and, thus unphysical (a big crunch) or time is going backward since the universe is observed to be expanding. Plotting this solution, we have
MatterDominatedSolutionPic
The second solution goes to zero at \eta=1/3, so our picture breaks down [in the sense that we believe that a(0)=0], but this is to be expected since the composition of the universe changed over time and this is only part of the total solution. None the less, it serves as a useful example of the Taylor series solution method.

The second solution looks like:
MatterDominatedSolutionPic1

The first 10 terms of the Taylor series for this solution is given by

Series[(-1 + 3 \eta)^(2/3)/2^(2/3), {\eta, 1, 10}]


1+(\eta-1)-\frac{1}{4} (\eta-1)^2+\frac{1}{6} (\eta-1)^3-\frac{7}{48} (\eta -1)^4 \\ +\frac{7}{48} (\eta-1)^5-\frac{91}{576} (\eta -1)^6+\frac{13}{72}(\eta-1)^7-\frac{247}{1152} (\eta-1)^8 \\   +\frac{2717}{10368}(\eta-1)^9-\frac{13585}{41472} (\eta-1)^{10}+O\left((\eta -1)^{11}\right)

Now using our procedure outlined above, we have to write our rhs as a function. To increase efficiency, we have the function have an index that denotes its derivative. The function will remember any derivatives computed, and we only have to compute the first derivative of the last derivative to get out next derivative.

f[a, \eta, 0] = a[\eta];
f[a, \eta, 1] = Sqrt[1/a[\eta]];
f[a_, \eta_, kder_] := f[a, \eta, kder] = D[f[a, \eta, kder-1], \eta]


We haven’t written our function in recursive form, so we will use a “Do” loop to build up the results.

aRules = {a[1] -> 1, a'[1] -> 1};
nTerms = 10;
Do[
  newRule = (D[a[\eta], {\eta, k}] /. \eta -> 1) -> (f[a, \eta, k] /. \eta -> 1);
  newRule = newRule /. aRules;
  AppendTo[aRules, newRule], {k, 2, nTerms}];
Series[a[\eta], {\eta, 1, nTerms}] /. aRules


1+(\eta-1)-\frac{1}{4} (\eta-1)^2+\frac{1}{6} (\eta-1)^3-\frac{7}{48} (\eta -1)^4 \\ +\frac{7}{48} (\eta-1)^5-\frac{91}{576} (\eta -1)^6+\frac{13}{72}(\eta-1)^7-\frac{247}{1152} (\eta-1)^8 \\   +\frac{2717}{10368}(\eta-1)^9-\frac{13585}{41472} (\eta-1)^{10}+O\left((\eta -1)^{11}\right)

This gives the same result as the series solution computed from the analytic solution for the same number of terms.

{\bf Convergence}

Frequently, in scientific applications, only a finite number of coefficients c_n are known. Typically, as n increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity. In this case, several techniques have been developed, based on the fact that the coefficients of a Taylor series are roughly exponential with ratio 1/r where r is the radius of convergence.

For example, when the signs of the coefficients are ultimately periodic, (as they appear to be in out case) Mercer and Roberts propose the following test.[MR 1990] Define the associated sequence \left\{b_n\right\}_{n=1}^{\infty } by

b_n^2=\frac{c_{n+1} c_{n-1}-c_n^2}{c_n c_{n-2}-c_{n-1}^2}

Then \frac{1}{r}=\underset{n\to \infty }{\text{Lim}}\,b_n

Domb and Sykes [DS 1957] proposes two straight forward test, plotting the points \left\{\frac{1}{n},\frac{c_n}{c_{n-1}}\right\} and finding the asymptote. We try the D-S method first using the first 100 coefficients of the Taylor series.

cf = Part[Series[(-1 + 3 \eta)^(2/3)/2^(2/3), {\eta, 1, 100}], 3];
ListPlot[
  Transpose[
    {Range[-1+Length[cf]]^-1, Drop[cf,-1]/Drop[cf,1]}
  ]
]


DSpic
We see that the plot indicates a radius of convergence of approximately 2/3. We will compare the exact solution with the 100 term Taylor series over the range [1/3,2].
convergence
The blue curve is the exact solution. The yellow curve is the Taylor series solution. We see that the solution is fine until about 1.7 where it diverges. In any real use of this method, we would have to be very careful about convergence. We would have to set up our solution with overlapping expansions using the original function to evaluate the function and its first derivative and repeat our process over the next radius of convergence.

{\bf Conclusions}

There are many ways to solve the Friedman equations, from analytic, to numerical to Taylor series and, even Pade approximants. I have no proof, but using Pade approximants, there seems to be a much wider radius of convergence and far fewer derivatives need to be computed. All in all, this seems like a much better way to go.

I will write another blog entry using this method as well.

{\bf References}

[MR 1990] Mercer, G.N.; Roberts, A.J. (1990), “A centre manifold description of contaminant dispersion in channels with varying flow properties”, SIAM J. Appl. Math. 50 (6): 1547-1565, doi:10.1137/0150091

[DS 1957] Domb, C.; Sykes, M.F. (1957), “On the susceptibility of a ferromagnet above the Curie point”, Proc. Roy. Soc. Lond. A 240 (1221): 214-228, doi:10.1098/rspa.1957.0078

Image

The Friedman scale factor for the expansion of the universe

We, of the RCSG, have been watching Leonard Susskind’s 2013 lectures on cosmology. I was rather unhappy with Susskind’s lecture 6. I thought he should have been more ready to actually finish the equations he started and show some results, instead of waving his hands and saying: “we have everything we need, so we can do it…” Also, I thought his coordinate chart, showing our reception of CMB radiation, was a tad misleading, so using null geodesic results, I created my own. This, at least, satisfies me.

This post numerically integrates the Friedman equation for a particular choice of cosmological ΛCDM parameters to find a homogeneous isotropic cosmological standard model solution. The inputs are the parameters that specify a FRW model —the Hubble constant H0, and the three Ω’s for radiation, matter, and vacuum energy (Λ). My work assumes these parameters are such that the universe started with a big bang and the particular constants used in these example are for a flat universe or a universe with a very tiny negative curvature.


Solving the equation for the scale factor in the FRW metric. Starting with Chapter 26 equation 26.20 of T. Moore, A General Relativity Workbook, we have:

\left(\frac{da}{dt}\right)^2=\left(\frac{da T_{\text{H0}}}{dt}\right){}^2=\left(\frac{da}{d\frac{t}{T_{\text{H0}}}}\right){}^2=\left(\frac{da}{d\eta }\right)^2=a'(\eta )^2

where I have introduced the dimensionless time η which is time, t, divided by the Hubble time today. η=0 corresponds to the big bang and η=1 corresponds to today. So, I write for 26.20

a'(\eta )^2=\left(a(\eta )^2 \Omega _{\Lambda }+\frac{\Omega _{\text{Matter}}}{a(\eta )}+\frac{\Omega _{\text{Rad}}}{a(\eta )^2}\right)+\left(\Omega _k \equiv K T_{\text{H0}}^2\right)

I have used Moore’s definition of \Omega_k from equation 26.19. This is different from Susskind’s definition which seperates out the Sign[K] as a new parameter which takes on -1,0,1 as values. Here the sign of K is retained with \Omega_k


Arthur Karp posted an objection to my work when I sent a different version of this material out a month or two ago. I will not go into details, but simply refer to solutions that have been around in the astrophysical community since the late 70’s and early 80’s. These are refered to as the \LambdaCDM model. Joel Primack was one of the key collaborators with Jim Peebles and Martin Rees. Last July he spoke at the UCSC Philosophy of Cosmology workshop and presented the following slide: J Primack-History of Cosmic ExpansionShowing the results for the scale factor compared to the time since the Big-Bang. I have used Mathematica’s ParametricNDSolve function to produce a comparison slide. Here it is: My Version of Primack SlideAs you can see, my solutions for all cases are identical. This is not a proof, but I do consider it sufficient evidence that I have it right. Or, maybe, Art can correct Peebles, Primack, Rees and others and save the Cosmology community from itself. From now on, I will simply ignore any objections that Art comes up with. My real purpose is to provide a little more detailed information on solutions to the Friedman equation. These will appear in subsequent posts and deal with special cases (fairly well dealt with by Susskind) and with combination cases (Susskind provides hand-waving motivation, which although it does point in the right direction, the exact solutions exist and and are not all that complicated). After that I will show what I consider to be a better picture of how photons, emitted near the beginning of the universe, would travel to be intercepted by us at the present time. After that I will derive the cumulative density of galaxies as a function of redshift (another big hand wave by Susskind). Finally, I will start on the scalar field, lectures 8, 9, & 10. So, stay tuned.

Random musings on understanding cosmology…

I’m trying a new format for cosmology discussions. Hopefully, this will be easier and more bite-sized for any interested folks to read than the PDFs that I have been distributing…

There is a group of us who get together and discuss various issues in Cosmology, General Relativity and Quantum Mechanics. This blog is, at least initially, focused on those participants. Others may join in if they are interested, but the context of each blog entry will be, generally, the last set of discussions we had or lectures we are watching. I hope to learn something and have fun.

Cheers/Bob