Saturday, May 14, 2005

String cosmology at Columbia

On Friday 13th, Brian Greene and Maulik Parikh at Columbia University, 120th street/Broadway in the New York City, organized the
  • Fifth addition to the Northeastern conference on string cosmology

and it was a nice and successful event, I think. It's great if such an event is organized by someone who is not just a smart physicist, but also a likable celebrity. I plan to describe the lectures. For the time being, let me just enumerate them:

  • Lee Smolin: Loop quantum gravity: principles and phenomenology
  • David Spergel: What cosmological observations tell us about dark energy & inflation?
  • Savdeep Sethi: Matrix Big Bang
  • Gary Shiu: Search for realistic vacua in the landscape

Lee Smolin - LQG

I will try to minimize my comments because most readers know my opinions about LQG anyway. Lee Smolin prepared a very complete overview of loop quantum gravity. He presented the results of the work of 100+ people - one of his first transparencies contained an extensive list of 100+ LQG researchers. He also defined the symbols "R" and "S" that mean "rigorous" and "supergravity" (or more generally, "S" stands for a result that can be generalized).

Lee started with some elementary things - such as the definition of the Wilson loop. He had simplified the formula for the Wilson loop so that the astrophysicists would not be distracted by unnecessary mathematical equations. Then he claimed (...) that the LQG quantization of gauge fields works better for gravity than it does for Yang-Mills theory. He argued (...) that the background independence - whatever it exactly means - is important and realized in LQG "all the way down to the level of mathematical rigor" - a kind of phrase that appeared many times in the talk. He stated that the Hilbert space realizing some commutators (the commutator of the flux through a surface with a Wilson line is proportional to the Wilson line) is (...) unique. (You may object that the infinite-dimensional separable spaces are always isomorphic - but your objections will fail because the LQG Hilbert space is not separable.)

What about the perturbative non-renormalizability of general relativity? "Physicists will scream id they have not heard - or if they have heard but have not understood - what happens with the perturbative non-renormalizability," he said. Because I don't belong to either of these categories, I am not gonna scream right now. Instead, I will repeat what Lee thinks is an argument. He said that the non-renormalizability is an illusion because the calculations rely on two things, namely

  1. smoothness of space all the way down to the Planck scale
  2. local Lorentz invariance,

neither of which holds in LQG. Smolin, of course, said that neither of these things has to hold (...) in quantum gravity in general. And these two wrong assumptions invalidate the loop perturbative calculations, he believes. (Once again, I am not gonna write what are the correct answers to these questions; the purpose of this text is to summarize Lee's talk.)

LQG may be written as a topological action with constraints. At this moment Alan Guth arrived (their train from Boston had a delay). Lee Smolin also said that the methods of LQG may be (...) applied to other theories, such as "d=11" supergravity, and he has flashed a transparency with some relevant formula for 2 seconds so that it was impossible for those who have not studied this specific proposal to decide whether it was meaningful.

At that moment, the "theorem" mentioned above occured again. The theorem says, roughly speaking, that if you want to construct a particular form of LQG that satisfies some particular assumptions (all of which are false in string theory), you will find a unique Hilbert space that can be used for this particular version of LQG. Paul Steinhardt asked which theories fall into this category affected by the theorem. Lee Smolin answered that all theories including gravity with arbitrary matter in any dimension that people ever studied and so forth, but the skepticism did not diminish at all and several participants repeated the question in one way or another.

Lee explained that the loop quantization is inequivalent to the Fock space quantization. He claimed that the volume operator is well-defined (...) and contains contributions from the spin network vertices with at least 4 legs. He outlined the path-integral (spin foam) quantization, and speculated whether topology can change in LQG (without a clear answer). He listed various gauge groups that should be relevant for various gravitational theories - SO(2,1) for 3D gravity, SU(2) for 4D gravity, OSp(1/32) for 11D supergravity, and so on. Steinhardt asked whether the gauge theory description captures the higher-order quantum corrections, and there was no coherent answer. Savdeep Sethi asked whether GR may be decoupled from other physical phenomena, and there was no writable answer either.

Someone asked whether Thiemann's new version of string theory - LQG methods applied to the worldsheet - has been validated or invalidated. Lee Smolin, who had said that all string theorists should work on Thiemann's paper a year ago, realized after 2 minutes which paper by Thiemann the participant had referred to. Lee said that he was not convinced by Thiemann's work because it did not seem to contain the physics that justifies the research in string theory (such as the graviton).

Lee also mentioned that Rovelli has allegedly represented the spin foam in terms of matrix models, and listed several LQG papers whose composition seemed rather random. Someone asked whether there was a Lagrangian description of LQG and whether it was equivalent to the Hamiltonian approach - something that seems untrue because the spin foam (Lagrangian) has a discrete time while the Hamiltonian approach requires a continuous time. Lee answered they were equivalent (...) but offered no arguments.

Lee Smolin sketched how loop quantum cosmology is done, and admitted that it is technically incompatible with loop quantum gravity because the loop methods are applied to the minisuperspace truncation of gravity, instead of an analysis of the minisuperspace limit of the full loop quantum gravity which was obtained by discretizing the full gravity. He also admitted that no questions relevant for physics can yet be answered by LQG because its low energy limit is not known. Paul Steinhardt asked how can you ever try to answer anything about cosmology without knowing the low-energy limit of your theory, and I did not understand Lee's answer.

Lee then said that most important problems have been solved - the exact Hamiltonian evolution in loop quantum cosmology; its FRW limit; the absence of singularities. He also discussed antifriction and showed a formula for the LQG estimate of corrections to an inflationary formula. Most participants thought that the formula was incorrect already on dimensional grounds, and the disagreement about the meaning of some letters such as "k" has never been resolved.

Lee Smolin also discussed the black holes - everything about the black holes has been resolved in LQG, he argued, and Olaf Dreyer solved all problems about the black hole entropy, Lee said. Finally, he discussed phenomenology - the methods how to distinguish a relativistic Universe from a non-relativistic or "doubly-special relativistic" Universe proposed by LQG. It was a very nicely composed and complete talk that would have undoubtedly convinced many audiences - not sure about this one because the participants knew quite a lot about cosmology and gravity.

David Spergel - What cosmological observations tell us about dark energy and inflation?

David Spergel's talk was very interesting. The main point he wanted to convey is that cosmology is becoming a precise science - one that can explain huge amounts of data with a pretty good accuracy using a model that depends on 5-6 parameters only. In fact, the situation is the best situation we could dream about because not only we can validate the standard model of cosmology, but we seem to have a sufficient accuracy to test the predictions that go beyond this model.

This includes the questions "How the Universe began?" and various concrete questions that arise in string theory.

People are observing (or may observe) three main types of modes in the cosmic microwave background:

  • scalar modes
  • vector modes
  • tensor modes

When you look at the scalar modes, which are the most visible ones, you can determine the power spectrum and the higher moments. You may Fourier transform the energy density "rho(x,t)" into "A(k,t)", and the simplest model can express this quantity in terms of the same quantity at "t=0"

  • A(k,t) = A(k,0) T(k,t)

where "T(k,t)" is the transfer function. David Spergel emphasized that most of the volume of the visible Universe has the redshift "z" much greater than one - something that you may find counterintuitive.

The vector modes are much less known and they can arise from cosmic strings that are created according to various hybrid inflationary models. All the tensor modes are - as of today - measured indirectly, and it will be so in the near future, by

  • WMAP
  • PLANCK + balloons

Spergel discussed what is the maximal "l" (orbital angular momentum) that we can detect with various available tools:

  • WMAP, 1 year: l=300
  • WMAP, 6 years: l=600 (those six years are already planned; they are preparing the final report on the 3rd year of WMAP, and David said that people should not be impatient because an imperfect report would not help the community)
  • PLANCK: l=1500
  • IDEAL: l=2000 (this is the realistic maximum we can ever get)

The tensor modes - gravity waves - are always observed indirectly. He mentioned various future surveys that are intended to give us better data, such as:

  • Loeb + Zaldarriaga - the measurement of galaxies of high redshift "z" much greater than one using the 21 cm line of the Hydrogen atom
  • Melnick et al. - Cosmic inflation probe - the details of this project are not known, but it will probably be another observation of many galaxies
  • Square kilometer array

and so forth. David Spergel said that we can measure the "10^{-2}" errors today, we will be able to do better in the future, and it is plausible to realize finite-cost projects that will measure the errors up to "10^{-6}". It will become possible to observe galaxies with redshift around "z=7". His punch line was that

  • if there is a good and convincing prediction from string theory or another theory about the 10^{-5} errors, they can test them.

As you can see, this is an example where experiments are ahead of theory. We simply do not have convincing predictions for this kind of observations so far.

David said that the "low k" observations will probably never improve because there are so few of these modes. He discussed the non-gaussianities and their origin in non-linearities. He asserted that they were almost completely absent in the cyclic models, and small but non-trivial in most inflationary models. Galaxies generate non-gaussianities, too

Dark energy

Concerning the positive cosmological constant, he argued that the evidence is now very strong and comes from many sources. For example, Seljak has shown that "w" from the equation of state is most likely constant, by combining Sloan, Supernovae, and WMAP.

David showed various graphs that agree about its values. There is also strong evidence that the dark matter must be non-baryonic. What is dark energy? He sketched various recent speculations that it was not really a cosmological constant. He also addressed the anthropic explanation of the cosmological constant by a transparency informing that the best place to discuss creationist science is Kansas. (Later, he also entertained us with a story about the Spanish inquisition.) The primary measurement that determines whether the cosmological constant is there and whether it behaves properly is the function "H(z)", Hubble's constant as a function of the redshift. Note that once you measure this function, you may calculate

  • delta eta = int dz / H(z)

Then it's important to know the distance which is obtained either as

  • luminosity distance - from supernovae (standard candles)
  • angular momentum distance, d = delta eta / (1+z)

Conclusions: Cosmology now offers not only a beautiful agreement between theory and experiments, but also the capacity to probe physics at the fundamental scale.


  1. Hi Lubos,

    I was wondering why I didn't see you here at Harvard at Vafa's talk yesterday, didn't guess that it was because you were down at Columbia!


  2. He wanted to be sure he didn't miss Lee Smolin's talk.