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Cyclic cosmology: motivation still unknown

Cyclic cosmology remains one of the research directions whose motivation and value looks utterly incomprehensible to your humble correspondent. Originally, I only wanted to write about another very weak article in Nude Socialist about Abhay Ashtekar's cyclic loop quantum cosmology but then I decided to be more general after squeehunter asked a question about the infinite cyclic cosmology.

The purpose of theoretical work

Let me begin with an explanation of my perspective on the value of a theorist's work. Good results in theory resemble the key and the lock:



You see, something fits together. The lock may be a set of experimental and observational data and the key is a theory with its calculations for the given situation. They often lead to the same result. The patterns agree. The lock may be opened.

Alternatively, the key and the lock may be two assumptions in our previous successful theoretical understanding of the world that were thought to be independent. A new insight may show that they are not independent, after all. They may become two manifestations of a deeper logic, of a more fundamental starting point. Theoretical work may find out that they only looked independent because the people in the past didn't think about the consequences of their own theories carefully enough.




A good piece of theoretical work may discover and correct some mistakes in a previous work or add a missing solution to the previous equations or laws that eliminates a discrepancy or a puzzle. Again, the idea of the key and the lock comes to mind.

To summarize, good theoretical work has some well-defined goals, directly or indirectly motivated by the experiments, that are answered more correctly, more accurately, more rationally, more inevitably, with fewer independent assumptions, more controllably, or more completely in the new framework than in the old one.

Science is not about surreal stories that someone says just for the sake of themselves. Although some people may like a similar form of arts - I don't - it is certainly not good a template for science. In the text below, you will probably get more familiar with my feeling that cyclic cosmology belongs to this category.

Historical origins: Albert Einstein

For millenia, people used to think that the Universe lasted from minus infinity to plus infinity. 


The reason was completely analogous to the reason why they used to believe that the Earth was flat: when you observe a sufficiently small region of spacetime, both space (and the Earth's surface) and time (and the life of the Universe) simply look infinitely large and uniform. Newton's dynamics encapsulated this paradigm by treating time as a real number (while the latitude and longitude were already known to take values in a compact set): as other real numbers, it is between minus infinity and plus infinity.

However, many centuries ago, people became familiar with the compactness of the Earth's surface. And Albert Einstein found the right equations - general relativity - that implied, for the first time, that the broad structure and size of the Universe couldn't have been constant in time. Well, Alexander Friedmann realized that the size of the Universe behaved just like the height of Newton's apple: it couldn't sit at a fixed value because it was accelerating.

Einstein could have treated this result rationally. He could have realized that the static character of the Universe was just a prejudice that was unsupported at very long timescales while he had a theory - a rational argument - indicating that the Universe was not static. He could have predicted the expansion of the Universe. Well, he hasn't because he wasn't quite God. Even when he learned about the correct answer, he still wanted to preserve the wrong one.

For this purpose, he also invented the concept of a cosmological constant. Its value was chosen in such a way that the acceleration trying to shrink the Universe was compensated by a repulsing force distributed across the Universe - the negative pressure arising from a positive cosmological constant, if you allow me to use the modern terminology. Much later, in 1998, the cosmlogical constant was indeed found to exist, but the value was different than the value needed for the Einstein static Universe.

(It was also 60-123 orders of magnitude smaller than the most straightforward expectations based on explicit quantum calculations in field theories, a puzzle known as the cosmological constant problem.)

It was demonstrated by Einstein that one could arrange the cosmological constant for the acceleration to be zero at one moment. However, the arrangement was unstable: a small fluctuation was enough for the size of the Universe to exponentially fly away from the critical value, in one direction or the other. The Einstein static Universe was dead.

But I believe that it was still the same prejudice about the infinite life of the Universe (in both directions) that led Albert Einstein to propose one more theory in the 1930s: the cyclic cosmology. At every moment, the Universe may be expanding or contracting, but these two alternating phases may continue indefinitely, Einstein thought.

Correct me if I am wrong, but it is my understanding that the dogma about the infinite extent of time is the only justification of the cyclic cosmology at this point. And it is an irrational one, too. There's nothing wrong with a Universe whose time is bounded in one direction or both. As long as the lifetime is longer than the epochs of the cosmic life that have already been observed, the theory about a "beginning" is just fine.

However, the physicists used to be able to think about the basic consequences of their theories. The oldest cyclic cosmology predicted that the cycles can't have a constant duration. Why? Because of the second law of thermodynamics. The entropy of the Universe keeps on increasing. So at the beginning of the new cycle, it is larger than at the beginning of the previous cycle.

Something about the world is changing, anyway. The increasing entropy of the Universe will make the newer cycles longer than the older ones. If you move into the past, the cycles become shorter and shorter. You can actually sum their proper lifetimes and you obtain a finite answer, again. The Universe did have a beginning, anyway.

Big Bang and inflation

The prediction of general relativity was pretty much indisputable at the very first moment when Friedmann used Einstein's equations to analyze a uniform cosmology. Nevertheless, people needed many decades to settle the question. Hubble found the red shift proving that the galaxies were drifting away from each other. The Universe was expanding. Other people found the cosmic microwave background, the correct abundance of light elements predicted by nucleosynthesis, and other things that showed that the Universe used to be very small, indeed.

The Big Bang cosmology became an established piece of science. Nevertheless, the question whether there was anything outside our current "cycle" (that is probably bounded in the past only, because of the positive cosmological constant), remained open. Both answers are a priori equally good. However, I hope that we are already wiser today and we realize that there exists no rational argument indicating that "something must exist" outside the present cycle.

The Big Bang cosmology has some awkward features such as the horizon problem. Its causal structure doesn't make it possible for distant regions on the skies to communicate with each other. This fact makes it shocking that the temperature of the cosmic microwave background is so nicely constant in all directions. This problem, together with many other "surprises" such as the observed absence of magnetic monopoles and other exotic objects, motivates inflationary cosmology.

Inflation solves these problems in the good old lock-and-key fashion. It assumes a small number of assumptions that are consistent with the previous results of science, but it can actually solve some nontrivial problems - in fact many of them.

Modern cyclic cosmologies

Are the modern cyclic cosmologies (and I will choose a better name later) being studied for the same reasons? I have never had this feeling. Because of a chronological overlap, the cyclic cosmology is sometimes presented as a competitor to inflation. But it just doesn't look like the horizon problem and other problems are the driving forces that motivate the cyclic researchers.

Moreover, I don't think that the cyclic cosmology has a new solution to any of these problems. It seems that its champions sometimes say that the cyclic cosmology solves these problems but these statements look like hot air to me. They remind me of the loop quantum gravity people's assertions that they could calculate the black hole entropy. These proclamations began in the middle 1990s when string theory computed the correct microscopic entropy for the first time. The loop quantum gravity people wanted to be competitive except that they have never been. Their arguments never worked. The claims that "they could also do it" always remained a sociological issue, a wishful thinking.

In the same way, I haven't heard any convincing explanation of a new mechanism by which the cyclic cosmology solves the horizon problem. The most promising comments that came close to realizing these goals were methods to incorporate the idea of inflationary cosmology: there is an effective scalar field whose fate follows the same fate as the inflaton. It really is an inflaton.

But this is an idea of the inflationary cosmology, not an idea of the cyclic cosmology: inflationary cosmology doesn't tell you how you should call the inflaton or what is the shape of extra dimensions of space. Instead, it provides us with a working mechanism that explains certain observed features of the Universe. Any realization of this idea is an example of inflation. And, by the way, it is Alan Guth, Andrei Linde, and a few others - and not the cyclic community - that should get credit if this idea is correct.

So it is my understanding that the cyclic cosmology doesn't offer any new ideas to actually explain some puzzling features of the observed data. That's already too bad. On the other hand, the cyclic cosmology surely contains many ideas that go "beyond" their realization of the inflation. Unfortunately, these additional ideas have nothing to do with the observational and theoretical puzzles of the previous theories, as far as I can say.

It is very important to realize that these additional ideas can be freely separated from the inflationary ideas. And we should also realize that it is only the inflationary ideas, and not the additional cyclic ideas, that are supported by some evidence rooted in observations.

Motivation behind postmodern cyclic cosmology

In the context of cosmology, let me reserve the adjective "modern" for the standard Big Bang cosmology, including the cosmological constant, and the inflationary cosmology. "Postmodern" cosmology is composed out of newer theories that are meant to revoke some "modern" stuff. That's a purely chronological definition but let me admit that there are other reasons why I use the word "postmodern", too.

The motivation behind the postmodern cyclic cosmologies looks even more incomprehensible to me than the motivation behind Einstein's post-classic cyclic attempts. Einstein at least presented his assumptions about the everlasting character of the Universe explicitly enough for others to notice. I don't think that most of the contemporary proponents of the cyclic cosmology are doing the same thing.

So I think that the motivation is pretty much identical to Einstein's motivation - they are just hiding it. They have a problem with the Universe that is created at some point. Well, I, for one, have no problems with that. God comes and She says: "Let there be light, QCD, W,Z bosons, quantum gravity, and leptons and quarks." (Well, She actually said "Let the stringy world compactify on M(y) manifold" but that would be too abstract for too many readers.) What's wrong with that?

Do they think that the cyclic cosmology makes the Universe more well-behaved or more predictable?

Well, I don't see any of these things. For example, let us ask: Does the cyclic character make the Universe more well-behaved and "natural"? Well, that would be the case if the Universe were an ordinary classical manifold. Indeed, if you imagine that it is, any boundary or a singular point makes it "more awkward".

Fine. So why do I disagree that the cyclic cosmology improves things "aesthetically"? Simply because I realize that when the Universe becomes very dense and nearly singular, the intuition of classical geometry simply breaks down. And it is replaced by something that is not "just" classical geometry. From the viewpoint of long distances, the resulting physics that is fully compatible with the exact laws of physics may still look singular.

There are lots of similar examples in string theory. For example, you might think that Nature abhors A_N singularities (a quotient of C^2 by Z_{N+1} with a higher-dimensional conical singularity at the origin). You might think that M-theory or type IIA string theory will show their "muscles" and "new forces" that will prevent the Universe from developing an A_N singularity.

Except that they don't. Instead, they confirm that this singularity is perfectly fine and it occurs in physics. And they tell you what the degrees of freedom are. There is indeed a singular point in space that looks just like what you would expect from the singular classical geometry, if your space resolution were worse than the string/Planck length (and, in fact, even if the resolution were better). And they also tell you that there is an SU(N+1) supersymmetric gauge theory supported by the singular locus. It's perfectly fine. It's highly constrained by mathematics. It's beautiful but the truth is different from the most naive expectation based on non-singular manifolds. By the way, type IIB string theory has very different degrees of freedom living near the same singularity. There's a whole segment of science that studies what happens. It is determined in all cases but you can't guess the right answer without knowing the maths.

In a similar way, some people could expect that strings with boundaries shouldn't exist because they're ugly and singular. (I actually believed these things when I was a high school student.) Except that open strings do exist, after all. Whether you like it or not, theories with open strings - such as type I theory - are as perturbatively consistent as theories with closed strings. There is a higher number of topologies of diagrams at each order but it is an extra work for you, not a problem of the theory. And non-perturbatively, these objects (open strings) are still found as resonances in the spectrum (even if they're non-BPS, like in type I).

Spacetimes shouldn't have boundaries either, right? Except that it may bave boundaries. For example, boundaries in 11-dimensional M-theory exist and one can show that they carry a beautiful E_8 gauge supermultiplet (gluons and gluinos), the kind of beautiful structure whose infinitesimal portion is accessible to certain surfer dudes. The actual consistency in quantum gravity works in such a way that the boundaries are very constrained. But the constraint doesn't tell you that the boundaries can't exist or that they carry no physics: instead, they carry an E_8 gauge supermultiplet. It would be hard to guess the right answer but it can be demonstrated by profound calculations.

That's the whole point of theoretical physics that one often needs deep maths and calculations, rather than just common sense and emotions, to determine the right answers to many questions about the world.

So I believe that the "obvious" need to make the time continue - by the rules of beauty - is simply misguided because it makes very naive assumptions about the physics near singularities, assumptions that are almost certain to be incorrect because they have been shown incorrect in so many similar cases. But is the theory of a longer, cyclic, evolving Universe more predictive because the properties of our current cycle are being linked to a distant past?

I don't see that either. If the size of the Universe ever shrinks close to the Planck length (or even much longer distances), all the observers and their experimental gadgets are destroyed. From an operational perspective, it is very questionable whether the "next epoch" should be considered a continuation of the previous epoch. From a purely experimental standpoint, it is a vacuous statement because no experimenter can ever measure both of them. In a certain sense, such a connection between the two cycles is equivalent to reincarnation. ;-) 

But even from a theoretical viewpoint, I think that the "merger" of the two cycles is a highly problematic procedure. Near the singularity, when the densities are huge, the "time" surely has very different characteristics than in the situations where things are almost constant and naturally extrapolated from one moment to the following moment. By the way, analogous reasons also make it difficult to identify the time in two different Universes that are connected by tunneling (bubble nucleation) on the landscape. Much like the "spatial geometry" refers to different degrees of freedom of different vacua, so does time.

The only way how you would convince me that it makes physical sense to connect the cycles of the cyclic cosmology (or to study the "pre-history" of our Universe before it was nucleated as a bubble in another Universe) would be for you to write down the rules to calculate (at least in principle) some features of the new Universe as a function of the features of the old Universe or Universes. Unless and until you will do it, I would consider the previous (and following) cycles to be unphysical. Observers and instruments are destroyed, time is redefined, and unitarity may break down. It is not clear whether you are allowed to use the same "time" to continue your story past the seemingly singular point.

Versions of postmodern cyclic cosmology

We have explained that the motivation behind the postmodern cyclic cosmology remains unknown. However, many of the old problems that make the picture unlikely are here with us to stay. Most importantly, the entropy is still increasing and the cycles still tend to evolve.

The braneworld pictures try to solve this entropy problem by throwing the extra new entropy away. They only "recycle" a small portion of the parent Universe. The entropy of this portion is pretty much equal to the entropy of the parent Universe. So if this portion becomes the new daughter Universe, the daughter may resemble its parent.

Still, I don't see what is good about this scenario. There is nothing I observe about the real world, directly or indirectly, which would indicate that our Universe or our cosmological cycle had to have a parent whose macroscopic characteristics were essentially identical to those of our Universe. Such an equality looks like a case of a huge fine-tuning - fine-tuning that is moreover unjustified by any observations I am aware of. The addition of such an unmotivated fine-tuning into a theoretical framework certainly makes it less attractive and less likely to be true. I just don't understand why anyone would believe such a hypothesis.

The fine-tuning becomes more plausible if the characteristics of the newly born Universe are based on special values, namely a vanishing entropy. I think it is probably correct to imagine that the entropy of a newly born Universe was zero almost by definition - a natural "strengthening" of the second law of thermodynamics (by a maximal extrapolation of time into the past) and a conceivable consequence of the Hartle-Hawking (pure) wave function of the Universe and similar pictures.

On the other hand, if the entropy were zero to start with, the inclusion of such a uniquely determined Universe into some Universe that had a nonzero entropy looks immensely futile. If the entropy of our newborn Universe were zero, it would have been described by a unique microstate. So it would be obvious that such a unique microstate couldn't depend on a previous high-entropy Universe because the latter were inevitably non-unique (much like all high-entropy objects).

The detailed knowledge about the parent Universe would be inevitably useless for learning anything about ours. And I think that even statistical predictions would be impossible because causally disconnected portions of the Universe can't share any statisticians (or reach the thermal equilibrium).

The braneworld and ekpyrotic versions of the cyclic cosmology assume certain types of collisions of branes. Well, I have no problem with a colliding space-filling brane. However, I think that it is impossible to "guess" what branes were colliding and what the details of these collisions were. It seems clear to me that we must use independent arguments to figure out what the branes in our world have to be - if there are any - and then to analyze their possible collisions. We don't have any other "direct" data about the collisions of the space-filling branes in our Cosmos so any attempt to guess any details of these collisions is a hopeless guesswork.

You simply can't guess these things. And I think that there exists no positive argument that such a collision had to occur. To assume that it didn't occur is at least as natural an approach as to assume that it did. I would prefer Occam's razor and I would only look at these hypothetical collisions at the moment when someone would present a reason why such a hypothetical event could be interesting or related to some known physics.

Two years ago, Lauris Baum and Paul Frampton (a celebrated TRF reader) proposed a new method to solve the "entropy problem" of the cyclic cosmologies. They have a captivating story with a Big Rip, ready for a Hollywood movie (the Universe shrinks or expands so brutally that it is torn to pieces), but the physics point of this story remains unclear to me, too. To clean their Universe, they also require pressure to be smaller than -rho, much like in the Big Rip pictures in general. That violates the dominant energy condition and is effectively equivalent to a bizarrely superluminal propagation of signals.

That's a pretty high price to pay for ... a movie and for the laws of physics (and continuous space) that disappear earlier than we expected (without being replaced by any other laws). Sorry, Paul, but I am simply not getting it! At least, I agree that your Big Rip story is fun and you seem to appreciate the actual problems of the cyclic cosmology (increasing entropy etc.).

Infinite in the past vs infinite in the future

It may be a good place to mention one important principle of rational reasoning - a kind of logical causality. Some people construct certain models that have a certain "desired" behavior in the far future. But you can never use an assumption about the future as your starting point because we don't know the future. The only way to determine whether our Universe will end up with a Big Crunch or whether it will expand indefinitely (yes!) or whether it will undergo infinitely many cycles is to calculate the future from the laws that have been determined by our observations and analyses of the past! 

The future is always determined by the past (plus quantum randomness). Even gnoseologically, our knowledge about the future is also determined by our knowledge of the past.

Please, leave the decision whether the world should live indefinitely to Nature. It is very clear that whatever Her answer is, She can realize it smoothly. Whatever the past is and whatever Her dynamical laws to evolve are, She always knows what to do with the world and your attempts to help Her by guaranteeing that the future will or will not look in one way or another are just plain ludicrous. So the only remotely logical constraints may refer to the past. Again, both temporally finite and infinite Universe in the past are conceivable. 

And the second law of thermodynamics surely favors a temporally finite Universe with a beginning in the past because the entropy is increasing and the "number of elementary macroevents" (where one elementary macroevent is defined to increase the entropy by one) has thus been finite. There's no good way to "throw away" the entropy. If you postulate that the entropy has macroscopically decreased at any moment, it either means that 
  1. you assume a gigantically unlikely event (entropy drop) that makes your theory unlikely and unpredictive or that 
  2. you follow an incomplete description of Nature because you are just overlooking many degrees of freedom that must still live somewhere
Cyclic loop quantum cosmology

Finally, let me say a few words about Ashtekar's version of this cyclic story, promoted by Nude Socialist. Well, this particular fairy-tale seems to unify pretty much all myths and silly naive assumptions about all these interrelated questions. And their work connects them in a fashion that looks utterly irrational to me.

First of all, they believe that "singularities are bad". As we have explained many times, singularities are points where older, approximate theories break down. But what happens after they break down is pretty much an open question, at least a priori. A deeper theory may try to protect the world against such singularities. (Einstein also used to believe that something would prevent the stars from a collapse into black holes because the latter looked ugly to him: but such a new force would violate the rules of GR and locality.) But more often, the detailed microscopic confirms that something like these singularities does exist - approximate theories are guaranteed to see them because at their level of approximations, singularities are real. 

But the more complete theories tell us more accurately what happens with the singularities. Unpredictability and foggy indeterminate forms are replaced by a new predictive picture with finite numbers.

It is simply not true that "everything looking like a singularity must be prohibited" is the only possible outcome of a more detailed analysis based on a deeper theory. The singularities may also be confirmed - and they often are. Also, the whole question may be shown vacuous because the degrees of freedom that exhibited a singular behavior in the approximate theory may be replaced by completely new, possibly nonlocal degrees of freedom that don't admit such a singular behavior at all.

Ashtekar et al. suffer from this irrational singularophobia.

More seriously, they also misunderstand what it means to give a rational argument for anything. When they play with the silly equations of loop quantum cosmology that lead to some outcomes, they already think that it is a physical result.

Well, it is definitely not. Loop quantum gravity is a dumb, Lorentz-breaking, flat-space-denying toy model. And loop quantum cosmology is not even derived from loop quantum gravity. It is an even more silly application of the same oversimplified naive rules of loop quantization to the framework of the oversimplified FRW equations of cosmology. It is clear that a discretized version of the FRW cosmology must exhibit one of the few possible kinds of behavior when the Universe becomes very small.

But neither of them is a priori better than others. The only way how one of them could become better is to find rational arguments that one answer is more correct - or more likely to be correct - than others. For example, you would have to show that the young Universe actually follows some particular discrete or other rules - or rules that are in the same "universality class" as your theory and give the same qualitative answer to a question.

Obviously, they have nothing that is even remotely close to that: there are no rational or independent arguments, directly or indirectly rooted in observations or in careful mathematical classification or analysis of theoretical possibilities, that would indicate that loop quantum cosmology has anything to do with reality. What they have is a ludicrously oversimplified and randomly chosen toy model (or many of them) which is an oversimplified version of another oversimplified theory that can already be seen to be naive enough to have nothing to do with this world. We are supposed to be excited when a character in a silly discrete game bounces off the corner of a chessboard. Wow, the bishop can walk back and forth, isn't it amazing?

Well, didn't we just define the rules of the game so that the bishop can walk back and forth? When the bishop can move in this way, does it really mean that our Universe is doing the same thing?

And they want to trust its answers in contexts that require a much more careful and accurate theoretical analysis than the contexts where their theory already fails brutally (because it fails to reproduce the very existence of smooth space, its rotational and Lorentz symmetries, the existence of particles and forces, and all low-energy field equations and other equations that summarize everything we know about Nature). In order to say something about quantum gravity near the Big Bang or the Big Crunch, you actually need a theory that is more accurate a description of reality than the established low-energy effective field equations, not less accurate!

There exists not a single check - direct or indirect experimental check or an internal consistency check - that would indicate that what they are doing is anything else than a meaningless and random pseudo-mathematical masturbation. On the other hand, there seem to exist many checks that show that what they are doing is simply wrong. 

What these people completely lack is a broader perspective, a separation from the problems they try to solve, or "levity". They can never look at the set of "a priori possible theories and answers" from the bird's perspective. They can never see that some theories would give some answers while others would give different answers and it is a priori not clear who is right. Instead, they eagerly, quickly, and irrationally adopt whatever random assumptions that become popular in their community at a given moment. They quickly parachute into a random unhospitable valley in their landscape (a desert), chew the sand, and spend most of their lives by arguing that it is the ultimate paradise.

This is just not a way to do science. It's just pure nonsense, Prof Ashtekar. And it is simply bad that the media are promoting this nonsense. Cyclic cosmology is a highly ill-motivated, controversial, unjustified, and mathematically shallow sub-discipline of contemporary cosmology. But the cyclic loop quantum cosmology is worse: it is a really shameful corner of the cyclic cosmology where the rules of science, logic, and even ethics break down because its proponents seem to pretend that rational arguments exist where they don't exist and they claim priority in (not established) ideas that have been discussed in a similar fuzzy way for 75 years or more.

And that's the memo.

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reader Anonymous said...

Wow thanks. I didn't get to read the whole thing but your post is very useful to those of us who hear about these things but can't make sense of them.

I agree that making a cyclical universe because it's more eloquent is a bad reason. It only has to do with our idea of eloquent. The universe doesn't abide by our ideals.


reader Anonymous said...

Hey Lubos, it's just me again with another question of the layman. What's your response to this theory in the fast comments section? I would really like to know.

"Hi Lubos and Dmitry,

I think you are right regarding many of the problems of cyclic cosmology (and don't get me started on LQG/LQC), but there is actually a nice way to get rid of the extreme fine tuning of initial conditions without further ingredients, that is without postulating another potential like Justin proposed (this is work by Jean-Luc Lehners which he presented about a month ago at a workshop here and it is not on the arxiv yet - so I write the next paragraphs from memory): As Dmitry explained, the issue is that the fields have to be really close to the ridge during the contracting phase, so one might conclude that this model needs a lot of fine tuning.

However, if there is at least 60 e-folds of CC domination in the expanding phase, one can show that if there is a small patch going through a bounce, then the size of a suitable patch for the next bounce will increase (if measured at the corresponding time). Thus, one ends up with an eternally cycling universe, without any new ingredients. Most of our current Hubble horizon will of course not make it though the next cycle (thus only a small fraction is recycled which has bearings on the entropy problem), but all you need is a tiny fraction of todays Hubble volume (about a mm^3) with the proper initial condition anyway. The majority of the universe (not just our hubble patch) is hospitable in this setup, in contrast to eternal inflation where the majority of the universe is still inflating and only some bubbles are hosptitable. This model also has some interesting consequences for the CC (we had a long discussion about this with Neil, Paul and Jean Luc, so this is work in progress).

Anyway, I gotto go and do some work now,
cheers,
Thorsten Battefeld"

Thanks as always.


reader Anonymous said...

Come on Lubos, blow this guy's theory out of the water like you always do or else I'll worry that him and his crazy cyclical model is right!

(sorry if you're busy)