Friday, July 02, 2010

Why now problem: why it's not a problem

Originally posted on July 1st in the morning.

Daniel Holz wrote a short article about an interesting topic:
Casting aside Copernicus
He was provoked by a talk by John Moffat who proposes a new adjective for the standard model of cosmology: it is "anti-Copernican", he claims. Why?

Because we apparently live at a special time when the mass stored in the dark energy is comparable to the dark matter. That's "anti-Copernican" because there should be nothing special about us.

Let's look at all these things much more accurately.

Copernican principle

For long millennia, people were dividing the world to two approximately equal parts: "we" and "they". (Many people are still doing so today.) In this scheme of things, we're pretty special - whoever "we" are. Our nation is special, our species is special, our planet is special, our Solar System (once we believe that one exists) is special, our Galaxy (once we believe it's not just milk on the skies) is special, and so on.

This attitude is very natural. We may say that it originates in egotism. We may also say that it respects Occam's razor - we just don't need too many other nations, species, planets, stars, and galaxies, do we? (The usual anthropic talking point says that people have also thought that our Universe with the well-known coupling constants is special - but there are many of them. But we won't talk about these speculative questions in this text.)

However, we learned that our families and nations are just examples among many others. Our species is just one species among millions of species - although a pretty successful one. And there are many planets, stars, solar systems, galaxies, and so on. I don't want to repeat all these well-known clichés.

Does it have to be like that?

Well, we can invent many philosophies that are consistent with these observations. And we may try to extrapolate these philosophies. However, all such extrapolations may fail - and they almost always fail at some point. It is ultimately the scientific theories that can successfully predict empirically observable data that have the exclusive right to rationally inform us about the validity of all these philosophies.

Nevertheless, the philosophies are fun to think about and they may meaningfully organize many general facts in our minds.

Both the "small Universe with a finite, limited number of species and stars etc." as well as a "huge Universe with a nearly infinite number of species and stars" may be "natural", depending on the (more) exact meaning of the adjective "natural" that we happen to choose, and depending on the logic we find important or persuasive.

First, as I have mentioned, a "small number of species or people or planets", of order one, is natural because of Occam's razor, because of general notions of "minimality", and because numbers of order one are "natural" in the usual technical sense. If you imagine that there's a simple enough formula that produces these dimensionless numbers, the result has to be of order one - i.e. pretty small.

Second, the "large numbers" may be viewed as natural because there are many "very high numbers". So you can imagine that there are "natural" distributions dominated by high numbers. You will find that the normalizable distributions with the very large - namely divergent - expectation value are actually pretty special - think about something like "p(n) = c/n^2" which is normalizable but the sum of "p(n).n" diverges. So this argument about "large numbers being more likely" is contrived.

However, you will also find rational reasons why the number of people, species, or stars should be large. It's a pretty complicated task to construct an intelligent human being such as yourself - a reader of The Reference Frame ;-) - and lots of atoms, species, failed attempts, failed blogs, their confused readers, and numerous dead stars are needed to make this dream come true, aside from many other pre-requisites. ;-)

If you love these philosophical arguments and you read the arguments above, you should get more humble. It's a damn shaky foundation for big claims about the Universe. There's no universally justifiable reason why any of these philosophies about naturalness should be dominant. The empirical data always beat philosophy if you actually have them.

Uniform Universe

Another natural, "modern" interpretation of the Copernical principle is the homogeneity of the Universe. The Universe looks qualitatively identical from all points in the Universe. The detailed constellations will be different but there will be constellations seen from any planet orbiting any star.

This seems to be true. Is it inevitable? Well, it is surely an experimental fact that the "cubes" of the Universe whose side is about 600 million light years - about 100 times shorter than the size of the visible Universe - look almost identical. There are many hierarchies in the Universe - clusters of clusters of galaxies etc. - but all of them are smaller than 600 million light years or so.

From a very general viewpoint, it doesn't have to be so. The Universe could have many different regions where all basic features of the Universe look very different - after all, that's what defines the multiverse and many people believe in it.

However, unless you have a good reason or a predictive framework that tells you how these "new regions" look like (yes, string theory may be viewed as such a framework, but it doesn't directly imply that the multiverse has to exist i.e. that the "landscape of possibilities" has to be physically realized), you should use Occam's razor and cut all the unnecessary fantasies.

An advantage of a uniform Universe is that the basic laws hold everywhere. If you want to understand its qualitative shape, you may just "copy and paste" our region of the Universe many times. In this sense, a large uniform world is as natural - and simply defined - as a very small world. You just repeat the same pattern many times. Of course, you need the copies to be "not quite identical" if you actually want to use them to obtain new things.

The translational and rotational symmetry of the Universe - including its rough shape at cosmological distances - have certainly been assumptions that have guided the research in most of the modern cosmology. They still seem to be satisfied. And the inflationary cosmology explains that a pretty robust, natural mechanism is very efficient to produce huge volumes of the homogeneous Universe.

If some observations clearly said that it's not the case, we would have to change our opinion, of course.

Uniformity broken in time

For example, the history of our Universe is not "uniform in time" in the same sense as it is "uniform in space", so the simplest generalization of the Copernican principle from space to time is instantly falsified.

The expansion of the Universe shows that the galaxies used to be closer. The Universe was denser, hotter in the past. Many things have been changing as a function of the time from the Big Bang. This evolution is what cosmology is all about. And the Big Bang itself is the ultimate "non-uniformity in time" - it's an absolutely exceptional moment in time, namely the first one in which our visible Universe began its existence.

Does this fact violate something important that science rests upon? Well, it surely doesn't. Space and time are related by the Lorentz symmetry - but the spatial and temporal coordinates still differ, e.g. by the sign of the metric tensor components.

While these signs don't spoil many properties of space and time - usually properties of a quantitative character - they surely do spoil the analogies referring to many qualitative properties, usually those that are comprehensible to mathematically untrained eyes. It is enough to look in the real world - or consider these things deeply in theory - to realize that the uniformity of the Universe in space but not time is one example.

The Universe is simply not uniform in time although it is uniform in space. In certain natural coordinates, inflation or the current expansion is increasing the volume of the Universe. And because the total mass carried by dust is pretty much conserved, the density of dust decreases during inflation - or any cosmological expansion.

We can certainly see many differences between space and time that result from inflation - and other things. For example, the energy-momentum vectors of physical objects need to be time-like. That's required by causality - you can't change your past. Causality combined with this "positivity of the squared mass" also tells you that you should better take purely spatial slices of your spacetime. Many things such as probabilities like to be fundamentally positive which means that one has to treat e.g. time-like and space-like components of the gauge field's excitations differently.

With the space-like slices, the de Sitter space produced by inflation seems to be exponentially growing as a function of the remaining coordinate which is time and many things depend on time, too. There's just no doubt that time and space differ in some important qualitative properties. None of these differences contradicts the Lorentz symmetry of the (local) laws of physics. Mathematics of the physical Universe still respects the symmetry but different signs often lead to different qualitative consequences. Many of them are deep and universal.

We may only apply arguments referring to "naturalness" if we have a sufficiently well-defined framework to parameterize "all conceivable theories" and the "likelihood that a theory will have certain properties". The generalized Copernican principle that is vague enough not to recognize time and space is surely not sufficiently well-defined.

I am not going to include other, "squeezed" versions of this graph because such depictions make not only the intersection look singular; they make other things look singular as well. So they're obscuring all things which means that they're not good tools to present a paradox even if it were real.

"Why now" problem

Finally, I am getting to the main topic that gave this article its name, the "coincidence problem". The graph above shows the fraction of the total energy carried by dark energy and by dark matter (which also includes the visible matter). Somewhat counterintuitively, the present is on the left, the Big Bang is on the right side. So these are "billions of years ago".

About 4 billion years ago, the dark energy represented 50% of the total energy. So it's not exactly "us" who see the most democratic distribution among the energy species. We are separated by a factor of 14/10 from that "critical age". So the multiplicative error margin is something like log_{10}(14/10) = 0.15 decadic orders of magnitude or 1/3 of the base-e "e-folded order of magnitude" (also known as 33 E-percent).

That's a somewhat interesting but not a terribly impressive agreement. The meaningful time when "something occurs" is between 10^{0} Planck times (shorter times are physically meaningless due to the effects of quantum gravity) and 10^{70} Planck times (when the matter gets way to diluted).

So if you imagine that the "reasonable times from the Big Bang" are uniformly distributed on the log axis - between the exponent being 0 and 70 - then the probability that we're just 0.15 orders of magnitude away from the "right point" is merely 0.15/70 = 0.002 or so. The odds are 1 in 500.

You may be intrigued by this successful "hit". There can be something special about "our time" - or the time when intelligent civilizations are beginning to rise. But you should appreciate two things that make the success less amazing:
  1. There are many pairs of quantities (analogous to the two densities) that can be compared.
  2. The optimum [approximate] moment for life to arise is indeed given by some calculations - and the results of calculations always refer to points where "two quantities are [nearly] equal".
So you shouldn't be too surprised.

Let me say a few words about both points. The first point emphasizes that in our "proof that our time is special", we arbitrarily chose two quantities - the fractions of energy carried by dark matter and dark energy, respectively. We could have chosen many other densities of various things (dust, radiation, visible matter, or densities of entropy carried by various types of objects), temperatures, and so on.

If you identify 20 quantities like that, there are (20 choose 2) = 190 different pairs, and the 1-in-700 luck we have encountered becomes just 1-in-4 luck if you're allowed to search through many pairs.

Second, there indeed exist arguments why life should thrive at this special point of the Universe. For example, if you believe Weinberg's anthropic estimate for the cosmological constant, it also solves the "Why now" problem. Why?

It's because Weinberg (tautologically) claims that the cosmological constant in our Universe can't be too high - above a certain bound CC9 - because if it were higher, the Universe would expand too quickly, the density would drop too early, galaxies wouldn't be clumped, and stars wouldn't start to shine which would be pretty unpleasant for life (although not "demonstrably lethal" for all kinds of conceivable life).

Weinberg says more than this tautology: he claims that it's natural for the CC to be of the same order as CC9 because the previous paragraph describes the most important selection criterion.

If that's so, you can see why the CC will be comparable to the conventional matter density around the same time when the living forms are created. When the Universe got a few billion years old, the accelerated expansion driven by the cosmological constant became an important "threat" for the life of the galaxies.

It's natural for life to use the bulk of the time interval when it's possible, i.e. when it is not destroyed by a threat, so it is not shocking that it began billions of years after the Big Bang, too - somewhat before the destruction of chances to create new galaxies and life. This calculation is not terribly accurate. Of course, if you want a more accurate calculation of "our birthday" or "your birthday", you need a more accurate model for cosmology, astrophysics, geology, and biology - and a more accurate and more objective definition of "us" and "yourself", too. ;-)

Needless to say, even if the anthropic explanation of the cosmological constant is not the best one, I can still use the arguments above because the cosmological constant is what it is even if there's a better calculation of its value than Weinberg's hand-waving. I must just replace the short but foggy arguments about the "survival" by the superior calculation of the value of the cosmological constant that someone may find in the future. But the equality will continue to result from the argument.

The shock and awe are gone

When you combine both points - the point that many things could have agreed, but the agreement is not too precise; and the point that the optimization problems really do predict that it's natural for various things to be of the same order at the best moments - you will probably lose your excitement about the coincidence problem.

The coincidence problem can only be defined vaguely and approximately. And at this level, it is well explained by the vague and approximate arguments above. There's no good reason to look for a much more accurate explanation of a much less precise would-be problem or a much less accurate would-be fine-tuning problem. Only well-defined and sufficiently unlikely coincidences deserve well-defined and very accurate explanations.

And that's the memo.

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