Tachyons and the Big Bang

Let's start with a general description of the tachyons, their history, their meaning, and their role in string theory; we will get to the current questions whether the tachyons are important for cosmology later.

There have been many twists and turns in the history of the tachyons. As soon as Albert Einstein discovered his special relativity 100 years ago, it became clear that no particle, no object, and no carrier of information can travel faster than light. The reason is that an observer in a different reference frame would realize that such an object not only travels faster than light but it in fact travels backwards in time. Lorentz transformations make superluminal propagation equivalent to propagation backwards in time. That would destroy causality; it would allow you to kill your father before he met your mother - and such a possibility would transform our Universe into a pile of logical absurdities and oxymorons.

In other words, the four-momentum of acceptable particles must be a time-like vector. Let's choose the convention in which "p squared" is positive in these good cases. Particles with negative values of "p squared" are banned. They are called "tachyons" - the Greek word "tachos" means speed and the name is apt because the tachyons have to be very fast; they would always move with superluminal velocities.

In quantum field theory it is still useful to think of tachyons as particles with space-like four-momenta but it is not the most realistic description of their behavior. Instead, the typical configuration involves a time-like imaginary four-momentum of a tachyon field - a configuration of the field that exponentially grows with time instead of oscillating. The potential energy "m squared times phi squared (over two)" for a scalar field "phi" has a minimum if "m squared" is positive - the regular massive particles - and it has a maximum for negative values of "m squared".




The latter case corresponds to tachyons. As you can see, tachyons signal instabilities akin to Columbus' egg standing on its tip. For example, the Higgs field in the Standard Model would be a tachyon if the symmetry remained unbroken. However, the Universe with such a Higgs field would be unstable much like the egg. It would spontaneously choose a random direction and "fall"; the speed of the fall would be growing exponentially, at least for a little while.

The key question is what happens with the egg if you make it stand on its tip but it eventually falls. The ordinary egg will find another, stable position. Whether or not this occurs for a tachyon in a physical theory depends on the details. The final fate of the Universe with a tachyon - a counterpart of the egg - depends on the existence of a minimum of the tachyonic potential. For example, the potential has a minimum in the Standard Model (a whole sphere of minima whose points are, however, equivalent due to the original gauge symmetry). This minimum describes the spontaneously broken electroweak symmetry - a situation in which our world lives today. If you expand the potential energy around this point, you will find out that the squared masses are never negative. Higgses becomes massive particles and the tachyons disappear if the theory is described as a small perturbation around a stable point - around a minimum of the potential as opposed to a maximum (or a saddle).

Let's turn our attention to string theory. The original "bosonic" string theory found in the late 1960s and early 1970s lived in 26 dimensions. Its closed strings could behave not only as gravitons or (infinitely types of) massive particles but also as tachyons. Regular closed strings are free to move throughout the space. In the case of the closed strings that vibrate as gravitons, the previous sentence implies that gravity is always a property of the whole spacetime; string theory agrees with general relativity that gravity is dynamics of spacetime's geometry. In the case of the closed string tachyons, the same sentence means that these tachyons signal an instability of the whole spacetime. Something brutal has to happen with the whole Cosmos - for example, it can lose most of its dimensions.

People in the 1970s tried to get rid of the tachyons; to pretend that they were not there; to find a minimum of the potential. All these attempts have failed. A well-known story from the history of physics is that the "bosonic" string theory in 26 dimensions was superseded by superstring theory in 10 dimensions in the middle 1970s. The latter has no tachyons; it is a consequence of spacetime supersymmetry that allows you to write the energy "E" as "Q squared" where "Q" is another operator, namely the supersymmetry generator; this makes it obvious that the energy can't be negative which would have to be the case for a specific tachyonic particle. (Superstring theory also predicts the existence of fermions which is pretty useful to match the real world, but matching the real world of particle physics is not the topic of this text.)

It is fair to say that the tachyons played very little role in the string theory research between 1975 and 1998. Much progress in these two decades relied upon spacetime supersymmetry - especially the supersymmetric calculational revolution sparked by Seiberg in the early 1990s. Tachyons were viewed as an uninteresting disease of unrealistic or toy model versions of string theory.

In 1998 or so, people decided to return to the question of the tachyons. However, they did not need to return to 26 dimensions; tachyons may be found in superstring theory, too. It's enough if you compactify string theory in certain ways that break supersymmetry (such a breaking is not a sufficient condition) or if you include unstable configurations of branes.

Ashoke Sen was the first person who has understood the fate of some string-theoretical tachyons. Consider two (nearly) coincident parallel D-branes as found by Joe Polchinski; a D-brane or a Dp-brane is a p-dimensional membrane defined by the property that open strings are allowed to end on it so that some of their coordinates (namely those transverse to the D-brane) have Dirichlet boundary conditions "x=x_0" at the endpoints (which is where "D" comes from). Rotate one of these two D-branes by 180 degrees in a "mixed" 2-plane (spanned by 1 parallel direction and 1 transverse direction); this changes the D-brane into an anti-D-brane because the direction of the arrow (or the volume form) is inverted. Its charge - something like a higher-dimensional winding number - thus gets reversed, too. What you get is a brane-antibrane pair.

Much like the positronium, the bound state of an electron and a positron, the brane-antibrane pair is unstable because these two guys may annihilate with each other. How do you see it in string theory? Once you have two D-branes, it is always allowed for an open string to terminate on one D-brane with one end and the second D-brane with the other end. These open strings describe physical quantities that these two D-branes share; in the case of intersecting branes, such open strings want to shrink near the intersection of the D-branes because the string can then be shorter (and henceforth lighter). This is why the resulting fields arising from such open strings are localized at the locus of the intersection. The ground state of such a string is a priori a tachyon; incidentally, note that tachyons are always excitations of a scalar field in string theory as well as any other consistent physical theory. For example, you won't be able to write down a Hermitean mass term for a Dirac fermion with an imaginary mass.

If these two D-branes are really identical, the tachyon is removed by the superstringy GSO (Gliozzi-Scherk-Olive) projection and it is not a part of the physical spectrum. Two parallel D-branes are stable. On the other hand, if one of these D-branes is an anti-D-brane, the sign of the GSO projection is inverted. The projection preserves exactly those particles that are banned in the "brane-brane" case which includes the tachyon. Now, because the open strings are attached to the D-branes, their excitations describe fields that are geometrically attached to the D-branes. A negative squared-mass scalar field is a sign of instability; but because this field is only defined within the D-branes, it is an instability of the D-branes only. It cannot directly destroy the rest of the spacetime.

Ashoke Sen conjectured that the potential energy for this tachyon has a minimum and one can exactly say what is the value at the minimum. The energy density at the minimum is exactly the energy density at the maximum (where you found the tachyon) minus the total tension of the D-branes. This means that once the tachyonic scalar field rolls from the maximum to the minimum, the D-branes are destroyed completely and their full energy "E=mc^2" is released. Such a scenario has many implications. For example, it quantitatively predicts the energy at the minimum which has been tested in string field theory - a formalism that describes (open) string theory as a quantum field theory with infinitely many fields. They have found, numerically, roughly 99.9999% of the right value of the energy difference and everyone was happy. The numerical calculations were needed in the cubic string field theory; in the boundary string field theory, analytical proofs were possible.

A more sophisticated gedanken experiment involves a brane-antibrane pair whose antibrane (for example) has a non-trivial gauge field so that the two branes can't annihilate completely. After the annihilation, something (a lower-dimensional object) is left. All possible objects that are left are classified by certain topological data. This line of reasoning showed us that the allowed D-brane charges are not classified by homology - the possible cycles onto which the D-branes are simply wrapped - but by something else, namely K-theory that is physically closer to topologically non-trivial configurations of gauge fields. K-theory is similar to homology but it differs in details; for example, new D-branes conserved "modulo k" have been found. A link with another mathematical field has emerged.

More convoluted thinking along these lines has led people to conjecture that not only K-theory but also the "derived categories of coherent sheaves" are useful and relevant for understanding of D-brane charges as well as some of their dynamics. The string theory community is a community of mathematically oriented physicists, but most of them are not mathematically oriented enough to transform coherent sheaves into the most exciting object to study. But you should remember that those who propose these coherent sheaves and other notions from category theory - Paul Aspinwall, Dave Morrison, Michael Douglas, among others - as a new language (and maybe not just language) for string theory are very smart people and there can be something deep in it although I have personally not understood what it should be.

Despite many attempts, I still have not understood the basic question whether the category-theoretical description is just a different language to describe known features of string theory; or whether it implies new physics that is not contained in the CFT; in the latter case, I also don't understand whether the compatibility of this category theory with the "usual" string theory is a conjecture or a proven fact. The status of these ideas remains unclear and it is likely that they're valid in the perturbative realm only. (D-branes can only be singled out among other branes in the weakly coupled regime.)

It is fair to say that these developments have convinced all of us that the fate of the open string tachyons has been understood completely. It was natural to look at the closed string tachyons. Because the bulk tachyons seem to signal an instability of the whole Universe and the whole theory and the fate of the Universe that collapses completely seems uncertain, people focused on the closed string tachyons that are localized much like the open string tachyons. How can you localize closed string tachyons? They can be localized if they appear in a twisted sector of an orbifold - a fact that Adams, Polchinski, and Silverstein (APS that does not mean American Physical Society) exploited in 2001 or so.

Consider the "C/Z_N" orbifold - a two-dimensional plane whose points are identified if they are related by a rotation by "360.k/N" degrees around the origin for integer values of "k". In field theory this would be equivalent to a new field theory defined on a cone (recall how can you produce cones by gluing a wedge of paper appropriately). In string theory, you also reduce the number of independent "points" and "fields", but you must compensate this reduction by adding twisted sectors. The twisted sectors are new sets of closed strings that are not exactly closed in the original plane "C" but they are closed up to a rotation by "360.w/N" where "w" runs between "0" and "N-1" and labels one of the "N-1" twisted sectors (plus "1" untwisted sector for "w=0").

The twisted strings would be long (and heavy) unless they are localized near the origin - the fixed point of the rotation. That's analogous to the open strings living near the intersection; the open string case is actually a worldsheet orbifold of the closed string case. The punch line is that the twisted closed strings describe fields that are localized at the origin.

This APS example is a non-supersymmetric orbifold and you will find a lot of tachyons in the twisted sectors. What do these tachyons tell us? They tell us that string theory does not like the sharp tips of the cones. The tachyons condense near the tip which smears out the tip of the cone which makes the tip nice and round. Although this picture does not offer truly quantitative predictions analogous to Sen's energy difference, there are many qualitative consequences of such a story that can be checked and have been checked.

Tachyons made of winding closed strings are analogous to the twisted closed strings. Their squared mass is the sum of the tachyonic basic squared mass plus the squared circumference of the circle (multiplied by the squared stringy tension). This means that these modes only become tachyonic if the circle onto which the closed strings are wrapped is short enough. It has been shown that they have the capacity to change the topology of the space. Unlike the case of Sen's open string tachyons and unlike the supersymmetric topology changing processes, this scenario itself does not offer quantitative predictions either but it is a nice and self-consistent picture that shows you how non-supersymmetric "handles" may be removed - a picture proposed by Adams plus four more authors.

As you can see, these considerations are inevitably becoming less quantitative, more uncertain and kind of frustrating, but potentially also deeper conceptually. Recently it has been proposed by John McGreevy and Eva Silverstein - who were building on some comments by Polyakov; calculations of some correlators of tachyons by Strominger and Takayanagi; the general philosophy to study time-dependent background, especially time-dependent tachyon fields, in the early 2000s; the stories explained above - that the bulk closed string tachyons are an important key to understand the initial singularity in string theory.

They study various backgrounds in string theory where a tachyon - or a tachyon-like operator in the conformal field theory (whose main component is a relevant operator) - has a vacuum expectation value that is large at the moment of the singularity but exponentially decreases later. Such a tachyonic condensate behaves much like the Liouville wall in two-dimensional string theory. Recall that the Liouville wall is useful in two-dimensional string theory because it acts as an additional potential that repels the strings from the strongly coupled region. John and Eva are using their tachyonic potential in a similar way: the tachyon near the singularity "turns off the lights" and it repels strings and all physical phenomena for that matter. The authors believe that this solves the problems and infinities of the initial singularity.

I don't really understand these claims and let me try to specify what I don't understand about them.

First of all, even in the Liouville case itself, the Liouville wall is not something one should be proud about. It is a method how perturbative string theory may remain internally consistent without telling you what is the physics at strong coupling: the places where the coupling is strong have also a powerful Liouville wall that does not really allow the observers to probe these regions. It's closing your eyes. My personal belief is that physics at strong coupling in two-dimensional string theory is not really uniquely well-defined and the folklore about the uniqueness of quantum gravity does not really apply in 2 spacetime dimensions (not even 3 spacetime dimensions) because of the topological nature of gravity in these low dimensionalities. Quantum gravity is only tough if at least 4 dimensions are rather large and flat; this is where one expects uniqueness and where string theory can show (and, in some cases, does show) its predictive muscles.

The situation in John's and Eva's setup is even more difficult. In the two-dimensional case, the tachyonic condensate eventually becomes infinitely large in the infinitely strongly coupled region and the screening is "perfect". The tachyon of John and Eva has always a finite vacuum expectation value as long as the coefficient is finite. Some of their qualitative conclusions depend on the coefficient's being infinite which seems as a confused order of limits to me. The problems they claim to have solved are not solved for any finite value of the tachyon which may always be viewed as a small perturbation of the original singular background. The larger value of the tachyonic field they choose, the more thoroughly they may solve some problems, but the more their solution is similar to simply cutting the cosmology at some moment "t=BBC", an abbreviation that I will explain below.

We may forget about these non-perturbative worries, of course, and ask whether we have removed all dangerous singularities from the perturbative expansion of string theory. My worries continue. Some of my primary questions are the following ones:

Are their conformal field theories really conformal? A reason to doubt this statement is that the expanding Universe near the singularity looks like a conical singularity, and much like a regular cone, it has a deficit angle and violates the Ricci flatness by a delta-function at the very tip of the cone. There exist orbifolds in string theory that admit a nonzero deficit angle, but they are very special and moreover a small condensation of the tachyon leads to a destruction of the whole cone, not just the stringy neighborhood of the tip that is needed in John's and Eva's setup.

If there are no conformal field theories of this type, the discussion would end and the idea would be ruled out. So let us assume that there are such conformal field theories in which the tachyon is only condensed in the stringy vicinity of the singularity. The next pressing question is:

How unique is this CFT? It is unlikely to be completely unique - John and Eva themselves offer many examples how the initial singularity could look like. The real question is really: How many continuous adjustable parameters each of these backgrounds has? Zero, finite number, or infinitely many? I think that this question must be answered before some correlators are computed because the correlators are only physically meaningful if the answer to the previous question is "zero" or, in the worst case, "finite number".
There are no predictions if infinitely many parameters are undetermined.


The most interesting point of their paper - as far as I can say - is that they show that the two-point functions have a thermal behavior whose temperature is "kappa/pi" where "kappa" is the rate of the tachyon's decrease; the tachyonic vev is "C.exp(-kappa.X0)". Note that this could be a universal temperature in the whole Universe. If such things worked with a small number of adjustable parameters - or a larger number of parameters that do not change the gross behavior - one would not even need inflation to explain the uniform temperature of the CMB radiation in the visible Cosmos. The temperature would simply be related to some universal properties of the tachyon near the Big Bang.

Unfortunately, I think - especially after discussions with Nima - that such a scenario is extremely unlikely. The uniform character of the CMB temperature also depends on the flatness of the Universe at the very beginning. John and Eva have not really showed that the Universe with flat spatial slices near the beginning is preferred in any way. It is critical to decide whether all shapes can lead to CFTs or whether some of them are preferred. Only if the choice of the initial configuration is more or less unique, we have solved something. The main problem of a naked singularity is that predictions are impossible because of infinitely many choices. A singularity can emit anything.

It seems that John and Eva declare that the tachyon provides us with a BBC - these letters do not stand for "British (or Bin Laden's?) Broadcasting Company" ;-) but rather "Big Bang Cutoff" - and the times before "t=BBC" - such as "t=0" - are unphysical. If this is what they mean, the prescription does not solve anything because we may equally well call "t=BBC" to be the new physical Big Bang. The calculational strategy is then nothing else than extrapolation of arbitrary initial conditions at "t=BBC" to "t=TODAY". We could have always thought that this is the only thing that physics can do. If we follow this strategy, we close our eyes to avoid the moments before "t=BBC". Moreover, such a philosophy has nothing whatsoever to do with the tachyons.

Once again, the only hope would be to show that the CFT and its non-perturbative completion near the Big Bang is essentially unique and it gives us a stringy counterpart of the Hartle-Hawking wavefunction. Note that it is allowed for the non-perturbative phenomena to help us to make this choice unique - in analogy with various non-perturbative terms that stabilize some moduli in flux vacua and elsewhere. It is tollerable that the CFTs are degenerate but non-perturbative effects require us to choose a specific one. At any rate, my feeling is that if the initial singularity is not unique, the rules of the game are ill-defined and all correlators that one calculates in these theories are physically meaningless.

Some people could say that "string theory won't be capable to predict some things but only some other things". I find these statements either incorrect or vague because no one has found a key in string theory to separate physical questions to "predictable" and "unpredictable" and those who claim to have found such a key have not shown any arguments that their key is better than zillions of other keys to divide quantities to "predictable" and "unpredictable" differently. My prediction is that such a key will never be found in string theory because of its "everything-or-nothing" nature.

As far as the very beginning and the "initial conditions" of the Universe go, physics will either be able to make predictions, or it won't. Note that the inflationary era makes most of the details of the "initial conditions" almost exactly irrelevant for the present observations. But still, if you assume a finite inflationary era, nothing is changed qualitatively about the question whether the conditions before the inflation can be studied scientifically.

The statement that the tachyon condensation is an essential key to understand the Planckian/stringy cosmology is an interesting, bold, and extraordinary statement, and it requires extraordinary evidence.