# String Theory Cosmology: Review

November 3, 2009 13 Comments

A big complicating factor in understanding string cosmology is understanding string theories. String theories and M theory appear to be limiting cases of some bigger, more fundamental theory. Until that’s sorted out, anything we think we know today is potentially up for grabs.

That being said, there are some basic issues in string theory cosmology:

1. Can string theory make any cosmological predictions relevant to Big Bang physics?

2. What happens to the extra dimensions?

3. Is there Inflation in string theory?

#### Low energy string cosmology

The baryonic matter that makes up the nuclei of atoms seems to provide only a small fraction of the total mass in the Universe.

Most of the mass in our Universe appears to occur in the form of dark matter, which is most likely made up of some exotic particle or particles that interact very weakly and have a very large mass.

String theories require supersymmetry for quantum consistency, and supersymmetric theories require bosons and fermions to come in pairs, because the supercharge operator turns bosons into fermions and vice versa.

So supersymmetric theories are good places to look for exotic matter in the form of fermionic superpartners of bosonic particles that carry forces.

In the Standard Model of particle physics, recall there is a spontaneously broken symmetry that gives mass to the weak interaction gauge bosons through the Higgs potential. The Standard model contains three massive gauge bosons, two charged and one neutral, and a massive neutral Higgs field.

The Minimal Supersymmetric Standard Model (MSSM) is a supersymmetric version of the Standard Model. The weak interaction gauge bosons and Higgs fields in the MSSM have fermionic superpartners, and the neutral superpartners are called neutralinos. A neutralino would make a good candidate for for dark matter, because it couples with weak interaction strength but should have a high mass.

But this is true only as long as it is stable. A neutralino would be stable if there were nothing of lower mass that it could decay into, i.e. it is the Lightest Supersymmetric Particle (LSP), and if something called R-parity is conserved.

The experimental limits on supersymmetric particle masses say that any neutralino LSP out there must have a mass greater than 40 GeV. A neutralino of that mass could give

and that’s already in the right ballpark for the observed amount of dark matter out there.

But the success of such a model depends on whether supersymmetry can be broken at the right scale. Supersymmetry breaking has other cosmological implications, such as a cosmological constant with a value that can run away from the very small, but nonzero, value that has recently been observed in the redshifts of supernovae. So this is far from a settled problem.

#### Cosmology and string duality

The standard Big Bang cosmology assumes that the Universe began expanding from a state that was very hot, very small, and very highly curved. The Big Bang model agrees so well with observation that it is therefore commonly assumed that any cosmological era that preceded the Big Bang must have involved a Universe that was even hotter and even smaller and more highly curved, until we reach the Planck scale and the Planck temperature, where our ability to describe geometry runs into fundamental quantum limits where gravity is strongly coupled and can no longer be treated as a fixed classical substrate in which particles or strings interact.

But string theory complicates such a naive monotonic extrapolation backwards through time, temperature and curvature, because in string theory there are symmetries that can obscure the difference between large and small distance, large and small curvature, and large and small coupling strength.

One such symmetry is T-duality. Recall that with strings quantized in a flat spacetime background, if one dimension is wrapped into a circle of radius R, by identifying x^{i} with x^{i} + 2pR, there are two new kinds of modes added to the spectrum: modes with quantized momentum going around the circle with quantum number n, and modes that wrap around the circle with winding number w. The total mass squared of the string then depends on these two numbers

This formula has a symmetry under the exchange

This is T-duality. The self dual point is where

At the self-dual point, extra massless fields enter the dynamics that reflect an enhanced group of symmetries.

T-duality has been applied to pre-Big Bang cosmology to build a model that is probably wrong, but interesting to study nonetheless.

A cosmological solution to the vacuum Einstein equations that is homogeneous but not isotropic is the Kasner metric, which can be written as

The set of exponents {p_{i}} as constrained above have the properties that they are all smaller than one, and they can’t all have the same sign. If n of the exponents are positive so that the Universe expands as time increases in those n directions, then the remaining D-n exponents are negative, and the Universe shrinks in those directions as time increases.

String theory has a scalar field called the dilaton, and the Kasner metric in this case extends to

Again, directions with p_{i} positive expand as time increases, and those with p_{i} negative contract as time increases. Notice that in this case, isotropic solutions are allowed where p_{i} = ± D^{-1/2}.

For every solution with some set of exponents and dilaton {pi, f(t)}, there is a dual solution with {pi’,f'(t)} given by

So expanding solutions and contracting solutions are dual to one another.

This duality symmetry has led to an interesting proposal for pre-Big Bang cosmology where the stringy Universe starts out flat, cold and very large instead of curved, hot and very small. This early Universe is unstable and starts to collapse and contract until it reaches the self dual point, where it heats up and starts to expand to give the expanding Universe we observe today.

One advantage to this model is that it incorporates the very stringy behavior of T duality and the self dual point, so it is a very inherently stringy cosmology. Unfortunately, the model has failings in both the technical and observational categories, so it’s no longer considered a viable model for string cosmology

A big complicating factor in understanding string cosmology is understanding string theories. String theories and M theory appear to be limiting cases of some bigger, more fundamental theory. Until that’s sorted out, anything we think we know today is potentially up for grabs.

That being said, there are some basic issues in string theory cosmology:

1. Can string theory make any cosmological predictions relevant to Big Bang physics?

2. What happens to the extra dimensions?

3. Is there Inflation in string theory?

#### Low energy string cosmology

The baryonic matter that makes up the nuclei of atoms seems to provide only a small fraction of the total mass in the Universe.

Most of the mass in our Universe appears to occur in the form of dark matter, which is most likely made up of some exotic particle or particles that interact very weakly and have a very large mass.

String theories require supersymmetry for quantum consistency, and supersymmetric theories require bosons and fermions to come in pairs, because the supercharge operator turns bosons into fermions and vice versa.

So supersymmetric theories are good places to look for exotic matter in the form of fermionic superpartners of bosonic particles that carry forces.

In the Standard Model of particle physics, recall there is a spontaneously broken symmetry that gives mass to the weak interaction gauge bosons through the Higgs potential. The Standard model contains three massive gauge bosons, two charged and one neutral, and a massive neutral Higgs field.

The Minimal Supersymmetric Standard Model (MSSM) is a supersymmetric version of the Standard Model. The weak interaction gauge bosons and Higgs fields in the MSSM have fermionic superpartners, and the neutral superpartners are called neutralinos. A neutralino would make a good candidate for for dark matter, because it couples with weak interaction strength but should have a high mass.

But this is true only as long as it is stable. A neutralino would be stable if there were nothing of lower mass that it could decay into, i.e. it is the Lightest Supersymmetric Particle (LSP), and if something called R-parity is conserved.

The experimental limits on supersymmetric particle masses say that any neutralino LSP out there must have a mass greater than 40 GeV. A neutralino of that mass could give

and that’s already in the right ballpark for the observed amount of dark matter out there.

But the success of such a model depends on whether supersymmetry can be broken at the right scale. Supersymmetry breaking has other cosmological implications, such as a cosmological constant with a value that can run away from the very small, but nonzero, value that has recently been observed in the redshifts of supernovae. So this is far from a settled problem.

#### Cosmology and string duality

The standard Big Bang cosmology assumes that the Universe began expanding from a state that was very hot, very small, and very highly curved. The Big Bang model agrees so well with observation that it is therefore commonly assumed that any cosmological era that preceded the Big Bang must have involved a Universe that was even hotter and even smaller and more highly curved, until we reach the Planck scale and the Planck temperature, where our ability to describe geometry runs into fundamental quantum limits where gravity is strongly coupled and can no longer be treated as a fixed classical substrate in which particles or strings interact.

But string theory complicates such a naive monotonic extrapolation backwards through time, temperature and curvature, because in string theory there are symmetries that can obscure the difference between large and small distance, large and small curvature, and large and small coupling strength.

One such symmetry is T-duality. Recall that with strings quantized in a flat spacetime background, if one dimension is wrapped into a circle of radius R, by identifying x^{i} with x^{i} + 2pR, there are two new kinds of modes added to the spectrum: modes with quantized momentum going around the circle with quantum number n, and modes that wrap around the circle with winding number w. The total mass squared of the string then depends on these two numbers

This formula has a symmetry under the exchange

This is T-duality. The self dual point is where

At the self-dual point, extra massless fields enter the dynamics that reflect an enhanced group of symmetries.

T-duality has been applied to pre-Big Bang cosmology to build a model that is probably wrong, but interesting to study nonetheless.

A cosmological solution to the vacuum Einstein equations that is homogeneous but not isotropic is the Kasner metric, which can be written as

The set of exponents {p_{i}} as constrained above have the properties that they are all smaller than one, and they can’t all have the same sign. If n of the exponents are positive so that the Universe expands as time increases in those n directions, then the remaining D-n exponents are negative, and the Universe shrinks in those directions as time increases.

String theory has a scalar field called the dilaton, and the Kasner metric in this case extends to

Again, directions with p_{i} positive expand as time increases, and those with p_{i} negative contract as time increases. Notice that in this case, isotropic solutions are allowed where p_{i} = ± D^{-1/2}.

For every solution with some set of exponents and dilaton {pi, f(t)}, there is a dual solution with {pi’,f‘(t)} given by

So expanding solutions and contracting solutions are dual to one another.

This duality symmetry has led to an interesting proposal for pre-Big Bang cosmology where the stringy Universe starts out flat, cold and very large instead of curved, hot and very small. This early Universe is unstable and starts to collapse and contract until it reaches the self dual point, where it heats up and starts to expand to give the expanding Universe we observe today.

One advantage to this model is that it incorporates the very stringy behavior of T duality and the self dual point, so it is a very inherently stringy cosmology. Unfortunately, the model has failings in both the technical and observational categories, so it’s no longer considered a viable model for string cosmology.

#### Inflation vs. the giant brane collision

Inflation is still the preferred cosmological model of astrophysicists. But efforts to derive a suitable inflationary potential from the low energy limit of superstring theory have met with many obstacles. The dilaton field would seem to be an obvious candidate for the inflaton, but in perturbative low energy string theory the dilaton has no potential, the field is massless and couples to gravity solely through its kinetic energy, which is positive and would slow down the expansion of the Universe rather than speed it up.

String theories contain other scalar field called moduli, but the moduli are also massless in perturbative string theory, and their nonperturbative potentials are still unknown. Any nonperturbative physics that fixes stable minima for these fields controls the supersymmetry breaking scale, the sizes of compactified dimensions, the value of the cosmological constant, and the dynamics of the inflaton field, and that’s why deriving a string theory inflationary model has been such a challenge.

But inflationary models suffer from a conceptual inadequacy in that they are constructed using a combination of relativistic quantum field theory and classic general relativity. String theory is a theory of quantum gravity. And so string theory ought to be able to describe cosmology on a more fundamental level than inflationary models are capable of describing.

The discovery of extended fundamental structures in string theory called D-branes has brought forth some startling new ideas for the structure of spacetime. The first such model by Horava and Witten started with M-theory in eleven spacetime dimensions, compactified on a 6-dimensional Calabi-Yau space, leaving four space dimensions and time. The four space dimensions are bounded by two three-dimensional surfaces, or branes, separated by some distance R between the three-branes in the fourth direction. One of those three-branes, called the visible brane, can be seen as the three-dimensional world on which we live. The other three-dimensional brane is called the hidden brane, and we never see it. The volume V of the Calabi-Yau space varies from the visible brane to the hidden brane, and each brane has a different set of E_{8} gauge supermultiplets living on it, with the gauge couplings of fields living on the visible and hidden branes related by

This model is an effective five-dimensional theory, because the value of R is large compared to the size of the Calabi-Yau space.

This Horava-Witten world is not a cosmological model, but this picture has been applied to cosmology with interesting and controversial results. The latest version of braneworld cosmology is the giant brane collision model, also known as the Ekpyrotic Universe, or the Big Splat.

The Ekpyrotic Universe starts out as a cold, flat, static five-dimensional spacetime that is close to being a supersymmetric BPS state, meaning a state invariant under some special subalgebra of the supersymmetry algebra. The four space dimensions of the bulk are bounded by two three-dimensional walls or three-branes, and one of those three-branes makes up the space that we live on.

But how does the Universe evolve to give the Big Bang cosmology for which there is so much observational evidence? The Ekpyrotic theory postulates that there is a third three-brane loose between the two bounding branes of the four dimensional bulk world, and when this brane collides with the brane on which we live, the energy from the collision heats up our brane and the Big Bang occurs in our visible Universe as described elsewhere in this site.

This proposal is quite new, and it remains to be seen whether it will survive careful scrutiny.

#### The problem with acceleration

There is a problem with an accelerating Universe that is fundamentally challenging to string theory, and even to traditional particle theory. In eternal inflation models and most quintessence models, the expansion of the Universe accelerates indefinitely. This eternal acceleration leads to some contradictions in the mathematical assumptions made about spacetime in the fundamental formulations of quantum field theories and string theories.

According to the Einstein equation, for the usual case of a four-dimensional spacetime where space is homogeneous and isotropic, the acceleration of the scale factor depends on the energy density and the pressure of the “stuff” in the Universe as

The equation of state for the “stuff” in the Universe, combined with the Einstein equation, tells us that

The boundary of the region beyond which an observer can never see is called that observer’s event horizon. In cosmology, the event horizon is like the particle horizon, except that it is in the future and not in the past. In the class of spacetimes we’ve been looking at, the amount of the future that an observer at some time t_{0} would be able to see were she or he to live forever is given by

This tells us that an accelerating Universe will have a future event horizon, because

From the point of view of human philosophy or the internal consistency of Einstein’s theory of relativity, there is no problem with a cosmological event horizon. So what if we can’t ever see some parts of the Universe, even if we were to live forever?

But a cosmological event horizon is a major technical problem in high energy physics, because of the definition of relativistic quantum theory in terms of the collection of scattering amplitudes called the S Matrix. One of the fundamental assumptions of quantum relativistic theories of particles and strings is that when incoming and outgoing states are infinitely separated in time, they behave as free noninteracting states.

The presence of an event horizon implies a finite Hawking temperature and the conditions for defining the S Matrix cannot be fulfilled. This lack of an S Matrix is a formal mathematical problem not only in string theory but also in particle theories.

Well done. Excellent post, Bruceleeeowe. I’ve much to say but sharing with you in short. Although there are much evidences to support the theory big bang and inflationary universe. But still astrophysicists are facing problems with singularity physics of black hole where all physics fails. Yet there are problems that can’t be explained better them of too. Is that correct?

Correct… Nelson. I was thinking about this too.

I’m not interested in mathematics. I’ve always struggled with math. Anyway nice article

I don’t try to think what is correct. I believe in what my teachers tought me in school days but looking for something new. I liked yours math. Good math.

Hey, I read a lot of blogs on a daily basis and for the most part, people lack substance but, I just wanted to make a quick comment to say GREAT blog!…..I”ll be checking in on a regularly now….Keep up the good work! 🙂

Hey very nice blog!!….I’m an instant fan, I have bookmarked you and I’ll be checking back on a regular….See ya 🙂

This has been on my mind for some time….. and I agree with you to some degree.

frokostordning

Pingback: Where Is The Missing Dark Matter? « Bruceleeeowe's Blog

Part 3

Looking at the process in detail, it is clear that long periods of time would be detrimental, rather than helpful to this fraudulent, totally contrived and deliberately misleading so-called “theory” of the origin of life which was created, among other reactionary reasons (see below), in order to provide a false pretext for NASA to carry out extremely costly and entirely unnecessary and useless space ventures using the primitive form of space travel, rocketry, to outlying planetary satellites under the false pretext of “searching for life,” wherever there might be water (!) discovered by spectral analysis for example, in order to keep their jobs and obtain continued government funding and to serve as yet one more pretext to divert money from social spending. The most recent ridiculous “projects’ in this series are the Obama plans to revisit the Moon to build a moon base and take a trip to Mars and trying to privatize—just like Bush—what should properly be part of the existing government, even after a Socialist Revolution in the United States! Privatization of government agencies reflects both the one-way dynamic of capitalism and its true inherent anarchy. Instead there should be a concerted attempt to develop (or back-engineer) the electromagnetic anti-gravity engine used by the UFO’s. This requires nuclear power and a structural material which is super-conducting at room temperature. That material exists and is known as the nanotube form of carbon, the hardest known material ever discovered, far harder than diamonds. But it is highly likely that there is too much money invested in rocketry the most primitive form of space travel, which is also backed by the oil industry. This is further proof of the almost entirely one-way dynamic of capitalism leading to Fascism, barbarism and finally the end of civilization; and now with the advent of the Runaway Greenhouse Effect, the end of all life on Earth. We need a Socialist Revolution here in the United States. In the final analysis all wars are won and lost on morale and every movement begins with the call. This analysis is part of that call.

William H. Depperman, Coordinator

United Front Against Racism

And Capitalism-Imperialism

New York, N.Y.

Revised May 21, 2010