Matrix Cosmology

An Introduction

Miao Li

University of Science and Technology Institute of Theoretical Physics Chinese Academy of Science

1st Asian Winter School, Phoenix Park

Contents: 1. A toy model

2. Matrix description

3. A class of generalizations

4. More generalizations

5. Quantum computations

Motivations:

String theory faces the following challenges posed by cosmology:

1. Formulate string theory in a time-dependent background in general.

2. Explain the origin of the universe, in particular, the nature of the big bang singularity.

3. Understand the nature of dark energy.

……

None of the above problems is easy.

1. A toy model

Recently, in paper

hep-th/0506180,

Craps, Sethi and Verlinde consider the “simple” background:

This background is not as simple as it appears, sincethe Einstein metric

has a null singularity at . The spacetime Looks like a cone:

lightcone time

CSV shows that perturbative string description breaksdown near the null singularity. In fact, the scatteringamplitudes diverge at any finite order.

I suspect that string S-matrix does not exist.

Nevertheless, CSV shows that a variation of matrixTheory can be a good effective description.

In the 11 dimensional perspective, the metric is

locates in a finite distance away in terms ofthe affine parameter if we follow a null geodesic.

If , then

These quantities blow up at .

More comments on the singularity later.

★ String vertex operator

With a constant dilaton, a vertex operator assumesThe form

with the on-shell condition:

With , we need to attach a factor

to the vertex operator

The on-shell condition for k is the same as before. The vertex operator blows up at

★ Scattering amplitudes

Blows up whenever 2g-2+n>0.

Thus, the string perturbative S-matrix is ill-defined.

2. Matrix description

CSV propose to use a matrix model to describe thePhysics in this background, since

● The background preserves half of all SUSY

if

● There is a decoupling argument a la Seiberg and Sen.

★ The proposal

In the IIA matrix string model, for a sector with a Fixed longitudinal momentum

Where , the matrix action

The Yang-Mills coupling constant is related to thestring couping constant through

Now with

We simply have

Thus,

So near the “big bang” singularity, the SYM is a freenonabelian theory. On the other hand, near

the theory tends to a CFT with an orbifold target space.

Strings are in the twisted sectors.

More details:

● For , in the gauge

● For

Residual gauge symmetry is permutations of theeigen-values of the matrices

3. A class of generalizations

In hep-th/0506260, I showed that the CSV model is a special case of a large class of models.

In terms of the 11 dimensional M theory picture, themetric assumes the form

where there are 9 transverse coordinates, groupedinto 9-d and d .

This metric in general breaks half of supersymmetry.Next we specify to the special case when both f and g are linear function of :

If d=9 and one takes the minus sign in the above, weget a flat background.

The null singularity still locates at .

Again, perturbative string description breaks downnear the singularity. To see this, compacitfy one spatial direction, say , to obtain a string theory.Start with the light-cone world-sheet action

We use the light-cone gauge in which , wesee that there are two effective string tensions:

As long as d is not 1, there is in general no plane wave vertex operator, unless we restrict to the specialsituation when the vertex operator is independent of . For instance, consider a massless scalar satisfying

The momentum component contains a imaginaryPart thus the vertex operator contains a factor

diverging near the singularity.

Since each vertex operator is weighted by the stringcoupling constant, one may say that the effectivestring coupling constant diverges. In fact, the effective Newton constant also diverges:

We conjecture that in this class of string background,there is no S-matrix at all.

However, one may use D0-branes to describe the theory, since the Seiberg decoupling argument applies.

We shall not present that argument here, instead,We simply display the matrix action. It containsthe bosonic part and fermionic part

This action is quite rich. Let’s discuss the generalconclusions one can draw without doing anycalculation.

Case 1.

The kinetic term of is always simple, but the kinetic term of vanishes at the singularity, thisimplies that these coordinates fluctuate wildly. Also,coefficient of all other terms vanish, so all matricesare fully nonabelian.

As , the coefficients of interaction terms blowup, so all bosonic matrices are forced to be Commuting.

Case 2.

At the big bang, are independent of time, andare nonabelian moduli if d>4. There is no constraint on other commutators of bosonic matrices.

As , if d>4, all matrices have to be commuting. For d<4, are nonabelian.

4. More generalizations

Bin Chen in hep-th/0508191 considers the followingMore general background

where

This class of backgrounds all preserve half of SUSY

Bin Chen’s background has to satisfy only one diff.equation. However, it is not clear whether one canwrite down a matrix model.

Das and Michelson in hep-th/0508068 study a background appears to be a special case of Bin Chen:

Das and Michelson claim that one can write downa matrix model for this background.

It is interesting that these authors noted that, a String which appears to be weakly coupled at latertimes is actually a fuzzy cylinder at early times.

Das, Michelson, Narayan and Trivedi in hep-th/0602107 constructed a model in IIB stringTheory which is a deformation of , thiswork overlaps with the work of Chu and Ho.

Ishino and Ohta in hep-th/0603215 study the matrix string description of the following background:

Again, all functions are functions of only u. Theyare subject to a single equation

Finally, Chu and Ho in hep-th/0602054 considerthe following class of time-dependent deformationof the AdS solution:

Where

Also subject to a single equation.

Chu and Ho propose that string theory in this background is dual to a generalized super Yang-Mills theory in 3+1 dimension with both time-dependent metric and time-dependent coupling.

5. Quantum computations

To check whether these matrix descriptions are reallycorrect, we need to compute at least the interactionbetween two D0-branes. This is done inhep-th/0507185

by myself and my student Wei Song.

There, we use the shock wave to represent the background generated by a D0-brane which carriesa net stress tensor .

In fact, the most general ansatz is

for multiple D0-branes localized in the transverse space , but smeared in the transverse space . The background metric of the shock wave is

with

The probe action of a D0-brane in such a backgroundis

with

We see that in the big bang, the second term in thesquare root blows up, thus the perturbative expansionin terms of small v and large r breaks down.

The breaking-down of this expansion implies the breaking-down the loop perturbation in the matrix calculation. This is not surprising, since for instance,some nonabelian degrees of freedom become lightat the big bang as the term

in the CSV model shows.

Therefore, it is of no surprising that some Computations done so far have not correctly reproduce the previous result.

In hep-th/0512335, Wei Song and myself usedMatrix model to compute interaction between twoD0-branes, we find a null static potential, howeverthere is a complex term, signaling an instability.

Craps, Rajaraman and Sethi in hep-th/0601062 also computed the interaction at the one loop level,and found a different result.

They found a static potential decays at later times.

Why these results are different? Possible answers:

1. Results depend sensitively on the method ofcalculation: initial conditions can be subtle.

2. D0-branes and associated potential are not goodobservables.

Conclusion: Time-dependent backgrounds are beasts hard totame in string theory.