Matrix Cosmology

Miao Li

Institute of Theoretical Physics

Chinese Academy of Science

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.

Recently, in paper

hep-th/0506180,

Craps, Sethi and Verlinde consider the “simple” background in which the string frame metric is flat, While the dilaton has a linear profile in a light-likedirection:

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 pertubative 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 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 matriceare 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.

To check whether these matrix descriptions are reallycorrect, we need to compute at least the interactionbetween two D0-branes. This calculation is carried out only on the supergravity side in

hep-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 localizedin 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.

Conclusions: We are only seeing the emergence of an exciting direction in constructing matrix theory for a realisticcosmology.