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/ , 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-like direction:

This background is not as simple as it appears, since the Einstein metric has a null singularity at. The spacetime Looks like a cone: lightcone time

CSV shows that pertubative string description breaks down near the null singularity. In fact, the scattering amplitudes diverge at any finite order. I suspect that string S-matrix does not exist. Nevertheless, CSV shows that a variation of matrix Theory can be a good effective description.

In hep-th/ 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, the metric assumes the form where there are 9 transverse coordinates, grouped into 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, we get a flat background. The null singularity still locates at.

Again, perturbative string description breaks down near 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, we see 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 special situation when the vertex operator is independent of. For instance, consider a massless scalar satisfying The momentum component contains a imaginary Part thus the vertex operator contains a factor diverging near the singularity.

Since each vertex operator is weighted by the string coupling constant, one may say that the effective string 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 contains the bosonic part and fermionic part This action is quite rich. Let’s discuss the general conclusions one can draw without doing any calculation.

Case 1. The kinetic term of is always simple, but the kinetic term of vanishes at the singularity, this implies that these coordinates fluctuate wildly. Also, coefficient of all other terms vanish, so all matrice are fully nonabelian. As, the coefficients of interaction terms blow up, so all bosonic matrices are forced to be Commuting.

Case 2. At the big bang, are independent of time, and are 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 really correct, we need to compute at least the interaction between two D0-branes. This calculation is carried out only on the supergravity side in hep-th/ by myself and my student Wei Song. There, we use the shock wave to represent the background generated by a D0-brane which carries a 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 background is with We see that in the big bang, the second term in the square root blows up, thus the perturbative expansion in 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 light at 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 realistic cosmology.