Matrix Cosmology An Introduction

1. 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

2. Contents: • A toy model • Matrix description • A class of generalizations • More generalizations • Quantum computations

3. Motivations: • String theory faces the following challenges posed • by cosmology: • 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. • ……

4. None of the above problems is easy. 1. A toy model One and half years ago, in paper hep-th/0506180, Craps, Sethi and Verlinde consider the “simple” background:

5. 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

6. CSV shows that perturbative 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.

7. In the 11 dimensional perspective, the metric is locates in a finite distance away in terms of the affine parameter if we follow a null geodesic. If , then These quantities blow up at .

8. More comments on the singularity later. ★String vertex operators With a constant dilaton, a vertex operator assumes the form with the on-shell condition:

9. 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

10. ★Scattering amplitudes Blows up whenever 2g-2+n>0. Thus, the string perturbative S-matrix is ill-defined.

11. 2. Matrix description CSV propose to use a matrix model to describe the Physics in this background, since ● The background preserves half of all SUSY if ● There is a decoupling argument a la Seiberg and Sen.

12. ★ The proposal In the IIA matrix string model, for a sector with a Fixed longitudinal momentum Where , the matrix action

13. The Yang-Mills coupling constant is related to the string couping constant through Now with We simply have Thus,

14. So near the “big bang” singularity, the SYM is a free nonabelian theory. On the other hand, near the theory tends to a CFT with an orbifold target space. Strings are in the twisted sectors.

15. More details: ● For , in the gauge ● For Residual gauge symmetry is permutations of the eigen-values of the matrices

16. 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, the metric assumes the form where there are 9 transverse coordinates, grouped into 9-d and d .

17. 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 .

18. 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:

19. 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.

20. 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.

21. 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.

22. 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 matrices are fully nonabelian. As , the coefficients of interaction terms blow up, so all bosonic matrices are forced to be Commuting.

23. 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.

24. 4. More generalizations Bin Chen in hep-th/0508191 considers the following More general background where This class of backgrounds all preserve half of SUSY

25. Bin Chen’s background has to satisfy only one diff. equation. However, it is not clear whether one can write down a matrix model. Das and Michelson in hep-th/0508068 study a background appears to be a special case of Bin Chen:

26. Das and Michelson claim that one can write down a matrix model for this background. It is interesting that these authors noted that, a String which appears to be weakly coupled at later times is actually a fuzzy cylinder at early times. Das, Michelson, Narayan and Trivedi in hep-th/0602107 constructed a model in IIB string Theory which is a deformation of , this work overlaps with the work of Chu and Ho.

27. Ishino and Ohta in hep-th/0603215 study the matrix string description of the following background: Again, all functions are functions of only u. They are subject to a single equation

28. Finally, Chu and Ho in hep-th/0602054 consider the following class of time-dependent deformation of the AdS solution: Where Also subject to a single equation.

29. 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.

30. 5. Quantum computations To check whether these matrix descriptions are really correct, we need to compute at least the interaction between two D0-branes. This is done 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 carries a net stress tensor .

31. 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

32. 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.

33. 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.

34. 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 used Matrix model to compute interaction between two D0-branes, we find a null static potential, however there is a complex term, signaling an instability.

35. 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 of calculation: initial conditions can be subtle. 2. D0-branes and associated potential are not good observables.

36. Conclusion: Time-dependent backgrounds are beasts hard to tame in string theory.