1. Gauge Theories with time-dependent couplings and AdS cosmologies Sumit R. Das A.Awad, S.R.D, K. Narayan and S. Trivedi, PRD 77 (2008) 046008. A.Awad, S.R.D, S.Nampuri, K. Narayan and S. Trivedi, hep-th/0807.1517 S.R.D., A. Ghosh, Jae-Hyuk Oh and S. Trivedi, to appear

2. Gauge Theories with time-dependent couplings and AdS cosmologies Sumit R. Das S.R.D, J. Michelson, K. Narayan and S. Trivedi, PRD 74 (2006) 026002 S.R.D, J. Michelson, K. Narayan and S. Trivedi, PRD 74 (2007) 026002 A.Awad, S.R.D, K. Narayan and S. Trivedi, PRD 77 (2008) 046008. A.Awad, S.R.D, S.Nampuri, K. Narayan and S. Trivedi, hep-th/0807.1517 S.R.D., A. Ghosh, Jae-Hyuk Oh and S. Trivedi, to appear

3. Usual AdS/CFT IIB string theory in asymptotically space-times is dual to large-N expansion of =4 SYM theory on the boundary with appropriate sources or excitations. The usual relationship between the dimensionless parameters on the two sides are Where is the string coupling, is the square of the Yang-Mills coupling, is the string length and is the AdS length scale

4. Usual AdS/CFT IIB string theory in asymptotically space-times is dual to large-N expansion of =4 SYM theory on the boundary with appropriate sources or excitations. The usual relationship between the dimensionless parameters on the two sides are Usual notions of space-time are valid only in the regime where supergravity approximation is valid, i.e.

5. Usual AdS/CFT IIB string theory in asymptotically space-times is dual to large-N expansion of =4 SYM theory on the boundary with appropriate sources or excitations. The usual relationship between the dimensionless parameters on the two sides are Usual notions of space-time are valid only in the regime where supergravity approximation is valid, i.e. For generic values of the parameters, the gauge theory hopefully continues to make sense, though there is no interpretation in terms of General Relativity.

6. Usual AdS/CFT IIB string theory in asymptotically space-times is dual to large-N expansion of =4 SYM theory on the boundary with appropriate sources or excitations. The usual relationship between the dimensionless parameters on the two sides are Usual notions of space-time are valid only in the regime where supergravity approximation is valid, i.e. For generic values of the parameters, the gauge theory hopefully continues to make sense, though there is no interpretation in terms of General Relativity. Could we make use of this fact to understand singularities ?

7. In this talk : I will describe one particular approach. Some parts of our work overlap with Chu and Ho (2006,2007) Other approaches of interest – 1. Hertog and Horowitz (2005); Craps, Hertog and Turok (2007,2008) 2. Bak, Gutperle and Hirano (2003,2007); Bak, Gutperle and Karch (2007)

8. A Scenario At early times, start with the ground state of the gauge theory with large ‘t Hooft coupling. The physics is now well described by supergravity in usual t x z boundary Intial Time Slice

9. A Scenario Now turn on a time dependent source in the Yang-Mills theory which deforms the lagrangian. t x z boundary Intial Time Slice J(t)

10. A Scenario Now turn on a time dependent source in the Yang-Mills theory which deforms the lagrangian. This source will be specified for all times. t x z boundary Intial Time Slice J(t)

11. A Scenario Now turn on a time-dependent source in the Yang-Mills theory which deforms the lagrangian. This corresponds to turning on a non-normalizable mode of the supergravity in the bulk, thus deforming the original t x z boundary Intial Time Slice Sugra mode

12. A Scenario The gauge theory evolves according to the deformed (time dep.) hamiltonian t x boundary

13. A Scenario The gauge theory evolves according to the deformed (time dep.) hamiltonian At sufficiently early times the supergravity background evolves according to the classical equations of motion t x z boundary

14. A Scenario At later times, the curvatures or other invariants of supergravity start becoming large If we nevertheless insist on the supergravity solution we encounter a singularity at some finite time Beyond this time, it is meaningless to evolve any further. t x z boundary Spacelike singularity

15. A Scenario However, the gauge theory could be still well defined at this time. And if we are lucky enough the gauge theory may be evolved beyond this point If not, we want to understand what is precisely the obstruction in the gauge theory. t x z boundary Spacelike singularity

16. Models implementing this Scenario We will try to implement this scenario by turning on sources in the gauge theory which correspond to time dependent couplings There would be no other source. We will choose the gauge theory coupling to be bounded everywhere and becoming vanishingly small at some time. t

17. Models implementing this Scenario We will try to implement this scenario by turning on sources in the gauge theory which correspond to time dependent couplings There would be no other source. This raises the hope that we could be able to analyze the gauge theory in some controlled fashion. t

18. Poincare Patch solutions The simplest supergravity solutions are deformations of the Poincare patch of AdS. The metric is constrained to be FLAT on the boundary and a dilaton whose boundary value matches the gauge theory coupling. At early times this should be

19. Null Solutions The best controlled solutions of this type are those with null rather than spacelike singularities Where is the dilaton which is a function of alone. This function may be chosen freely..in particular we can choose this function of the desired form. For example,

20. Null Solutions The best controlled solutions of this type are those with null rather than spacelike singularities Where is the dilaton which is a function of alone. This function may be chosen freely..in particular we can choose this function of the desired form. These solutions maintain ½ of the supersymmetries.

21. t y z boundary Null singularity There is a singularity at with a suitable choice of the function, e.g. with Null geodesics can reach this point in a finite affine parameter.

22. t y z boundary Null singularity There is a singularity at with a suitable choice of the function, e.g. with Null geodesics can reach this point in a finite affine parameter. Tidal forces between such geodesics diverge.

23. t y z boundary Null singularity There is a singularity at with a suitable choice of the function, e.g. with Null geodesics can reach this point in a finite affine parameter. Tidal forces between such geodesics diverge. However constant w surfaces, in particular the boundary at w=0 are FLAT

24. t y z boundary Null singularity There is a singularity at with a suitable choice of the function, e.g. with Null geodesics can reach this point in a finite affine parameter. Tidal forces between such geodesics diverge. However constant w surfaces, in particular the boundary at w=0 are FLAT The only source in the gauge theory is a dependent coupling.

25. These solutions are in fact related to where by coordinate transformations This is an example of the general fact that a Weyl transformation on the boundary is equivalent to a special class of coordinate transformations in the bulk - the Penrose-Brown-Hanneaux (PBH) transformations.

26. A more general class In fact there is a more general class of solutions of the following form The 4d metric and the dilaton are functions of the four coordinates and the 5-form field strength is standard. This is a solution if and Thus a solution of 3+1 dimensional dilaton gravity may be lifted to be a solution of 10d IIB supergravity with fluxes.

27. We will consider solutions of this type where the 4d metric is conformal to flat space The connection between Weyl transformations on the boundary and PBH transformations then ensures that there is a different foliation of the AdS space-time in which the boundary is flat – and all we have is a nontrivial dilaton. We will define the dual gauge theory to live on this flat boundary.

28. Kasner-like Solutions The easiest form of time dependent solution is the lift of a usual 4d Kasner universe This has a spacelike curvature singularity at t=0. The effective string coupling vanishes here – as required. However the coupling diverges at infinite past and future.

29. Nevertheless it is instructive to see what the dual gauge theory looks like. This can be explicitly worked out for In this case the 4d metric is conformal to flat space

30. Nevertheless it is instructive to see what the dual gauge theory looks like. This can be explicitly worked out for In this case the 4d metric is conformal to flat space The exact PBH transformation may be written down and the metric which has a flat boundary is

31. Time dependence with bounded couplings We need to obtain solutions which approach the Kasner solution near a space-like singularity, but also asymptote to standard anti-de-Sitter with constant dilaton at early times – so that the coupling of the dual gauge theory is always bounded. One such solution

32. Time dependence with bounded couplings We need to obtain solutions which approach the Kasner solution near a space-like singularity, but also asymptote to standard anti-de-Sitter with constant dilaton at early times – so that the coupling of the dual gauge theory is always bounded. One such solution Flat space-time in Milne coordinates

33. Time dependence with bounded couplings We need to obtain solutions which approach the Kasner solution near a space-like singularity, but also asymptote to standard anti-de-Sitter with constant dilaton at early times – so that the coupling of the dual gauge theory is always bounded. One such solution Singularity is at

34. We should be able to find PBH transformations to a foliation of this spacetime which leads to a flat metric.

35. In this case, we have not been able to determine the exact form of the transformations. However all one needs is the transformation near the boundary. We will, therefore, determine this in an expansion around the boundary.

36. We should be able to find PBH transformations to a foliation of this spacetime which leads to a flat metric. In this case, we have not been able to determine the exact form of the transformations. However all one needs is the transformation near the boundary. We will, therefore, determine this in an expansion around the boundary. The metric near the boundary becomes The boundary metric is now explicitly flat.

37. Behavior near singularity The solution we have approaches the Kasner solution near the singularity at

38. We are in fact considering dilaton driven cosmology in 3+1 dimensions and can use the standard BKL analysis For a large class of initial conditions the dilaton always approaches the Kasner value.

39. Properties of the gauge theory Even though the theory lives on flat space, the dilaton factor is in front of the kinetic term and diverges at the time of bulk singularity. Normally one would absorb the coupling factor by a field redefinition so that only nonlinear terms involve the coupling, The field redefinition will introduce extra stuff involving the derivative of the dilaton

40. This could lead to singular pieces in terms which are quadratic in the fields. This is because in such terms appear without any accompanying factor of The nonlinear terms can involve only factors of While can be arranged to be finite at the singularity, since vanishes (or becomes small) here, is necessarily large or even infinite.

41. The quadratic part of the gauge field lagrangian becomes

42. For a null dilaton, we can fix a light cone gauge. Then all these extra terms vanish, and the kinetic term is standard. Furthermore the constraints become identical to the standard case.

43. The quadratic part of the gauge field lagrangian becomes For a null dilaton, we can fix a light cone gauge. Then all these extra terms vanish, and the kinetic term is standard. Furthermore the constraints become identical to the standard case. The nonlinear terms involve factors of and, but these are well behaved and small near.

44. The quadratic part of the gauge field lagrangian becomes For a null dilaton, we can fix a light cone gauge. Then all these extra terms vanish, and the kinetic term is standard. Furthermore the constraints become identical to the standard case. The nonlinear terms involve factors of and, but these are well behaved and small near. We expect that perturbation theory works near the singularity – as far as we can see there is no problem in loop diagrams for correlations near

45. The quadratic part of the gauge field lagrangian becomes For a null dilaton, we can fix a light cone gauge. Then all these extra terms vanish, and the kinetic term is standard. Furthermore the constraints become identical to the standard case. The nonlinear terms involve factors of and, but these are well behaved and small near. We expect that perturbation theory works near the singularity – as far as we can see there is no problem in loop diagrams for correlations near More significantly there is no particle production, because of a null isometry. It appears that the gauge theory provides a smooth evolution in the light front time.

46. For time dependent dilaton we similarly fix a gauge. However now these terms lead to a tachyonic time dependent mass term for the transverse components. Since for the dilaton is of the form with, we will study the problem for arbitrary In this case, the tachyonic mass term increases without bound as we approach the singularity : TIME DEPENDENT BACKGROUNDS

47. t = -100 t = -1

48. At the classical level this means that the redefined field is driven to infinitely large values in a finite time for generic initial conditions.

49. At the free level, the mode decomposition for a (redefined) transverse gauge field component At early times we have usual oscillatory behavior While near the field blows up Unless we have very fine tuned initial conditions

50. Thus, even though the coupling constant becomes small near the singularity at, the nonlinear terms are not small there We need to perform an analysis in the full interacting theory. For this, we need to go back to the original variables At the quadratic level, this has a finite limit as As the divergence from the Hankel function is cancelled by the prefactor.

51. Schrodinger Picture We need to ask what happens to an initial state as we evolve towards This is best done in the Schrodinger picture. The following toy quantum mechanics model captures the essential physics We may think of as a momentum mode of a transverse component of the gauge field, and the potential includes interactions. The dilaton is of the form

52. For reasons which will become clear soon, we will regulate the dilaton profile near, e.g. Or

53. For reasons which will become clear soon, we will regulate the dilaton profile near, e.g. Or The Schrodinger equation is As the potential term dominates and the leading time dependence can be easily obtained

54. The solution is of the form

55. For the time dependent coefficient of the phase factor behaves as

56. The solution is of the form For the time dependent coefficient of the phase factor behaves as Thus when strictly, and the phase of the wavefunction oscillates wildly, and a meaningful continuation beyond t = 0 does not exist When there is a continuation.

57. The solution is of the form For the time dependent coefficient of the phase factor behaves as Thus when strictly, and the phase of the wavefunction oscillates wildly, and a meaningful continuation beyond t = 0 does not exist When there is a continuation. For any nonzero the wave function can be of course continued regardless of the value of

58. The solution is of the form For the time dependent coefficient of the phase factor behaves as Thus when strictly, and the phase of the wavefunction oscillates wildly, and a meaningful continuation beyond t = 0 does not exist When there is a continuation. For any nonzero the wave function can be of course continued regardless of the value of

59. The wild oscillations of the phase of the wavefunctional results in diverging expectation values for physical quantities. For example the expectation value of the momentum conjugate to behaves as

60. If we restrict to the free theory, the wave function can be of course obtained completely, and the above conclusion can be verified. From another point of view, we can calculate the amount of particle production in the free theory due to a regulated time dependent dilaton. For this diverges when and is finite when. However our conclusion about the behavior of the wavefunction near is completely general – this is valid for the full interacting theory.

61. More significantly, when an infinite amount of energy is pumped into the system as we approach, regardless of the value of.

62. This simply follows from the form of the hamiltonian Since the potential term dominates near we have In the phase factor of the wave function cancels, and the time dependence is basically governed by the factor. For a generic state is nonzero and diverges. Taking into account the fact that there are degrees of freedom we have typically For this is slower than the supergravity answer which may be obtained from the holographic stress tensor,

63. There are some special states in which vanishes, and the contribution to comes from the kinetic term. It turns out that for these subdominant terms are finite - otherwise they diverge. Furthermore for large the fluctuations are suppressed.

64. For generic states, our conclusions are independent of the details of the state. This is important – we have analyzed the gauge theory in the region. For early and late times, differs significantly from the power law behavior we have considered.

65. The coupling profile is Starting from a vacuum state at, the state would evolve into some non-trivial state by the time we enter the small t region. From this point onward we can use the above analysis. This is where we looked

66. In the regulated theory, a finite but large amount of energy density is produced as However now the system may be continued meaningfully beyond t = 0. For the coupling begins to increase, and energy is now extracted from the system When we reach late times, a large amount of energy would have been extracted, leaving us with a finite energy density – for a generic initial state. There are of course very finely tuned special states for which no energy is left – this is a time symmetric state. However a general state will not be time-symmetric even though the background is.

67. Meaning of regulator We have performed the analysis for a general, and for a dilaton profile which departs from the power law behavior near the “singularity” at

68. Meaning of regulator We have performed the analysis for a general, and for a dilaton profile which departs from the power law behavior near the “singularity” at For the specific supergravity solutions we discussed,

69. Meaning of regulator We have performed the analysis for a general, and for a dilaton profile which departs from the power law behavior near the “singularity” at For the specific supergravity solutions we discussed, In the gauge theory we can turn on any source we like. The modification of the dilaton profile would modify the supergravity solution.

70. Meaning of regulator We have performed the analysis for a general, and for a dilaton profile which departs from the power law behavior near the “singularity” at For the specific supergravity solutions we discussed, In the gauge theory we can turn on any source we like. The modification of the dilaton profile would modify the supergravity solution. However the modification is only in the region near.

71. Meaning of regulator We have performed the analysis for a general, and for a dilaton profile which departs from the power law behavior near the “singularity” at For the specific supergravity solutions we discussed, In the gauge theory we can turn on any source we like. The modification of the dilaton profile would modify the supergravity solution. However the modification is only in the region near. Here supergravity is not reliable anyway – the dual is some stringy background : we do not know how to describe that. Since this modification is only near the singularity, we can keep it unchanged at early times – hence maintain our overall scenario.

72. The Likely Outcome It is not yet possible to estimate the net amount of energy which is produced during the entire time evolution, since the theory becomes strongly coupled again at late times. However, regardless of the amount, this energy is expected to thermalize at late times. The coupling variation is very slow at late times, and the system will always have sufficient time to thermalize. At such late times, the coupling is again strong and supergravity intution would be applicable. Since the gauge theory is defined on, the supergravity dual of a thermal state is always a black brane – i.e. a horizon is formed with a different singularity inside it.

73. In the background we have considered, supergravity would lead us to believe that all observers would crash into a space-like singularity at. For a regulated dilaton, this not a singularity in the technical sense, but bulk curvatures become Planck scale anyway. The gauge theory allows us to address the behavior of the wavefunction near. This appears to predict that one class of observers can avoid a singularity and remain outside the horizon of the final black brane.

74. Can we avoid Black Hole Formation ? The inevitability of black brane formation is tied to the fact that our boundary was always. The dual of a thermal state on flat space-time is always a black brane If we could implement our scenario for a gauge theory which is instead defined on, this could be avoided if the amount of energy produced is small enough. Because of Hawking-Page transition, for small enough energies the dual would be thermal AdS rather than a black hole. This would involve obtaining supergravity backgrounds of our type in global AdS.

75. De Sitter foliations (S.R.D.,A. Ghosh, Jae-Hyuk Oh, S. Trivedi –in progress) We have not been able to find such solutions in global coordinates. However we have recently found solutions whose boundaries are de Sitter space-times at early/late times – so we have a gauge theory living on 3+1 dimensional de Sitter space with a time dependent coupling. Since the field theory is Weyl invariant, and de Sitter space-time is conformal to, properties of this theory can be obtained from one which is defined on, after being careful about anomalous transformation properties of the energy momentum tensor.

76. What we used so far…… Consider a 5 dimensional metric of the form The 4d metric and the dilaton are functions of the four coordinates and the 5-form field strength is standard. This is a solution if and Thus a solution of 3+1 dimensional dilaton gravity may be lifted to a solution of 10d IIB supergravity with fluxes.

77. Now note the following.. Consider a 5 dimensional metric of the form The 4d metric and the dilaton are functions of the four coordinates and the 5-form field strength is standard. This is a solution if Thus a solution of 3+1 dimensional dilaton gravity with a positive cosmological constant may be lifted to a solution of 10d IIB supergravity with fluxes.

78. When vanishes, this is solved by What we have is with foliations which are 3+1 dimensional de Sitter space,.

79. When vanishes, this is solved by What we have is with foliations which are 3+1 dimensional de Sitter space,. For a nonzero dilaton, consider the ansatz

80. When vanishes, this is solved by What we have is with foliations which are 3+1 dimensional de Sitter space,. For a nonzero dilaton, consider the ansatz Einstein equations then yield

81. On the other hand, the dilaton equation of motion yields where is a constant of integration.

82. On the other hand, the dilaton equation of motion yields where is a constant of integration. It turns out that the two other equations lead to a single equation, which combined with the form of the dilaton above, becomes

83. On the other hand, the dilaton equation of motion yields where is a constant of integration. It turns out that the two other equations lead to a single equation, which combined with the form of the dilaton above, becomes This is the equation of motion of a particle moving in a potential with an energy May be solved in terms of Elliptic functions

84. The various solutions correspond to oscillating cosmologies, or those with a single big bang or big crunch, depending on the value of the constant. AdS Solution Bounce Big Crunch Big Bang and Big Crunch V(H) H

85. In our scenario solutions with represent cosmologies which start off as at some time, chosen to be and have a spacelike singularity at a finite time The behavior of the solution near is independent of the constant and agrees with the behavior of the solution. Near the singularity, the conformal factor goes to zero linearly, with the coefficient proportional to The dilaton, however, becomes universal – and identical to the symmetric Kasner solution

86. H(t) t Solutions with a Big Crunch

87. We have started an analysis of the gauge theory defined on such boundaries. Hopefully we will be able to determine whether the outcome is anything other than a black hole !!

88. We have started an analysis of the gauge theory defined on such boundaries. Hopefully we will be able to determine whether the outcome is anything other than a black hole !! Thank you

89. Free theory wavefunctionals Consider the state which is the vacuum of the harmonic oscillator at early times The wave functional may be easily calculated : Where

90. At early times, Thus the wavefunctional behaves as This is the standard harmonic oscillator ground state wave function.

91. It is straightforward to check that as Thus the wavefunctional behaves as

92. It is straightforward to check that as Thus the wavefunctional behaves as Therefore for, the phase of the wave functional has a smooth limit, whereas for the phase oscillates wildly as we approach the limit. This prevents a meaningful continuation of the wave functional beyond, reinforcing the conclusion we reached in the Heisenberg picture.