2 which the equations of motion that are time-reversal in- variant lead to an evolution which singles out an arrow of time. At first sight one might think this is impossible but one can do it with unbounded Hamiltonians. Let me first describe the idea with a simple example. Con- sider an ‘inverted’ harmonic oscillator q ( t ) obeying the equations of motion a future theory of gravity — possibly a model for quan- tum gravity — to eliminate the singularity (effectively leading to (( ρ + 3 p ) < 0). Could it be that such a theory might solve the problem of arrow of time? This is un- likely. To see this, let us ask what kind of dynamics we expect in such a ‘final’ theory. The dynamics will cer- tainly get modified at Planck epoch but away from it we expect some effective equations (possibly with quantum corrections) to govern the evolution of an (effective) ex- pansion factor. The solutions could, for example, have a contracting phase (and bounce) or could start from a Planck size universe at t = −∞ , just to give two non- singular possibilities. While we do not know these equa- tions or solutions, we can be confident that they will still be time-reversal invariant because quantum theory, as we know it, is time-reversal invariant. So without a choice for initial conditions (now possibly at t = −∞ ), we still cannot explain how the cosmological arrow of time emerges. Since quantum gravity is unlikely produce an arrow of time, it is worthwhile to attempt to understand this problem in the (semi)classical context. For generic initial conditions, there is no relationship be- tween q (0) and ˙ q (0). Hence at late times (i.e, t ≫ ω − $^{1}$), we get one part of the solution being selected out This equation is invariant under t → − t so one would have thought that no arrow of time will emerge from the dynamics unless we impose it in the initial conditions. The general solution to this equation is There is another peculiarity related to the quatum regime of the universe which does not seem to have at- tracted the attention it deserves. The conventional wis- dom is that the universe was a quantum system with order unity quantum fluctuations near Planck time. But during the evolution it seems to have become more and more classical! This is an extraordinary feature and again needs explanation. In general, systems with bounded Hamiltonians do not become more and more classical 3 spontaneously under dynamical evolution. One suspects that this peculiarity is related to the origin of cosmic arrow of time. A more complicated “solution” to arrow of time issue, which is sometimes envisaged, is as follows: Consider a very inhomogeneous initial condition in some t = t$_{i}$ hy- persurface and assume that certain regions behave like ‘local’ Friedmann models with ˙ a > 0 and other regions have ˙ a < 0 so that no global arrow of time can be defined from the expansion. Further assume that the dynamics of these patches are independent of each other and we just happen to be in a patch with ˙ a > 0 thereby ‘solv- ing’ the problem. There are several difficulties with this scenario. It is impossible to prove that, for a generic ini- tial condition, the patches will evolve independently in a nonlinear theory of gravity and — in fact — it is very unlikely to be true. Even defining a sensible ‘local expan- sion factor’ for a ‘local patch’ without assuming special symmetries is impossible. Such scenarios are often in- voked in the context of inflationary models but not with mathematically rigorous proofs. It will be nicer and simpler if we can find a way by leading to an “expansion” and an arrow of time! Note that the solution in Eq. (4) is time-reversal invariant in the sense that q ( t ) = q ( − t ) if we let ˙ q (0) → − ˙ q (0) when we do t → − t . But once we have chosen any generic solution with some uncorrelated q (0) and ˙ q (0), late time dynamics picks out an arrow of time. (Of course, there are special initial conditions like q (0) = − ˙ q (0) ω − 1 or q (0) = 0 = ˙ q (0) for which this will not happen but these are special choices and not generic.) One can easily see that this behaviour arises for a wide class of Hamiltonians that are unbounded. It is not nec- essary that the potential energy has to be unbounded. If the kinetic energy term has the ‘wrong’ sign so that the Lagrangian has a form like L = − (1 / 2) ˙ q 2 − V ( q ) with a V which is positive and unbounded from above, say, we will again have an instability and late time evolution of q will give an arrow of time. Incredibly enough, it is known for decades [2] that the expansion factor a ( t ) does have such a wrong sign in the kinetic energy term in the Hilbert action and hence rep- resents an unstable mode. Our analysis suggests that it is this cosmic instability which we call expansion, and for timescales larger than Planck time, it picks out an arrow of time. Clearly, same feature will occur even in any effective theory of (semi)classical gravity once a ( t ) acquires an unstable dynamics. This is guaranteed to happen because any sensible theory will approach Fried- mann model at times larger than Planck time and — in the Friedmann limit — a ( t ) is an unstable model. There are two other nice features in this attempt to connect the cosmic arrow of time with the wrong-sign ki- netic energy term for a ( t ) and the consequent instability. First, the idea tells us that it is very special to cosmol- ogy. Unbounded Hamiltonians with negative kinetic en- ergy term are taboo in laboratory scale physics for good reasons; one does not want run-away solutions in the lab 3 It is possible to quantify the degree of classicality of the system using, e.g., the Wigner function and specifying how sharp is the phase space trajectory. ¨ q = ω $^{2}$q (3) q ( t ) = q (0) cosh ωt + ˙ q (0) ω − $^{1}$sinh ωt (4) q ( t ) ≈ 1 2 ( q (0) + ˙ q (0) ω − $^{1}$) e ωt ∝ e ωt (5)