Why Does the Universe Expand ? T. Padmanabhan ∗ IUCAA, Post Bag 4, Ganeshkhind, Pune - 411 007, India (Dated:) The purpose of the paper is five-fold: (a) Argue that the question in the title can be presented in a meaningful manner and that it requires an answer. (b) Discuss the conventional answers and explain why they are unsatisfactory. (c) Suggest that a key ingredient in the answer could be the instability arising due to the ‘wrong’ sign in the Hilbert action for the kinetic energy term corresponding to expansion factor. (d) Describe how this idea connects up with another peculiar feature of our universe, viz. it spontaneously became more and more classical in the course of evolution. (e) Provide a speculative but plausible scenario, based on the thermodynamic perspective of gravity, in which one has the hope for relating the thermodynamic and cosmological arrows of time. PACS numbers: Observations show that our universe, at sufficiently large scales, is homogeneous and isotropic and is de- scribed by a Friedmann model. The dynamics of such a model is contained in a single function of time a ( t ), called expansion factor. 1 Other degrees of freedom in the metric tensor, needed to describe the universe at smaller scales, do not play any role at sufficiently large scales. There- fore, one can model the universe as a dynamical system governed by differential equations determining the evolu- tion of a ( t ). These, obtained from Einstein’s equations, are: and where ρ and p are the energy density and pressure of matter. (The units are chosen with c = ℏ = 1 so that L 2 P = G ). We will assume, for simplicity, that there is only one component of matter with some equation of state p = p ( ρ ). These two equations imply (when ˙ a = 0) that d ( ρa $^{3}$) = − pd ( a $^{3}$). To determine the dynamics from Eq. (2), we need to specify (˙ a, a, ρ ) at some fiducial time t = t$_{fid}$ such that they are consistent with the constraint Eq. (1). This will allow us to solve the equations and determine the dynamics of the universe for both t < t$_{fid}$ and t > t$_{fid}$ . Equation (1) and Eq. (2) are invariant under the time reversal t → − t . It turns out that, to match with obser- vations, we have to choose ˙ a > 0 at t = t$_{fid}$ > 0 (say, at the current epoch), thereby breaking the time-reversal invariance of the system. This, by itself, is not an issue ∗ Electronic address: paddy@iucaa.ernet.in 1 To be precise we also have to specify the curvature of spatial slices by k = 0 , ± 1 but we will set k = 0 for simplicity. None of the key results in the paper depend on this choice, which, in any case, is favored observationally. 2 The phrase ‘cosmological arrow of time’ means different things to different people; I will use it in the sense that a ( t ) being a monotonic function of t with ˙ a > 0, gives a direction for t from the evolution of a . In the above discussion we assumed ( ρ + 3 p ) > 0 thereby leading to a singularity. Since physically mean- ingful theories must be nonsingular, we certainly expect In a laboratory system, we can usually take another copy of the system and explore it with a time-reversed choice of initial conditions, because the time can be spec- ified by degrees of freedom external to the system. We cannot do it for the universe because we do not have extra copies of it and — equally importantly — there is noth- ing external to it to specify the time. So the problem, as stated, is specific to cosmology. To see why, let us first consider the case of ( ρ +3 p ) > 0 for all t . The choice ˙ a > 0 implies that we are assuming that the universe is expanding. Then Eq. (1), Eq. (2) show that the universe will expand at all times in the past and will have a singularity ( a = 0) at some finite time in the past (which we can take to be t = 0 without loss of generality). The mathematical structure of Eq. (1) and Eq. (2) prevents us from specifying the initial conditions at t = 0. We are therefore forced to take ˙ a > 0 at some t = ϵ > 0, thereby breaking time reversal symmetry. The universe expands at present because we choose it to expand at some epoch in the past for reasons we do not know. This expansion, in turn, leads to an arrow of time such that either a or t can be used as a time coordinate — which is precisely what a cosmologist does in using redshift z = (1 /a ) − 1 to specify the cosmic epoch. Why do we have to choose the solution with ˙ a > 0 at some epoch?. This is the essence of expansion problem [1]. An alternative way of asking the same question is: How come a cosmological arrow of time 2 emerges from equations of motion which are time-reversal invariant ? for a physical system in the lab. We know that specific solutions to dynamical equations need not respect the symmetries of the equations. But, for the universe, this is an issue. ( ˙ a a ) 2 = 8 πL 2 P 3 ρ ( t ) (1) ¨ a a = − 4 πL 2 P 3 ( ρ + 3 p ) (2) arXiv:1001.3380v1 [gr-qc] 19 Jan 2010