State-to-State Cosmology: A New View on the Cosmological Arrow of Time and the Past Hypothesis

Abstract

Cosmological boundary conditions for particles and fields are often discussed as a Cauchy problem, in which configurations and conjugate momenta are specified on an “initial” time slice. But this is not the only way to specify boundary conditions, and indeed in action-principle formulations we often specify configurations at two times and consider trajectories joining them. Here, we consider a classical system of particles interacting with short range two body interactions, with boundary conditions on the particles’ positions for an initial and a final time. For a large number of particles that are randomly arranged into a dilute gas, we find that a typical system trajectory will spontaneously collapse into a small region of space, close to the maximum density that is obtainable, before expanding out again. If generalizeable, this has important implications for the cosmological arrow of time, potentially allowing a scenario in which both boundary conditions are generic and also a low-entropy state “initial” state of the universe naturally occurs.

Appendices

Appendix A: Numerical Method

We consider N particles subject to a force field F, and want to solve

$$\begin{aligned} m \frac{d^2 x_i}{dt^2} = F_i(\{x_j\}) \end{aligned}$$

(A1)

where m is the mass, which we will take to be unity for notational simplicity. $i = 1,2\dots \,N d$ where N is the number of particles and d is the spatial dimension. We impose boundary conditions that at $t=0$ and $t=T$, all the $x_i$’s are given.

Note that we do not attempt to minimize the action, because it is minimal for a trajectory satisfying Newton’s second law only for sufficiently short times [21]; in general, it is only a stationary point.

We replace the continuous time dependence with a discretization into M time points. The time step between adjacent time points is $\Delta t \equiv T/M$. Eq. (A1) is replaced with a discretized second derivative

$$\begin{aligned} D^2 x_i = F_i(\{x_j\}) \end{aligned}$$

(A2)

where

$$\begin{aligned} D^2 x(m) = \frac{x(m+1)-2x(m)+x(m-1)}{\Delta t^2}. \end{aligned}$$

(A3)

We will minimize the following objective function O, because its minimum is a solution to Eq. (A2), that is, the discretized version of Newton’s second law,

$$\begin{aligned} O \equiv \sum _{m=1}^M \sum _{i=1}^{Nd} [ D^2 x_i(m) - F_i(\{x_j(m)\})]^2. \end{aligned}$$

(A4)

There are two methods for doing this. The first is to apply the Fletcher-Reeves conjugate gradient algorithm [22] directly to these $(M-2)d N$ degrees of freedom. Because of numerical rounding in double precision, this by itself is often not enough to achieve a well converged minimum. To overcome this numerical impediment, we devise a hybrid method similar to the shooting method but iterative.

We consider iterating Eq. (A2) in time. Given $x_i(m)$ and $x_i(m-1)$, this equation allows us to determine $x_i(m+1)$. This is known as the Newton-Störmer-Verlet method [23]. We also divide up the complete time interval of M points into segments, each of length $M_S$. And we only iterate along individual segments. We iterate this discretization of Newton’s law for $M_S$ time slices starting from the first two adjacent time slices in a particular segment S. Therefore instead of having the original number of variables to minimize, we have many less, because the only degrees of freedom are the points that are initially given, as all other points are determined by iteration. For every $M_S$ time points, we attempt to match the two starting points of a segment, with the two final points of the previous segment. This is illustrated in Fig. 6. The black dots at the beginning of a segment represent the degrees of freedom that will be adjusted so that the beginning and end of overlapping adjacent segments agree. To make the notation less cumbersome, we will suppress the particle indices, and write the coordinates of a segment starting at time slice K as $[x(K),x(K+1),\dots ,x(K+M_S),x(K+M_S+1)$. All points are generated deterministically from the initial two time points, x(K) and $x(K+1)$ iterating Eq. (A2). The adjacent segment starts $M_S$ time slices later and its coordinates are labeled $[y(K+M_S),y(K+M_S+1),\dots ,y(K+2M_S),y(K+2M_S+1)$. To get these segments to match, we would like $x(K+M) = y(K+M)$ and $x(K+M+1) = y(K+M+1)$. To achieve this iteratively with only two separate segments, we minimize some combination of quadratic differences

$$\begin{aligned}&O_K(x,y) \equiv \nonumber \\&\quad (x(K+M) - y(K+M))^2 + (x(K+M+1) - y(K+M+1))^2\nonumber \\&\quad +\lambda [(y(K+M+1)-y(K+M))\nonumber \\&\quad -(x(K+M+1)-x(K+M))]^2 \end{aligned}$$

(A5)

where $\lambda$ is a constant that determines how much weight is put in matching derivatives as opposed to function values. It could be set to zero, but it was found that convergence was faster with non-zero values, such as $\lambda = 10$.

With multiple segments we minimize the objective function for this hybrid shooting method $O_{hybrid}$

$$\begin{aligned} O_{hybrid} \equiv \sum _{K} O_K \end{aligned}$$

(A6)

using the same Fletcher Reeves conjugate gradient algorithm as above [22].

Fig. 6

An illustration of the hybrid shooting, and minimization procedure employed. The black dots represent the time slice for the degrees of freedom being minimized. The number of time points in each segment is $M_S$

As with the local minimization method, there is a limit to the precision of trajectories. To improve on the minimum found, we use the final path obtained with segments of length $M_S$ as starting points for a new minimization iteration, where we double the path length of each segment from $M_S$ to $2 M_S$. Now the number of points where matching is required is halved. This continues until the segment is the full number of time slices in the system.

The code used is available at https://github.com/joshdeutsch/StateToState.

Appendix B: Calculation Cluster Entropy

We depict a multi-clump trajectory in Fig. 7. The thick gray lines represent the surface regions between neighboring clumps. The width of these regions is the mean free path l. Particles that are initially in the grey regions are able to travel to either side of the surface, meaning that they have a choice of which clump to join. Particles deep inside a clump, that is, not in the grey regions, must stay inside their initial region traveling towards the center. The linear dimension $R_M$ of a clump is related to the number of particles in a clump M, through the number density n and the volume $\propto R_M^d$, implying that $R_M \sim (M/n)^{1/d}$. Here we are considering M sufficiently large so that $R_M > l$.

Fig. 7

A representation of a multi-clump trajectory where the clumps of size $R_M$ are larger than the mean free path l. Particles starting deep inside of a clump must stay in it, but particles within a distance l of the surface can choose to go to one of two clumps

The number of particles in the surface region involves its thickness l, its surface area, and the density. Therefore for a single clump, the number of particles in this surface is $\sim n l R_M^{d-1} = n l (M/n)^{(d-1)/d}$. The total number of particles in all surface regions (all of the grey area in Fig.  7) is N/M times this quantity, which is $(N/M) n l R_M^{d-1} = N l (M/n)^{-1/d}$. Each particle in this region has two choices of where to go. Therefore the entropy of these surface regions is $~\sim N l (M/n)^{-1/d} \log 2$. The scaling of this with M and the total number of particles N, is $N/M^{1/d}$. As expected, the larger M, the smaller the surface area, and therefore the smaller this contribution. We should also include the different configurations of these surfaces. For example if $d=2$, then the entropy of these surfaces is bounded from above by assuming that surfaces are random walks. In that case the surface configurational entropy is also proportional to the surface area, giving an equal or a lesser contribution to the entropy. The same statement should hold in higher dimension. Therefore, taking into account the different surface configurations will not alter the dependence of the entropy of M.

We conclude that as a function of M, the number of clusters ${\mathcal{N}}_{{\mathcal{C}}} (M) \sim \exp (cNl(M/n)^{{ - 1/d}} )$. where c is a constant. This decreases with increasing M and is extensive in N, as claimed in Sect. 5.3.