Modeling the Past Hypothesis: A Mechanical Cosmology

Abstract

There is a paradox in the standard model of cosmology. How can matter in the early universe have been in thermal equilibrium, indicating maximum entropy, but the initial state also have been low entropy (the “past hypothesis"), so as to underpin the second law of thermodynamics? The problem has been highly contested, with the only consensus being that gravity plays a role in the story, but with the exact mechanism undecided. In this paper, we construct a well-defined mechanical model to study this paradox. We show how it reproduces the salient features of standard big-bang cosmology with surprising success, and we use it to produce novel results on the statistical mechanics of a gas in an expanding universe. We conclude with a discussion of potential uses of the model, including the explicit computation of the time-dependent coarse-grained entropies needed to investigate the past hypothesis.

Appendices

Appendix A: Model Lagrangian

The proposed $\text {model}$ ($N=1$ for simplicity; $N\ne 1$ follows trivially) is derived from the Einstein-Hilbert and geodesic actions as follows:⁵

$$\begin{aligned} S=S_{EH} + S_{Geo}= \int d^4x \sqrt{-g} R -m\int d\tau \sqrt{g_{\mu \nu }\dot{x}^\mu \dot{x}^\nu }, \end{aligned}$$

(A1)

where the dot denotes $\frac{d}{d\tau }.$ Choose coordinates such that $\tau =t$ and insert the $\text {FLRW}$ metric $ds^2={\mathcal {N}}^2dt^2-a^2(t)\Big ((1-kr^2)^{-1}dr^2+r^2(d\theta ^2 +\sin ^2\theta \;{\varphi }^2)\Big )$.

Then,

$$\begin{aligned} \begin{aligned} S&=V\int dt \,{\mathcal {N}}a^3 \left( \frac{6}{16\pi G} \left( \frac{\ddot{a}}{a{\mathcal {N}}^2} + \frac{{\dot{a}}^2}{a^2{\mathcal {N}}^2}+\frac{k}{a^2}\right) -2\Lambda \right) \\ &\quad -m\int dt \sqrt{{\mathcal {N}}^2 - a^2 \left( \frac{|\dot{\varvec{r}}|^2}{1-k{|{\varvec{r}}|}^2}+r^2 {\dot{\theta }} ^2 + r^2 \sin ^2 \theta {\dot{\phi }} ^2\right) }, \end{aligned} \end{aligned}$$

(A2)

indicating

$$\begin{aligned} \begin{aligned} L&=V\frac{1}{8\pi G}\left( \frac{3\ddot{a} a^2}{{\mathcal {N}}} + \frac{{3\dot{a}}^2a}{{\mathcal {N}}}+3ka{\mathcal {N}}-\Lambda a^3{\mathcal {N}}\right) \\ &\quad -m \sqrt{{\mathcal {N}}^2 - a^2\left( \frac{|\dot{{\varvec{r}}}|^2}{1-k{|{\varvec{r}}|}^2}+r^2 {\dot{\theta }} ^2 + r^2 \sin ^2 \theta {\dot{\phi }} ^2\right) }. \end{aligned} \end{aligned}$$

(A3)

Now, we want to get rid of the $\ddot{a}$, so we realize that $\frac{d}{dt} (a^2 \dot{a})= 2a{\dot{a}}^2 + a^2 \ddot{a}$. So, $a^2 \ddot{a}= \frac{d}{dt} (a^2 \dot{a})-2a{\dot{a}}^2$. We can insert this and gauge away the total time derivative, which is a boundary term on the time integration, to arrive at

$$\begin{aligned} \begin{aligned}&L=V\frac{1}{8\pi G}\left( -\frac{{3{\dot{a}}^2}{a}}{{\mathcal {N}}}+{3k}{a}{\mathcal {N}}+\Lambda a^3{\mathcal {N}}\right) \\ {}-m&\sqrt{{\mathcal {N}}^2 - a^2\left( \frac{|\dot{{\varvec{r}}}|^2}{1-k{|{\varvec{r}}|}^2}+r^2 {\dot{\theta }} ^2 + r^2 \sin ^2 \theta {\dot{\phi }} ^2\right) }. \end{aligned} \end{aligned}$$

(A4)

The Hamiltonian is obtained via a Legendre transformation, and in the discussion of the it and its equations of motion, we have set the comoving volume V to be 1 for simplicity (Fig. 11).

Fig. 11

Comparison between the central potential (top) and simulated potential in 2 dimensions, Eq. (12) for $\alpha =4$ and $a=1$. Both are periodic on square intervals, but the simulated potential is also smooth at the boundary, leading to continuity of forces

Appendix B: Mechanical Equation of State

To compare the thermodynamic equation of state with the mechanical equations of motion, consider the case with negligible interactions:

$$\begin{aligned} \rho = \frac{1}{a^3}\left( \sum _i\sqrt{m_i^2+\frac{|\varvec{p}_i|^2}{a^2}}\right) \end{aligned}$$

(B5)

$$\begin{aligned} P= \frac{1}{3}\left( \frac{1}{a^3}\sum _i \frac{|\varvec{p}_i|^2/a^2}{\sqrt{m_i^2+\frac{|{\varvec{p}}_i|^2}{a^2}}} \right) . \end{aligned}$$

(B6)

In the HR limit, $\frac{|{\varvec{p}}|/a}{m}\gg 1,$ so

$$\begin{aligned} \rho \sim \sum \frac{1}{a^3}\frac{|{\varvec{p}}|}{a}\sim \sum \frac{|{\varvec{p}}|}{a^4} \end{aligned}$$

(B7)

$$\begin{aligned} P \sim \sum \frac{1}{3}\left( \frac{|\varvec{p}|^2}{a^5}\frac{a}{|{\varvec{p}}|} \right) \sim \sum \frac{1}{3}\frac{|{\varvec{p}}|}{a^4}. \end{aligned}$$

(B8)

The equation of state is $P=w \rho ,$ indicating correctly that $w=1/3$. In the NR limit, $\frac{|{\varvec{p}}|/a}{m}\ll 1,$ so to first order,

$$\begin{aligned} \rho \sim \sum \frac{m}{a^3} \end{aligned}$$

(B9)

and $P \ll 1$, indicating that $w \sim 0$.