User:Jim Pivarski

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The Big Bang as a Manifold

In the framework of general relativity, the universe is described by the Friedmann-Lemaître-Robertson-Walker metric (FLRW or FRW), which is homogeneous in space but depends on a scale factor $a(t)$ (another good reason to drop all but one space dimension). Assuming zero spatial curvature (a good approximation, especially in the early universe), a Lorentz-invariant interval ds is given by

$ds^{2}=c^{2}\,dt^{2}-a(t)^{2}\left(dx^{2}+dy^{2}+dz^{2}\right)$

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with $a(t)$ determined from a model of the universe’s contents. At early times, the dominant contribution to Einstein’s stress-energy tensor was radiation from electromagnetic waves and highly relativisitc particles, which yields a scale factor proportional to $t^{1/2}$. As the universe cooled, wavelengths redshifted and non-relativistic matter became dominant, evolving the power law to $a(t)\propto t^{2/3}$. The supernova evidence we have been discussing at lunch indicates that at very recent times (the past 7 billion years or so), a new component is taking over the cosmological evolution, giving $a(t)$ a positive second derivative. If this is due to a cosmological constant, $a(t)\propto \exp(kt)$. (See the Hitchhiker’s Guide to the Early Universe for more details and derivations.)

This scale factor imposes an intrinsic curvature on space-time. As an intuitive model, I imagine there being more space points per time slice as time increases, as though the space-time manifold were made like a knitted cap: each new row (time slice) contains more stitches (space) than the last. The result is a surface which does not lie flat because there isn’t enough room to accomodate all those extra stitches (see Crochet Models of Hyperbolic Spaces). In the case of the universe, $a(t)$ prescribes how many stitches to knit per time-row $t$.

The attached picture presents a manifold with two local coordinates: time and one space dimension. Time ranges from zero to infinity, but space is represented by a connected loop from $^{-}\pi$ to $^{+}\pi$. Though it is possible that our universe is topologically connected on scales larger than we can observe (a giant torus), many cosmologists assume that it is simply infinite. In that case, we could extend the domain of the space variable to ($-\infty$, $\infty$) without changing the picture. Instead of connecting space to itself, we imagine that it’s passing through itself over and over, the same way that a Klein bottle in 3D is forced to poke through itself (an immersion, rather than an embedding).

The pictured manifold represents the early universe, so it is radiation-dominated with $a(t)\propto t^{1/2}$. Naïvely, we can draw it by constructing a surface of revolution around the $z$ axis with radius $r(z)=kz^{1/2}$, but this assumes that time, the local coordinate on the surface, is parallel to the $z$ axis. The unit vector in the time direction is always tangent to the surface, and the time at a given point is the arclength from the origin to that point, so we actually have to solve

$\displaystyle r(z)=k\left(\int _{0}^{z}{\sqrt {1+\left({\frac {dr}{dz'}}\right)^{2}}}\,dz'\right)^{1/2}$

which yields

$\displaystyle z+C=\left({\frac {2}{k^{2}}}\right)\left({\frac {r}{2}}{\sqrt {r^{2}-\left({\frac {k^{2}}{2}}\right)^{2}}}-{\frac {\left({\frac {k^{2}}{2}}\right)^{2}}{2}}\ln \left|r+{\sqrt {r^{2}-\left({\frac {k^{2}}{2}}\right)^{2}}}\right|\right)$

as an implicit solution. This has a strange feature: radii below $k^{2}/2$ are not defined (on the real $z$ axis, anyway), so our embedded manifold would have a circular hole near the bottom. This radius does not correspond to $t=0$, and there is no discontinuity in the space-time coordinates here; it is a problem with our embedding. The manifold wants to curve more than it can in 3D. Fortunately for our visualization, $k$ is an arbitrary constant, determined by how large we think “$2\pi$” of space is, relative to the time interval pictured. We can make the hole as small as we like by assuming the universe is very, very large. Furthermore, when $r/k^{2}\gg 1$, the above solution approaches

$\displaystyle z+C\approx {\frac {r^{2}}{k^{2}}}$

so the surface of revolution is $r(z)\approx kz^{1/2}$ after all.

To make this visualization more useful pedagogically, I drew some milestones of cosmic history on the surface (not to scale). The first stars are timelike streaks that appear near the top, the cosmic microwave background is a space-filling plasma that stops interacting at $t$ = 400,000 years, nuclei are synthesized in the first few minutes, and before that, the shape of the manifold changes during inflation. Inflation is a very early era in which the scale factor evolved as $a(t)\propto \exp(kt)$ (at least $t^{p}$ with $p\geq 1$ to solve the horizon problem); it looks like a bee stinger on our manifold. It also covers the hole in our embedding, because $a(t)\propto \exp(kt)$ gives us a surface of revolution with radius $r(z)\approx \exp(kt)$ for small $t$ (it has a similar problem at large $t$).

The very last point on the picture is the Beginning of Time. I put a question mark after this phrase because I don’t know of any reason why inflation could not start at $t=-\infty$. If there’s a reason why inflation must be limited, or something different should be expected before inflation, let me know!