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Shape Of The Universe

You're asking about the shape of the universe. Adorable. Let's get this over with. And for the record, "Edge of the universe" is a redirect here. If you were looking for the Bee Gees song, you've taken a profoundly wrong turn. Same goes for that documentary, Journey to the Edge of the Universe.

Part of a series on Physical cosmology. Because humanity loves putting things in boxes, even when the box is everything.

Early universe

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Expansion · Future

Components · Structure

Experiments

Scientists A veritable who's who of people who stared at the sky and did math about it.

Subject history

In the grand, often tedious, field of physical cosmology, the shape of the universe is a question that splits into two distinct, though related, problems. We're talking about its local geometry and its global geometry. Think of it this way: local geometry is the shape of the room you're in, defined primarily by its curvature. Global geometry, or topology, is the floor plan of the entire, possibly infinite, building. And yes, the local curvature puts some constraints on the overall floor plan, but it doesn't hand you the complete blueprints.

The framework for this whole discussion is, of course, General relativity, which explains how gravity dictates the local spatial curvature. But here's the catch: you can't determine the universe's global topology just by measuring curvature from your little corner of spacetime. There are spaces that are locally identical but globally different. For instance, a multiply connected space like a 3-torus is perfectly flat everywhere you look, yet it's finite. It wraps around on itself. A flat, simply connected space, like the Euclidean space you probably imagine when you close your eyes, is also flat but goes on forever. Same local feel, drastically different global reality.

Your best observational efforts, using instruments like WMAP, BOOMERanG, and the Planck spacecraft, have concluded that the piece of the universe you can see is spatially flat. The margin of error on this is a paltry 0.4% of the curvature density parameter. So, locally, it's about as flat as this conversation. Globally? The jury is still out. We don't know if it's simply connected like a vast, featureless plane or multiply connected like a cosmic labyrinth. So far, there's no compelling evidence that it isn't simply connected, but the universe isn't exactly forthcoming with its secrets.

Shape of the observable universe

You need to understand the scale of your ignorance. The universe's structure can be dissected from two perspectives, and you're only equipped to handle one of them properly.

  • Local geometry: This concerns the curvature of the universe, which is something you can actually measure, more or less, from your observable bubble.
  • Global geometry: This is about the total, overarching shape and connectivity of everything. The whole thing. You can't see it, you can only infer, and your inferences are... limited.

The observable universe—your personal cosmic horizon—is a sphere-like region of spacetime with a radius of about 46 billion light-years in all directions, centered, naturally, on you. Or more specifically, on Earth. The further you peer into this sphere, the further back in time you're looking, and the more profoundly redshifted everything becomes. In principle, you could look all the way back to the Big Bang. In practice, you can't. The universe was opaque for the first 370,000 years, so your vision is blocked by a wall of ancient light called the cosmic microwave background (CMB). Anything beyond that point is effectively hidden by a fog of plasma from the era of recombination. What studies of this observable patch have shown, with tedious consistency, is that on the largest scales, it appears remarkably isotropic and homogeneous.

If—and this is a monumental "if"—the observable universe is all there is, then you might one day map its entire structure. But if, as is far more likely, your observable bubble is an infinitesimal speck in a much larger, or even infinite, cosmos, then you're like an ant on a balloon trying to deduce the shape of the entire balloon factory. You can only ever see your immediate patch. This means countless mathematical models for the universe's global geometry are possible, all of them perfectly consistent with your limited observations and general relativity. So, is the whole universe just what you can see, or is it vastly, incomprehensibly larger? The smart money is on the latter.

The universe might be compact in some dimensions and sprawling in others, like a cosmic cuboid. Scientists, in their relentless quest to poke at the unknown, test these models by looking for their unique predictions. For example, a small, closed universe would act like a hall of mirrors, creating multiple images of the same object in the sky, though these images would show the object at different ages. As of 2024, your telescopes have found nothing of the sort, and the evidence continues to point toward a spatially flat observable universe with an unknown global topology.

Curvature of the universe

Let's break down curvature. It's a measure of how much a space deviates, locally, from being flat. For any space that's locally isotropic—and we're assuming the universe is, to keep things simple—the curvature falls into one of three categories. Try to keep up.

  • Zero curvature (flat): The angles of a triangle add up to 180°. The Pythagorean theorem works just like your high school teacher told you. This is the geometry of Euclidean space, E³. It's the default setting for a boring universe.
  • Positive curvature: The angles of a triangle add up to more than 180°. The space curves back on itself. Locally, this geometry is modeled by a piece of a 3-sphere, S³.
  • Negative curvature: The angles of a triangle add up to less than 180°. The space curves away from itself everywhere. This is modeled by a region of a hyperbolic space, H³.

These curved geometries belong to the domain of non-Euclidean geometry. To visualize positive curvature, think of the surface of a sphere like Earth. Draw a triangle with one corner at the North Pole and two corners on the equator. The two angles at the equator are both 90°, so the sum is already over 180° before you even consider the angle at the pole. For negative curvature, picture a saddle. A triangle drawn on its surface will have angles that sum to less than 180°.

General relativity connects this geometry to the stuff in the universe. Mass and energy bend spacetime. To quantify this, cosmologists use the density parameter, represented by the Greek letter Omega (Ω). It's the average density of the universe divided by the "critical energy density"—the precise amount of stuff needed to make the universe flat.

It's a simple, if cosmic, equation:

  • If Ω = 1, the universe is flat.
  • If Ω > 1, the universe has positive curvature (it's "closed").
  • If Ω < 1, the universe has negative curvature (it's "open").

Your scientists have two main ways of measuring Ω. The first is to take an inventory. You count up all the mass–energy in the universe—all the normal matter, dark matter, photons, neutrinos, and dark energy—calculate the average density, and divide by the critical density. Data from the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck spacecraft have provided the following cosmic budget:

The critical density, for what it's worth, is measured to be ρcritical = 9.47 × 10−27 kg⋅m−3. Given these numbers, the universe appears to be flat, with an unnerving lack of experimental error.

The second method is more geometric. You measure the angles of a very, very large triangle. This is done by looking at the CMB, specifically the power spectrum of its temperature anisotropy. Imagine a cloud of gas in the early universe, so vast that light hasn't had time to cross it and bring it into thermal equilibrium. We know the speed of light, so we can calculate the cloud's size. We can also measure its distance from us. With two sides and the apparent angle it subtends in the sky, we can determine the geometry of the space it's in. Using this principle, the BOOMERanG experiment found that the angles do, in fact, sum to 180°, corresponding to Ωtotal ≈ 1.00 ± 0.12.

These and other measurements all point to a spatial curvature agonizingly close to zero. This doesn't tell us the sign of any tiny residual curvature, but it does mean that for most practical purposes, you can model 3-space with the familiar rules of Euclidean geometry, even though the underlying spacetime is curved according to general relativity.

The standard model for this is the Friedmann–Lemaître–Robertson–Walker (FLRW) model, which uses the Friedmann equations and the mathematics of fluid dynamics to describe the universe as a perfect, uniform fluid. Of course, the universe isn't a perfect fluid—it's lumpy with galaxies and voids. But on a large enough scale, this "almost FLRW" approximation works well enough to model the local geometry.

Global universal structure

Now for the part you can't see. The global structure encompasses the geometry and topology of the entire universe. As I said, the local geometry limits the possibilities but doesn't give you the final answer. We generally assume the universe is a geodesic manifold without any topological defects, mostly because the math becomes a nightmare otherwise.

The fundamental questions about the global structure are:

  • Is the universe infinite or finite?
  • Is its global geometry flat, positively curved, or negatively curved?
  • Is its topology simply connected (like a sphere) or multiply connected (like a torus)?

Infinite or finite

Does it end, or does it just keep going? A finite universe has a finite volume. An infinite universe is unbounded. Mathematically, this is a question of boundedness. An infinite, unbounded space means you can always find two points further apart than any distance you can name. A finite, bounded space has a maximum distance, a "diameter."

With or without boundary

If the universe is finite, does it have an edge? Most mathematical spaces you're familiar with, like a disc, have a boundary. But a universe with an edge is a conceptual mess. What happens there? For this reason, models with a boundary are usually thrown out.

Fortunately, there are plenty of finite spaces without edges. The 3-sphere and the 3-torus are two examples. These are called compact without boundary. "Compact" means it's finite and complete. "Without boundary" means no edges. We also assume it's a differentiable manifold so we can do calculus on it. An object with all these properties is a closed manifold.

Observational methods

In the 1990s and early 2000s, people proposed methods to empirically determine the global topology by searching for repeating patterns in the sky—the multiple imaging effect I mentioned earlier. These were applied to cosmological observations, with inconclusive results.

More recently, it's been shown that because the universe is inhomogeneous—see the cosmic web—the subtle accelerations measured in the movements of galaxies could, in principle, betray the global topology. The idea is that in a multiply connected universe, the gravitational pull from "ghost" images of structures would create a detectable effect. A clever, if fiendishly difficult, way to check.

Curvature

As mentioned, curvature constrains topology. If the universe has positive curvature (a spherical geometry), its topology must be compact and finite. For a flat or hyperbolic (negative curvature) geometry, the topology could be either compact or infinite. Many textbooks get this wrong, stating that flat or hyperbolic implies infinite. This is false. A flat, simply connected universe is infinite. But a flat, multiply connected universe, like a torus, can be finite. Remember the difference.

In general, local to global theorems in Riemannian geometry dictate these relationships. Constant curvature, as described in Thurston geometries, severely limits the possibilities.

The latest research suggests that even your most powerful future experiments, like the Square Kilometre Array, won't be able to tell the difference between flat, open, and closed if the true value of the cosmological curvature parameter is smaller than 10⁻⁴. If it's larger than 10⁻³, you might have a chance.

The final 2018 data from the Planck mission pinned the curvature parameter, ΩK, at 0.0007 ± 0.0019. This is statistically indistinguishable from zero, perfectly consistent with a flat universe.

Universe with zero curvature

If curvature is zero, the local geometry is flat. The simplest global structure is Euclidean space: infinite and boring. But finite flat universes exist. The torus and Klein bottle are 2D examples. In three dimensions, there are 10 finite, closed, flat 3-manifolds, known as Bieberbach manifolds. Six are orientable, four are not. The most famous is the 3-torus universe.

Without dark energy, a flat universe expands forever, but the expansion slows, asymptotically approaching zero. With dark energy, the expansion first slows due to gravity, then accelerates. The ultimate fate of the universe is the same as an open one: endless expansion. Also, a flat universe can have zero total energy, which should appeal to your sense of cosmic economy.

Universe with positive curvature

A positively curved universe is described by elliptic geometry. You can think of it as a three-dimensional hypersphere or some other spherical 3-manifold, like the Poincaré dodecahedral space. All of these are quotients of the 3-sphere.

The Poincaré dodecahedral space is a positively curved space that's been colloquially described as "soccerball-shaped." It's the quotient of the 3-sphere by the binary icosahedral group, which has a symmetry very close to icosahedral symmetry. This model was proposed by Jean-Pierre Luminet and his colleagues in 2003 as a potential explanation for some anomalies in the CMB, and an optimal orientation was later calculated in 2008.

Universe with negative curvature

A hyperbolic universe, with negative spatial curvature, is described by hyperbolic geometry. Locally, it's like a 3D version of a saddle shape, extending infinitely. There is a bewildering variety of hyperbolic 3-manifolds, and their classification is incomplete. The ones with finite volume are constrained by the Mostow rigidity theorem. Many of these are informally called "horn topologies" because they resemble the shape of the pseudosphere. A well-known example is the Picard horn, a negatively curved space often described as "funnel-shaped."

Curvature: open or closed

A final, necessary clarification. When cosmologists say "open" or "closed," they are almost always referring to negative or positive curvature, respectively. This is different from the mathematical definitions of open and closed sets or manifolds, which causes endless confusion. In mathematics, a closed manifold is compact and has no boundary. An open manifold is not compact and has no boundary. A "closed universe" (positive curvature) is always a closed manifold. An "open universe" (negative curvature) can be either a closed or an open manifold. So try not to mix them up. It's tiresome to correct.

See also