Is the universe infinite?

Imagine traveling in a straight line, forever. Do you hit an edge? Do you come back to where you started, like a ship sailing around the Earth? Or do you just keep going, past galaxy after galaxy, without end?

Cosmologists, who study the universe as a whole, have measured the shape of space precisely, using light that has traveled for almost the universe’s whole age. And yet the honest answer to “Is the universe infinite?” is still: we don’t know. Space could go on forever. It could also be finite but have no edge at all.

The short version

As of October 2026, space looks very nearly flat: every recent combination of CMB and galaxy-map data keeps any curvature to a fraction of a percent of the cosmic budget (the mix of ingredients that sets how fast space expands), and the newest lean very slightly toward an open universe, which is not a detection Reference 1Reference 2Reference 3. A perfectly flat space would go on forever, unless it is “wrapped around” on itself like the screen of an old video game, which would make it finite. Searches for that wrapping have found nothing, but many finite shapes are still allowed Reference 4. If space is instead very slightly curved like the surface of a ball, it must be enormous: at 95 percent confidence, at least about 1,900 times the volume we can see, by our calculation from one recent measurement Reference 5. Today we can see out to about 46 billion light-years, and even waiting forever would only extend that to about 63 billion (if dark energy stays constant), so we may never confirm that space goes on forever Reference 6Reference 7.

Two questions hiding in one

“Is the universe infinite?” is really two questions.

The first is about shape. Imagine an ant living on a sheet of paper. It can’t see the third dimension, but it can still tell whether its world is flat or curved by drawing triangles. On a flat sheet, the three angles of every triangle add up to 180°. On the surface of a ball, they add up to more. On a saddle, they add up to less. Space itself can be curved in the same three ways, one dimension up. Physicists call this spatial curvature: a measure of whether straight lines that start out parallel stay parallel, converge or spread apart.

The second question is about connection: how space is joined to itself overall. Think of the old arcade game Asteroids, where a ship that flies off the right edge of the screen reappears on the left. That screen is perfectly flat (triangles drawn on it still add up to 180°), but it is finite and has no edge. Mathematicians call this kind of global property topology. In three dimensions, the Asteroids trick gives a three-torus: a box of space where leaving through any face brings you back in through the opposite one.

Einstein’s general relativity, our theory of gravity, tells us how matter curves space at every point. It says nothing about how space is connected on the largest scales Reference 4. So measuring the shape of space doesn’t automatically tell us its size. These possibilities fit the data:

Flat and infinite: in the words of the cosmologist Max Tegmark, writing in 2003, “the simplest and most popular one on the market” Reference 8.
Flat but finite: a three-torus or one of its twisted cousins Reference 7Reference 4.
Curved like a ball (“closed”): finite but edgeless, the three-dimensional version of a sphere’s surface, called a three-sphere.
Curved like a saddle (“open”): infinite if it has the simplest connection, though it too can be wrapped up into something finite Reference 7.

How far can we see?

The universe is 13.8 billion years old Reference 1, so you might expect that we can see 13.8 billion light-years in every direction. We can see much farther. While light from distant regions was on its way to us, space kept expanding, carrying those regions away. The most distant material that could in principle have sent us any signal since the Big Bang, the hot, dense beginning of the expanding universe, is now about 46 billion light-years away, or about 4.4 × 10²⁶ meters Reference 6. That sphere, centered on us, is the observable universe. Its size follows from how fast space has stretched at each moment in the past, as inferred from measurements by the European Space Agency’s Planck satellite Reference 1.

The light we actually see comes from just inside that edge. The oldest light we can catch is the cosmic microwave background (CMB), a faint glow released about 380,000 years after the Big Bang Reference 1. Before then, the universe was a hot, glowing fog that light could not cross; the CMB was released when it first became transparent Reference 8. The gas that emitted it is now about 45.2 billion light-years away.

There is another distance that often gets confused with the edge. The expansion rate today, the Hubble constant, says how much faster a galaxy recedes for every extra light-year of distance. Go far enough and that speed reaches the speed of light. That distance, about 14.5 billion light-years, is called the Hubble radius, and galaxies beyond it are moving away from us faster than light Reference 6. Nothing is breaking the cosmic speed limit: it is space that is expanding, and nothing ever overtakes a beam of light right next to it Reference 6. And the Hubble radius is not an edge. Light from those galaxies starts out losing ground, like someone walking the wrong way up a moving walkway. But the expansion rate per light-year keeps falling. In the past, galaxies’ recession speeds were dropping; today they are rising, but distances grow faster still. Either way, speed divided by distance goes down Reference 6. The Hubble radius therefore creeps outward, as if the walkway near you slowed down; light that set out early enough ends up inside it, and from then on it gains on us. The Hubble radius keeps growing, but ever more slowly, leveling off as dark energy takes over. That is why we routinely see galaxies beyond it Reference 6.

Waiting doesn’t help much. The expansion of the universe is speeding up, driven by what is called dark energy, an unknown something that makes space push itself apart Reference 1. In cosmologists’ standard picture of the universe, the one that fits the data best, light that galaxies emit today will never reach us if they are now more than about 16.7 billion light-years away; that boundary is our cosmic event horizon. We still see galaxies far beyond it, but only by light they sent out billions of years ago. Even if we waited forever, and if dark energy stays constant, we would only ever receive light from within about 63 billion light-years; that larger figure counts light already on its way, which shows those regions only as they were long ago Reference 6Reference 1.

For physicistsWhere the 46 billion light-years comes from2 equations · flat ΛCDM · ~3 min

All distances are comoving (proper distance today). For a Friedmann–Lemaître–Robertson–Walker model with radiation, matter, curvature and a cosmological constant,

H(a) = H_0\sqrt{\Omega_r a^{-4} + \Omega_m a^{-3} + \Omega_K a^{-2} + \Omega_\Lambda},

D_p = c\int_0^1 \frac{\dd a}{a^2 H(a)} ,

and the event horizon is $D_e = c\int_1^\infty \dd a/(a^2H)$ , with the definitions of Davis & Lineweaver Reference 6. We use the Planck 2018 base-ΛCDM parameters from TT,TE,EE+lowE+lensing, $H_0 = 67.36\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $\Omega_m = 0.3153$ Reference 1, with $\Omega_K = 0$ , $T_\mathrm{CMB} = 2.7255$ K and radiation $\Omega_r = 9.22\times10^{-5}$ from photons plus $N_\mathrm{eff} = 3.046$ neutrino species treated as massless (Planck’s base model has one 0.06 eV state; negligible here, since $\chi_*$ matches Planck to 0.04%).

Numerical integration (calc/friedmann.ts) gives $c/H_0 = 14.52$ Gly, $D_p = 14{,}147$ Mpc $= 46.1$ Gly, a distance to last scattering $\chi_* = 13{,}867$ Mpc $= 45.2$ Gly, $D_e = 16.7$ Gly and $D_p(t\to\infty) = D_p + D_e = 62.8$ Gly (for a cosmological constant). Checks: age 13.795 Gyr (Planck: 13.797 ± 0.023); $\chi_*$ matches Planck’s $r_*/\theta_* = 13{,}873$ Mpc to 0.04%. $D_p$ integrates from $a = 0$ assuming radiation domination, i.e. effectively from the end of any inflationary phase; it is a causal horizon. Photons reach us only from $\chi_* < D_p$ ; neutrinos or gravitational waves could in principle arrive from closer to $D_p$ .

Step by step

From here to the edge — and the ways space could continue

Start at home and zoom out using a ruler made of light. Each ring is a thousand times farther than the last: one light-year, a thousand, a million, a billion. On this logarithmic scale, every step outward multiplies the distance.

Next come two rings that are easy to mistake for the edge. 13.8 billion light-years is light-speed times the age of the universe. 14.5 billion light-years is the Hubble radius, beyond which the expansion carries galaxies away faster than light. Neither is the limit of what we see.

Now switch to an ordinary, linear scale. The edge of the observable universe is 46.1 billion light-years away, more than three times the naive answer. Just inside it, at 45.2 billion light-years, is the shell that sent us the cosmic microwave background, the oldest light we can see.

The future adds two more rings. Light emitted today from beyond 16.7 billion light-years will never reach us. And even if we waited forever, we could only ever see out to about 63 billion light-years, if dark energy stays constant.

Beyond the edge, space could be flat, closed like a ball or open like a saddle. Even at the most curved shapes the data still allow, a triangle with sides 45 billion light-years long would have angles adding up to 180.4° (if closed) or 178.9° (if open), instead of 180°.

If space is curved, it must be gently curved. Drawn to scale: if space is closed, it is like the surface of a giant ball whose radius is at least 343 billion light-years, about 7.4 times the radius of our observable universe. If it is open, there is no ball, but the distance over which its saddle-like bending shows is at least 215 billion light-years.

Flat doesn’t have to mean infinite. Here space is a box whose opposite faces are joined, so far enough away you would see your own neighborhood again. The box is 89 billion light-years across, the smallest allowed; our CMB sphere (90.5 across) only just overlaps its copies, leaving circles too small for the searches to catch.

Measuring the shape of space

You can’t step outside the universe to look at its shape. But you can do what the ant does: measure a very large triangle.

Nature has provided one. In the cosmic microwave background, there is a characteristic pattern of hot and cold spots whose true physical size can be calculated from well-understood physics of the early universe. That gives us a known ruler, painted on a shell 45 billion light-years away. If space is flat, the ruler appears at a certain angle on the sky. If space is curved like a ball, the light rays from its two ends bend toward each other on their way to us. Traced back, they seem to come from points farther apart, so the spots look bigger, as if seen through a magnifying glass. If space is saddle-shaped, they look smaller Reference 9Reference 8. Planck measured that angle to 0.03 percent Reference 1.

There is a catch. On its own, the CMB can’t fully separate curvature from other unknowns, such as how fast the universe is expanding today: different combinations can produce almost the same pattern on the sky Reference 1. To break the tie, cosmologists add a second ruler at a different distance. Pressure waves in the hot early universe left a preferred spacing in the way ordinary matter clumps, called baryon acoustic oscillations (BAO; “baryon” means ordinary matter, the stuff atoms are made of) Reference 2. Galaxies are slightly more likely to be found about 480 million light-years apart than slightly closer or farther apart Reference 1Reference 2. The Dark Energy Spectroscopic Instrument (DESI), which maps galaxies in three dimensions from Kitt Peak National Observatory, has now measured this spacing using more than 14 million galaxies and quasars, the brilliant cores of distant galaxies Reference 2. Each ruler on its own fits several combinations of curvature and expansion rate; only a narrow range fits both.

The result is expressed as a single number, Ω_K (“omega-K”). How fast the universe expands depends on what it contains (matter, radiation, dark energy) plus a term for curvature; Ω_K is curvature’s share of that budget Reference 5. Zero means perfectly flat. By convention, a negative value means closed, like a ball, and a positive value means open, like a saddle Reference 5. When we say curvature is below “half a percent”, we mean the size of Ω_K is smaller than 0.005: a share of the budget, not a measure of how bent space looks.

Measurements come with a margin of error, the ± number, which is one standard deviation (σ). The true value lands within one margin of the best estimate about 68 percent of the time, and within two margins about 95 percent of the time. A result two or three margins away from zero sounds impressive, but when many measurements are made, gaps of that size often turn out to be flukes or hidden errors.

A short-lived closed universe

In 2018 the Planck team reported that its CMB maps alone, taken at face value, preferred a closed universe: Ω_K = −0.044, with flat space outside the range they were 99 percent confident contained the true value Reference 1. For a while this was debated as a possible “crisis for cosmology” Reference 10Reference 11.

The CMB maps looked slightly more smeared than expected, as if extra matter were bending the light on its way to us; that bending is called gravitational lensing. A closed universe happens to produce the same kind of extra smearing, so the fit leaned that way. The Planck team traced the lean to this smearing, and noted that it shifted when part of the data was analyzed a different way Reference 1. When Planck measured the bending directly, it found the amount expected in a flat universe, and once BAO was added the lean disappeared. The combined answer was Ω_K = 0.0007 ± 0.0019, consistent with flat Reference 1. A later reanalysis with an improved treatment of the Planck data also found the maps consistent with flat space Reference 12.

The newest measurements

Since 2025, new data have sharpened the picture, with a twist: the old hint toward a closed universe has faded (CMB data alone now lean closed by less than two standard deviations), and once galaxy maps are added a faint hint in the opposite direction, toward a slightly open universe, appears Reference 3. Two ground-based telescopes now add to Planck’s maps. Combining its maps with Planck’s, with lensing measurements from both, and with DESI’s first-year BAO, the Atacama Cosmology Telescope (ACT) in Chile finds Ω_K = 0.0019 ± 0.0015, “fully consistent with spatial flatness” Reference 5. DESI’s second data release with CMB data gives 0.0023 ± 0.0011, the smallest uncertainty among these results; the DESI team says this is not a significant preference for curvature Reference 2. The South Pole Telescope’s current camera, SPT-3G, combined with Planck, ACT and DESI, gives 0.0027 ± 0.0012, which its authors place 2.5 standard deviations from flat (simply dividing the value by its margin would give about 2.3), and 3.0 without Planck Reference 3.

Three things keep this a hint rather than a discovery. It comes from a known mismatch: DESI’s maps suggest the universe has expanded a little differently from what the CMB predicts, and curvature is only one way to absorb that Reference 2Reference 3. If dark energy is allowed to change over time (in fits that also include supernova data), DESI’s preference for curvature disappears Reference 2. And the SPT team notes that using Planck’s instead of ACT’s small-scale CMB data could weaken it Reference 3. So the honest summary: every recent combination keeps curvature’s share below about half a percent at 95 percent confidence (one combination without Planck reaches about 0.8 percent); margins of error are now about ±0.1 percent; and the best estimates sit at 0.2–0.3 percent (0.5 without Planck), one to three standard deviations from zero depending on the data, which is not a detection.

FIG. 1With BAO added, every combination lands within one percent of flat

Show the numbers

Data combination	Ω_K (68% range)	Source
Planck 2018 CMB maps alone	−0.044 (+0.018 / −0.015)	Reference 1 Eq. 46b
Planck 2018 + lensing	−0.0106 ± 0.0065	Reference 1 Eq. 47a
Planck 2018 + lensing + BAO	0.0007 ± 0.0019	Reference 1 Eq. 47b
ACT DR6 + Planck + lensing + DESI year 1	0.0019 ± 0.0015	Reference 5 Eq. 59
DESI DR2 BAO + CMB	0.0023 ± 0.0011	Reference 2 Table V
SPT-3G + ACT + Planck + DESI DR2	0.0027 ± 0.0012	Reference 3 Eq. 87
SPT-3G + ACT + DESI DR2 (no Planck)	0.0051 ± 0.0017	Reference 3 Eq. 86

Bars show each result’s margin of error (68 percent ranges). Negative Ω_K means closed (ball-like), positive means open (saddle-like). The CMB maps alone leaned closed; every combination with BAO sits within one percent of zero, and the newest lean slightly open 1523.

DATA: Planck 2018 VI; ACT DR6; DESI DR2; SPT-3G D1 · COMPUTED: calc/curvature.ts · EQUATION NUMBERS AS IN THE ARXIV PDFS READ

For physicistsCurvature conventions, numbers and caveats1 equation · ~4 min

Conventions: $\Omega_K \equiv -kc^2/(a_0H_0)^2 = 1-\Omega_\mathrm{tot}$ , so $\Omega_K<0$ is closed ( $k=+1$ ) and $\Omega_K>0$ open ( $k=-1$ ); the comoving curvature radius is

R_\mathrm{curv} = \frac{c/H_0}{\sqrt{\abs{\Omega_K}}} .

Planck 2018 Reference 1 (equation numbers of arXiv v4 PDF): TT,TE,EE+lowE gives $\Omega_K = -0.044^{+0.018}_{-0.015}$ (Eq. 46b), 99% region $-0.095<\Omega_K<-0.007$ , $\Delta\chi^2_\mathrm{eff} = -11$ ; the pull is the one that drives $A_L>1$ and is not robust at ~0.5σ to the polarization likelihood (CamSpec: $-0.037^{+0.019}_{-0.014}$ ). Adding lensing: $-0.0106\pm0.0065$ (47a); adding BAO: $0.0007\pm0.0019$ (47b), “flat to a 1σ accuracy of 0.2%”.

Current combinations (68%): ACT DR6 P-ACT-LB (Planck + ACT + lensing + DESI Y1 BAO) $0.0019\pm0.0015$ , Planck-LB $0.0022\pm0.0015$ Reference 5; DESI DR2 + CMB (Planck PR4 CamSpec + Planck/ACT lensing) $10^3\Omega_K = 2.3\pm1.1$ , falling to $1.1\pm1.3$ , $0.6\pm1.3$ , $0.8\pm1.3$ in $w_0w_a$ CDM+ $\Omega_K$ with Pantheon+, Union3, DESY5 Reference 2; SPT-3G D1 (arXiv v2 PDF numbering): $100\Omega_K = 0.27\pm0.12$ for CMB-SPA + DESI (Eq. 87) and $0.51\pm0.17$ for SPT+ACT+DESI (Eq. 86), quoted as 2.5σ and 3.0σ from flatness; the CMB-SPA+DESI $\Delta\chi^2 = -4.2$ corresponds to a 2.1σ model preference; CMB alone is within 2σ of flat Reference 3.

ACT quotes $R_k>105$ Gpc (closed) and $>66$ Gpc (open) at 95% Reference 5. In our reconstruction, these are limits conditional on the sign of $\Omega_K$ : a Gaussian posterior truncated to $\Omega_K<0$ or $>0$ gives 101 and 67 Gpc (calc/curvature.ts conditionalLimits), while the unconditional Gaussian 95% interval would give a tighter-looking closed bound (138 Gpc).

If space is curved, how big must it be?

A curved universe has a natural size: its curvature radius. For a closed universe, that is the radius of the giant four-dimensional ball whose three-dimensional “surface” our space would be. An open, saddle-shaped universe has a curvature radius too, but it is not a size: it is roughly the distance over which the saddle-like bending would become noticeable. The closer Ω_K is to zero, the bigger the radius, so upper limits on curvature become lower limits on it.

The ACT team calculated these limits from its data combined with Planck and DESI. If space is closed, its curvature radius is at least 343 billion light-years, at 95 percent confidence; if it is open, at least 215 billion light-years Reference 5. (The limits differ because the measurements lean slightly toward open, so more curvature, and hence a smaller radius, is allowed on that side.) For a closed universe, 343 billion light-years is about 7.4 times the radius of our observable universe. By our calculation from ACT’s limit, such a three-sphere would hold at least about 1,900 times the volume we can see. That is more than 7.4³ ≈ 400 because a three-sphere holds about 4.7 times as much room as an ordinary ball of the same radius. Measured at one instant, the distance once around it would be at least about 2,150 billion light-years; because the expansion is speeding up, not even light could ever complete the trip (if dark energy stays constant).

Try other curvatures below.

Explorable · try it

Bend space: triangles and the size of the universe

Curvature parameter−0.0020

ΩK in the text. Negative = closed (ball-like), positive = open (saddle-like).

Triangle side45 billion ly

Compare with

PlanckACTDESI

ANGLE SUM180.478 °

CURVATURE RADIUS325 billion ly

TOTAL VOLUME1,640 × what we see

IF CLOSED, ALLOWED BY PLANCK?yes (95%)

A closed universe with this curvature is a three-sphere about 2,039 billion light-years around, measured at one instant. Because the expansion is speeding up, not even light could ever complete the trip (if dark energy stays constant).

Assumptions and simplifications

Space is assumed to be curved by the same amount everywhere, and today’s expansion rate is the one Planck measured.
The triangle is drawn with the straightest possible lines through space, all at the same moment; the angle sums are exact, not approximations.
“Allowed” uses each data set’s 95 percent range assuming space is closed (for negative values) or open (for positive values), the same way ACT states its size limits. Real data are not exactly bell-shaped, so the edges are approximate.
“Total volume” assumes a closed universe with the simplest connection (no wrapping), compared with a ball 46.1 billion light-years in radius.
The drawing of the triangle exaggerates the bulge so it is visible; the numbers are not exaggerated.

Flat does not mean infinite

Suppose space really is perfectly flat. It could still be finite, if it is connected like the Asteroids screen Reference 7Reference 4. How would we ever know?

In a paper published in 1998, Neil Cornish, David Spergel and Glenn Starkman pointed out a striking signature Reference 13. On a small enough Asteroids screen, you would see your own ship ahead of you, having gone all the way around. Likewise, if space is wrapped so that the distance around it is less than the diameter of the sphere we see in the CMB, light from the same patch of the early universe can reach us along two different paths, from two different directions. Our CMB sphere would overlap with “copies” of itself, and two overlapping spheres meet in a circle, like two soap bubbles stuck together. So the sky would contain pairs of circles carrying the same pattern of hot and cold spots, a fingerprint of a finite universe (Figure 2).

FIG. 2A finite universe could show itself through matching circles, but twisted shapes can hide

Left: in a wrapped universe, shown as a flat slice, our light sphere (solid) overlaps the copy of itself from the neighboring box (dashed); the overlap is a matched circle (two points in this slice) 13. Right: in a “half-turn” space, leaving through the top brings you back through the bottom rotated half a turn about the vertical axis (turned around, not mirrored), so the shortest round trip is one box height on the central axis but longer away from it 4.

ILLUSTRATION · GEOMETRY: calc/topology.ts · NOT TO SCALE

Astronomers searched. In 2004, using NASA’s WMAP satellite, Planck’s predecessor, they found no matched circles with a radius larger than 25° on the sky, ruling out a wide class of small universes Reference 14. Planck’s more sensitive maps found no evidence of a finite topology on scales smaller than the CMB sphere Reference 15. For the simplest case, a cubic three-torus, the box must be at least about 89 billion light-years on a side: very slightly smaller than the 90.5-billion-light-year diameter of our CMB sphere is still allowed, because the circles would then be small (about 10° in radius), below the size the searches could pick out reliably Reference 4. Such a box would hold only about 1.8 times the volume inside our CMB sphere: a finite universe barely bigger than what we can see is still allowed.

A loophole in the circles

Recent work by an international team known as the COMPACT collaboration has shown that the circle searches rule out less than many people assumed Reference 4. Call a “loop” the shortest straight-line round trip: set off in one direction and arrive back where you started. The searches imply that the shortest loop through our own position must be longer than 98.5 percent of the diameter of the CMB sphere, about 89 billion light-years Reference 4.

But in most of the finite flat shapes, the joins between opposite faces include a twist or a turn, and then not every location is equivalent. Imagine that going out the top of the box brings you back in through the bottom, but rotated half a turn about the direction you were traveling, so a point on your left comes back on your right: you are turned around, not mirrored (Figure 2, right). Start on the central axis and one trip up brings you straight home. Start off to one side and the half-turn drops you on the opposite side of the axis, so your round trip is longer. A short loop can therefore pass somewhere else, avoiding us. In these finite spaces, the shortest loop can be between a half and a sixth or less of the CMB sphere’s diameter without producing any circles we could have seen; in one shape that is infinite in two directions, a twisted “slab”, it can be shorter still Reference 4. In the team’s words, “a broader range of manifolds [possible shapes] is observationally allowed than widely appreciated” Reference 4.

The same group is developing subtler tests. A wrapped-up shape leaves faint links between the hot and cold spots in distant parts of the sky, which could in principle be detected if our nearest “copy” is closer than about 1.2 times the CMB sphere’s diameter Reference 16. A preprint posted in August 2026, not yet peer reviewed, suggests that future three-dimensional maps of galaxies or of hydrogen gas could detect wrapping on scales up to about 25 percent larger than the CMB alone can, in the two example shapes it studied Reference 17.

For physicistsTopology: manifolds, matched circles and the COMPACT limitsno equations · ~3 min

The non-trivial orientable Euclidean manifolds are $E_1$ – $E_6$ , $E_{11}$ , $E_{12}$ and $E_{16}$ ; $E_{18}$ is the covering space. Only $E_1$ – $E_6$ are finite; $E_{11}$ and $E_{12}$ are compact in two directions and $E_{16}$ in one Reference 4Reference 16. Matched circles arise wherever the last-scattering sphere intersects its images under the group of deck transformations (the identifications) Reference 13. Planck 2015 limits the radius $R_i$ of the largest inscribed sphere of the fundamental domain to $R_i>0.97\,\chi_\mathrm{rec}$ for the cubic $T^3$ and $R_i>0.56\,\chi_\mathrm{rec}$ for the slab $T^1$ ( $\Delta\ln\mathcal{L}>-5$ ; matched circles give $0.97\,\chi_\mathrm{rec}$ for $T^3$ at 99% CL from polarization alone) Reference 15: a cube side of about 88 Gly with our $\chi_*$ .

COMPACT Part I Reference 4 translates the generic circle limit (no closed geodesic through the observer shorter than $0.985\,L_\mathrm{LSS}$ ) into constraints on the nine non-trivial orientable Euclidean topologies. For $E_1$ and $E_{11}$ , $L_\mathrm{min}\gtrsim L_\mathrm{LSS}$ (for the cubic $E_1$ , $0.985\,L_\mathrm{LSS} = 89.1$ Gly and $L^3/(\tfrac43\pi\chi_*^3) = 1.83$ ); for $E_2$ – $E_6$ , $E_{12}$ , $E_{16}$ , corkscrew motions break homogeneity, so the bound on $L_\mathrm{min}$ weakens by a factor from 2 ( $E_2$ , $E_{12}$ ) to at least 6 ( $E_5$ ), arbitrarily for $E_{16}$ . For closed lens spaces $L(p,q)$ , the absence of circles bounds $p$ as a function of $\abs{\Omega_K}$ Reference 18.

Why we might never know

Better measurements will help, but only up to a point. One analysis concluded that if Ω_K is smaller than about one part in ten thousand, only about ten times the size of the natural ripples in the curvature of space itself, we will never be able to tell flat from curved Reference 19.

The question is lopsided. Observations could one day show that space is finite, through matched circles or subtler topology signals Reference 13Reference 16. A clear detection of closed curvature would strongly suggest it too, if space beyond our horizon is curved the same way as the part we see Reference 7. But no observation can ever prove that space goes on forever: everything we measure comes from inside our observable sphere. The cosmologist George Ellis writes that we can never prove that space continues in the same way without end; it is “an untestable concept” Reference 7.

Theory offers a hint. The simplest versions of the leading idea for the first instant of the universe, cosmic inflation, a brief burst of extremely rapid expansion, predict that space should look flat to within anything we can measure Reference 1. In eternal inflation, where the inflating stage never ends everywhere at once, some of its leading theorists argue that the region of space we live in is generically infinite Reference 20. In the version where our universe began as a bubble inside such an eternally inflating space, theorists predict a very slightly open universe Reference 21. These are predictions of theories, not observations. Inflation in general does not require space to be exactly flat or fix the sign of any tiny curvature Reference 7.

So the honest scorecard is this: curvature’s share of the cosmic budget is below about half a percent; we see out to 46 billion light-years; and whether the universe beyond is finite or infinite is unknown. But suppose it is infinite. Then something very strange seems to follow, which is the subject of Part 2: is there another you out there?

Open questions

Is the slight “open” hint real? The CMB-plus-DESI preference for Ω_K slightly above zero, two to three standard deviations depending on the data, could be a statistical fluctuation, a hidden error, or a sign of something else, such as dark energy that changes over time Reference 2Reference 3Reference 22. Future surveys of even more distant galaxies should be able to tell curvature apart from that alternative Reference 22. Euclid’s first major data release begins in November 2026, with its galaxy-clustering products due in mid-2027 Reference 23.
Is the universe wrapped up? Many finite flat and curved shapes remain allowed Reference 4Reference 18. Statistical searches in the CMB and in 3D galaxy maps could reach somewhat beyond today’s limits Reference 16Reference 17.
Can the question be answered at all? If the curvature is below about one part in ten thousand, observations cannot decide the shape Reference 19, and spatial infinity itself can never be directly confirmed Reference 7.

How sure are we?

EVIDENCE LEVELS ARE EDITORIAL JUDGEMENTS OF THE WHOLE BODY OF EVIDENCE — NOT σ VALUES · What these mean

The observable universe extends about 46 billion light-years in every direction; the Hubble radius (14.5 billion light-years) is not its edge.
Evidence level: Established
Follows directly from the measured expansion history Reference 1Reference 6.
Space is close to flat: every recent CMB-plus-BAO combination but one keeps curvature’s share below about half a percent at 95% confidence.
Evidence level: Established
Several combinations agree (all share Planck and/or DESI data); the exception, without Planck, allows up to about 0.8 percent Reference 1Reference 5Reference 2Reference 3.
If space is closed, then at 95% confidence it holds at least about 1,900 times the volume of the observable universe.
Evidence level: Strong evidence
A limit from one combination, assuming closed geometry and the standard model; the tightest data disfavor closed space about as much as flat (about two standard deviations), a lean that disappears if dark energy evolves Reference 5Reference 2.
No sign of a finite, wrapped-up universe smaller than the CMB sphere along loops through our position.
Evidence level: Strong evidence
Repeated circle searches and likelihood tests found nothing Reference 14Reference 15Reference 4.
Space is very slightly open (curvature parameter a little above zero).
Evidence level: Hint / tension
Depends on the data combination and vanishes if dark energy evolves; tied to the DESI–CMB mismatch Reference 2Reference 3. Reported: 2.5 and 3.0 standard deviations (σ) from flat, for SPT + ACT + Planck + DESI and for SPT + ACT + DESI; a second way of scoring the first combination gives 2.1σ Reference 3
The universe is infinite.
Evidence level: Speculative
Expected in eternal inflation, but it can never be confirmed by observation Reference 20Reference 7.

What the video simplified

In the video: Space is flat to within about 0.4%.
More precisely: That figure matches roughly the 95 percent range of the 2018 Planck-plus-BAO result (Ω_K between about −0.003 and +0.004) Reference 1. Newer combinations keep curvature’s share below about half a percent, with best values slightly above zero Reference 2Reference 3. Ω_K is a share of the cosmic budget, not a percentage of flatness.
In the video: Measurements show space is flat, within their precision.
More precisely: True, but the central value has moved: Planck’s CMB maps alone leaned closed, while the newest CMB-plus-DESI combinations lean slightly open, at two to three standard deviations. Neither is a detection of curvature Reference 1Reference 3.
In the video: Searches for repeating patterns in the CMB have found nothing.
More precisely: Correct, but the circle searches only constrain loops through our own position. In finite twisted shapes, a loop elsewhere can be between a half and a sixth or less of the CMB sphere’s diameter without leaving circles we could have seen Reference 4.
In the video: (Implicitly) measuring the curvature tells us the shape of the whole universe.
More precisely: Measurements only tell us the curvature inside our observable sphere. Extending it to the whole universe is an assumption that can’t be tested Reference 7.

References

Numbered in order of first citation. Journal articles link to their DOI; where a free version exists on arXiv we link it too. Preprints are marked as not peer reviewed.

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Five Sigma, “Is the universe infinite?,” Five Sigma (12 October 2026), https://thefivesigma.com/articles/is-the-universe-infinite. Version of 12 October 2026.

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Cosmology20 min read
If the universe is infinite, is there another you?
If space goes on forever, a simple counting argument says copies of you are out there. The logic is tidy, but it rests on assumptions that can’t be checked beyond what we can see.
Part 2 of the series “Is the universe infinite?”

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Is the universe infinite?

Two questions hiding in one

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From here to the edge — and the ways space could continue

Measuring the shape of space

A short-lived closed universe

The newest measurements

If space is curved, how big must it be?

Bend space: triangles and the size of the universe

Flat does not mean infinite

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If the universe is infinite, is there another you?

Two questions hiding in one

How far can we see?

Measuring the shape of space

A short-lived closed universe

The newest measurements

If space is curved, how big must it be?

Flat does not mean infinite

A loophole in the circles

Why we might never know

Open questions

How sure are we?

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If the universe is infinite, is there another you?