Deep divefrom Episode 6Cosmology13 October 202624 min

Is Space Flat? A Ruler-Free Curvature Test From DESI

A deep dive into one paper: what was known before, what the researchers did, what the numbers say and what they don't.

The paper: Stress-testing the spatially flat Universe: nonparametric spatial curvature determination from DESI DR2 (arXiv preprint)

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Transcript

The script the two AI voices read from. The times are worked out from the script, not heard, so a line can start a few seconds early or late.

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Cold open

Host 1You heard the trailer — this is the whole story. Welcome to the Hastrup Review of Science — where artificial intelligence reads real science, so you don't have to pretend you understood the paper. One paper today, and one very old question: is space itself flat, or is it curved?

Host 2The paper is called Stress-testing the spatially flat Universe: nonparametric spatial curvature determination from DESI D R two. It is by Hannah Turner, Suhail Dhawan, Alice Townsend and Peter Massey at the University of Birmingham, with Sunny Vagnozzi of the University of Trento. It was posted to arXiv on the twenty-second of September twenty twenty-six, formatted as a short letter for the Monthly Notices of the Royal Astronomical Society.

Host 1Formatted for, but not yet published in?

Host 2Right, the acceptance line in the version we read still says Accepted XXX, so treat it as a preprint. Every number today comes from that version.

Host 1And the claim, in two sentences?

Host 2Using only measurements of the relatively nearby universe, and without assuming any particular model of dark energy, they find a mild preference for a universe that is very slightly open rather than perfectly flat. Mild is the authors' word, and it is the right one.

Why this field exists

Host 1Start at the beginning. Why would anyone think space could be curved at all?

Host 2This is background, not the paper. Einstein's general relativity says matter and energy shape geometry, and in the nineteen-twenties Alexander Friedmann and Georges Lemaître worked out that a universe that is the same everywhere and in every direction can have exactly three shapes. Flat, closed like the surface of a sphere, or open like a saddle that keeps flaring outward.

Host 1What does that mean in practice for someone standing inside it?

Host 2In a flat universe, two laser beams fired side by side stay parallel forever, and the angles of a giant triangle add up to one hundred eighty degrees. In a closed one the beams slowly converge and the triangle adds up to more; in an open one they drift apart and it adds up to less. The effect only shows up over distances comparable to the whole observable universe, which is why it has to be measured with cosmology rather than with a protractor.

Host 1And the number the paper keeps talking about is omega K.

Host 2Omega K is the curvature parameter: zero means flat. The sign convention trips everyone up, so here it is once, slowly: a positive omega K means open, the saddle, and a negative omega K means closed, the sphere.

Host 1Positive is open. Got it, I think.

Host 2Positive is open. That will matter for every number we read today.

Host 1Why do cosmologists care so much about this one number?

Host 2Two reasons, and the paper names both. Curvature carries information about the very early universe, especially about inflation, and it also changes how the expansion history of the universe translates into the distances we measure.

Host 1Unpack the inflation part.

Host 2Again background: inflation is the idea that the universe went through an extremely rapid expansion in its first instant, and one of its classic selling points is that it stretches any initial curvature almost perfectly flat. So in the simplest versions, a clearly measurable curvature today would be an awkward fact to explain.

What was known before

Host 1So what did we know before this paper?

Host 2The most precise numbers come from the cosmic microwave background, the afterglow of the early universe. The paper notes that analyses including that data constrain omega K to roughly the one-in-a-thousand level, consistent with flat, but only after assuming a specific cosmological model.

Host 1Though not always consistent with flat, if I remember the story.

Host 2Correct. The paper recalls that the Planck satellite's twenty eighteen data on its own seemed to prefer a slightly closed universe, a negative omega K. That hint was debated, and it mostly went away once other data were added.

Host 1And now something has changed.

Host 2Now the Dark Energy Spectroscopic Instrument, DESI, has released large maps of galaxies, and the paper cites more than a dozen recent analyses where the same newer data mildly favour the opposite sign, a positive omega K, an open universe. On top of that there are hints that dark energy may change over time, plus a few other small tensions in matter density, neutrino masses and an early-universe quantity called the optical depth to reionization.

Host 1That sounds like a lot of mild tensions at once.

Host 2The authors' own word is confusion. Their point is that in each of those cases, how significant the result looks depends partly on what you assumed about the expansion history.

The open question

Host 1So what is the actual open question here?

Host 2Whether you can measure curvature in a way that is as purely geometrical as possible. No microwave background, no chosen dark energy model, and no dependence on the absolute distance scale that the Hubble tension is fighting about.

Host 1Remind us what the Hubble tension is.

Host 2Background: measurements of today's expansion rate disagree depending on method, roughly sixty-seven versus seventy-three kilometres per second per megaparsec, and nobody has settled why. Any curvature result that leans on one side of that fight inherits the fight.

Host 1And this team had tried before.

Host 2Yes. In twenty twenty-one, Dhawan, Justin Alsing and Vagnozzi introduced this exact method and applied it to the supernova data of the time, getting omega K of minus zero point zero three, give or take zero point two six. Essentially flat, with an error bar wide enough to fit almost any answer.

Host 1What was missing back then?

Host 2Good measurements of the expansion rate along the line of sight. They had only three such galaxy-survey points over a narrow range of distances, so they could only forecast what a big survey would do. This paper runs that forecast on real data.

The one idea

Host 1Give me the core idea before the machinery.

Host 2You have two independent things measured against distance, or more precisely redshift, which is how much light has been stretched by expansion. Supernovae give you relative distances, and a second probe gives you how the expansion rate changes with redshift.

Host 1And curvature connects them?

Host 2Exactly. If you know how the expansion rate changes, you can add it up along the light's path and predict distances, and that prediction differs between flat, open and closed geometry. Comparing the predicted distances with the supernova distances leaves curvature as the thing that must make them agree.

Host 1Which way does it go?

Host 2Background geometry: in an open universe the light from a distant source is spread over more area than in a flat one, so for the same expansion history it looks a bit fainter, as if farther away. If supernovae look systematically farther than the expansion history predicts for flat space, the fit leans open.

Host 1And the word relative keeps coming up.

Host 2It is the key design choice. They only use the shape of things, how distances and expansion rates change relative to each other, never their absolute size.

The ingredients

Host 1Walk me through the data, one ingredient at a time.

Host 2First, type one-A supernovae, exploding white dwarfs that are close to standard in brightness. Background: they are the same tool that revealed accelerating expansion in nineteen ninety-eight.

Host 1And they use three different collections.

Host 2Pantheon Plus, spectroscopically confirmed supernovae from redshift zero point zero one to two point two six. Union three, binned into twenty-two bins from zero point zero five to two point two six. And the Dark Energy Survey's five-year sample, recalibrated through a program called Dovekie, from zero point zero two five to one point one three, which they call DES Dovekie.

Host 1Why does Pantheon Plus lose its nearest supernovae?

Host 2They drop the low-redshift calibrator sample, the supernovae in galaxies with independent distance anchors. That only matters for absolute brightness, and this method deliberately does not need absolute brightness.

Host 1Second ingredient?

Host 2Baryon acoustic oscillations from DESI's second data release. Background: sound waves in the early universe left a preferred spacing in how galaxies cluster, a standard ruler of roughly one hundred forty-seven megaparsecs, and measuring that ruler along the line of sight tells you the expansion rate at that redshift.

Host 1How many points?

Host 2Six redshift bins, from zero point five one to two point three three. They drop DESI's nearest sample, at zero point two nine five, because it only gives an averaged distance rather than a line-of-sight one.

Host 1And the third?

Host 2Cosmic chronometers: fifteen measurements of the expansion rate from about zero point one eight to nearly two. Background: you take massive galaxies that stopped forming stars long ago, compare their ages at slightly different redshifts, and the age difference per step in redshift gives you the expansion rate directly.

Host 1Any quality filter on those?

Host 2They only use the chronometer points with a full covariance matrix, meaning the correlations between points and the systematic errors are included. That is one of the explicit upgrades over the twenty twenty-one paper.

Under the hood

Host 1Now the machinery. Nonparametric means what here?

Host 2It means they never write down a formula for how expansion evolves, no specific dark energy model. Instead they put a Gaussian process prior on the expansion rate, which in plain terms says only that the curve must be smooth, with the data deciding its shape.

Host 1How do you actually compute with a smooth curve of unknown shape?

Host 2They pin the curve down at a number of nodes, and the redshifts of those nodes are themselves free and averaged over. The smoothness assumption then becomes a joint bell-curve prior on the expansion rate at those nodes, using a standard squared-exponential kernel.

Host 1And how smooth is smooth?

Host 2That is set by two hyperparameters, an amplitude and a length scale, and they marginalize over both rather than fixing them. The paper stresses this, because several competing studies fix those values, which can make results look tighter than they are.

Host 1And then?

Host 2Everything is sampled at once: curvature, the supernova absolute brightness, the expansion curve at the nodes and the hyperparameters. They use Hamiltonian Monte Carlo in a tool called PyStan, with flat priors on curvature and on the supernova brightness.

Host 1You said absolute scales do not matter. How do they know?

Host 2They checked directly. They rescaled each dataset by arbitrary constants, which amounts to changing the supernova brightness, the overall expansion rate, or the ruler size, and got identical curvature results.

Host 1So the ruler size of one hundred forty-seven megaparsecs is just a placeholder?

Host 2Yes, they fix it to the standard-model value and it drops out. That is the whole case for why the Hubble tension does not leak in.

The stress tests

Host 1The title says stress-testing. What is being stressed?

Host 2Three things: which supernova sample you use, which expansion-rate probe you use, and whether the two cover the same redshifts. For each of the three supernova samples they run three combinations, so nine analyses in total.

Host 1Spell out the three combinations.

Host 2Supernovae plus DESI over the full range, which is the baseline. Supernovae plus DESI but keeping only supernovae above redshift zero point five one, where DESI's points start. And supernovae plus cosmic chronometers instead of DESI.

Host 1Why the overlap cut?

Host 2Below the first DESI point, the expansion curve is shaped by the supernovae alone. Cutting the nearby supernovae tests whether the answer is really coming from that low-redshift stretch, or from a mismatch between the near and far parts of a sample.

Results: the baseline

Host 1Numbers. The baseline first.

Host 2With Pantheon Plus, omega K is zero point two one nine, give or take about zero point one one. With DES Dovekie it is zero point three three zero, plus zero point zero nine eight or minus zero point one five zero. With Union three it is zero point two six zero, give or take about zero point one two five.

Host 1All positive. All open.

Host 2All positive, and they agree with each other well within one standard deviation. Flat, omega K of zero, sits roughly two standard deviations away from each central value.

Host 1Three samples agreeing sounds like three confirmations.

Host 2The authors explicitly say not to read it that way. All three share the same DESI data, so it is not three independent indications. And as background, the supernova compilations also share many of the same supernovae.

Host 1How does this compare to their twenty twenty-one result?

Host 2That earlier error bar was zero point two six; the baseline error bars now are around zero point one one to zero point one five. The paper describes it as roughly a factor of two to two and a half sharper.

Results: the overlap test

Host 1Now cut the nearby supernovae.

Host 2The constraints get weaker, as expected with less data. DES Dovekie gives zero point two two six and Union three zero point two zero zero, both give or take about zero point one seven to zero point one eight, so still positive and close to baseline.

Host 1And Pantheon Plus?

Host 2Pantheon Plus flips sign: minus zero point zero eight five, plus zero point one nine one or minus zero point two zero two. That is slightly closed, and comfortably consistent with flat.

Host 1Is that a contradiction?

Host 2No; the authors say there is no evidence of inconsistency between the samples, because the error bars are wide. But the open preference weakens in all three, most for Pantheon Plus, which tells you the signal leans partly on the low-redshift supernovae.

Host 1Does the cut cleanly isolate the DESI range?

Host 2No, and the paper is upfront about it. Even a supernova at redshift zero point six still has its predicted distance built from the expansion curve all the way from zero, so the low-redshift reconstruction never fully disappears from the calculation.

Results: swapping the probe

Host 1Third test, chronometers instead of DESI.

Host 2Pantheon Plus gives zero point two one nine again, give or take about zero point one two. Union three gives zero point two five zero, plus zero point one three four or minus zero point one five one.

Host 1And DES Dovekie?

Host 2Zero point one nine seven, but with a lopsided spread, plus zero point one eight three or minus zero point three zero four. That is wide enough that flat sits inside the central sixty-eight percent range, so on its own it is no preference at all.

Host 1So what does the chronometer test establish?

Host 2That the open lean in the full supernova samples is not a quirk of DESI specifically, since an independent way of measuring the expansion rate lands in the same place. It does not make the lean stronger.

Host 1Sum it up.

Host 2Eight of nine analyses put the central value on the open side, the only exception being Pantheon Plus with the overlap cut. The lean survives changing samples and probes, but it depends on having the full redshift lever arm.

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How it compares

Host 1Where does this sit among other recent curvature results?

Host 2Several similar model-independent analyses lean open too. Jiang and colleagues in twenty twenty-four found central values between zero point zero six seven and zero point one three five with uncertainties of about zero point zero nine, and Li and Zhang found zero point zero nine, give or take zero point zero nine, from DESI's first release alone.

Host 1Those are smaller than this paper's values.

Host 2Smaller central values, and slightly smaller error bars despite older DESI data. The paper attributes that to methodology, since that Gaussian process approach could also use DESI's transverse and averaged distances.

Host 1Anyone who got the other sign?

Host 2Yes, and it is the most interesting comparison. Favale and colleagues in twenty twenty-five, with DESI's second release, the Dark Energy Survey supernovae and chronometers, found minus zero point one four three, give or take zero point zero eight five, which leans closed.

Host 1Same data family, opposite answer. Why?

Host 2The paper's explanation is that Favale's team used the chronometers to set the absolute supernova brightness and the ruler size, whereas this paper uses only shape. The authors say that makes a direct comparison difficult and may contribute to the difference, which is a hedge, not a resolution.

Host 1And the microwave background analyses?

Host 2Those reach curvature to about one part in a thousand, two orders of magnitude tighter than this paper, but only by assuming a specific model. The paper notes a recurring finding there: giving dark energy more freedom can weaken or even remove the open-universe hint.

Why it matters

Host 1So what is the principle?

Host 2That you can check the geometry of space using consistency alone: two independently measured curves have to fit together, and curvature is the hinge. When the data get good enough, you no longer need to assume a model or an absolute scale to ask the question.

Host 1And what does that change, and for whom?

Host 2For cosmologists arguing about whether the recent open-universe hints are an artefact of forcing dark energy into too simple a formula, it is a useful data point. Since this method assumes no dark energy formula and the hint persists, the authors argue it is unlikely to be only that artefact.

Host 1There was also something about parameters that do not enter.

Host 2Matter density, neutrino masses and the optical depth play no role here; the first two are absorbed into the reconstructed expansion curve and the last is irrelevant to these probes. So the tangle of degeneracies between them, which complicates model-based analyses, does not touch this one.

Host 1Now the important part. What does this not mean?

Host 2It does not mean the universe is open. It is a lean of roughly two standard deviations, in a field where results at that level come and go regularly.

Host 1Does it mean inflation is in trouble, or the microwave background is wrong?

Host 2Neither, and the paper claims neither. The microwave background bounds are much tighter but model-dependent, and the authors say directly that the two cannot be compared head to head.

Host 1And is it independent of the other tensions?

Host 2Not necessarily, and the authors make that point themselves. A redshift-dependent quirk in the supernova or DESI data could push this analysis and the model-based ones in the same direction; what they can say for sure is that their result does not come from an assumed matter density.

Honest caveats

Host 1Caveats. Start with the biggest.

Host 2Supernova systematics that change with redshift. The paper names dust properties and how the exploding stars themselves may evolve over cosmic time, says these are hard to quantify with current data, and does not model them beyond the published covariances.

Host 1Second.

Host 2Nonparametric is not assumption-free, in the paper's own words. It still assumes the universe is the same in all directions on large scales, that curvature is a single constant, and that expansion history is smooth enough for a Gaussian process to describe.

Host 1That smoothness assumption sounds like it could do a lot of work.

Host 2It is the one that replaces a dark energy model, so it matters. A sharp feature in the expansion history would be smoothed away; the paper states that its independence from dark energy models holds insofar as the expansion is sufficiently smooth.

Host 1Anything in the numbers themselves?

Host 2One wording point. The results section says flat lies about two standard deviations from the baseline values, while the discussion and conclusions say the preference reaches up to about three in some cases. Reading the table ourselves, the strongest case, DES Dovekie with DESI, sits a bit over two lower error bars from zero, so we would describe all of this as about two.

Host 1And the overlap test.

Host 2It showed the signal leans on low-redshift supernovae, and Pantheon Plus flipped sign without them. That does not falsify anything, but it is exactly where a redshift-dependent calibration issue would hide.

Host 1Reproducibility?

Host 2The data are available on request, and the code will be released after publication. So right now nobody outside the team can rerun these nine analyses.

Host 1And the size of the claimed curvature itself?

Host 2This is our own back-of-envelope arithmetic, not the paper's. An omega K of zero point two five would mean a curvature radius only about twice the Hubble distance, a curvature far stronger than the microwave background analyses allow under the standard model, which is one reason these geometrical results are read as hints rather than measurements.

What to watch next

Host 1What should listeners watch for?

Host 2The authors' code release, so others can test the method's sensitivity to choices like the kernel and the node placement. And more independent work on redshift-dependent supernova systematics, which the paper flags as the key open issue.

Host 1And new data?

Host 2Background, not from the paper: DESI is still collecting, and later releases should tighten the expansion curve. New supernova samples from the Vera Rubin Observatory and, later, the Nancy Grace Roman Space Telescope should test exactly the low-redshift and high-redshift consistency this paper relies on.

Host 1What would change your mind either way?

Host 2If the open lean grows as errors shrink and survives across supernova samples with independent calibrations, it gets serious. If it shrinks toward zero, or tracks one sample's calibration, it was noise or systematics, which is how most two-standard-deviation hints end.

Sign-off

Host 1That is the deep dive: Stress-testing the spatially flat Universe, by Turner, Dhawan, Vagnozzi, Townsend and Massey. The link is in the show notes.

Host 2Short version: a model-free, calibration-free check finds space leaning slightly open, at about two standard deviations, with sensible caveats. Interesting, worth watching, not settled.

Host 1Every Friday this same feed carries the roundup episode: five new papers as trailers, and you pick which become deep dives. This has been the Hastrup Review of Science.