A New 200,000 Year Old Denisovan Genome

Bernard's blog does a good job of reviewing the recent publication of a 200,000 year old Denisovan genome. 

This Denisovan's life predates the emergence of modern humans from Africa, but overlaps with the existence of the earliest modern humans within Africa.

In a nutshell, the preprint is: Stéphane Peyrégne, et al., "A high-coverage genome from a 200,000-year-old Denisovan" bioRxiv (October 20, 2025). According to Bernard (via Google translate from French):

They sequenced the genome of molar Denisova 25. Initial results showed that the individual was male. Furthermore, the mitochondrial and Y chromosome haplogroups both belong to the Denisovan population.

The Latest Neutrino Oscillation Parameters

Background

Data from W and Z boson decays and from cosmology measurements, strongly favor a model with exactly three active neutrino flavors, as do requirements for mathematical consistent in the Standard Model of Particle Physics, which requires that each generation of Standard Model fermions (i.e. an up-like quark, a down-like quark, an electron-like charged lepton, and a neutrino) must be complete with four members. The lower bound on a fourth active neutrino mass is on the order of 45,000,000,000,000 meV, while we know that none of the other three active neutrino masses are more than about 900 meV, and we have strong indications that the largest of the three active neutrino masses is no more than about 70 meV.

When the Standard Model was first formulated in the 1970s, neutrinos were assumed to be massless fermions. Experiments proved that this couldn't be the case in 1998, and that neutrinos change flavors and oscillate between their three mass eigenstates. Since then, scientists have worked to determine their mass and additional properties arising from the fact that they have mass, which has culminated in a basic three favor, Dirac neutrino model of neutrino oscillation to which the data has been fit.

This neutrino oscillation behavior is characterized by two mass differences Δm(21) and Δm(32), whether the masses are in a "normal" or "inverted" hierarchy, and four parameters of the PMNS matrix: three of which describe the probability of each of the possible transitions between the three neutrino flavors, and one of which δCP describes charge parity violation (i.e. how those transition probabilities differ between neutrinos and antineutrinos).  

The two mass differences have been measured to decent precision. A normal mass hierarchy is favored by the experimental data, but not to terribly great statistical significance (the preference is close to two sigma). The three main mixing angles of the PMNS matrix have been measured to reasonable but modest precision, although it isn't entirely established if one of them is a bit less than 45º or a bit more than 45º (the data increasing favors a value that is a bit more than 45º). Attempts to measure δCP are very imprecise and generally can't entirely rule out the possibility that there is no CP violation in neutrino oscillation, the best fit values of measurements of δCP consistently favor near maximum CP violation in neutrino oscillations.

The world average measured value of those parameters are as follows (according to the Particle Data Group):

In addition, to fully characterize the properties of neutrinos in the basic three flavor, Dirac neutrino model to which experimental data is fitted, one needs to know the absolute rest mass of at least one of the neutrino mass eigenstates. The experimental upper bound on the mass of the lightest absolute neutrino mass eigenstate is about 800 meV. The experimental lower bound on the sum of the three neutrino mass eigenstates is on the order of 58 meV for a "normal" hierarchy of neutrino masses, and 110 meV. for an "inverted" hierarchy of neutrino masses Reasonably robust upper bounds on the sum of the three neutrino masses from cosmology models and astronomy measurements favor an upper bound for the sum of the three neutrino masses to around 130 meV, with some measurements putting it below the 110 meV cap allowed for an inverted hierarchy for neutrino masses with some more aggressive theoretical assumptions.

The New Paper

A new paper combines the latest data from two major neutrino physics collaborations (NOvA and T2K) to tighten up the precision of measurements of the Δm(32) and δCP parameters of neutrino oscillations, which is hard to do with a single collaboration's data, because the observables in each experiment depend upon more than one parameter, and it is hard to tell with just a single experiment's data, which parameter is driving those observables. But, since the mix of parameters that drive the observables in each experiment is different (in part, by design to allow for just this kind of combined data analysis), when the two collaborations' data are combined, these degeneracies in each individual collaboration's data can be minimized.

The new paper below improves the precision of the measurement of Δm(32) a bit, and also makes for a still very imprecise, but improved, measurement of δCP. 

The new combined measurement for Δm(32) is at the very low end of the current the world average plus or minus two sigma range.

The new paper rules out the possibility of zero CP violation in neutrino mixing at the 3 sigma level for an inverted neutrino mass hierarchy assumption, and at a roughly 2.4 sigma level of significance for a normal neutrino mass hierarchy assumption.

The landmark discovery that neutrinos have mass and can change type (or "flavor") as they propagate -- a process called neutrino oscillation -- has opened up a rich array of theoretical and experimental questions being actively pursued today. 

Neutrino oscillation remains the most powerful experimental tool for addressing many of these questions, including whether neutrinos violate charge-parity (CP) symmetry, which has possible connections to the unexplained preponderance of matter over antimatter in the universe. Oscillation measurements also probe the mass-squared differences between the different neutrino mass states (Δm^2), whether there are two light states and a heavier one (normal ordering) or vice versa (inverted ordering), and the structure of neutrino mass and flavor mixing. 

Here, we carry out the first joint analysis of data sets from NOvA and T2K, the two currently operating long-baseline neutrino oscillation experiments (hundreds of kilometers of neutrino travel distance), taking advantage of our complementary experimental designs and setting new constraints on several neutrino sector parameters. 

This analysis provides new precision on the Δm(32)^2 mass difference, finding 2.43+0.04−0.03 (−2.48+0.03−0.04) × 10^−3 eV^2 in the normal (inverted) ordering, as well as a 3σ interval on δCP of [−1.38π, 0.30π] ([−0.92π, −0.04π]) in the normal (inverted) ordering. The data show no strong preference for either mass ordering, but notably if inverted ordering were assumed true within the three-flavor mixing paradigm, then our results would provide evidence of CP symmetry violation in the lepton sector.

NOvA, T2K Collaborations, "Joint neutrino oscillation analysis from the T2K and NOvA experiments" arXiv:2510.19888 (October 22, 2025).

Further Neutrino Property Issues 

Additional parameters are needed if neutrinos are actually Majorana particles (i.e. if they are their own antiparticles), or if they oscillate with a "sterile" right handed neutrino which only interacts via neutrino oscillation (and not via the electromagnetic, weak, or strong forces of the Standard Model) which is often proposed as a way for Dirac neutrinos to acquire mass in what is called a see-saw mechanism. 

Most physicists believe that one of these two possibilities should be possible to give rise to a mechanism for mass generation in neutrinos. Mass generation via the Higgs mechanism is not a good fit for neutrinos since all neutrinos are "left handed" in parity, and all antineutrinos are "right handed" in parity, unlike all other Standard Model fermions which have both left and right parity versions of both their particles and their antiparticles, making four states possible.

The most definitive phenomenological parameter of Majorana neutrinos would be neutrinoless double beta decay, which has not been observed to high precision. But neutrinoless double beta decays involving Majorana neutrinos is a function of their absolute mass scale. It is more rare to the extent that neutrinos are less massive. And, current bounds on neutrinoless double beta decay are not so strong that this can be ruled out for reasonable neutrino mass scales, although it's getting close to that point. Majorana neutrinos would also have a more complicated oscillation behavior involving more mixing parameters than the Dirac neutrino model.

A Dirac neutrino model with a see-saw mechanism involving oscillation with a sterile neutrino would imply that the transition probabilities of the PMNS matrix parameters wouldn't be unitary. In other words, the probabilities of all the three flavor model transitions wouldn't add up to 100%, because some small percentage of neutrino oscillations would be to one or more sterile neutrino flavors. So far, the observed PMNS matrix parameters are consistent with unitarity. But since the measured parameters each have uncertainties, there is room within those uncertainties for transitions to an additional sterile neutrino flavor (or even to multiple sterile neutrino flavors). The upper bound on missing neutrino transition probabilities is quite low, however, and to make a see-saw mechanism work with such small experimentally allowed transition probabilities, the mass of a hypothetical sterile neutrino would have to be very high.

For the record, I don't like either solution and think that we need to find a "third way" mechanism for generating neutrino mass, although I don't know what it would be.

Because Deur Is Awesome, Even At His Day Job

Alexandre Deur's side hustle, described in the sidebar link, is his work on a gravitational explanation for dark matter and dark energy phenomena, which would solve several of the greatest unsolved problems in physics.

His day job is as a QCD physicist at Jefferson Lab, a U.S. Department of Energy particle physics facility in Newport News, Virginia. There, his progress in determining the value of the least accurately known Standard Model model coupling constant, and confirming that its running with energy scale is consistent with the Standard Model, is also a good thing. 

Unsurprisingly, the research done by him and his colleagues confirms that the Standard Model's running of the strong force coupling constant of quantum chromodynamics determined experimentally confirms the Standard Model over a huge range of energy scales (from hundreds of MeVs to about 14,000,000 MeV).

The strong force coupling constant is usually quoted with values converted using the beta-function that describes its running with energy scale in the Standard Model to the Z-boson mass of 91.188 ± 0.002 GeV (according to the Particle Data Group, inverse error weighted world average measurement). Its world average value converted to that energy scale is 0.1180(9).

The numerical values shown below are in a normalized scale, so the numerical value doesn't match the familiar number.

We discuss how the Bjorken sum rule allows access to the QCD running coupling αs at any scale, including in the deep infrared IR domain. The Bjorken sum data from Jefferson Lab, together with the world data on αs reported by the Particle Data Group, allow us to determine the running of α(s)(Q) over five orders of magnitude in four-momentum Q. We present two possible future measurements of the running of α(s)(Q) using the Bjorken sum rule: the first at the EIC, covering the range 1.5 < Q < 8.7 GeV, and the second at Jefferson Lab at 22 GeV, covering the range 1.0 < Q < 4.7 GeV.

A. Deur, "The strong coupling from the IR to the UV extremes: Determination of α(s) and prospects from EIC and JLab at 22 GeV" arXiv:2510.19556 (October 22, 2025) (Contribution to the proceedings of the "QCD at the Extremes" workshop, Sept. 1-11 2025).

The paper above discusses how proposed low energy experiments at the electron-ion collider at the Brookhaven Lab on Long Island, New York (EIC) and JLab would greatly reduce uncertainties in the measurement of the strong force coupling constant measurement at low energies (i.e. below 5,000 MeV) as shown by the chart below.

The Hunter-Gather To Bronze Age Transition In Kazakhstan

Bernard discusses a paper on ancient DNA from Kazakstan. It revealed that the hunter-gatherer population that persisted there until the late Neolithic era in Europe was roughly 95% replaced by Bronze Age early Indo-Europeans herders similar to the Sintashta and Andronovo cultures genetically. The paper is Haechan Gill, et al., "Ancient genomes from eastern Kazakhstan reveal dynamic genetic legacy of Inner Eurasian hunter-gatherers" (2025).

The paper also has many other secondary insights.

The samples from the current study are in yellow, with the Bronze age samples on the left, and in the MLBA clines, and the Neolithic samples in the Steppe HG cline on the right.

A Search For X17 Comes Up Empty And Assorted Astrophysics Papers

Today's preprint harvest was abundant and I have a little time to blog this morning.

An X17 paper

BESIII searched for an X17 boson and didn't find it. 

We report a direct search for a new gauge boson, X, with a mass of 17 MeV/c^2, which could explain the anomalous excess of e+e− pairs observed in the 8Be nuclear transitions. The search is conducted in the charmonium decay χcJ→XJ/ψ (J = 0,1,2) via the radiative transition ψ(3686)→γχcJ using (2712.4 ± 14.3) × 10^6 ψ(3686) events collected with the BESIII detector at the BEPCII collider. No significant signal is observed, and the new upper limit on the coupling strength of charm quark and the new gauge boson, ϵc, at 17 MeV/c^2 is set to be |ϵc| < 1.2 × 10^−2 at 90% confidence level. We also report new constraints on the mixing strength ϵ between the Standard Model photon and dark photon γ′ in the mass range from 5 MeV/c^2 to 300 MeV/c^2. The upper limits at 90% confidence level vary within (2.5−17.5) × 10^−3 depending on the γ′ mass.

BESIII Collaboration, "Search for a hypothetical gauge boson and dark photons in charmonium transitions" arXiv:2510.16531 (October 18, 2025).

Four astrophysics papers

There were several MOND or MOND-adjacent papers in today's preprints that I don't really have time to discuss at great length.

Stacy McGaugh, one of the leading members of the current generation of MOND researchers looks at pattern in the ordinary matter mass v. size relationship for galaxies in a large data set:

The mass-size relations of galaxies are generally studied considering only stars or only gas separately. Here we study the baryonic mass-size relation of galaxies from the SPARC database, using the total baryonic mass (Mbar) and the baryonic half-mass radius (R50,bar). We find that SPARC galaxies define two distinct sequences in the Mbar−R50,bar plane: one that formed by high-surface-density (HSD), star-dominated, Sa-to-Sc galaxies, and one by low-surface-density (LSD), gas-dominated, Sd-to-dI galaxies. The Mbar−R50,bar relation of LSD galaxies has a slope close to 2, pointing to a constant average surface density, whereas that of HSD galaxies has a slope close to 1, indicating that less massive spirals are progressively more compact. 

Our results point to the existence of two types of star-forming galaxies that follow different evolutionary paths: HSD disks are very efficient in converting gas into stars, perhaps thanks to the efficient formation of non-axisymmetric structures (bars and spiral arms), whereas LSD disks are not. 

The HSD-LSD dichotomy is absent in the baryonic Tully-Fisher relation (Mbar versus flat circular velocity Vf) but moderately seen in the angular-momentum relation (approximately Mbar versus Vf×R50,bar), so it is driven by variations in R50,bar at fixed Mbar. This fact suggests that the baryonic mass-size relation is the most effective empirical tool to distinguish different galaxy types and study their evolution.

Zichen Hua, Federico Lelli, Enrico Di Teodoro, Stacy McGaugh, James Schombert, "The baryonic mass-size relation of galaxies. I. A dichotomy in star-forming galaxy disks" arXiv:2510.17770  (October 20, 2025) (accepted by Astronomy & Astrophysics).

The creator of MOND muses in a public lecture about what a fundamental theory explaining MOND (a FUNDAMOND) has to look like:

In default of a fundamental MOND theory -- a FUNDAMOND -- I advocate that, alongside searching for one, we should try to identify predictions that follow from wide classes of MOND theories, if not necessarily from all. In particular, predictions that follow from only the basic tenets of MOND -- ``primary predictions'' -- are shared by all MOND theories, and are especially valuable. Such predictions permit us to test the MOND paradigm itself, or at least large parts of it, without yet having a FUNDAMOND. 

Concentrating on the deep-MOND limit, I discuss examples of either type of predictions. 

For some examples of primary predictions, I demonstrate how they follow from the basic tenets (which I first formulate). I emphasize that even predictions that pertain to the deep-MOND limit - namely, those that concern gravitating systems that have low accelerations everywhere -- require the full set of MOND tenets, including the existence of a Newtonian limit close to the deep-MOND regime. This is because Newtonian dynamics is a unique theory that all MOND theories must tend to in the limit of high accelerations, and it strongly constrains aspects of the deep-MOND regime, if the transition between the limits is fast enough, which is one of the MOND tenets.

Mordehai Milgrom, "The deep-MOND limit -- a study in Primary vs secondary predictions" arXiv:2510.16520 (a talk presented at the MOND workshop, Leiden, September 2025) (October 18, 2025).

The paper by Scholz below is an attempt to devise a "FUNDAMOND":

Under carefully chosen assumptions a single general relativistic scalar field is able to induce MOND-like dynamics in the weak field approximation of the Einstein frame (gauge) and to modify the light cone structure accordingly. 

This is shown by a Lagrangian model formulated in the framework of integrable Weyl geometry. It contains a Bekenstein-type (``aquadratic'') term and a second order term generating additional mass energy for the scalar field. Both are switched on only if the gradient of the scalar field is spacelike and below a MOND-typical threshold, like in the superfluid model of Berezhiani/Khoury. The mass term induces non-negligible energy and pressures of the scalar field and leads to gravitational light deflection compatible with MOND-ian free fall trajectories. In the weak field (Newton-Milgrom) approximation the Bekenstein term implies a deep MOND equation for the scalar field. In this model the external field effect of the MOND approach has to be reconsidered. This has important consequences for hierarchical systems like clusters, which may suffice for explaining their dynamics without additional dark matter

Erhard Scholz, "Einstein gravity extended by a scale covariant scalar field with Bekenstein term and dynamical mass generation" arXiv:2510.17704 (October 20, 2025).

Finally a notable dark matter search paper rules out a significant swath of dark matter particle parameter space, that most people assumed never existed (heavy charged dark matter):

There is a claim in the literature that charged dark matter particles in the mass range 100(qX/e)^2 TeV≤mX≤10^8(qX/e) TeV are allowed, based on arguing that heavy charged particles cannot reach the Earth from outside the magnetized region of the Milky Way (Chuzhoy-Kolb, 2009). We point out that this claim fails for physical models for the Galactic magnetic field. We explicitly confirm our argument by simulating with the software CRPropa the trajectories of heavy charged dark matter in models of the Galactic magnetic field.

Daniele Perri, Glennys Farrar, "The window on heavy charged dark matter was never open" arXiv:2510.17026 (October 19, 2025).

The Population Genetics Of Egypt Have Been Stable For A Long Time

An ancient DNA sample from ca. 2500 BCE in Egypt reveals a great deal of continuity in the population genetics of Egypt then and the population genetics of Egypt today. 

I didn't have a lot of time to look carefully at this study, but prior studies have shown a modest increase in sub-Saharan African admixture since then, due to the trans-Saharan slave trade in more recent time periods.

Ultralight Dark Matter

While ultra-light bosonic dark matter (ULDM) in a Bose-Einstein condensate (BEC) state could naturally account for the central core in some galaxies and resolve the core-cusp problem, the dark matter density distribution in the outer regions of galaxies remains less explored. We propose a trial wavefunction to model the ULDM distribution beyond the BEC core. We derive the corresponding rotation velocity curve, which shows excellent agreement with those of 12 dwarf spheroidal galaxies. The best-fit ULDM particle mass for each dwarf galaxy falls within a strikingly narrow range of m = (1.8−3.2) × 10^−23 eV.

Tian-yao Fang, Ming-Chung Chu, "Constraining Ultra-Light Dark Matter mass with Dwarf Galaxy Rotation Curves" arXiv:2510.12848 (October 14, 2025).

The best fit particle mass is in line with other studies and very close to the average mass-energy of a graviton, if they exist (and gravitons are, of course, bosons).

In general, ultralight bosonic dark matter proposals are are better fit to the data than any of the other dark matter particle models. 

Even warm dark matter, in the keV mass range, only barely improves upon failed cold dark matter and ultraheavy dark matter models. Self-interacting dark matter models have also not stood up well against the data from galaxy dynamics.

A Quantum Gravity Observation From Sabine

This basic idea has been floating around in quantum gravity circles for a while, but Hossenfelder's take is more cogent and careful than many of these attempts. Her model is basically a superdeterministic one.

I present a simple argument for why a fundamental theory that unifies matter and gravity gives rise to what seems to be a collapse of the wavefunction. The resulting model is local, parameter-free and makes testable predictions.

Sabine Hossenfelder, "How Gravity Can Explain the Collapse of the Wavefunction" arXiv:2510.11037 (October 13, 2025).

The conclusion states:

I have shown here how the assumption that matter and geometry have the same fundamental origin requires the time evolution of a quantum state to differ from the Schr¨odinger equation. This has the consequence that the ideal time evolutions which minimise the action are those with end states that are to good approximation classical. We can then identify these end states with the eigenstates of the measurement device. 

This new model therefore explains why quantum states seem to ‘collapse’ into eigenstates of the measurement observable, and how this can happen while preserving locality. Since the collapse process is governed by quantum gravitational contributions whose strength is known, the resulting model is parameter free. 

Collapse happens in this model whenever the accumulated phase difference between dislocated branches, τm|Φ12|, exceeds ∼ 1. The model’s phenomenology—notably the collapse itself—can be tested in roughly the same parameter range as other tests of the weak field limit of quantum gravity.

A Proposal To Explain The Neutrino Mixing Angles

Many papers try to explain fundamental constants in the Standard Model in terms of deeper relationships. This attempt to gain insight into the neutrino oscillation parameters is more thought provoking than most. 

We propose a geometric hypothesis for neutrino mixing: twice the sum of the three mixing angles equals 180∘, forming a Euclidean triangle. This condition leads to a predictive relation among the mixing angles and, through trigonometric constraints, enables reconstruction of the mass-squared splittings. 

The hypothesis offers a phenomenological resolution to the θ23 octant ambiguity, reproduces the known mass hierarchy patterns, and suggests a normalized geometric structure underlying the PMNS mixing. 

We show that while an order-of-magnitude scale mismatch remains (the absolute splittings are underestimated by ∼10×), the triangle reproduces mixing ratios with notable accuracy, hinting at deeper structural or symmetry-based origins. 

We emphasize that the triangle relation is advanced as an empirical, phenomenological organizing principle rather than a result derived from a specific underlying symmetry or dynamics. 

It is testable and falsifiable: current global-fit values already lie close to satisfying the condition, and improved precision will confirm or refute it. We also outline and implement a simple χ2 consistency check against global-fit inputs to quantify agreement within present uncertainties.

Mohammad Ful Hossain Seikh, "A geometrical approach to neutrino oscillation parameters" arXiv:2510.06526 (October 7, 2025).

Does Non-Perturbative QCD Have A Cosmological Constant Analog?

A new paper explores a potential parallel between non-perturbative quantum chromodynamics (the physics of the strong force that binds quarks into hadronic structures) and gravity. This isn't entirely surprising, as both are non-abelian gauge theories. And, it suggests that features like the cosmological constant may have a natural source in a non-abelian quantum gravity theory.

Einsteins gravity with a cosmological constant Λ in four dimensions can be reformulated as a λϕ^4 theory characterized solely by the dimensionless coupling λ∝G(N)Λ (G(N) being Newton's constant). The quantum triviality of this theory drives λ → 0, and a deviation from this behavior could be generated by matter couplings. Here, we study the significance of this conformal symmetry and its breaking in modeling non-perturbative QCD. The hadron spectra and correlation functions are studied holographically in an AdS(5) geometry with induced cosmological constants on four-dimensional hypersurface. 

Our analysis shows that the experimentally measured spectra of the ρ and a(1) mesons, including their excitations and decay constants, favour a non-vanishing induced cosmological constant in both hard-wall and soft-wall models. Although this behavior is not as sharp in the soft-wall model as in the hard-wall model, it remains consistent. Furthermore, we show that the correction to the Gell-Mann-Oakes-Renner relation has an inverse dependence on the induced cosmological constant, underscoring its significance in holographic descriptions of low-energy QCD.

Mathew Thomas Arun, Nabeel Thahirm, "On the role of cosmological constant in modeling hadrons" arXiv:2510.06380 (October 7, 2025).

A New Paper Argues For Dark Matter Over MOND

This paper argues for dark matter particles rather than modified gravity based upon observations of very low mass dwarf galaxies, although it has a very small sample size of just twelve galaxies.

A tight correlation between the baryonic and observed acceleration of galaxies has been reported over a wide range of mass (10^8 < Mbar/M⊙ < 10^11) - the Radial Acceleration Relation (RAR). This has been interpreted as evidence that dark matter is actually a manifestation of some modified weak-field gravity theory. 

In this paper, we study the radially resolved RAR of 12 nearby dwarf galaxies, with baryonic masses in the range 10^4 < Mbar/M⊙ < 10^7.5, using a combination of literature data and data from the MUSE-Faint survey. We use stellar line-of-sight velocities and the Jeans modelling code GravSphere to infer the mass distributions of these galaxies, allowing us to compute the RAR. We compare the results with the EDGE simulations of isolated dwarf galaxies with similar stellar masses in a ΛCDM cosmology. 

We find that most of the observed dwarf galaxies lie systematically above the low-mass extrapolation of the RAR. Each galaxy traces a locus in the RAR space that can have a multi-valued observed acceleration for a given baryonic acceleration, while there is significant scatter from galaxy to galaxy. 

Our results indicate that the RAR does not apply to low-mass dwarf galaxies and that the inferred baryonic acceleration of these dwarfs does not contain enough information, on its own, to derive the observed acceleration. 

The simulated EDGE dwarfs behave similarly to the real data, lying systematically above the extrapolated RAR. We show that, in the context of modified weak-field gravity theories, these results cannot be explained by differential tidal forces from the Milky Way, nor by the galaxies being far from dynamical equilibrium, since none of the galaxies in our sample seems to experience strong tides. As such, our results provide further evidence for the need for invisible dark matter in the smallest dwarf galaxies.

Mariana P. Júlio, et al., "The radial acceleration relation at the EDGE of galaxy formation: testing its universality in low-mass dwarf galaxies" arXiv:2510.06905 (October 8, 2025) (Accepted for publication in A&A).

How Flat Is The Universe?

The planet Earth is, to a good approximation, a perfect sphere. But, it isn't perfectly spherical.

Space-time in the universe as a whole is, to a good approximation, perfectly Euclidian. But, it has some curvature.

The magnitude by which the Earth differs from being a perfect sphere (in relative terms) is roughly similar to the magnitude by which the universe differs from being perfectly Euclidian. And, both on average and at the greatest extremes, Earth differs less from being perfectly spherical in relative terms, than the space-time of the universe differs from being perfectly Euclidean.