Beyond the Reach of the Test

Physics has theories that run ahead of any experiment: hidden dimensions, partner particles, a landscape of universes. What they promised, what was found when the tests caught up, and how physicists now argue over the rest.

An elegant answer and its assumptions

Hold two protons apart and ask which of two forces between them is stronger: their electric repulsion or their gravitational attraction. The electric force wins by a factor of about 10³⁶, a one followed by thirty-six zeros. Gravity holds planets and galaxies together only because matter is electrically neutral on the whole and mass always adds. Particle by particle it is absurdly weak, and nobody knows why.

In 1999 Lisa Randall and Raman Sundrum published an answer in a four-page paper.[1] Suppose there is a fifth dimension, a short interval with a four-dimensional world at each end, a “brane”. Suppose the space between them is curved in a particular way, so that every energy scale shrinks exponentially as one moves from one brane to the other. Gravity is concentrated near the first brane; we live on the second. Seen from our end, gravity is diluted by a factor that grows exponentially with the distance between the branes, and a modest distance is enough to produce the enormous ratio. The geometry is a solution of Einstein’s equations in five dimensions, with suitable sources of energy at each end. No absurdly large number has to be put in by hand.

It is a beautiful result, and it comes with a list of assumptions. There is an extra spatial dimension, for which there is no evidence. Our world is confined to a brane, which the original model assumes rather than derives. The geometry has the particular warped shape that produces the effect. And the distance between the branes has to be held fixed by something, which the first paper did not supply; Walter Goldberger and Mark Wise added a mechanism for it later the same year.[2] The result that explains the weakness of gravity depends on all four.

That is the shape of a whole class of proposals in fundamental physics. A real puzzle is answered by new structure that cannot yet be seen directly, and the answer is judged at first by its internal qualities: whether it is consistent, economical, beautiful, connected to other ideas. The worry about such theories is easy to state. Explaining one unexplained fact with several unverified assumptions may not be progress. The defence is also easy to state. Several of the best theories in physics began exactly this way and waited decades for their test.

This essay looks at both, with the record so far. It takes the best-known proposals of the past fifty years, asks what each predicted that could be checked, and reports what happened when experiments got there. Then it turns to the argument that has grown up among physicists and philosophers about how to judge the theories that experiments cannot yet reach. It does not try to settle that argument. The candidate quantum theories of gravity themselves, strings, loops and the rest, are described on this blog in The Smooth and the Grainy; here the subject is the evidence.

The problem behind the problem

A puzzle that is not a measurement

The weakness of gravity is an observation. The “hierarchy problem” that Randall and Sundrum set out to solve is something slightly different, and the difference matters for everything that follows.

The strength of gravity is set by the Planck mass, about 1.2 × 10¹⁹ GeV (a GeV is roughly the mass of a proton). The masses of the known particles are set by the Higgs field, whose particle weighs 125 GeV. The ratio of the two, squared, is about 10⁻³⁴. In the quantum theory of particles, the mass of the Higgs is not a fixed input: it receives corrections from every other particle it interacts with, and for a particle like the Higgs, those corrections grow with the highest energy at which the theory still holds. If the Standard Model of particle physics were valid all the way to the Planck scale, the corrections would be some 10³⁴ times larger than the mass we observe, and the starting value would have to cancel them to thirty-four decimal places. Nothing in the theory forbids this. It just looks like a coincidence of an extreme kind.

Gerard ’t Hooft gave this unease a precise form in 1980: a small number in a theory is “natural” only if setting it to zero would add a symmetry, so that the symmetry protects it.[3] The Higgs mass has no such protection in the Standard Model. A theory in which it did, or in which the corrections stopped growing just above the energies already explored, would be natural. That expectation is what made the 1990s and 2000s so confident that new particles would appear at the Large Hadron Collider (LHC), at energies around a thousand GeV, a TeV.

The argument had a real success behind it. In 1974 Mary Gaillard and Benjamin Lee used the same style of reasoning on the decays of kaons: a certain correction would be far too large unless a new particle cut it off below about 2 GeV. They estimated the mass of the charm quark that way, months before it was discovered, at about 1.5 GeV.[4] Gian Giudice, who reviewed the history in 2008, notes that this case, unlike others often cited, is historically accurate: it was the argument actually used before the discovery.[5]

The argument also has a famous failure. The energy of empty space, the cosmological constant, should by the same reasoning be of the order of the largest energies in the theory. The measured value is smaller than that estimate by a factor usually quoted as about 10¹²⁰.[6] No symmetry has been found that explains it, and naturalness, applied there, gets the answer wrong by the largest margin in physics.

So the hierarchy problem is not a disagreement between a theory and a measurement. It is a disagreement between a measurement and an expectation about how the constants of nature ought to look. Whether that expectation is a reliable guide is itself part of what the experiments below were testing.

The dimensions

Hidden dimensions, large and warped

The idea of a fifth dimension is older than quantum mechanics. In 1921 Theodor Kaluza showed that Einstein’s equations written in five dimensions produce, in four, both gravity and Maxwell’s electromagnetism, provided the extra dimension is somehow invisible.[7] Oskar Klein suggested in 1926 that it is curled up too small to see. The idea has never been confirmed and never been dropped, and string theory, which needs six extra dimensions, revived it in a larger form.

In 1998 Nima Arkani-Hamed, Savas Dimopoulos and Gia Dvali turned it into something testable.[8] Suppose the particles and forces we know are confined to our four-dimensional world, but gravity spreads into extra dimensions as well. Then gravity looks weak to us because most of it is elsewhere, and its true strength could be comparable to the other forces at TeV energies, which would make the hierarchy problem disappear. The size of the extra dimensions follows from the arithmetic. With a single one it would have to be as large as the solar system, which is ruled out. With two, it comes out at about a millimetre. Below that distance, the paper pointed out, gravity would stop following Newton’s inverse-square law and start falling off faster.

That was a prediction anyone with a sensitive enough balance could check, and the group at the University of Washington known as Eöt-Wash did. Their torsion balances measure the pull between test masses a fraction of a millimetre apart. In 2020 they reported Newton’s law holding down to 52 micrometres, about the width of a hair, and ruled out any gravitational-strength deviation with a range above 38.6 micrometres.[9] The version of the theory with two extra dimensions at a millimetre is gone. Versions with more, smaller extra dimensions survive, because they predict deviations at distances no balance can reach, and the colliders, which looked for the energy that would leak into the extra dimensions, have not seen it either.[10]

The Randall–Sundrum proposal was built to avoid those tests. Its extra dimension is small, and its signature is different: a series of heavy partners of the graviton, the hypothetical particle of gravity, with masses around a TeV and couplings strong enough to be produced at a collider. The paper called the consequences “new and dramatic”.[1] They would show up as narrow bumps in the number of photon pairs or lepton pairs at a particular energy. With the full data of the LHC’s second run, ATLAS excluded such a graviton below 4.5 TeV for the standard choice of coupling, and CMS below 2.47 to 4.78 TeV depending on the coupling.[11][10]

That does not kill the idea. The model has a free parameter that sets the mass of the first graviton, and it can be moved up. But each move reintroduces part of the problem the model was built to solve: if the new particles are at several TeV rather than one, the Higgs mass again needs some cancellation, smaller than before but no longer absent. The model has meanwhile found other uses. Its geometry is a slice of the curved anti-de Sitter space that appears in the best-understood link between gravity and quantum field theory, and it is used in models of dark matter. A 2021 paper by Adrián Carmona, Javier Castellano Ruiz and Matthias Neubert, reported on this blog in December 2024, used a warped dimension to explain the hierarchy, the masses of the fermions and the amount of dark matter together.[12] Like the original, it is a consistent possibility with no observation behind it yet.

Fig. 1 — Experiments reach directly to about 10⁶ GeV, counting the most energetic cosmic ray ever recorded. The proposals that were meant to be seen at the LHC were placed inside the shaded band; the energies of grand unification and quantum gravity lie far outside it. Sources: ATLAS 2021; Fly’s Eye 1995[13]; BICEP/Keck 2021; own arithmetic.

The partners

A partner for every particle

The most popular answer to the hierarchy problem was not geometry but symmetry. Supersymmetry pairs every known particle with a partner whose spin differs by a half: the electron with a “selectron”, the quarks with “squarks”, the gluon with a “gluino”. If the symmetry holds, the corrections to the Higgs mass from each particle are cancelled by those from its partner, and the cancellation is automatic. It was first studied for its own mathematical interest in the early 1970s. In 1981 Savas Dimopoulos and Howard Georgi proposed a version in which the partners were light enough to protect the Higgs and in which the three forces of particle physics meet at a single strength at very high energy.[14] The fact that the measured strengths of the forces do meet, in this version and not in the Standard Model alone, became one of its strongest arguments.

Supersymmetry also offered a dark matter particle, the lightest partner, and it is a necessary part of string theory as usually formulated. For thirty years it was the default expectation for what the LHC would find. For the partners to do their job without a large cancellation, the most important of them had to weigh no more than a few hundred GeV to about a TeV.[5]

The LHC found the Higgs boson in 2012 and, so far, nothing else. In the simplest models used to interpret the searches, ATLAS excludes gluinos below 2.3 TeV and the partners of the light quarks below 1.85 TeV.[15] Those limits assume particular decay patterns, and more elaborate spectra, with partners close in mass or decaying in unusual ways, escape them. Supersymmetry as a symmetry of nature is not excluded and cannot be by any finite collider, because its partners could be at any mass. What has been excluded is the version that was natural, the one whose partners had to be within reach.

The search also showed how the field behaves when a hint appears. In December 2015 ATLAS and CMS each reported a small excess of photon pairs at 750 GeV. Theorists wrote hundreds of papers explaining it, some 500 by one count; Physical Review Letters published four of them together with an editorial on the flood.[16][17] In August 2016, with more data, the excess was gone: a statistical fluctuation.[18] Critics took the episode as a sign of a field hungry for anything to explain; others pointed out that working out the possibilities quickly is exactly what theorists are for.

The proton

Waiting for a proton to die

Some predictions of very high-energy theories can be tested at low energy, because the effect is tiny but accumulates. The clearest case is grand unification. In 1974 Howard Georgi and Sheldon Glashow proposed that the strong, weak and electromagnetic forces are parts of a single force that splits at an energy around 10¹⁵ to 10¹⁶ GeV.[19] No machine will ever reach that. But the theory puts quarks and electrons in the same family, so a quark can occasionally turn into a positron, and the proton, which has been thought to be eternal, decays.

For the simplest version the predicted lifetime was about 10²⁹ years, give or take a factor of a hundred.[20] That is vastly longer than the age of the universe, but a tank holding thousands of tonnes of water contains some 10³³ protons, so a detector of that size should see many decays a year. Underground detectors were built in the 1980s for this purpose. They saw nothing, and the simplest grand unified theory was ruled out.[20] The detector built to succeed them, Super-Kamiokande in Japan, has watched 50,000 tonnes of water since 1996. By 2020 it had set the proton’s lifetime for the most expected decay above 2.4 × 10³⁴ years.[21] The minimal supersymmetric version of Dimopoulos and Georgi, which predicted a faster decay into a kaon, is also considered ruled out.[22][23]

Grand unification has survived in other versions, whose proton lifetimes are longer or harder to pin down, and the review of the Particle Data Group notes that outside supersymmetry the prediction is “hard to quantify”.[23] Hyper-Kamiokande, with about ten times the useful volume, is due to start in 2028[24] and aims to reach about 10³⁵ years.[23] The story so far: a theory of energies no collider can reach made a definite prediction, the prediction failed, and the idea continued in modified forms whose predictions are less definite.

The landscape

A theory with too many answers

String theory is the largest of these programmes and the one whose testability is most disputed. Its strengths and costs as a theory of quantum gravity are set out in The Smooth and the Grainy. The question here is narrower: what does it predict?

The six extra dimensions of string theory have to be curled up, and the way they are curled up, together with the fluxes of fields threading them, determines the particles and constants of the four-dimensional world that results. Around 2000 it became clear that the number of such arrangements is enormous. Raphael Bousso and Joseph Polchinski showed how a vast number of possible values of the vacuum energy could arise; later counts put the number of apparently consistent, roughly realistic vacua at “perhaps greater than 10⁵⁰⁰”.[25][26] Leonard Susskind called the collection the landscape.[6]

If all those vacua exist somewhere, as regions of a much larger universe produced by inflation, then the constants we measure may not be predictable at all. They would be accidents of where we are, constrained only by the requirement that observers can exist. That is the anthropic argument, and it has one result in its favour. In 1987 Steven Weinberg argued that if the cosmological constant varied from region to region, observers could only find themselves where it was small enough for galaxies to form, and that a value somewhat below that bound would help with puzzles of the time.[27] In 1998 two teams measuring distant supernovae found that the expansion of the universe is accelerating, which means a small positive cosmological constant.[28][29] Supporters count this as a successful prediction made in advance. Critics note that Weinberg’s own paper concluded that the bound was probably too loose to explain the smallness, and that an argument which accepts almost any value that permits observers forbids very little.

Joseph Polchinski, one of the architects of the landscape, made a deliberately informal Bayesian estimate in 2015: starting from even odds and updating on the observed cosmological constant and on what string theory implies, he arrived at a 94% probability that the multiverse exists. He added that “Probability 94% is two sigma; two sigma effects do go away”, against the five sigma required for the Higgs boson.[30]

The record

When the test came back

The case for patience rests on history. Einstein’s general relativity is the example most often given, and it deserves care, because one of its classic tests was not a prediction. The orbit of Mercury turns slowly around the Sun by an amount that Newton’s gravity could not fully account for; the discrepancy, about 43 seconds of arc per century, had been known since Urbain Le Verrier’s analysis of 1859. Einstein calculated it from his new theory in November 1915 and found the right number.[31] That was a powerful check, but it explained a known fact; it was not a forecast. The genuine predictions came after: light bent by the Sun, measured at the eclipse of 1919;[32] clocks running slower in stronger gravity, measured decades later; and gravitational waves, proposed in 1916 and detected in September 2015, ninety-nine years on.[33] The full history is in The Happiest Thought.

Other long waits ended well. Wolfgang Pauli proposed the neutrino in 1930 to rescue the conservation of energy in radioactive decay; Clyde Cowan, Frederick Reines and colleagues detected it at a nuclear reactor in 1956.[34] Peter Higgs and others proposed the field that bears his name in 1964; the particle was found in 2012, after forty-eight years and the largest machine ever built.[35][36][37]

Fig. 2 — Proposals from 1915 to 2000 and the year of a decisive test, or 2026 where there has been none. “Ruled out as first proposed” means the version stated at the time; modified versions may survive. The choice of proposals and the dates of the open ones are my own summary; sources in the text.

Figure 2 puts these beside the proposals of the previous sections. Two patterns are visible. Long waits are not unusual for ideas that were later confirmed, so the absence of a test after twenty or thirty years is not, by itself, evidence against an idea. And the proposals made since the 1970s to solve the hierarchy problem were tested sooner than they could have been, because they were built to be within reach, and when the test came, it came back empty.

There is also a quantum-gravity idea that met its test. Several approaches suggested that if space is grainy at the Planck scale, light of higher energy might travel very slightly slower than light of lower energy. In 1998 Giovanni Amelino-Camelia and colleagues pointed out that the effect would accumulate over cosmological distances and could be seen in the arrival times of photons from gamma-ray bursts.[38] In May 2009 the Fermi telescope recorded a burst about 7 billion light years away; a photon of 31 GeV arrived within the burst’s first second. The result required the scale of any such effect, in its simplest linear form, to lie above the Planck mass, beyond where the theories put it.[39] One prediction of quantum gravity was thus tested with a telescope, and the simplest version failed.

The sky gives one more indirect reach. If the early universe went through inflation, a brief exponential expansion, it should have left gravitational waves whose imprint can be seen in the polarisation of the cosmic microwave background. None has been found. The current limit, from the BICEP and Keck telescopes at the South Pole, puts the ratio of these waves to ordinary density fluctuations below 0.036,[40] which in the simplest models means inflation happened at an energy below about 1.4 × 10¹⁶ GeV, close to the scale of grand unification. Inflation has had confirmations too: it predicted that the fluctuations would be very slightly stronger on large scales than on small, and the Planck satellite measured just such a tilt.[41]

The debate

The argument about the argument

The usual test for whether a theory is scientific comes from Karl Popper: it must say something that an observation could prove false.[42] By that standard the landscape, and string theory in its present state, are in difficulty, and in December 2014 the cosmologists George Ellis and Joe Silk said so in Nature. Their article’s summary reads: “Attempts to exempt speculative theories of the Universe from experimental verification undermine science”.[43]

Among the views they criticised was that of the philosopher Richard Dawid, whose 2013 book String Theory and the Scientific Method argued that physicists can legitimately gain confidence in a theory before any test.[44] Dawid names three kinds of evidence. The first is that scientists have looked hard for alternatives and found none. The second is that theories with similar credentials have tended to succeed in the past. The third is that a theory built for one problem turns out, unexpectedly, to explain others.[45] String theory, on his reading, scores on all three: it is the only known consistent quantum theory of gravity that includes the other forces, it continues a line of theories that succeeded, and it explained things it was not built for, such as the entropy of black holes.

A year after the Nature article, Dawid, Ellis, Silk and others met in Munich at a workshop called “Why Trust a Theory?”; David Gross opened it, Polchinski’s contribution contained the 94% estimate, and the talks became a book.[46] The positions did not converge, but they were stated clearly.

Carlo Rovelli, who works on the rival loop approach to quantum gravity, answered Dawid that non-empirical arguments may guide research but cannot make a theory reliable. “No sensible person would entrust her life to a prediction of string theory,” he wrote; the distinction between established and speculative theories is “an essential ingredient of science”.[47] Sean Carroll argued the opposite case for the multiverse: theories are judged by how well they explain, by Bayesian reasoning and by their empirical success overall, and “‘falsifiability’ was never the way that scientific theories were judged”.[48] Sabine Hossenfelder’s 2018 book Lost in Math made the broader charge that criteria like naturalness and beauty, untested as guides, had led the field to expect discoveries that did not come.[49] The question of whether beauty is a reliable guide in physics is the subject of Invented, Then Unavoidable on this blog.

Among particle physicists the reckoning has been quieter. In 2017 Giudice, the same theorist who had set out the naturalness argument in 2008, described the period after the LHC’s first results as “the post-naturalness era”.[50] He used the old Greek sense of crisis, krisis, a turning point rather than a collapse: “What is changing is the feeling that the paradigm that so successfully led to the Standard Model may not be the right tool to make further progress.” As for what replaces it, he wrote: “Of course I have no idea.”

Lee Smolin’s version of the critique, that the field gave up Einstein’s insistence on background independence too early, is in an interview reposted on this blog, Are we approaching quantum gravity all wrong?.

The scorecard

What the record shows, and what it does not

Fig. 3 — Six proposals and the state of their tests in 2026. The “result” column refers to the version first proposed; most have modified successors. My own summary; sources in the text.

Three different situations hide under the phrase “untestable theory”, and they deserve different verdicts.

The first is a theory that has not been tested yet but says clearly what a test would show. Large extra dimensions in 1998, minimal grand unification in 1974, energy-dependent light speed in 1998: each made a definite statement, and each was checked within a few decades, by a torsion balance, a tank of water and a gamma-ray telescope. All three failed in their first form. That is science working as Popper described it, and the ideas behind them, even where they survive, are now more constrained.

The second is a theory that could be tested but can always move its predictions out of reach. Supersymmetry and warped dimensions are in this position: nothing forbids their partners from being heavier than any machine can make, and each step up costs part of the reason they were proposed. Here the evidence works slowly and by attrition. It is fair to say that the natural versions have been tested and not found; it is not fair to say the ideas have been refuted.

The third is a theory whose main claims concern regions or energies that may never be observed, like the landscape of string vacua or the full theory of quantum gravity. Here the debate between Dawid and his critics is real and unresolved. Its defenders point out that this has happened before, that atoms and neutrinos were also once beyond any experiment, and that the mathematics developed along the way has been useful elsewhere. Its critics point out that the waiting time is no longer decades but open-ended, and that confidence without a test is what the scientific method was designed to prevent.

Underneath the second and third sits the question the hierarchy problem raises. If the constants of nature are what they are, unexplained but not in need of explanation, then the problem was an expectation, and the LHC has shown that the expectation was wrong at least in its simplest form. If the constants do need a reason, the reason may lie at energies we cannot reach, or in a landscape of possibilities we cannot see. The experiments that could still change the picture are known: a heavier collider, Hyper-Kamiokande’s search for proton decay, the next measurements of the microwave background, and table-top experiments that ask whether gravity can entangle two masses, described in The Smooth and the Grainy. Each of them is a test of whether nature is natural.

On method and tools

This essay was written with Claude Opus 5.5 (Anthropic): human specification, editorial direction and critical review; machine research, drafting and figure generation. The subject came from a conversation Luis had about the Randall–Sundrum model; he asked for a neutral review of the proposals in fundamental physics that outrun experiment. The bibliographic records come from INSPIRE-HEP; the limits and quotations were checked in the full texts on arXiv (Lee et al. 2020; ATLAS and CMS 2021; Super-Kamiokande 2005 and 2020; Fermi 2009; Randall & Sundrum, Arkani-Hamed et al., Giudice 2008 and 2017, Polchinski, Rovelli, Carroll, Dawid 2017, Douglas & Kachru) and in the Particle Data Group review. The Ellis & Silk comment is paywalled and only its summary was read; the count of 750 GeV papers is from Physics World. The force ratio of two protons, the Higgs-to-Planck ratio, the collision energy of the Fly’s Eye cosmic ray on a proton at rest and the energy scale of inflation allowed by the BICEP/Keck limit (single-field slow roll) are my own arithmetic. Figures 2 and 3 are my summary, with the choices of proposal and date stated in their captions. scripts/beyond_test.py draws the figures and the cover, a scale of energies from 1 GeV to the Planck energy, lit where experiments have been.

Authored by: Luis Matos Ferreira — Physicist, Developer, Writer

Related essays on this blog

1. The Smooth and the Grainy — why gravity and quantum mechanics clash, five candidate theories, and a table-top test.
2. The Happiest Thought — how general relativity was found, and the tests that took a century.
3. An Act of Desperation — how quantum mechanics was assembled, from Planck to the Bell tests.
4. Invented, Then Unavoidable — mathematics, beauty and what the aesthetic criterion tracks.
5. Loops of Space — loop quantum gravity, in Luis’s 2024 notes.

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