To the editors:
Jean-Pierre Luminet’s essay brilliantly summarizes the current state of affairs concerning dark matter. He also examines a number of alternatives, including modified gravity. The latter discussion is mostly limited to Mordehai Milgrom’s modified Newtonian dynamics (MOND), with only a brief mention of Erik Verlinde’s emergent gravity model.1 In this singular focus, Luminet is not alone. Throughout much of the literature, MOND is presented as the archetypal modified gravity theory, despite its many shortcomings. In its original form, MOND is no more than a phenomenological modification of Newton’s second law and as such does not even respect basic conservation laws.
In actuality, the literature of modified gravity is far richer, and the search for modified theories of gravitation began far earlier. Foremost among the modern theories of modified gravity is the scalar-tensor theory of Pascual Jordan, Carl Brans, and Robert Dicke.2 This theory promotes the Newtonian gravitational constant to a dynamical scalar field, the value of which may vary in space and time. In its unmodified form, Jordan–Brans–Dicke theory runs afoul of precision tests that have since been performed using spacecraft in the solar system. Nonetheless, it still serves as a prototype for many new modified gravity theories.
The family of theories known as f(R) is closely related to Jordan–Brans–Dicke theory.3 This nomenclature refers to the presence of the Ricci curvature scalar R in the Lagrangian formulation of general relativity first presented by David Hilbert, i.e., the Einstein–Hilbert Lagrangian. It is possible to replace the scalar R with a function f(R), e.g., f(R) = R2 or f(R) = 1/R, and construct a sensible theory of gravitation. Most such theories can be shown to be fundamentally equivalent to Jordan–Brans–Dicke theory, but there are subtle differences.
Entire families of modified gravity theories, including Jordan–Brans–Dicke theory and f(R) theories, can be investigated using the parameterized post-Newtonian (PPN) framework. The roots of this approach go all the way back to the pioneering work of Arthur Eddington, who first used such a parameterization to measure how a relativistic theory of gravitation deviates from the Newtonian prediction.4 Modern versions of such parameterization also cover more exotic families of theories, including theories that introduce unusual features for the metrical field, such as torsion or nonmetricity.5 As many of the PPN parameters are measured using observations such as precision radio navigation or lunar laser ranging, a number of theories can be excluded, but some tantalizing candidates remain.
Another strong constraint on modified theories of gravitation comes from the recent multispectral observation of GW170817.6 The simultaneous arrival of a gravitational wave signal and signals in the electromagnetic spectrum from a neutron star merger yield strong constraints on so-called bimetric theories. These are theories in which not all constituent fields couple to the same metrical field.7 One such theory is Jacob Bekenstein’s relativistic generalization of MOND, known as tensor–vector–scalar gravity (TeVeS).8
Other challenges may be less stringent. The apparent absence of dark matter in the dwarf galaxy NGC 1052-DF2 is consistent with the view that most observed dwarf galaxies are tidally disturbed satellites of their hosts.9 As such, their velocity dispersions will appear much greater than warranted by their observed visible mass, thus mimicking dark matter. If this view is valid, NGC 1052-DF2 may be one of the few tidally undisturbed dwarf galaxies out there. This possibility is consistent with modified gravity theories that predict little or no deviation from Newtonian dynamics for such small galaxies.
At the same time, a viable modified gravity theory must do a lot more than account for galactic dynamics. Extremely precise measurements now exist for minute temperature deviations of the cosmic microwave background from its mean value.10
Computerized surveys that involve millions of galaxies offer statistics on the distribution of matter at various cosmic scales.11 These data sets are modeled with exquisite accuracy in the context of the standard, so-called concordance model of cosmology.12 In this collision, less cold dark matter plays a fundamental role governing structure growth. A modified gravity theory that does away with dark matter must offer a suitable alternate mechanism.
Constructing a modified theory of gravitation that satisfies all these constraints is incredibly challenging. It is easy to add terms to the Einstein–Hilbert Lagrangian to produce an alternative theory. But it is very difficult to construct an alternative theory that does not run afoul of all the observational data collected about galaxies and the universe since the days of Georges Lemaître and Edwin Hubble. Yet the continuing absence of any direct observation of dark matter provides a strong incentive for research in this direction.
A modified theory of gravitation may be able to tackle issues that challenge the concordance model. One such issue is the Hubble tension, the apparent discrepancy between values of the Hubble parameter, which measures cosmic expansion, derived from data obtained from our local neighborhood versus data derived from cosmological observations.13 Many modified theories of gravitation yield a different relationship between these values, which may be more consistent with observation. It is, of course, entirely possible that the Hubble tension is rooted in another issue altogether: the notion that the universe is not only homogeneous and isotropic on average, but that the Milky Way region is representative of the average density. If it turns out that the Milky Way is situated in a large cosmic void, the Hubble tension may vanish. Other resolutions, such as the presence of systemic bias in certain families of measurement, have also been offered by many authors.
There is also the issue of the cosmological constant, originally proposed by Einstein to create a stationary solution of his field equations. It was reintroduced after the discovery from the observed luminosity-distance relationship of type Ia supernovae that the expansion rate of the universe is increasing.14 But how is this constant interpreted? It has been known for more than a hundred years that it can be viewed as a medium with strong negative pressure, which in turn shows up as negative effective mass density in the Newtonian approximation of Einstein’s theory. Could such a medium exist? An obvious candidate, with the right equation of state, would be the zero-point energy of the quantum fields of the Standard Model of particle physics. But its calculated value is dozens of orders of magnitude higher than the value deduced from the observed acceleration. It would count as a major success for any modified gravity theory if it could offer an alternative explanation for cosmic acceleration, resolving or eliminating this embarrassing discrepancy, known as the cosmological constant problem.15
Unambiguous direct detection of dark matter would settle the question decisively. Until such a detection is confirmed, the possibility remains open that the dynamics attributed to dark matter is, in fact, due to deviations from the predictions of Einstein’s theory of relativity on the scale of galaxies and beyond. The search for a viable modified theory of gravity is a difficult challenge but it may not be futile.
Viktor Toth
Jean-Pierre Luminet replies:
I would like to thank Viktor Toth for his brilliant commentary. He brings additional theoretical and observational insights to the dark matter problem and, more specifically, to the modified theories of gravity that aimed at solving this problem without necessitating the existence of hypothetical particles of nonbaryonic matter. These particles, while predicted by various high energy physics theories, are yet to be detected experimentally.
Toth is right in pointing out that my paper essentially reduces the question of modified gravity to the primitive model—in both senses of the word—of Mordehai Milgrom. I can offer at least two justifications for what may appear to be an oversimplification of the problem.
The first is that my essay remains only a succinct summary of a vast problem that arises when analyzing the apparently abnormal movements of luminous matter. To explain these movements, I invoke either certain exotic forms of dark matter, or a modification of the laws of gravity, or—as I briefly mention at the end of my essay—a possible combination of the two. Much more development would be needed to treat the full problem exhaustively, as evidenced by the abundant literature on the subject.
The second reason is a personal bias. As a result of my training as a physicist in general relativity, I have always felt somewhat reticent toward the MOND model proposed by Milgrom. As Toth rightly points out, this model is purely phenomenological and does not respect even basic conservation laws. The theory has certainly been improved through its relativistic generalization proposed by Jacob Bekenstein and other researchers. But as far as I know, there is still no convincing MOND theory, i.e., one capable of accounting for the dynamics of celestial objects at different scales in a universal model. The change in gravity that could explain the kinematics of stars within galaxies is not the same as that to explain the dynamics of galaxies within their clusters. MOND models also do not provide a detailed account for gravitational lenses. Former colleagues from the Paris Observatory have pointed out that it is quite possible to reach the same success as MOND through a new kind of matter, dipolar dark matter, retaining general relativity for the law of gravity.16
Toth’s reply offers a remarkable panorama of the different theories of modified gravity. As such, his letter could have been the subject of a full-fledged essay. Among the theories he cites, the popularized version of Milgrom is undoubtedly the weakest.
As a relativistic physicist, I am obviously aware of the existence of the Jordan–Brans–Dicke tensor-scalar theory,17 in which the gravitational interaction is transmitted by a scalar field as well as the tensor field of general relativity. The gravitational constant G is no longer assumed to be constant, and its inverse, 1/G, is replaced by a field that can vary in space and time. Given the extremely small deviations from Albert Einstein’s classical general relativity predicted by Jordan–Brans–Dicke’s theory, it is generally considered that the two theories cannot be distinguished from each other by astronomical observations. Jordan–Brans–Dicke’s theory thus represents a minority point of view in modern theoretical physics. For this reason I did not mention it in my short essay. The same goes for the so-called f(R) theories pointed out by Toth.
The recent advent of gravitational astronomy allows for the detection of gravitational waves resulting from the coalescence of compact objects such as black holes and neutron stars. This development also creates new possibilities for testing the different theories of gravity with more precision. Toth mentions the multispectral observations of GW170817—an event attributed to the coalescence of a pair of neutron stars that resulted in a double gravitational and electromagnetic signal. Measuring the arrival times of these two signals—which seem to coincide with a high degree of precision—strongly constrains not only the classical theories of gravity referred to as bimetric,18 of which the MOND theory is a very particular case, but also some theories of quantum gravity, such as superstrings.19 I recently discussed these constraints as part of a book on the various approaches in quantum gravity.20 In a long chapter, I examine another essential problem mentioned by Toth: dark energy. Within the scientific community, dark energy is debated no less fiercely than dark matter. There is an enormous discrepancy between the experimental value of dark energy deduced from the acceleration of cosmic expansion and the theoretical value of quantum vacuum energy calculated within the framework of supersymmetry theory. I express the opinion, shared by some prestigious colleagues, that this discrepancy is a clear sign of the failure of the latter. This viewpoint is an alternative to the more common position, which is to dismiss the interpretation of dark energy in terms of the cosmological constant, originally introduced by Einstein for the wrong reasons. Georges Lemaître—the real founding father of modern relativistic cosmology21—was the first to associate the constant to the energy of the quantum vacuum in its fundamental state.
Toth also mentions a subject that has been hotly debated for the last few years known as the Hubble tension. Namely, the apparent discrepancy in measurements of the expansion rate of the universe as derived from local observations versus cosmological observations. Rather than posing a fundamental challenge to general relativity, which forms the basis for the current standard model of cosmology, I share Toth’s opinion that the tension may arise from some simplifying assumptions associated with the current model. Toth mentions the hypothesis of homogeneity and isotropy. This has only been tested on a very large spatial scale, but is already known to be violated on the scale of galaxy clusters. There is good reason to think that this tension might arise artificially from the hypothesis of a strictly zero spatial curvature. The tension could be eliminated by a slightly positive space curvature, which would be perfectly compatible with the latest observations of the Planck telescope.22 It would also have a better basis on a physico-mathematical level.23
What I particularly appreciate in this letter is that, while advocating theories of modified gravity to the detriment of nonbaryonic dark particle models, Toth admits it is difficult to obtain a viable modified gravity theory that is both mathematically coherent and experimentally verified. But this certainly does not render futile the research done in this subfield of fundamental physics, a body of work that my short essay may not have done justice to.
