A New Modified Gravity Theory To Explain Dark Matter and Dark Energy

The hypothesis of the scale invariance of the macroscopic empty space, which intervenes through the cosmological constant, has led to new cosmological models. They show an accelerated cosmic expansion after the initial stages and satisfy several major cosmological tests. No unknown particles are needed. 

Developing the weak-field approximation, we find that the here-derived equation of motion corresponding to Newton's equation also contains a small outward acceleration term. Its order of magnitude is about Newton's gravity ( being the mean density of the system and the usual critical density). 

The new term is thus particularly significant for very low density systems. A modified virial theorem is derived and applied to clusters of galaxies. For the Coma Cluster and Abell 2029, the dynamical masses are about a factor of 5–10 smaller than in the standard case. This tends to leave no room for dark matter in these clusters.

Then, the two-body problem is studied and an equation corresponding to the Binet equation is obtained. It implies some secular variations of the orbital parameters. The results are applied to the rotation curve of the outer layers of the Milky Way. Starting backward from the present rotation curve, we calculate the past evolution of the Galactic rotation and find that, in the early stages, it was steep and Keplerian. Thus, the flat rotation curves of galaxies appear as an age effect, a result consistent with recent observations of distant galaxies by Genzel et al. and Lang et al. 

Finally, in an appendix we also study the long-standing problem of the increase with age of the vertical velocity dispersion in the Galaxy. The observed increase appears to result from the new small acceleration term in the equation of the harmonic oscillator describing stellar motions around the Galactic plane. 

Thus, we tend to conclude that neither dark energy nor dark matter seems to be needed in the proposed theoretical context.

Andre Maeder "Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?" 849(2) The Astrophysical Journal 158 (November 10, 2017) (pre-print here).

From the introduction:

The group of invariances subtending theories plays a most fundamental role in physics (Dirac 1973). The Maxwell equations in absence of charge and currents are scale invariant, while the equations of General Relativity (GR) do not enjoy this additional property (Bondi 1990). We know that a general scale invariance of the physical laws is prevented by the presence of matter, which defines scales of mass, time and length (Feynman 1963). However, the empty space at large scales could have the property of scale invariance, since by definition there is nothing to define a scale. The real space is never empty in the Universe, however the properties of the empty space intervene through ΛE, the Einstein cosmological constant. It is true that the vacuum at the quantum level is not scale invariant, since some units of mass, length and time can be defined on the basis of the Planck constant. However, the large scale empty space differs by an enormous factor from the quantum scales. Thus, alike we may apply Einstein’s theory at large scales even if we cannot do it at the quantum level, we may make the scientifically acceptable hypothesis that the properties of the empty space represented by ΛE at macroscopic and astronomical scales are scale invariant. In this work (see also Maeder (2017a), hereafter called Paper I), we are exploring further consequences of the above hypothesis. The MOND theory has been noted to have this property (Milgrom 2009), but since this is a classical theory, it is not contained in a cosmological model.  

The consequences are far reaching, as shown by the cosmological models in Paper I which consistently include, through ΛE, the invariance of the empty space at macroscopic scales. These models clearly account for the acceleration of the cosmic expansion, without calling for some unknown particles of any kinds. Several cosmological tests have been performed, they concern the distance vs. redshift z relation, the magnitude–redshift m − z diagram, the plot of the density parameters Ωm vs. ΩΛ, the relations of the Hubble constant H0 with the age of the Universe and Ωm, the past expansion rates H(z) vs. z and the transition from braking to acceleration, and more recently the past temperatures of the CMB vs. redshifts (Maeder 2017b). All these tests are impressively satisfactory and they open the possibility that the so-called dark energy may be an effect of the scale invariance of the empty space at large scales. Therefore, it is scientifically reasonable to explore further consequences of the above hypothesis to see whether at some stage it meets severe contradictions with the observations or whether it continues to show agreement. We now especially consider the dynamical evidences of dark matter.  

As the internal dynamics of clusters of galaxies and the rotation of galaxies are at the origin of the claim for the existence of dark matter, we focus here on these dynamical problems. In Sect. 2, we study the Newtonian approximation of the geodesic equation consistent with the above key hypothesis. In Sect. 3, we examine the dynamical or virial masses of clusters of galaxies in the scale invariant context and apply our results to the Coma and Abell 2029 clusters. In Sect. 4, we study the scale invariant two-body problem and then discuss the outer rotation curve of the Galaxy. The case of galaxies at significant redshifts is also considered. Sect. 5 gives brief conclusions. In an Appendix, we examine the age - velocity dispersion relation of stellar groups in the Galaxy, in particular in the vertical direction where there is no consensus on the origin of the relation.

And the conclusion:

There is progressively an accumulation of tests supporting the hypothesis of the scale invariance of the empty space at large scales, see also Milgrom (2009). Firstly, there are the various cosmological tests (Maeder 2017a) mentioned in the introduction, as well as the test on the past CMB temperatures vs. redshifts (Maeder 2017b). Now, the studies of the clusters of galaxies, of the rotation curves of the Milky Way and of high redshift galaxies, as well as of the vertical velocity dispersion of stars in the Milky Way, all appear positive. The long standing problems of the dark energy (Maeder 2017a) and of the dark matter may possibly find some solutions in terms of scale invariance. In this context, it is noteworthy that it has been claimed that halos of dark matter particles are inconsistent with a large variety of astronomical observations and in particular given the absence in the data of evidence for dynamical friction on the motions of galaxies due to these particles (Kroupa 2015).  

These various results are encouraging and the hope is that they will stimulate future works. The list of problems that await further studies is long. In this context, we again stress a central point of methodology, the tests to be valuable need to be internally coherent and not use ”observations” implicitly involving in their derivations other cosmological models or mechanical laws.

Stern criticism of this approach is found at Backreaction. She puts the conclusion in the middle of her write-up, and follows it with analysis of why the math is wrong:

For those of you who merely want to know whether you should pay attention to this new variant of modified gravity, the answer is no. The author does not have a consistent theory. The math is wrong.

Maeder introduces a conformal prefactor in front of the metric. You can always do that as an ansatz to solve the equations, so there is nothing modified about this, but also nothing wrong. He then looks at empty de Sitter space, which is conformally flat, and extracts the prefactor from there.

He then uses the same ansatz for the Friedmann Robertson Walker metric (eq 27, 28 in the first paper). Just looking at these equations you see immediately that they are underdetermined if the conformal factor (λ) is a degree of freedom. That’s because the conformal factor can usually be fixed by a gauge condition and be chosen to be constant. That of course would just give back standard cosmology and Maeder doesn’t want that. So he instead assumes that this factor has the same form as in de Sitter space.

Since he doesn’t have a dynamical equation for the extra field, my best guess is that this effectively amounts to choosing a weird time coordinate in standard cosmology. If you don’t want to interpret it as a gauge, then an equation is missing. Either way the claims which follow are wrong. I can’t tell which is the case because the equations themselves just appear from nowhere. Neither of the papers contain a Lagrangian, so it remains unclear what is a degree of freedom and what isn’t. (The model is also of course not scale invariant, so somewhat of a misnomer.)

Maeder later also uses the same de Sitter prefactor for galactic solutions, which makes even less sense. You shouldn’t be surprised that he can fit some observations when you put in the scale of the cosmological constant to galactic models, because we have known this link since the 1980s. If there is something new to learn here, it didn’t become clear to me what.

I would argue that a form of equation that can seemingly fit cosmology data, galaxy rotation data and cluster data, even if the derivation is wrong and contains flaws due to lack of rigor, is still a meaningful step forward, because more modified gravity approaches lack that aspect than have it. But, time will tell.