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While there is overwhelming evidence for dark matter (DM) in galaxies and galaxy clusters, all searches for DM particles have so far proved negative. It is not even clear whether only one particle is involved or a combination of particles, their masses not precisely predicted. This non-detectability raises the possible relevance of modified gravity theories: MOND, MONG, etc. Here we consider a specific modification of Newtonian gravity (MONG) which involves gravitational self-energy, leading to modified equations whose solutions imply flat rotation curves and limitations of sizes of clusters. The results are consistent with current observations including that involving large spirals. This modification could also explain the current Hubble tension. We also consider the effects of dark energy (DE) in terms of a cosmological constant.

Over the past few decades, there has been a plethora of sophisticated experiments involving massive sensitive detectors trying to catch faint traces of the elusive Dark Matter (DM) particles. But so far all of these efforts have been to no avail. Most of these detectors are designed to look for Weakly Interacting Massive Particles (WIMPS) which are much heavier than the proton (several GeV), with there being no definite prediction for the masses of these heavy particles [^{−3} - 10^{−5} eV). Here again there is no definite theoretical prediction for the axion mass. A recent work [

So considering these negative results of all experiments so far, can we try to understand why we don’t see them? While evidence for DM as such is overwhelming, otherwise galaxies and galaxy clusters would fly apart but for the additional gravity they provide, the question is what type of particles constitutes them, or can there be alternate ideas to understand the enhanced gravity. Is only one type of DM particle involved or a combination? Theories do not predict what combination of particles or what type of particles will fix the ratio of DM to baryonic matter as about six. We suggest some reasons why we have not seen these elusive particles.

One such suggestion that has been made [

Another possibility is that the DM particles could have much weaker cross-sections and their masses may not be in the range assumed. This would lead to non-detection. A more drastic conclusion would be that the particles in the predicted mass range, fluxes and coupling may not exist. They may be a different kind of particle, interacting only gravitationally with higher masses, the fluxes would be smaller and number of events less.

Alternate Theories of GravityIn the absence of detection of DM particles so far, it is natural to explore alternate possibilities such as modification of Newtonian gravity that could explain the galaxy rotation curves and motion of clusters. There have been recent approaches in this direction [

The theory required an ad hoc introduction of a fundamental acceleration a 0 ≈ 10 − 8 cm / s 2 . When the acceleration approaches a 0 , the Newtonian law giving the field strength is modified as:

a = ( G M a 0 ) 1 / 2 r (1)

where a is the acceleration, r is the radial distance, M is the central mass. And this gives a constant velocity, i.e. flat rotation curve for the galaxies, with the constant velocity, (at the galactic outskirts) given by, v c = ( G M a 0 ) 1 / 4 .

These results can also be arrived at by considering a minimum acceleration given by [

a min = G M r max 2 (2)

Here r max is the radius of the structure corresponding to the minimum acceleration and it sets the limit for the size of large scale structures, which follows from Equation (2) as, r max = ( G M a min ) 1 / 2 , hence we get,

v c = ( G M a min ) 1 / 4 (3)

This velocity is independent of r beyond r max , which is consistent with observation. For Milky Way, this constant velocity (Equation (3)) ~300 km/s, same order as that observed. For clusters of galaxies, such as Virgo ( M ≈ 1.25 × 10 15 M ⊙ ) and Coma ( M ≈ 7 × 10 14 M ⊙ ) clusters, the velocity (from Equation (3)) is ~1500 km/s which is again in accordance with what is observed. The constraints on the size of large scale structures such as galaxies, clusters and super clusters, i.e. r max = ( G M a min ) 1 / 2 , closely matches with observations (

It was also discussed in recent papers that the requirement that the attractive gravitational binding self-energy density of large scale structures (such as galaxies, clusters, superclusters, etc.) should at least be equal to the background repulsive DE (cosmological constant Λ) density implies a mass-radius relation of the type [

M r 2 = c 2 G Λ ≈ 1 g / cm 2 (4)

for the observed value of Λ ~ 10 − 56 cm − 2 . Here M and r correspond to mass and radius of these structures. This relation holds true for primeval galaxies as well as those at present epoch [

Large scale structure | r_{lim} (in cm) | r_{obs} (in cm) |
---|---|---|

Virgo Cluster | 2.28 × 10^{24} | 7.09 × 10^{24} |

Coma Cluster | 2.16 × 10^{25} | 9.46 × 10^{24} |

Omega Centauri | 1.6 × 10^{18} | 8.13 × 10^{19 } |

Saraswati Supercluster | 1.15 × 10^{25} | 2 × 10^{26} |

Laniakea Supercluster | 2.58 × 10^{25} | 2.36 × 10^{26} |

Horologium Supercluster | 2.5 × 10^{25} | 5 × 10^{26} |

Corona Borealis Supercluster | 7.47 × 10^{25} | 3.1 × 10^{26} |

The flat rotation curves can also be explained by considering Modifications of Newtonian Gravity (MONG). By adding a gravitational self-energy term to the Poisson’s equation we get,

∇ 2 ϕ + K ( ∇ ϕ ) 2 = 4 π G ρ (5)

where ϕ ( ~ G M r ) is the gravitational potential and the constant K ~ G 2 / c 2 . The gravitational self-energy density is given by K ( ∇ ϕ ) 2 , and also contributes to the gravitational field along with the matter density ρ . For small values of the density ρ , (for e.g. at the outskirts of galaxies) we have,

∇ 2 ϕ + K ( ∇ ϕ ) 2 = 0 (6)

The solution of this equation yields,

ϕ = K ′ ln r r max (7)

where K ′ = G M r max is a constant. This gives the force of the form,

F = K ″ r (8)

where K ″ = ( G M a min ) 1 / 2 , is again a constant. The balance of centripetal force and gravitational force then gives, v 2 / r = K ″ / r .

This then implies the independence of v on r (i.e. flat rotation curve, v 2 = K ″ , which is a constant) for larger distances from the centre of the galaxy (i.e. for r > r max ). Including both gravitational self-energy and DE densities, the Poisson’s equation now takes the form,

∇ 2 ϕ + K ( ∇ ϕ ) 2 − Λ c 2 = 0 (9)

The general solution (for the potential ϕ ) can be written as:

ϕ = G M r + K ′ ln r r max + Λ r 2 c 2 (10)

We now make use of this general solution for different regimes of interest in the galaxy structure. Where matter density dominates, i.e. r < r max , we have ϕ ≈ G M r (solution of ∇ 2 ϕ = 4 π G ρ ) which gives a velocity varying linearly with distance. For r > r max , ( ∇ ϕ ) 2 term dominates, and ϕ goes as K ′ ln r r max , (giving a constant velocity) accounting for DM (solution of ∇ 2 ϕ + K ( ∇ ϕ ) 2 = 0 ). For r ≫ r max , ϕ goes as Λ r 2 c 2 , DE dominates (i.e. the cosmological constant term).

In the case of the Milky Way, the velocity flattens out beyond ~2 kpc, which is what is obtained from the above results (

In this connection, the extra term in the Poisson equation given by MONG, i.e. Equation (9) and its solution given by Equation (10) could have interesting consequences for current observations of super-spirals [

v = ( G M a min ) 1 / 4 ( ln r r max ) 1 / 2 (11)

where r max corresponds to the radius at which acceleration approaches a min . With r max = 20 kpc and super spiral extant r = r S S ≈ 200 kpc , this would give velocities ~450 km/s. In other words, the logarithmic term makes gravity stronger above r max (i.e., potential going as ln r , instead of 1/r), so that we do not need such colossal amounts of DM. This also implies a logarithmic correction to the Tully-Fisher relation.

The usual Friedmann equation now gets modified to:

R 2 ̇ R 2 = 8 π G ρ 3 + ( G M a 0 ) 1 / 2 ln R R max (12)

(R is the scale factor)

The second term can be seen as a modification in potential energy due to gravitational self-energy density in the usual balance between kinetic and potential energy terms, 8 π G ρ 3 (in the expanding Universe), i.e. the usual Newtonian analogue agreeing with the GR result.

With a 0 = 10 − 8 cm / s 2 , and the Universe having expanded at present to R > 10 28 cm the modified term will also contribute. With R ≈ 2 × 10 28 cm , ρ ≈ 10 − 29 g / cc , and the mass of the Universe, M = 2 π 2 R 3 ρ ≈ 10 56 g , the usual first term is ≈10^{21}, whereas the second term ≈10^{20}. This suggests that this extra term now manifesting itself would cause an increase of the expansion rate, i.e. a change in Hubble constant ( R ˙ 2 / R 2 ) by ~5%. This could perhaps account for the faster expansion rate seen at the present epoch.

Here we show that the modification of the gravitational field can provide an alternate explanation for the discrepancy in the value of the Hubble constant as implied by Planck observations of the CMBR in the early Universe and that deduced from other distance indicators in the present epoch [

The authors declare no conflicts of interest regarding the publication of this paper.

Sivaram, C., Arun, K., Prasad, A. and Rebecca, L. (2021) Non-Detection of Dark Matter Particles: A Case for Alternate Theories of Gravity. Journal of High Energy Physics, Gravitation and Cosmology, 7, 680-686. https://doi.org/10.4236/jhepgc.2021.72039