Oscillating cosmological force modifies Newtonian dynamics

In the Newtonian limit of general relativity a force acting on a test mass in a central gravitational field is conventionally defined by the attractive Newtonian gravity (inverse square) term plus a small repulsive correction (cosmological force), which is proportional to the slow acceleration of the universe expansion. In this paper we consider the effect of fast (quantum) oscillations of the universe scale factor on the cosmological force experienced by a test mass. Such fast oscillations were suggested recently by Wang et al. (Phys. Rev. D 95, 103504 (2017)) as a potential solution of the cosmological constant problem. The fast fluctuations of the cosmological scale factor induce strong changes to the current sign and magnitude of the average cosmological force, thus making it one of the potential probable causes of the modification of Newtonian dynamics in galaxy-scale systems.

The discrepancy between visible masses in galaxies and galaxy clusters and their dynamics predicted by the application of Newton's laws was first noticed by Fritz Zwicky almost hundred years ago [1]. The proposed solutions to this mystery range from dark matter halos around the galaxy centers [2] to various modifications of Newton's laws themselves [3,4]. However, despite almost hundred years of experimental and theoretical studies, the ultimate solution of this mystery remains elusive [5].
On the other hand, it was well established for quite a while [6,7] that in an expanding universe the equations of motion of a test body in a central gravitational field must be modified even in the Newtonian limit of general relativity. The easiest way to see it is to write down the metric that describes the spacetime in the vicinity of a point mass M placed in an expanding flat background [8]: where G is the gravitational constant, a(t) is the cosmological scale factor, and  is the comoving radial coordinate, which relates to the proper radial coordinate r as The geodesics corresponding to the line element (1) are defined by Introducing L as the constant angular momentum per unit mass the radial equation of motion for a test body in the Newtonian limit may be written as As we can see from Eq.(5), the Newtonian equation of motion in an expanding universe is modified by an additional small cosmological force [6,7] r a where m is the mass of the test body. The current value of a  is believed to be positive based on the experimental measurements of the universe deceleration parameter where H is the Hubble parameter. According to the Planck spacecraft data, in the current epoch 55 . 0   q [9], resulting in a small repulsive addition to the attractive Newtonian gravity term in eq. (5): At large distances from the center, the velocity distribution of the test bodies would be modified as 2 2 leading to an apparent upper limit on the radius R M of a gravitationally bound system at a given M: Assuming the mass of the Milky Way to be M~10 12 solar masses, the projected R M for the Milky Way should be about 3x10 6 light years, or about 30 times larger than its radius. This should mean that the so determined cosmological force plays very little role in the Milky Way dynamics, and it cannot be responsible for the observed flattening of the rotation curves in a typical galaxy [2]. Wang et al. [10] assumed the global metric of the universe to have the cosmology's standard Friedmann-Lemaître-Robertson-Walker (FLRW) form while allowing spatio-temporal inhomogeneity in the scale factor ) , ( x t a  . After solving the full coordinate-dependent Einstein field equations, they have obtained the following where  2 >0 for quantum fluctuations of the matter fields (see Eqs.(41,42) from [10]).
Due to the stochastic nature of these fast quantum fluctuations, is not strictly periodic. However, as demonstrated in [10], its effect on the gravitating system is still similar to a periodic function (see also further refinements of these arguments in [11]).
Let us evaluate the effect of these fast quantum fluctuations of the universe scale factor on the cosmological force described by Eqs. (5,6). Following the standard analysis of system dynamics under the influence of a fast oscillating force [12], let us consider a toy cosmological model in which the evolution of the scale factor of the universe is separated into the slow and fast components: In this decomposition a 0 (t) represents the observed slow cosmological evolution of the universe, while <<a 0 represents the typical amplitude of the scale factor fluctuations introduced in [10] (in order to simplify our consideration we have neglected spatial dependence of by replacing ) (x   with its spatial average). We will assume that  -1 is much faster than the typical time scales of galaxy evolution and the evolution of universe as a whole, since as suggested in [10], the physical origin of these fluctuations is due to quantum effects. Under these assumptions the radial equation of motion of a test body in a central gravitational field will be modified as follows: which indicates that the cosmological force experiences fluctuations. If these fluctuations are fast enough, the  2 term may not be neglected compared to the 0 a  term. Following the standard treatment in [12], the proper radial coordinate of the test body should be expressed as where (t) represents small oscillations of the test body with respect to its slowly evolving radial position r 0 (t). By separating the fast and slow motion in Eq.(14), and by expanding to leading order in small quantities, the amplitude of these small oscillations may be defined from thus leading to As usual (see [12]), we will assume that the average kinetic energy of the fast oscillations of the test body 2 / 2   m contributes to its effective potential energy (similar to the introduction of the effective potential energy for an inverted (Kapitza) pendulum [12]), so that the average effective cosmological force in Eq.(5) must be re- where A~1 is a dimensionless positive constant, which exact magnitude needs to be determined either from the future theory of quantum gravity, or from the astronomical observations. Such an attractive cosmological force will result in the following velocity distribution of the test bodies far from the compact central mass: and at very large distances from the center a slow linear velocity increase will be observed as a function of radial distance: However, the real value of A 1/2 may be considerably larger.
The difference between the measured rotation curve and the rotation curve calculated based on the Newtonian dynamics of visible matter is typically attributed to a spherical dark matter halo [2,5], which is assumed to exist around virtually every galaxy. If such a spherical halo has approximately constant density, its mass distribution would be leading to similar linear dependences of the gravitational and cosmological forces on the radial coordinate. This means that the effect of the dark matter halo and the effect of the cosmological force may mimic each other, and that the total amount of dark matter in a typical galaxy needs to be carefully re-examined. For example, Fig. 2 illustrates very simple analytical simulations of the rotation curve in M33, which were performed assuming different magnitudes of A 1/2 without any dark matter halo at all. While such simple analytical simulations cannot be perfect, calculations performed using A 1/2~0 .09 show reasonable agreement with the astronomical data (compare results in Fig.2 with data points for M33 in Fig. 1), which is quite a success for such a simplified model. To summarize, based on the potential solution to the cosmological constant problem recently suggested in [10,11], which assumes fast quantum mechanical oscillations of the cosmological scale parameter, we have re-examined the sign and the magnitude of the cosmological force which modifies Newtonian dynamics around a central gravitating body in an expanding universe. The cosmological force in this model appears to be attractive, which means that it can mimic the effects of a spherical dark matter halo around a galaxy center. Therefore, the total amount of dark matter (if any) in a typical galaxy needs to be carefully re-examined by taking into account the effect of the cosmological force.
Since the long-distance behaviour of the cosmological force is supposed to be universal, we propose to carefully re-evaluate the galaxy rotation curves at very large distances from the galactic centers. If these rotation curves show signs of a universal linear increase far from the galaxy center, such an effect may turn out to become a very important observational evidence of macroscopic quantum gravity.
Finally, since sign of the cosmological force due to scale factor fluctuations appears to be always attractive, the popular Big Rip scenarios [6] will also need to be re-examined.