Modifications of Newtonian dynamics from higher moments of quantum spin connection in precanonical quantum gravity

TL;DR

The work investigates whether Milgrom-style MOND dynamics can originate from quantum gravity without dark matter by exploiting precanonical quantization and spin-connection foam (SCF). By analyzing not only the variance but also the third and fourth moments of the geodesic equation in the non-relativistic static limit, the authors derive qMOND potentials expressed through Gauss and Appell hypergeometric functions, and identify corresponding MOND interpolating functions. They show that the second and fourth moments reproduce MOND with an a0 tied to the SCF fluctuation scale, while the third moment yields a generalized MOND (mMOND) with an explicit $r$-dependent $a_0(r)$, predicting an almost-flat rotation curve scaling as $v(r)\propto r^{-1/18}$. The results offer a first-principles quantum-gravity bridge to MOND phenomena at galactic scales and motivate further work beyond the point-mass, non-relativistic static approximations to assess observational viability.

Abstract

Building upon previous work that derived an alternative to (galactic) dark matter in the form of Modified Newtonian Dynamics (MOND), with a specific theoretical interpolating function, from the motion of a non-relativistic test particle in the gravitational field of a point mass immersed in the non-relativistic static limit of the spin connection foam -- which represents the quantum analogue of Minkowski spacetime within precanonical quantum gravity -- we now show the consequences of using higher moments (third and fourth) of the corresponding geodesic equation with a random spin connection term. These higher moments lead to more general quantum modifications of the Newtonian potential (qMOND potentials expressed in terms of Gauss and Appell hypergeometric functions), more general (steeper) MOND interpolating functions, and a new modification of MOND at low accelerations (mMOND) that features an almost-flat asymptotic rotation curve $\propto r^{-1/18}$, which is expected to operate at approximately the same galactic scales as MOND.

Figures (9)

• Figure 1: The qMOND potential $\Phi^{(2)}(r)$ for $\mu=0.1$, ${\bar{a}{}=0.01}$: (a) $\Phi^{(2)}(r)$; (b) $\Phi_{\mathrm{Newton}}(r)=-\frac{\mu}{r}$.
• Figure 2: Rotation curve $v(r)$ in the qMOND potential $\Phi^{(2)}(r)$ for ${\mu=0.1}$, ${\bar{a}{}=0.01}$: (a) The calculated orbital velocity curve, $v(r)$; (b) The asymptotic linear velocity, $v_{\mathrm{asymp}}(r)=\sqrt{\bar{a}r}$; (c) The Newtonian (Keplerian) curve, $v_{\mathrm{Kepler}}(r)=\sqrt{{\mu}/{r}}$; (d) The velocity minimum, $v_{m}=\sqrt[4]{2 \bar{a} \mu}$, located at radius $r_{m}=\sqrt{{\mu}/{\bar{a}}}$; (e) The line representing the minimum velocity plus 5%, $v_m + 5\%$; (f) The flat rotation curve band, with a 10% error margin.
• Figure 3: Rotation curve $v^{(4)}(r)$ derived from the $\Phi^{(4)}$ potential for $\mu=0.1$ and $\bar{a}=0.01$: (a) The calculated orbital velocity curve, $v^{(4)}(r)$; (b) The asymptotic linear velocity, $v^{(4)}_{\mathrm{asymp}}(r)=\sqrt{\bar{b}r}$; (c) The Newtonian (Keplerian) curve, $v_{\mathrm{Kepler}}(r)= {\sqrt{\mu/r}}$; (d) The velocity minimum, $v^{(4)}_{m}=\left( 2\mu^2 \bar{b}^2 + \frac{10}{3}\mu^2\bar{a}^2 \right)^{1/8}$, located at radius $r^{(4)}_{m}=\sqrt{{\mu}/{\bar{b}}}$; (e) The line representing the minimum velocity plus 5%, $v^{(4)}_m + 5\%$; (f) The flat rotation curve band, with a 10% error margin; (g) The orbital velocity curve from the 2nd moment, $v(r)$.
• Figure 4: Comparison of MOND interpolating functions: (a) The interpolating function derived from the 4th moment, $\mu^{(4)}(x)$; (b) The interpolating function from the 2nd moment (previous result), $\mu^{(2)}(x)$; (c) The simple IF commonly used in MOND phenomenology, $\mu_{\mathrm{simple}}={x}/{(1+x)}$; (d) The deep-MOND asymptote ($\mu \sim x$ for $x \ll 1$); (e) The Newtonian asymptote ($\mu \sim 1$ for $x \gg 1$).
• Figure 5: qMOND potential $\Phi^{(3)}(r)$ for ${\mu=0.1}$, ${\bar{a}{}=0.01}$: (a) $\Phi^{(3)}(r)$; (b) $\Phi_{\mathrm{Newton}}(r)=-\frac{\mu}{r}$; (c) the asymptote at $r\rightarrow \infty$: $\Phi^{(3)}_{\rm{asymp}}(r) \rightarrow \sqrt[3]{45 \mu \bar{a}^2 r} - \frac{\sqrt[4]{\frac{5}{3}} \Gamma \left(-\frac{1}{12}\right) \Gamma \left(\frac{3}{4}\right) \sqrt{\mu \bar{a}}}{\Gamma \left(-\frac{1}{3}\right)}$.