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Gravity: Where Do We Stand? pp 337–366Cite as

The Acceleration Scale, Modified Newtonian Dynamics and Sterile Neutrinos

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Abstract

General relativity is able to describe the dynamics of galaxies and larger cosmic structures only if most of the matter in the universe is dark, namely, it does not emit any electromagnetic radiation. Intriguingly, on the scale of galaxies, there is strong observational evidence that the presence of dark matter appears to be necessary only when the gravitational field inferred from the distribution of the luminous matter falls below an acceleration of the order of \( 10^{-10}\) m s\(^{-2}\). In the standard model, which combines Newtonian gravity with dark matter, the origin of this acceleration scale is challenging and remains unsolved. On the contrary, the full set of observations can be neatly described, and were partly predicted, by a modification of Newtonian dynamics, dubbed MOND, that does not resort to the existence of dark matter. On the scale of galaxy clusters and beyond, however, MOND is not as successful as on the scale of galaxies, and the existence of some dark matter appears unavoidable. A model combining MOND with hot dark matter made of sterile neutrinos seems to be able to describe most of the astrophysical phenomenology, from the power spectrum of the cosmic microwave background anisotropies to the dynamics of dwarf galaxies. Whether there exists a yet unknown covariant theory that contains general relativity and Newtonian gravity in the weak field limit and MOND as the ultra-weak field limit is still an open question.

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Notes

  1. 1.

    We use the standard parametrization for the Hubble constant today \(H_0=100 \, h\) km s\(^{-1}\) Mpc\(^{-1}\).

  2. 2.

    A year later, Bekenstein and Milgrom [20] showed that, in spherical symmetry, Eq. (1) is equivalent to a modified theory of gravity where the standard Poisson equation is replaced by

    $$ \nabla\cdot\left[\mu\left(\vert\nabla\Phi\vert\over a_0\right)\nabla\Phi\right] = 4\pi G\rho, $$
    (2)

    where \(\Phi \) is the gravitational potential and \(\rho \) the mass distribution. A more recent modified theory of gravity that reproduces Eq. (1) is the quasi-linear formulation of MOND (QUMOND) [21]. This theory has the advantage of involving only linear differential equations and one nonlinear algebraic step. See [15] for a comprehensive review of the various formulations of MOND as modified dynamics or modified gravity.

  3. 3.

    In the standard model, the cluster virial mass scales as \(M_{\rm vir} \propto \rho_c(z) \Delta_c(z) R^3 \), where \(R\) is the cluster size in proper units (not comoving), \(\rho_c(z)=3H^2(z)/(8\pi G)\) is the critical density of the universe and \(\Delta_c(z)\) is the cluster density in units of \(\rho_c(z)\). A widely used approximation is

    $$\Delta_c(z)=18\pi^2 + \left\{ \begin{array}{ll} 60 w - 32 w^2, &\Omega_m\le 1,\quad \Omega_\Lambda= 0\\ 82 w - 39 w^2, &\Omega_m+\Omega_\Lambda=1, \end{array}\right.$$
    (4)

    where \(w=\Omega_m(z)-1\) [97]. Now, \(\rho_c(z)\) scales with redshift \(z\) as \(\rho_c(z)\propto E^2(z) = \Omega_m(1+z)^3 + (1-\Omega_m-\Omega_\Lambda )(1+z)^2+ \Omega_\Lambda \). The cluster size thus scales as \(R\propto M_{\rm vir}^{1/3} \Delta_c^{-1/3}(z) E^{-2/3}(z)\), and the temperature as \(T\propto M_{\rm vir}/R \propto M_{\rm vir}^{2/3} \Delta_c^{1/3}(z) E^{2/3}(z)\).

  4. 4.

    In this section, we use units where the speed of light \(c=1\).

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Acknowledgements

We thank Benoit Famaey and Stacy McGaugh for useful suggestions and for providing us with Figs. 1, 2, 3 and 4. We thank Ana Laura Serra for a careful reading of the manuscript and Luisa Ostorero for enlightening and encouraging discussions. AD gratefully acknowledges partial support form INFN grant PD51 and PRIN-MIUR-2008 grant 2008NR3EBK_003 “Matter-antimatter asymmetry, dark matter and dark energy in the LHC era”. GWA is supported by the Claude Leon Foundation and a University Research Committee Fellowship from the University of Cape Town. This research has made use of NASA’s Astrophysics Data System.

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  1. Dipartimento di Fisica, Università degli Studi di Torino, and Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, Via P. Giuria 1, 10125, Torino, Italy

    Antonaldo Diaferio

  2. Astrophysics, Cosmology and Gravity Centre, University of Cape Town, Private Bag X3, 7700, Rondebosch, South Africa

    Garry W. Angus

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Diaferio, A., Angus, G. (2016). The Acceleration Scale, Modified Newtonian Dynamics and Sterile Neutrinos. In: Peron, R., Colpi, M., Gorini, V., Moschella, U. (eds) Gravity: Where Do We Stand?. Springer, Cham. https://doi.org/10.1007/978-3-319-20224-2_10

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