Large Scale Structure in Bekenstein’s Theory of Relativistic Modified Newtonian Dynamics

A relativistic theory of modified gravity has been recently proposed by Bekenstein. The tensor field in Einstein’s theory of gravity is replaced by a scalar, a vector, and a tensor field which interact in such a way to give modified Newtonian dynamics (MOND) in the weak-field nonrelativistic limit. We study the evolution of the Universe in such a theory, identifying its key properties and comparing it with the standard cosmology obtained in Einstein gravity. The evolution of the scalar field is akin to that of tracker quintessence fields. We expand the theory to linear order to find the evolution of perturbations on large scales. The impact on galaxy distributions and the cosmic microwave background is calculated in detail. We show that it may be possible to reproduce observations of the cosmic microwave background and galaxy distributions with Bekenstein’s theory of MOND.

Figure 1

The relative energy densities in $\varphi$ (thick solid line), radiation (dotted line), matter (dashed line), and $\Lambda$ (dot-dashed line) for ${\mu }_0=5$ as a function of the scale factor ($a$ is in arbitrary units). Note that the energy density in the scalar field tracks the dominant form of energy at each instance in time.

Figure 2

The effect of the MOND parameters on the power of spectrum of the CMB. Top panel: ${\mu }_0=200$, $l_B=100\mathrm{Mpc}$, and $K=1$ (solid), 0.1 (dotted), and 0.08 (dashed); Middle panel: ${\mu }_0=200$, $K=0.1$, and $l_B=1000\mathrm{Mpc}$ (solid), 100 Mpc (dotted), and 10 Mpc (dashed); Bottom panel: $K=0.1$, $l_B=100\mathrm{Mpc}$, and ${\mu }_0=1000$ (solid), 200 (dotted), and 150 (dashed).

Figure 3

The effect of the MOND parameters on the power of spectrum of the baryonic density fluctuations. Top panel: ${\mu }_0=200$, $l_B=100\mathrm{Mpc}$, and $K=1$ (solid), 0.1 (dotted), and 0.08 (dashed); Middle panel: ${\mu }_0=200$, $K=0.1$, and $l_B=1000\mathrm{Mpc}$ (solid), 100 Mpc (dotted), and 10 Mpc (dashed); Bottom panel: $K=0.1$, $l_B=100\mathrm{Mpc}$, and ${\mu }_0=1000$ (solid), 200 (dotted), and 150 (dashed).

Figure 4

The angular power spectrum of the CMB (top panel) and the power spectrum of the baryon density (bottom panel) for a MOND universe (with $a_0\simeq 4.2\times {10}^{-8}\mathrm{cm}/{\mathrm{s}}^2)$ with ${\Omega }_{\Lambda }=0.78$ and ${\Omega }_{\nu }=0.17$ and ${\Omega }_B=0.05$ (solid line), for a MOND universe ${\Omega }_{\Lambda }=0.95$ and ${\Omega }_B=0.05$ (dashed line) and for the $\Lambda \mathrm{CDM}$ model (dotted line). A collection of data points from CMB experiments and Sloan are overplotted.