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Galactic space-times in modified theories of gravity

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Abstract

We study Bertrand space-times (BSTs), which have been proposed as viable models of space-times seeded by galactic dark matter, in modified theories of gravity. We first critically examine the issue of galactic rotation curves in general relativity, and establish the usefulness of BSTs to fit experimental data in this context. We then study BSTs in metric f(R) gravity and in Brans–Dicke theories. For the former, the nature of the Newtonian potential is established, and we also compute the effective equation of state and show that it can provide good fits to some recent experimental results. For the latter, we calculate the Brans–Dicke scalar analytically in some limits and numerically in general, and find interesting constraints on the parameters of the theory. Our results provide evidence for the physical nature of BSTs in modified theories of gravity.

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Notes

  1. The literature on the subject is vast, and we refer the reader to the standard references [4–7] on the subject.

  2. These are solved at the radius of the circular motion. By a slight abuse of notation, we denote this by r as well.

  3. In the language of Perlick [10], the metric of Eq. (11) is a special case of what he has called Bertrand space-times of Type II, there being another version of the metric that supports closed, stable orbits at each point, called BSTs of type I. Since we will always be dealing with the metric of Eq. (11) in this paper, we will simply call this metric as the BST.

  4. We will choose a positive sign for \(\lambda \). This is dictated by the fact that a negative \(\lambda \) seems to render BSTs in f(R) gravity unphysical. This will be explained in more details in the next section.

  5. We will momentarily see that for the JNW space-time, a similar physicality condition dictates that \(\lambda \) is negative.

  6. In this section, the same values of \(\alpha \) and D will be chosen in sequel and we will not mention this further.

    Fig. 5
    figure 5

    Galactic potential as a function of the radial distance for BSTs in GR (dashed red) and f(R) gravity (solid blue) (see text for details) (color figure online)

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  7. The curve for \(\lambda = 0\) almost coincides with the the one shown for \(r \ge 0.5\). Expectedly, they differ significantly for very small values of r, but this is not shown here.

    Fig. 6
    figure 6

    Energy density of matter for BSTs in f(R) gravity (see text for details)

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  8. The field \(\phi \) appearing in this section is distinct from (and should not be confused with) that appearing in Sect. 3.

  9. The quantities \(\rho \), \(P_i\) refer to the matter part of the Lagrangian of Eq. (55), and is obtained from the matter part of Eq. (56). This will be understood in what follows, and we will avoid using a subscript, as this clutters up the notation.

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Acknowledgments

It is a pleasure to thank Sayan Kar for valuable comments.

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Authors and Affiliations

  1. Department of Physics, Indian Institute of Technology, Kanpur, 208016, India

    Dipanjan Dey, Kaushik Bhattacharya & Tapobrata Sarkar

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  1. Dipanjan Dey
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  2. Kaushik Bhattacharya
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Correspondence to Tapobrata Sarkar.

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Dey, D., Bhattacharya, K. & Sarkar, T. Galactic space-times in modified theories of gravity. Gen Relativ Gravit 47, 103 (2015). https://doi.org/10.1007/s10714-015-1945-x

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