Galaxy rotation curves via conformal factors
Abstract
We propose a new formula to explain circular velocity profiles of spiral galaxies obtained from the Starobinsky model in the Palatini formalism. It is based on the assumption that the gravity can be described by two conformally related metrics: one of them is responsible for the measurement of distances, while the other, the so-called dark metric, is responsible for a geodesic equation and therefore can be used for the description of the velocity profile. The formula is tested against a subset of galaxies taken from the HI Nearby Galaxy Survey (THINGS).
1 Introduction and motivation
Since there exist many issues that recently appeared in fundamental physics, astrophysics and cosmology which cannot be explained by General Relativity (GR) [1, 2, 3], one looks for other approaches which allow one to understand their mechanism. Classical GR is a well-posed theory. Many astronomical observations tested GR and have confirmed that it is the best matching theory that we have had so far for explaining gravitational phenomena. Unfortunately, GR is not enough to describe many unsolved problems such as late-time cosmic acceleration [4, 5] (which one explains by existing an exotic fluid called Dark Energy introduced to the standard Einstein’s field equations as cosmological constant), the Dark Matter puzzle [6, 7, 8, 9, 10, 11, 12, 13, 16], inflation [17, 18], and the renormalization problem [19].
There are two main ideas competing for an explanation of the Dark Matter problem: geometric modification of the gravitational field equations (see e.g. [13, 14, 15]) or going beyond the Standard Model of elementary particles and introducing weakly interacting particles, which have failed to be detected [20]. In fact, these two ideas do not contradict each other and can be combined in some future successful theory. The existence of Dark Matter is mainly indicated by anomalies in the observed galactic rotation curves. It interacts only gravitationally with visible matter and radiation, and also it has effects on the large-scale structure of the universe [21, 22].
Many interesting and promising models have faced the Dark Matter problem. The most famous one is Modified Newtonian Dynamics (MOND) [23, 24, 25, 26, 27, 28, 29, 30]. It has predicted many galactic phenomena and hence it is widely used by astrophysicists. Closely related is the so-called Tensor/Vector/Scalar (TeVeS) theory of gravity [31, 32] which is, roughly speaking, the relativistic version of MOND. Another approach is to consider Extended Theories of Gravity (ETGs)—one modifies the geometric part of the field equations [33, 34, 36]. There are also attempts to obtain a MOND result from ETGs; see for example [37, 38, 39, 40, 41, 42]. Another interesting proposal for explaining rotation curves is by using Weyl conformal gravity [43, 44, 45]. It should be noted that there is a model based on a quantum effective action and large-scale renormalization group effects [46, 47], later on constrained by Solar System tests [48]. It provides, up to the first nontrivial order, a conformal transformation of the spacetime metric and a logarithmic term in the modified Newtonian potential.
We would like to emphasize that GR is a very special case of the formalism described. One assumes at the very beginning that the connection \(\tilde{\varGamma }\) is a Levi-Civita connection of the metric g (the 1-form A is zero) which as a result means to treat the action of the theory as just metric-dependent. One may also treat the Einstein–Hilbert action as the one depending on two independent objects, that is, the metric g and the connection \(\tilde{\varGamma }\). This approach is called the Palatini formalism. Considering the simplest gravitational Lagrangian, linear in the scalar curvature R, the Palatini approach leads to the dynamical result that \(\tilde{\varGamma }\) is a Levi-Civita connection of the metric g. It is not so in the case of more complicated Lagrangians appearing in ETGs. Moreover, as shown in [52], all Palatini connections of the form (2) are singled out by the variational principle.
Our aim is to show how important the EPS interpretation can be for an explanation of the galaxy rotation curves. It turns out, as expected, that at the end we deal with an expression which consists of the Newtonian part and some modification which depends on the theory one wants to study in the EPS approach. As the simplest example which we want to examine is the Starobinsky quadratic Lagrangian [17] in the Palatini formalism, which currently reaches very good results in the cosmological applications [53, 54, 55]. The starting point will be the standard geodesic equation from which we will derive the rotational velocity. It will be shown that the velocity can be written as GR plus extra terms coming from the conformal factor. Its usefulness is tested on a sample of six HSB galaxies. The conclusions and future ideas will be presented in the last part of the paper.
The metric signature convention is \((-,+,+,+)\).
2 Velocities via conformal factors
2.1 An example: Starobinsky model
In principle there exists an entire class of gravity theories [49, 51] that are conformally related with Einstein general relativity via Eq. (8). Our aim is to explain the observed galaxy rotation curves using Eqs. (13) and (7) without assuming the existence of Dark Matter.
In what follows we would like to propose a model that fits well the astronomical observed data on galaxy rotation curves. Our analysis is performed on a subset of galaxies obtained from THINGS: the HI Nearby Galaxy Survey catalogue [61, 62], which is a high spectral and spatial resolution survey of HI emission lines from 34 nearby galaxies.
Any new model must take into account and reproduce the observed flatness of the galaxy rotation curves. At short distances (at least of the order of the size of the solar system) the velocity should have as a limit the Newtonian result \(v^2(r)=\mathrm{GM}/r\). This imposes some constraints on the functions A(r) and B(r).
Best-fit results according to Eq. (32) using the parametric mass distribution (28). These numerical values correspond to rotation curves presented in Fig. 2. Col. (1) galaxy name; Col. (2) total gas mass, in units of \(10^{10}\,M_{\odot }\), given by \(M_\mathrm{gas}=4/3M_{HI}\), with the \(M_{HI}\) data taken from [61]; Col. (3) measured scale length of the galaxy in kpc; Col. (4) galaxy luminosity in the B-band, in units of \(10^{10}\,L_{\odot }\), calculated from [61]; Col. (5) presents the best-fit results for the predicted total mass of the galaxy \(M_0\) (in \(10^{10}\,M_{\odot }\) units); col. (6) gives the predicted core radius \(r_c\) in kpc; Col. (7) reduced \(\chi ^2_r\); and Col. (8) the stellar mass-to-light ratio in units of \(M_{\odot }/L_{\odot }\). Note: all six galaxies are of type HSB
Galaxy | \(M_\mathrm{gas}\) | \(R_0\) | \(L_B\) | \(M_0\) | \(r_c\) | \(\chi ^2_r\) | M / L |
---|---|---|---|---|---|---|---|
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) |
NGC 3031 | 0.48 | 2.6 | 3.049 | 14.86 | 2.10 | 4.88 | 4.71 |
NGC 3521 | 1.07 | 3.3 | 3.698 | 38.45 | 3.69 | 1.84 | 10.10 |
NGC 3627 | 0.11 | 3.1 | 3.076 | 8.68 | 2.25 | 0.45 | 2.78 |
NGC 4736 | 0.05 | 2.1 | 1.294 | 0.53 | 0.59 | 2.41 | 0.37 |
NGC 6946 | 0.55 | 2.9 | 2.729 | 78.19 | 5.09 | 2.18 | 28.44 |
NGC 7793 | 0.12 | 1.7 | 0.511 | 18.24 | 3.36 | 4.82 | 35.45 |
The corresponding values for \(M_0\) and \(r_c\) are presented in Table 1. We observe very good agreement between the data points and the fitted continuous black curve. Unfortunately, after plotting the Newtonian curves using the same values of \(M_0\) and \(r_c\) from Table 1 we have found that there is almost no difference (see Fig. 1) between the Newtonian curve and the one derived using Eq. (32). Furthermore, the mass-to-light M / L values inferred from the fit in Table 1 are also too large compared to what is expected based on stellar population synthesis models [60].
3 Conclusions
In this paper we have considered the possible explanation of the observed galaxy rotation curves by the assumption that the metric and the connection are independent objects in the spirit of EPS formalism. We studied the case when the connection is a Levi-Civita connection of a metric conformally related to the metric which is responsible for the measurement of distances and angles. Due to that interpretation, masses moving in a gravitational field should follow geodesics determined by the connection, providing different equations of motion. It turns out that the rotational velocity formula obtained under this formalism differs from the Newtonian one by the presence of extra terms coming indirectly from the conformal factor of the metrics. This term is treated as a deviation from the Newtonian limit of General Relativity.
In Sect. 2.1 we used Palatini gravity, which is a representation of the EPS formalism, and as a working example we took the Starobinsky Lagrangian \(f(\hat{R})=\hat{R}+\gamma \hat{R}^2\) in order to derive a rotational velocity formula given by the expression in Eq. (32) for a star moving in a circular trajectory around the galactic center. Our results are presented in Table 1 together with Figs. 1 and 2. Although the galaxy masses resulting from the fitting of the data sub-sample proved to be too high, giving thus rise to unsatisfactory values for the mass-to-light ratio, nonetheless we have showed that the approach of obtaining galaxy rotation curves via conformal factors can be valid. It should also be noticed that we have used a very simple matter distribution (28) in order to be able to obtain an expression for the energy density \(\rho (r)\). More complex distributions could possibly give corrections which would provide different mass-to-light M / L values. This is one of the tasks of our future work.
Thus, we would like to briefly conclude by saying that the approach of obtaining galaxy rotation curves using two conformally related metrics can be valid and deserves further investigations. Furthermore, by trying other Lagrangians and other gravity models in the future, it is definitely possible to improve the findings reported here.
Notes
Acknowledgements
This work made use of THINGS, “The HI Nearby Galaxy Survey” (Walter et al. 2008). We would like to thank Professors Fabian Walter and Erwin de Blok for helping us in obtaining the RC data from the THINGS catalogue.
AW is partially supported by the grant of the National Science Center (NCN) DEC- 2014/15/B/ST2/00089. CS was partially supported by a grant of the Ministry of National Education and Scientific Research, RDI Programme for Space Technology and Advanced Research—STAR, project number 181/20.07.2017. We appreciate S. Odintsov, D.C. Rodrigues and S. Vagnozzi for drawing our attention to their papers. This paper is based upon work from COST action CA15117 (CANTATA), supported by COST (European Cooperation in Science and Technology)
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