A GENERALIZATION OF THE POINCARÉ-BIRKHOFF THEOREM

Dwarf galaxies without dark matter: constraints on modified gravity

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa3653 ◽

2020 ◽

Vol 501 (1) ◽

pp. 254-260

Author(s):

Ali Rida Khalifeh ◽

Raul Jimenez

Keyword(s):

Dark Matter ◽

Modified Gravity ◽

Virial Theorem ◽

Dwarf Galaxies ◽

Strong Support ◽

Gravity Models ◽

Birkhoff Theorem ◽

Astronomical Surveys ◽

Stringent Test ◽

Modified Gravity Models

ABSTRACT The discovery of 19 dwarf galaxies without dark matter (DM) provides, counterintuitively, strong support for the ΛCDM standard model of cosmology. Their presence is well accommodated in a scenario where the DM is in the form of cold dark particles. However, it is interesting to explore quantitatively what is needed from modified gravity models to accommodate the presence of these galaxies and what extra degree of freedom is needed in these models. To this end, we derive the dynamics at galaxy scales (Virial theorem) for a general class of modified gravity models. We distinguish between theories that satisfy the Jebsen–Birkhoff theorem, and those that do not. Our aim is to develop tests that can distinguish whether DM is part of the theory of gravity or a particle. The 19 dwarf galaxies discovered provide us with a stringent test for models of modified gravity. Our main finding is that there will always be an extra contribution to the Virial theorem coming from the modification of gravity, even if a certain galaxy shows very small, if not negligible, trace of DM, as has been reported recently. Thus, if these and more galaxies are confirmed as devoid (or negligible) of DM, while other similar galaxies have abundant DM, it seems interesting to find modifications of gravity to describe DM. Our result can be used by future astronomical surveys to put constraints on the parameters of modified gravity models at astrophysical scales where DM is described as such.

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Generalizations of the Poincaré Birkhoff fixed point theorem

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700010650 ◽

1977 ◽

Vol 17 (3) ◽

pp. 375-389 ◽

Cited By ~ 10

Author(s):

Walter D. Neumann

Keyword(s):

Fixed Point ◽

Lower Bound ◽

Fixed Point Theorem ◽

Point Theorem ◽

Periodic Points ◽

Open Problems ◽

Number Of Components ◽

Birkhoff Theorem

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

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BIRKHOFF THEOREM AND ERGOMETER: MEETING OF TWO CULTURES

International Journal of Modern Physics B ◽

10.1142/s0217979208050322 ◽

2008 ◽

Vol 22 (25n26) ◽

pp. 4572-4578 ◽

Cited By ~ 2

Author(s):

M. HOWARD LEE

Keyword(s):

Statistical Mechanics ◽

Two Cultures ◽

Ergodic Hypothesis ◽

Birkhoff Theorem ◽

Physical Measurements ◽

The Two Cultures

There are two approaches to understanding Boltzmann's ergodic hypothesis in statistical mechanics. The first one, purely mathematical, goes by way of theorems while the second one relies on physical measurements. By its own nature the former is universal whereas the latter is specific to a system. By all account they seem orthogonal to each other. But should not they meet at the end? If, for example, both conclude that the hypothesis is not valid in a given system, should not their conclusions be compatible? We illustrate in this work how the two cultures meet in the physics of ergodicity.

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