# Galaxy Rotation Curves in Covariant Hořava-Lifshitz Gravity

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## Abstract

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## 1. Introduction

## 2. Static and Spherically Symmetric Solutions of Covariant HL Gravity

#### 2.1. Action

#### 2.2. Constraints and Equations of Motion

- The variation with respect to A gives ${R}^{\left(3\right)}=0$, or equivalently:$$r{f}^{\prime}+f-1=0$$$$f\left(r\right)=1-\frac{2B}{r}$$
- The variation with respect to ν gives:$${\Theta}^{ij}{\nabla}_{i}{\nabla}_{j}\nu +{\Theta}^{ij}{K}_{ij}=0$$
- The variation with respect to N gives the Hamiltonian constraint:$${\int}_{0}^{\infty}dr\frac{{r}^{2}}{\sqrt{f\left(r\right)}}({K}_{ij}{K}^{ij}-{K}^{2}+V)=0$$
- The variation with respect to n gives:$${f}^{\prime}\left(r\right)n\left(r\right)=0$$
- The variation with respect to f gives:$${A}^{\prime}+\frac{A}{2r}\left(\right)open="("\; close=")">1-\frac{1}{f}+4\frac{fn{\left(\right)}^{\sqrt{r}}\prime}{}\sqrt{r}$$

#### 2.3. Solutions

- n ≠ 0 and A = 0: leads to the Schwarzschild solution in the Painleve-Gullstrand coordinates ($n\propto {r}^{-1/2}$), and the Hamiltonian constraint is automatically satisfied;
- n ≠ 0 and A ≠ 0: corresponds to the situation with multiple solutions for A and n, on which this article focuses. Since $B=0$, we have $f=1$ and $V=0$, such that the Equations (11) and (13) give respectively:$$\begin{array}{ccc}\hfill {\int}_{0}^{\infty}dr{\left(r{n}^{2}\right)}^{\prime}& =& 0\hfill \\ \hfill r{A}^{\prime}+2{\left(r{n}^{2}\right)}^{\prime}& =& 0\hfill \end{array}$$$${n}^{2}\left(r\right)=\frac{C}{r}-\frac{1}{2}A\left(r\right)+\frac{1}{2r}{\int}_{0}^{r}d\rho \phantom{\rule{3.33333pt}{0ex}}A\left(\rho \right)$$$${\int}_{0}^{\infty}drr{A}^{\prime}\left(r\right)=0$$

## 3. Galaxy Rotation Curves

#### 3.1. Gravitational Potential

#### 3.2. Navarro, Frenk and White Profile

Galaxy | a | v_{200} (km/s) | r_{200} (kpc) |
---|---|---|---|

NGC 2403 | 10.9 ± 0.6 | 106.1 ± 1.9 | 88.4 ± 2.6 |

NGC 3198 | 11.2 ± 0.43 | 104.0 ± 0.7 | 86.6 ± 1.6 |

NGC 3521 | 14.0 ± 12.6 | 122.5 ± 20.4 | 102.0 ± 18 |

#### 3.3. Pseudoisothermal Profile

Galaxy | R_{c} (kpc) | ρ_{0} (10^{−3}M_{⊙}pc^{−3}) | r_{200} (kpc) |
---|---|---|---|

NGC 2403 | 2.51 ± 0.32 | 59.1 ± 14.3 | 88.4 ± 2.6 |

NGC 2841 | 1.36 ± 0.75 | 674.8 ± 736.4 | 159.0 ± 3.7 |

NGC 3621 | 5.88 ± 0.32 | 13.0 ± 1.1 | 106.6 ± 4.0 |

#### 3.4. Vacuum Energy Contribution of the Auxiliary Field

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Hořava, P. Quantum gravity at a Lifshitz point. Phys. Rev. D
**2009**, 79. [Google Scholar] [CrossRef] - Alexandre, J. Lifshitz-type quantum field theories in particle physics. Int. J. Mod. Phys. A
**2011**, 26, 4523. [Google Scholar] [CrossRef] - Padilla, A. The good, the bad and the ugly .... of Hořava gravity. J. Phys. Conf. Ser.
**2010**, 259, 012033. [Google Scholar] [CrossRef] - Mukohyama, S. Hořava-Lifshitz cosmology: A review. Class. Quantum Gravity
**2010**, 27, 223101. [Google Scholar] [CrossRef] - Sotiriou, T.P. Hořava-Lifshitz gravity: A status report. J. Phys. Conf. Ser.
**2011**, 283, 012034. [Google Scholar] [CrossRef] - Sotiriou, T.P.; Visser, M.; Weinfurtner, S. Quantum gravity without Lorentz invariance. J. High Energy Phys.
**2009**, 2009, 033. [Google Scholar] [CrossRef] - Charmousis, C.; Niz, G.; Padilla, A.; Saffin, P.M. Strong coupling in Hořava gravity. J. High Energy Phys.
**2009**, 2009, 070. [Google Scholar] [CrossRef] - Hořava, P.; Melby-Thompson, C.M. General covariance in quantum gravity at a Lifshitz point. Phys. Rev. D
**2010**, 82, 064027. [Google Scholar] [CrossRef] - Abdalla, E.; da Silva, A.M. On the motion of particles in covariant Hořava-Lifshitz gravity and the meaning of the A-field. Phys. Lett. B
**2012**, 707, 311–314. [Google Scholar] [CrossRef] - Lin, K.; Wang, A. Static post-Newtonian limits in non-projectable Hořava-Lifshitz gravity with an extra U(1) symmetry. ArXiv E-Prints
**2013**. [Google Scholar] - Mukohyama, S. Dark matter as integration constant in Hořava-Lifshitz gravity. Phys. Rev. D
**2009**, 80, 064005. [Google Scholar] [CrossRef] - Cardone, V.F.; Radicella, N.; Ruggiero, M.L.; Capone, M. The Milky Way rotation curve in Hořava-Lifshitz theory. ArXiv E-Prints
**2010**. [Google Scholar] - Cardone, V.F.; Capone, M.; Radicella, N.; Ruggiero, M.L. Spiral galaxies rotation curves in the Hořava-Lifshitz gravity theory. ArXiv E-Prints
**2012**. [Google Scholar] - Romero, J.M.; Bernal-Jaquez, R.; Gonzalez-Gaxiola, O. Is it possible to relate MOND with Hořava Gravity? Mod. Phys. Lett. A
**2010**, 25, 2501. [Google Scholar] [CrossRef] - Alexandre, J.; Farakos, K.; Tsapalis, A. Liouville-Lifshitz theory in 3 + 1 dimensions. Phys. Rev. D
**2010**, 81, 105029. [Google Scholar] [CrossRef] - Dutta, S.; Saridakis, E.N. Overall observational constraints on the running parameter λ of Hořava-Lifshitz gravity. J. Cosmol. Astropart. Phys.
**2010**, 2010, 013. [Google Scholar] [CrossRef] - Lin, K.; Mukohyama, S.; Wang, A. Solar system tests and interpretation of gauge field and Newtonian prepotential in general covariant Hořava-Lifshitz gravity. Phys. Rev. D
**2012**, 86, 104024. [Google Scholar] [CrossRef] - Da Silva, A.M. An alternative approach for general covariant Hořava-Lifshitz gravity and matter coupling. Class. Quantum Gravity
**2011**, 28, 055011. [Google Scholar] [CrossRef] - Klusoň, J. Hamiltonian analysis of nonrelativistic covariant restricted-foliation-preserving diffeomorphism invariant Hořava-Lifshitz gravity. Phys. Rev. D
**2011**, 83, 044049. [Google Scholar] [CrossRef] - Blas, D.; Pujolas, O.; Sibiryakov, S. On the extra mode and inconsistency of Hořava gravity. J. High Energy Phys.
**2009**, 2009, 029. [Google Scholar] [CrossRef] - Bogdanos, C.; Saridakis, E.N. Perturbative instabilities in Hořava gravity. Class. Quantum Gravity
**2010**, 27, 075005. [Google Scholar] [CrossRef] - Koyama, K.; Arroja, F. Pathological behaviour of the scalar graviton in Hořava-Lifshitz gravity. J. High Energy Phys.
**2010**, 2010, 061. [Google Scholar] [CrossRef] - Blas, D.; Pujolas, O.; Sibiryakov, S. Comment on “Strong coupling in extended Hořava-Lifshitz gravity”. Phys. Lett. B
**2010**, 688, 350–355. [Google Scholar] [CrossRef] - Papazoglou, A.; Sotiriou, T.P. Strong coupling in extended Hořava-Lifshitz gravity. Phys. Lett. B
**2010**, 685, 197–200. [Google Scholar] [CrossRef] [Green Version] - Alexandre, J.; Pasipoularides, P. Spherically symmetric solutions in covariant Hořava-Lifshitz gravity. Phys. Rev. D
**2011**, 83, 084030. [Google Scholar] [CrossRef] - Greenwald, J.; Satheeshkumar, V.H.; Wang, A. Black holes, compact objects and solar system tests in non-relativistic general covariant theory of gravity. J. Cosmol. Astropart. Phys.
**2010**, 2010, 007. [Google Scholar] [CrossRef] - De Naray, R.K.; McGaugh, S.S.; de Blok, W.J.G. Mass models for low surface brightness galaxies with high resolution optical velocity fields. Astrophys. J.
**2008**, 676, 920. [Google Scholar] [CrossRef] - Lokas, E.L.; Mamon, G.A. Properties of spherical galaxies and clusters with an nfw density profile. Mon. Not. R. Astron. Soc.
**2001**, 321, 155–166. [Google Scholar] [CrossRef] - Navarro, J.F.; Frenk, C.S.; White, S.D.M. The structure of cold dark matter halos. Astrophys. J.
**1996**, 462, 563–575. [Google Scholar] [CrossRef] - De Blok, W.J.G.; Walter, F.; Brinks, E.; Trachternach, C.; Oh, S.-H.; Kennicutt, R.C., Jr. High-resolution rotation curves and galaxy mass models from THINGS. Astron. J.
**2008**, 136, 2648. [Google Scholar] [CrossRef] - Lynden-Bell, D. Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc.
**1967**, 136, 101–121. [Google Scholar] [CrossRef] - Dutta, S.; Saridakis, E.N. Observational constraints on Hořava-Lifshitz cosmology. J. Cosmol. Astropart. Phys.
**2010**, 2010, 013. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Alexandre, J.; Kostacinska, M.
Galaxy Rotation Curves in Covariant Hořava-Lifshitz Gravity. *Galaxies* **2014**, *2*, 1-12.
https://doi.org/10.3390/galaxies2010001

**AMA Style**

Alexandre J, Kostacinska M.
Galaxy Rotation Curves in Covariant Hořava-Lifshitz Gravity. *Galaxies*. 2014; 2(1):1-12.
https://doi.org/10.3390/galaxies2010001

**Chicago/Turabian Style**

Alexandre, Jean, and Martyna Kostacinska.
2014. "Galaxy Rotation Curves in Covariant Hořava-Lifshitz Gravity" *Galaxies* 2, no. 1: 1-12.
https://doi.org/10.3390/galaxies2010001