Modified Teleparallel theories of gravity Author: Supervisor: Sebasti´anBahamonde Dr. Christian G. B¨ohmer A thesis submitted in conformity with the requirements for the degree of PhD in Applied Mathematics Department of Mathematics Faculty of Mathematical & Physical Sciences University College London September 06, 2018 i Abstract Teleparallel gravity is an alternative formulation of gravity which has the same field equations as General Relativity (GR), therefore, it also known as the Teleparallel equivalent of General Relativity (TEGR). This theory is a gauge theory of the trans- lations with the torsion tensor being non-zero but with a vanishing curvature tensor, hence, the manifold is globally flat. An interesting approach for understanding the late-time accelerating behaviour of the Universe is called modified gravity where GR is extended or modified. In the same spirit, since TEGR is equivalent to GR, one can consider its modifications and study if they can describe the current cosmological observations. This thesis is devoted to studying several modified Teleparallel theories of gravity with emphasis on late-time cosmology. Those Teleparallel theories are in general different to the modified theories based on GR, but one can relate and clas- sify them accordingly. Various Teleparallel theories are presented and studied such as Teleparallel scalar-tensor theories, quintom models, Teleparallel non-local gravity, and f(T;B) gravity and its extensions (coupled with matter, extensions of new GR and Gauss-Bonnet) where T is the scalar torsion and B is the boundary term which ◦ is related with the Ricci scalar via R = −T + B. This thesis was completed under the supervision of Christian Bohmer¨ . ii Impact statement The new results showed in this thesis are based on eleven different published scientific papers. Basically, this thesis aimed at two directions, establishing new research into some questions within modified gravity and cosmology, in particular, for theories based on Teleparallel gravity. There is no foreseen impact outside of academia at this stage. One impact of this thesis was the possibility of determining a certain relationship between standard modified theories of gravity based on General Relativity (GR) and other modifications based on TEGR. Many of these theories were considered in isolation in the past and their relationship with other similarly looking theories was only made implicitly. Further, in this thesis, it was showed how two very popular ◦ modified theories coming from these two different approaches, f(R) (based on GR) and f(T ) gravity (based on TEGR), are related and could construct the fundamental theory underlying them both. Another impact of this thesis is related to constructing a viable modified Telepar- allel theory which could describe the evolution of the Universe without evoking any cosmological constant. This tells us that modifications of Teleparallel gravity could be good potential candidates for describing the dark energy problem. An additional impact related to a quantum gravity approach was also shown in iii this thesis. There was proposed a new non-local Teleparallel theory of gravity which is the first attempt to integrate this fundamental physical concept from quantum gravity into Teleparallel gravity. Moreover, in different studies related to non-locality (based on GR), researchers found that exponential type of couplings were able to achieve a renormalisable theory of gravity. In all of those studies, researchers put those terms by hand. In one work presented in this thesis, it was shown that these exponential couplings appear naturally by using first physical principles with the symmetries of the Lagrangian. Therefore, this study had an impact in resolving this aspect about non-locality. Preface Acknowledgments I would like to thank all the people who influenced and helped me throughout my PhD studies. Firstly, I would like to thank my supervisor Christian B¨ohmerfor his fantastic su- pervision over these four years. He has been an inspiration to me, not only as an excellent researcher but also as a person. It was very easy to be his student due to his supporting and easy going way of thinking. He has given me much good advice to become a better researcher and always been very objective and helpful. I also thank him for giving me the opportunity of doing independent research in other subjects and with other researchers. Secondly, I would like to thank all my international collaborators who have helped me to learn many different subjects and gave me the opportunity to publish different papers. Explicitly, I would like to thank in alphabetical order, the following collabo- rators: Salvatore Capozziello, Mir Faizal, Eduardo I. Guendelman, Mubasher Jamil, Francisco Lobo, Sergei D. Odintsov and Muhammad Zubair. Thirdly, I would like to thank Angela Cooper who has helped me a lot proofreading my thesis and for giving me very good feedback. I hope that she can be recover soon. I would also like to thank my friends who have supported me in this journey. Special thanks to my friends: Konstantinos F. Dialektopoulos, Manuela Irarr´azabal,Martin Krˇsˇs´ak,Catalina Maturana, Mihai Marciu, Victor Montenegro, Gabriel Saavedra, iv v Pietro Servini and Belgin Seymenoglu. I would also like to thank all my family for supporting me not only in my PhD, but in all my studies, specially my parents Hern´anand Pilar, my brothers Crist´obaland Joaqu´ınand my grandparents Fridia, Ester and Mois´es.Without them, I would not have been able to follow my dreams to study the subject that I love in a foreign country. I would also like to thank the love of my life, Maria-Jos´eVera who has been the most important person throughout this PhD. Without her love, support and caring, it would not have been possible to finish this PhD and also my life would not been so complete as it is. All the experiences shared with her gave me the power to continue and also to live the best years of my life. Finally, I would like to acknowledge financial support from the Chilean government for supporting me through my doctoral scholarship Becas Chile (CONICYT). To my family. vii List of Publications during my PhD 1. S. Bahamonde, C. G. Boehmer, S. Carloni, E. J. Copeland, W. Fang and N. Tamanini, \Dynamical systems applied to cosmology: dark energy and mod- ified gravity," [arXiv:1712.03107 [gr-qc]] (to appear in Physics Reports). 2. A. Dutta, S. Gangopadhyay, S. Bahamonde and M. Faizal, \The Effect of Modified Dispersion Relation on Dumb Holes," Int. J. Mod. Phys. D 27 (2018) 1850113, [arXiv:1805.01294 [gr-qc]]. 3. S. Bahamonde, \Generalised nonminimally gravity-matter coupled theory," Eur. Phys. J. C 78 (2018) no.4, 326, [arXiv:1709.05319 [gr-qc]]. 4. S. Bahamonde, M. Marciu and P. Rudra, \Generalised teleparallel quintom dark energy non-minimally coupled with the scalar torsion and a boundary term," JCAP 1804 (2018) no.04, 056, [arXiv:1802.09155 [gr-qc]]. 5. S. Bahamonde, M. Zubair and G. Abbas, \Thermodynamics and cosmolog- ical reconstruction in f(T;B) gravity," Phys.Dark Univ. 19 (2018) 78-90, [arXiv:1609.08373 [gr-qc]]. 6. A. Banerjee, K. Jusufi and S. Bahamonde, \Stability of a d-dimensional thin- shell wormhole surrounded by quintessence," Grav.Cosmol. 24 (2018) 1, [arXiv:1612.06892 [gr-qc]]. 7. S. Bahamonde, C. G. Boehmer and M. Krˇsˇs´ak, \New classes of modified teleparallel gravity models," Phys. Lett. B 775 (2017) 37, [arXiv:1706.04920 [gr-qc]]. 8. S. Bahamonde, S. Capozziello and K. F. Dialektopoulos, \Constraining Gen- eralized Non-local Cosmology from Noether Symmetries," Eur. Phys. J. C 77 (2017) no.11, 722, [arXiv:1708.06310 [gr-qc]]. viii 9. S. Bahamonde, C. G. Boehmer and P. Neff. \Geometrically nonlinear Cosserat elasticity in the plane: applications to chirality", Journal of Mechanics of Ma- terials and Structures 12 (2017) 689-710 [arXiv:1705.04868 [math-ph]] 10. S. Bahamonde, S. Capozziello, M. Faizal and R. C. Nunes, \Nonlocal Teleparal- lel Cosmology," Eur. Phys. J. C 77 (2017) no.9, 628, [arXiv:1709.02692 [gr-qc]]. 11. M. Zubair, S. Bahamonde and M. Jamil, \Generalized Second Law of Ther- modynamic in Modified Teleparallel Theory," Eur. Phys. J. C 77 (2017) no.7, 472, [arXiv:1604.02996 [gr-qc]]. 12. M. Azreg-A¨ınou, S. Bahamonde and M. Jamil, \Strong Gravitational Lens- ing by a Charged Kiselev Black Hole," Eur. Phys. J. C 77 (2017) no.6, 414, [arXiv:1701.02239 [gr-qc]]. 13. D. Momeni, M. Faizal, A. Myrzakul, S. Bahamonde and R. Myrzakulov, \A Holographic Bound for D3-Brane," Eur. Phys. J. C 77 (2017) no.6, 391, [arXiv:1608.08819 [hep-th]]. 14. S. Bahamonde, S. D. Odintsov, V. K. Oikonomou and P. V. Tretyakov, \De- celeration versus Acceleration Universe in Different Frames of F (R) Gravity,", Phys. Lett. B 766 (2017) 225-230, [arXiv:1701.02381 [gr-qc]]. 15. S. Bahamonde and S. Capozziello, \Noether Symmetry Approach in f(T;B) teleparallel cosmology," Eur. Phys. J. C 77 (2017) no.2, 107, [arXiv:1612.01299 [gr-qc]]. 16. N. S. Mazhari, D. Momeni, S. Bahamonde, M. Faizal and R. Myrzakulov, \Holographic Complexity and Fidelity Susceptibility as Holographic Informa- tion Dual to Different Volumes in AdS," Phys. Lett. B 766 (2017) 94, [arXiv:1609.00250 [hep-th]]. ix αβ 17. M. Zubair, F. Kousar and S. Bahamonde, \Thermodynamics in f(R; RαβR ; φ) theory of gravity," Phys. Dark Univ. 14 (2016) 116, [arXiv:1604.07213 [gr-qc]]. 18. S. Bahamonde and C. G. B¨ohmer,\Modified teleparallel theories of gravity: Gauss{Bonnet and trace extensions," Eur. Phys. J. C 76 (2016) no.10, 578, [arXiv:1606.05557 [gr-qc]]. 19. S. Bahamonde, U. Camci, S. Capozziello and M. Jamil, \Scalar-Tensor Telepar- allel Wormholes by Noether Symmetries," Phys. Rev. D 94 (2016) 084042 [arXiv:1608.03918 [gr-qc]].