DOCSLIB.ORG

Milgrom's Law and Lambda's Shadow: How Massive Gravity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Milgrom's Law and Lambda's Shadow: How Massive Gravity Journal of the Korean Astronomical Society preprint - no DOI assigned 00: 1 ∼ 4, 2015 June pISSN: 1225-4614 · eISSN: 2288-890X The Korean Astronomical Society (2015) http://jkas.kas.org MILGROM'S LAW AND Λ'S SHADOW: HOW MASSIVE GRAVITY CONNECTS GALACTIC AND COSMIC DYNAMICS Sascha Trippe Department of Physics and Astronomy, Seoul National University, Seoul 151-742, Korea; [email protected] Received March 11, 2015; accepted June 2, 2015 Abstract: Massive gravity provides a natural solution for the dark energy problem of cosmology and is also a candidate for resolving the dark matter problem. I demonstrate that, assuming reasonable scaling relations, massive gravity can provide for Milgrom’s law of gravity (or “modified Newtonian dynamics”) which is known to remove the need for particle dark matter from galactic dynamics. Milgrom’s law comes with a characteristic acceleration, Milgrom’s constant, which is observationally constrained to a0 1.1 10−10 ms−2. In the derivation presented here, this constant arises naturally from the cosmologically≈ × required mass of gravitons like a0 c√Λ cH0√3ΩΛ, with Λ, H0, and ΩΛ being the cosmological constant, the Hubble constant, and the∝ third cosmological∝ parameter, respectively. My derivation suggests that massive gravity could be the mechanism behind both, dark matter and dark energy. Key words: gravitation — cosmology — dark matter — dark energy 1. INTRODUCTION On smaller scales, the dynamics of galaxies is in ex- cellent agreement with a modification of Newtonian dy- Modern standard cosmology suffers from two criti- namics (the MOND paradigm) in the limit of small ac- cal issues: the dark matter problem and the dark celeration (or gravitational field strength) g, expressed energy problem. Canonical ΛCDM cosmology (e.g., in Milgrom’s law Bahcall et al. 1999) requires about 95% (e.g., Ade et al. 2014) of the mass/energy content of the universe to be g = µ(g/a ) g (1) provided by exotic dark components for which physi- N 0 cal counterparts have not been identified. Dark matter (Milgrom 1983a) with gN being the Newtonian field and dark energy are commonly assumed to be unre- strength, a0 denoting Milgrom’s constant, and µ be- lated: dark matter is identified with as yet undiscovered ing a transition function with µ = 1 for g/a0 1 and new exotic elementary particles, whereas dark energy is ≫ µ = g/a0 for g/a0 1. Such a modification implies linked to a cosmological constant, Λ, which is inserted that the ratio of dynamical≪ and luminous masses of a into Einstein’s field equations and which triggers an ac- stellar system, the mass discrepancy Mdyn/M0, exceeds celerated expansion of the universe on its largest scales. unity and becomes a function of acceleration (or field ΛCDM cosmology is based on the assumption that strength) when assuming Newton’s law of gravity. In gravitation is Einsteinian on all scales. Modified the- the “deep MOND” limit g/a0 1, the circular speed ories of gravity might be able to describe the universe ≪ vc of stars in rotation-supported dynamical systems (es- without dark components; this idea has initiated a vast pecially disk galaxies) is number of works exploring multiple models of gravita- tion (see Clifton et al. 2012 for an exhaustive review). 4 vc = GM0 a0 (2) A quantum-field theoretical modified theory of gravity, first proposed by Fierz & Pauli (1939), is massive grav- with G being Newton’s constant (Milgrom 1983a,b,c); arXiv:1506.00730v1 [astro-ph.CO] 2 Jun 2015 ity in which gravitation is mediated by virtual bosons, 2 2 2 note that g = vc /r = GMdyn/r and gN = GM0/r for gravitons, that have a very small (compared to all other circular orbits with radius r. For pressure-supported elementary particles) non-zero mass. Massive gravi- systems (especially elliptical galaxies and galaxy clus- tons, being virtual exchange particles, have a limited ters) with 3D velocity dispersion σ, the mass–velocity life time governed by Heisenberg’s uncertainty relation relation takes the form for energy and time. If the mass of the graviton is 4 4 such that its life time is 1/Λ, the decay of gravity σ = GM0 a0 (3) induces an accelerated expansionp of the universe like 9 the one actually observed – thus providing an elegant (Milgrom 1984, 1994). Equations (1, 2, 3), with resolution of the dark energy problem (recently, e.g., −10 −2 a0 1.1 10 ms , naturally provide for the Cardone et al. 2012; Clifton et al. 2012; Hinterbichler ≈ × 2012; Volkov 2012; Tasinato et al. 2013; De Felice et al. fundamental scaling laws of galactic kinematics, 2013; de Rham 2014). specifically the baryonic Tully–Fisher, baryonic Faber–Jackson, mass discrepancy–acceleration Corresponding author: S. Trippe (MDA), and surface mass density–acceleration 1 2 Trippe relations, plus the asymptotic flattening of rota- the cosmic expansion history requires Tg 1/Λ, re- ≈ tion curves and the occurrence of “dark rings” in sulting in p galaxy clusters (Sanders 1994, 2010; Rhee 2004a,b; ~ mg √Λ . (8) McGaugh 2004, 2005a,b; Milgrom & Sanders 2008; ≈ c2 McGaugh 2011; Gentile et al. 2011; Cardone et al. According to the standard “cosmic triangle” formalism 2 2011; Famaey & McGaugh 2012; Trippe 2014; (e.g., Bahcall et al. 1999), Λ = 3H0 ΩΛ, resulting in Walker & Loeb 2014; Wu & Kroupa 2015; Milgrom ~ 2015; Chae & Gong 2015). Milgrom’s law eliminates mg 2 H0 3ΩΛ ; (9) particle dark matter from galactic dynamics together ≈ c p −1 −1 with its substantial difficulties (cf., e.g., Kroupa 2012, for H0 70kms Mpc and ΩΛ 0.7 (e.g., ≈ −35 −2 ≈ 2015). Ade et al. 2014), Λ 1.1 10 s and thus mg As pointed out by Trippe (2013a,b,c, 2015), massive 4 10−69 kg 2 10≈−33 eV×c−2. ≈ gravity can, at least in principle, provide Milgrom’s law ×A natural choice≈ × for a quantum-physical mass scale, together with an expression for the function µ in agree- x in our case, is the Planck mass ment with observations. Following up on this ansatz, I investigate the link of galactic dynamics to the cosmic ~ c x mP = (10) expansion history. A physical connection is established ≡ r G by the graviton mass mg √Λ. Milgrom’s constant is with G being Newton’s constant; combination of Equa- ∝ −61 given by a0 c√Λ cH0√3ΩΛ, with c, Λ, H0, and tions (9) and (10) leads to mg/mP 2 10 . The ∝ ∝ ≈ × ΩΛ being the speed of light, the cosmological constant, choice for the scale y is less obvious; for our purpose, we the (present-day) Hubble constant, and the (present- require a scale characteristic for gravitationally bound day) third cosmological parameter, respectively. Ac- dynamical systems. A good choice (to be discussed in cordingly, massive gravity might offer a natural alter- Section 3) is the magnitude of the gravitational poten- native explanation for the phenomena conventionally tial, associated with dark matter and dark energy. GMdyn ǫ = (11) rc2 2. CALCULATIONS (cf. also Baker et al. 2015). As ǫ is dimensionless, we Massive gravity implies (Trippe 2013a,b,c, 2015) that can supply y with the unit of a length by multiplying ǫ any luminous (“baryonic”) mass M0 is the source of a and the Planck length spherical graviton halo with mass density ~ G −2 lP = 3 (12) ρg(R)= mg ng(R)= mg ηM0 R (4) r c so that y ǫl . where m is the graviton mass, n is the graviton parti- P g g Inserting≡ Equations (8, 10, 11, 12) into Equation (6) cle density, R is the radial coordinate, and η is a scaling −2 leads to factor. The proportionality ρg M0R follows from ∝ M r2 (i) consistency with the classical force law in the New- dyn =1+4 πfc √Λ . (13) tonian limit, and (ii) the inverse-square-of-distance law M0 GMdyn of flux conservation (but see also the discussion in Sec- Using the classical expression for gravitational field tion 3). For a circular orbit of radius r around M , the 0 strength, g = GM /r2, one finds total, dynamical mass experienced by a test particle dyn follows from integrating ρg(R) from 0 to r, M c √Λ dyn =1+4 πf (14) M0 g Mdyn = M0 + Mg = M0 (1+4 πηmg r) (5) or, using the expression given by Equation (9), with Mg being the mass contributed by the graviton halo. One can bring this expression into the more in- Mdyn cH0 √3ΩΛ =1+4 πf . (15) tuitive form M0 g M m r dyn =1+4 πf g (6) Comparison of Equations (14, 15) to the expres- M0 x y sion following from the “simple µ function” (e.g., Famaey & McGaugh 2012) of MOND, where f is a dimensionless number, x is a mass scale, and y is a length scale (with η f/(xy)). Mdyn a0 ≡ =1+ , (16) With gravitons being virtual exchange particles, their M0 g effective mass can be estimated from (cf., e.g., Griffiths shows that the expressions are equivalent, with Mil- 2008) Heisenberg’s uncertainty relation for energy and grom’s constant being given by a =4πfc√Λ. Match- time, 0 2 ing this expression with the empirical value a0 1.1 mgc Tg ~ (7) −10 −2 ≈ × × ≈ 10 ms requires a (universal) value of f 0.0088 ≈ with Tg being the life time of gravitons and ~ being the (or 4πf 1/9). It is straightforward to see that in the ≈ reduced form of Planck’s constant. Consistency with limit g a0, Equation (16) leads to Equation (2). ≪ How Massive Gravity Connects Galactic and Cosmic Dynamics 3 3. DISCUSSION sive gravity – which has not been found yet.
Load more

Recommended publications
  • Conformally Coupled General Relativity
    universe Article Conformally Coupled General Relativity Andrej Arbuzov 1,* and Boris Latosh 2 ID 1 Bogoliubov Laboratory for Theoretical Physics, JINR, Dubna 141980, Russia 2 Dubna State University, Department of Fundamental Problems of Microworld Physics, Universitetskaya str. 19, Dubna 141982, Russia; [email protected] * Correspondence: [email protected] Received: 28 December 2017; Accepted: 7 February 2018; Published: 14 February 2018 Abstract: The gravity model developed in the series of papers (Arbuzov et al. 2009; 2010), (Pervushin et al. 2012) is revisited. The model is based on the Ogievetsky theorem, which specifies the structure of the general coordinate transformation group. The theorem is implemented in the context of the Noether theorem with the use of the nonlinear representation technique. The canonical quantization is performed with the use of reparametrization-invariant time and Arnowitt– Deser–Misner foliation techniques. Basic quantum features of the models are discussed. Mistakes appearing in the previous papers are corrected. Keywords: models of quantum gravity; spacetime symmetries; higher spin symmetry 1. Introduction General relativity forms our understanding of spacetime. It is verified by the Solar System and cosmological tests [1,2]. The recent discovery of gravitational waves provided further evidence supporting the theory’s viability in the classical regime [3–6]. Despite these successes, there are reasons to believe that general relativity is unable to provide an adequate description of gravitational phenomena in the high energy regime and should be either modified or replaced by a new theory of gravity [7–11]. One of the main issues is the phenomenon of inflation. It appears that an inflationary phase of expansion is necessary for a self-consistent cosmological model [12–14].
    [Show full text]
  • Ads₄/CFT₃ and Quantum Gravity
    AdS/CFT and quantum gravity Ioannis Lavdas To cite this version: Ioannis Lavdas. AdS/CFT and quantum gravity. Mathematical Physics [math-ph]. Université Paris sciences et lettres, 2019. English. NNT : 2019PSLEE041. tel-02966558 HAL Id: tel-02966558 https://tel.archives-ouvertes.fr/tel-02966558 Submitted on 14 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Prepar´ ee´ a` l’Ecole´ Normale Superieure´ AdS4/CF T3 and Quantum Gravity Soutenue par Composition du jury : Ioannis Lavdas Costas BACHAS Le 03 octobre 2019 Ecole´ Normale Superieure Directeur de These Guillaume BOSSARD Ecole´ Polytechnique Membre du Jury o Ecole´ doctorale n 564 Elias KIRITSIS Universite´ Paris-Diderot et Universite´ de Rapporteur Physique en ˆIle-de-France Crete´ Michela PETRINI Sorbonne Universite´ President´ du Jury Nicholas WARNER University of Southern California Membre du Jury Specialit´ e´ Alberto ZAFFARONI Physique Theorique´ Universita´ Milano-Bicocca Rapporteur Contents Introduction 1 I 3d N = 4 Superconformal Theories and type IIB Supergravity Duals6 1 3d N = 4 Superconformal Theories7 1.1 N = 4 supersymmetric gauge theories in three dimensions..............7 1.2 Linear quivers and their Brane Realizations...................... 10 1.3 Moduli Space and Symmetries............................
    [Show full text]
  • Cosmologies of Extended Massive Gravity
    Cosmologies of extended massive gravity Kurt Hinterbichler, James Stokes, and Mark Trodden Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA (Dated: June 15, 2021) We study the background cosmology of two extensions of dRGT massive gravity. The first is variable mass massive gravity, where the fixed graviton mass of dRGT is replaced by the expectation value of a scalar field. We ask whether self-inflation can be driven by the self-accelerated branch of this theory, and we find that, while such solutions can exist for a short period, they cannot be sustained for a cosmologically useful time. Furthermore, we demonstrate that there generally exist future curvature singularities of the “big brake” form in cosmological solutions to these theories. The second extension is the covariant coupling of galileons to massive gravity. We find that, as in pure dRGT gravity, flat FRW solutions do not exist. Open FRW solutions do exist – they consist of a branch of self-accelerating solutions that are identical to those of dRGT, and a new second branch of solutions which do not appear in dRGT. INTRODUCTION AND OUTLINE be sustained for a cosmologically relevant length of time. In addition, we show that non-inflationary cosmological An interacting theory of a massive graviton, free of solutions to this theory may exhibit future curvature sin- the Boulware-Deser mode [1], has recently been discov- gularities of the “big brake” type. ered [2, 3] (the dRGT theory, see [4] for a review), al- In the second half of this letter (which can be read lowing for the possibility of addressing questions of in- independently from the first), we consider the covariant terest in cosmology.
    [Show full text]
  • Massive Supergravity
    1 Massive Supergravity Taichiro Kugo Maskawa Institute, Kyoto Sangyo University Nov. 8 { 9, 2014 第4回日大理工・益川塾連携 素粒子物理学シンポジウム in collaboration with Nobuyoshi Ohta Kinki University 2 1 Introduction Cosmological Constant Problem: Higgs Condensation ∼ ( 100 GeV )4 QCD Chiral Condensation ∼ ( 100 MeV )4 (1) These seem not contributing to the Cosmological Constant! =) Massive Gravity: an idea toward resolving it However, Massive Gravity has its own problems: • van Dam-Veltman-Zakharov (vDVZ) discontinuity Its m ! 0 limit does not coincides with the Einstein gravity. • Boulware-Deser ghost − |{z}10 |(1{z + 3)} = 6 =|{z} 5 + |{z}1 (2) hµν i N=h00;N =h0i massive spin2 BD ghost We focus on the BD ghost problem here. 3 In addition, we believe that any theory should eventually be made super- symmetric, that is, Supergravity (SUGRA). This may be of help also for the problem that the dRGT massive gravity allows no stable homogeneous isotropic universe solution. In this talk, we 1. explain the dRGT theory 2. massive supergravity 2 Fierz-Pauli massive gravity (linearized) Einstein-Hilbert action p LEH = −gR (3) [ ] h i 2 L L −m 2 − 2 = EH + (hµν ah ) (4) quadratic part in hµν | 4 {z } Lmass = FP (a = 1) gµν = ηµν + hµν (5) 4 In Fierz-Pauli theory with a = 1, there are only 5 modes describing properly massive spin 2 particle. : ) No time derivative appears for h00; h0i in LEH !LEH is linear in N; Ni. Lmass ∼ If a = 1, the mass term FP is also clearly linear in N h00 ! =) • Ni can be solved algebraically and be eliminated.
    [Show full text]
  • Three-Dimensional Teleparallel Chern-Simons Supergravity Theory
    Three-dimensional teleparallel Chern-Simons supergravity theory Ricardo Caroca∗, Patrick Concha∗, Diego Pe˜nafiel‡, Evelyn Rodr´ıguez, ∗Departamento de Matem´atica y F´ısica Aplicadas, Universidad Cat´olica de la Sant´ısima Concepci´on, Alonso de Ribera 2850, Concepci´on, Chile. ‡Facultad de Ciencias, Universidad Arturo Prat, Iquique, Chile. [email protected], [email protected], [email protected], [email protected], Abstract In this work we present a gauge-invariant three-dimensional teleparallel supergravity theory using the Chern-Simons formalism. The present construction is based on a supersymmetric extension of a particular deformation of the Poincar´ealgebra. At the bosonic level the theory describes a non-Riemannian geometry with a non-vanishing torsion. In presence of supersym- metry, the teleparallel supergravity theory is characterized by a non-vanishing super-torsion in which the cosmological constant can be seen as a source for the torsion. We show that the teleparallel supergravity theory presented here reproduces the Poincar´esupergravity in the van- ishing cosmological limit. The extension of our results to N = p + q supersymmetries is also explored. arXiv:2103.06717v1 [hep-th] 11 Mar 2021 1 Introduction Teleparallel gravity is an alternative theory of gravity known to be considered equivalent to General Relativity. However, they are conceptually quite different. In particular, the teleparallel formulation of gravity is described by a vanishing curvature and a non-vanishing torsion which characterizes the parallel transport [1–5]. In such case, the geometry is no more Riemannian but corresponds to the so-called Riemannian-Cartan (Weizenb¨ock) geometry. In three spacetime dimensions, there has been an interest in exploring black hole solutions and boundary symmetries of gravity theories with torsion [6–12].
    [Show full text]
  • Cosmological Solutions from Quasi-Dilaton Massive Gravity
    Saurabh Kumar, Department of Physics, Case Western Reserve University, Cleveland, Ohio. Co-authors: Stefano Anselmi (IAP & Observatoire de Paris), Diana Lopez-Nacir (CERN), Glenn Starkman (CWRU). Einstein’s GR is a massless spin-2 theory. Linearized GR + Mass Term (Fierz-Pauli Action): 5 degrees of freedom (2 in GR). vDVZ Discontinuity: Take m 0, you can’t recover GR! (van Dam & Veltman Nucl. Phys. B 1970, Zakharov JETP Lett. 1970) SAURABH KUMAR (CWRU) TeVPA August 2017 2 Non-linear Kinetic but Linear Potential Term: leads to BD Ghost (6th dof). (Boulware, Deser PRD 1972) dRGT theory is ghost free in all orders of interactions: (de Rham, Gabadadze & Tolley PRL 2011, Hassan & Rosen PRL 2012) SAURABH KUMAR (CWRU) TeVPA August 2017 3 dRGT theory does not admit FLAT FLRW solutions! (Guido D’Amico et.al. PRD 2011) Alternatives: Bi-metric gravity, (Hassan and Rosen, JHEP 2012) Graviton Mass a field, (Huang et.al, PRD 2012) Background coupled to a Scalar Field: Extended Quasi-Dilaton Massive Gravity (EQDMG) (Felice & Mukhoyama, Phys. Lett. B, 2014) SAURABH KUMAR (CWRU) TeVPA August 2017 4 Non-linear Kinetic Term of Graviton Kinetic Term of Quasi-Dilaton Metric-Quasi- Dilaton Interaction SAURABH KUMAR (CWRU) TeVPA August 2017 5 SUCCESS We reproduce observed expansion history. (Anselmi et.al. PRD 2015) Dark Energy Matter Radiation FAILURE Pathology: Scalar Perturbations are unstable: Wrong sign kinetic term for large wavenumbers. (Gumrukcuoglu et. al. PRD 2013, Guido D’Amico et.al. PRD 2013) SAURABH KUMAR (CWRU) TeVPA August 2017 7 Pathology can be treated by introducing a new coupling term to the fiducial metric.
    [Show full text]
  • Exotic Gauge Theories of Spin-2 Fields
    Imperial College of Science, Technology and Medicine Department of Physics Exotic Gauge Theories of Spin-2 Fields Nicholas Andrew Ondo Supervised by Dr. Andrew J. Tolley Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in in Physics of Imperial College London and the Diploma of Imperial College, October 2017 Abstract There has been a resurgence of interest in theories of massive spin-2 fields, owing to the recent discovery of ghost-free self-interactions. In addition to reviewing the historical and recent progress in this subject, I discuss my contributions, including the derivation of the complete decoupling limit of dRGT ghost-free massive gravity, proving no-go theorems on ghost-free interactions for charged spin-2 fields, updating the method of Dimensional Deconstruction for fermions to obtain massive supersym- metric gauge theories, and my progress towards supergravity theories with non-zero graviton mass. i ii Statement of Originality All of original research presented in this thesis are works that were done either by myself or alongside contributors. My original contributions to these works represent a substantial contribution. Any non-original works are appropriately cited to the best of the author's knowledge. iii iv Publications (*)\ The Complete Decoupling Limit of Ghost-free Massive Gravity" by Nicholas A. Ondo, Andrew J. Tolley. Published in JHEP 1311 (2013) 059. [ARXIV:1307.4769] (*)\ Interactions of Charged Spin-2 Fields" by Claudia de Rham, Andrew Matas, Nicholas A. Ondo, and Andrew J. Tolley. Published in Class.Quant.Grav. 32 (2015) no.17, 175008. [ARXIV:1410.5422] (*)\ Deconstructing Supergravity, I: Massive Supermultiplets" by Nicholas A.
    [Show full text]
  • THE STRUCTURE of GRAVITY and SPACE-‐TIME International
    THE STRUCTURE OF GRAVITY AND SPACE-TIME International Workshop 6-7 February 2014 Lady Margaret Hall, Oxford, UK Part of the ‘Establishing the Philosophy of Cosmology’ project The aim of this workshop is to review the state of the art of different models of gravity and to engage a debate on fundamental questions on the nature of gravity and the nature of space-time. Preference will be given to ample open discussion and to exploration of new territories. We will focus on a few metric and fundamental theories of space-time and gravity, to discuss what the theoretical-philosophical inputs and phenomenological outputs are for each, and what we empirically can and currently do know about them. 1 What are the similarities and differences? What scales enter? Are they ‘natural’? Interrelated? What limits do current observations place? What are the future experimental prospects? What theoretical and technical improvement is needed? What are the philosophical implications? Some aspects we will consider are not limited to i) Horizons ii) Lorentz violation iii) Equivalence principle iv) Entropy - thermodynamics v) Gravitational collapse, black holes vi) Lensing vii) Gravitational waves, inflation viii) Thermal/temporal evolution of the Universe, geometrodynamics, causal structure Participation is by invitation only, and the number of participants will be limited to approximately 40 people. The participants come from a wide range of backgrounds, and we will accordingly aim for inclusiveness in terms of content level, yet encourage necessary technical detail as appropriate. Organisers: Joe Silk (Oxford), Simon Saunders (Oxford), Martin Sahlen (Oxford), Khalil Chamcham (Oxford), David Sloan (Cambridge) 2 WEDNESDAY 5 FEBRUARY 18.00 Welcome reception: Denys Wilkinson Building, Keble Road, Level 5.
    [Show full text]
  • Massless and Massive Multi-Gravity in a Multi-Fold Universe
    Cite as: Stephane H Maes, (2020), ”Massless and Massive Multi-Gravity in a Multi-fold Universe”, viXra:2010.0095v1, https://vixra.org/pdf/2010.0095v1.pdf, https://shmaesphysics.wordpress.com/2020/06/30/ massless-and-massive-multi-gravity-in-a-multi-fold-universe/, June 19, 2020. Massless and Massive Multi-Gravity in a Multi-fold Universe Stephane H. Maes1 June 30, 2020 Abstract: In a multi-fold universe, gravity emerges from Entanglement through the multi-fold mechanisms. As a result, gravity-like effects appear in between entangled particles that they be real or virtual. Long range, massless gravity results from entanglement of massless virtual particles. Entanglement of massive virtual particles leads to massive gravity contributions at very smalls scales. Multi-folds mechanisms also result into a spacetime that is discrete, with a random walk fractal structure and non-commutative geometry that is Lorentz invariant and where spacetime nodes and particles can be modeled with microscopic black holes. All these recover General relativity at large scales and semi-classical model remain valid till smaller scale than usually expected. Gravity can therefore be added to the Standard Model. This can contribute to resolving several open issues with the Standard Model. In this paper, we discuss some of the properties and implications of the massive gravity contributions. In particular we will discuss it similarities and differences with what is usually meant by massive gravity in conventional physics and derived modern bigravity theories.. Instead of being a large-scale theory, where massive gravity can support a limited range or even a repulsive behavior, multi-fold massive gravity is here a mostly small-scale effect with almost no larger scale impact other than through entangled virtual neutrino pairs.
    [Show full text]
  • Cosmic Tests of Massive Gravity
    Cosmic tests of massive gravity Jonas Enander Doctoral Thesis in Theoretical Physics Department of Physics Stockholm University Stockholm 2015 Doctoral Thesis in Theoretical Physics Cosmic tests of massive gravity Jonas Enander Oskar Klein Centre for Cosmoparticle Physics and Cosmology, Particle Astrophysics and String Theory Department of Physics Stockholm University SE-106 91 Stockholm Stockholm, Sweden 2015 Cover image: Painting by Niklas Nenz´en,depicting the search for massive gravity. ISBN 978-91-7649-049-5 (pp. i{xviii, 1{104) pp. i{xviii, 1{104 c Jonas Enander, 2015 Printed by Universitetsservice US-AB, Stockholm, Sweden, 2015. Typeset in pdfLATEX So inexhaustible is nature's fantasy, that no one will seek its company in vain. Novalis, The Novices of Sais iii iv Abstract Massive gravity is an extension of general relativity where the graviton, which mediates gravitational interactions, has a non-vanishing mass. The first steps towards formulating a theory of massive gravity were made by Fierz and Pauli in 1939, but it took another 70 years until a consistent theory of massive gravity was written down. This thesis investigates the phenomenological implications of this theory, when applied to cosmology. In particular, we look at cosmic expansion histories, structure formation, integrated Sachs-Wolfe effect and weak lensing, and put constraints on the allowed parameter range of the theory. This is done by using data from supernovae, the cosmic microwave background, baryonic acoustic oscillations, galaxy and quasar maps and galactic lensing. The theory is shown to yield both cosmic expansion histories, galactic lensing and an integrated Sachs-Wolfe effect consistent with observations.
    [Show full text]
  • Theoretical Aspects of Massive Gravity
    Theoretical Aspects of Massive Gravity Kurt Hinterbichlera;1 a Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA Abstract Massive gravity has seen a resurgence of interest due to recent progress which has overcome its traditional problems, yielding an avenue for addressing important open questions such as the cosmological constant naturalness problem. The possibility of a massive graviton has been studied on and off for the past 70 years. During this time, curiosities such as the vDVZ discontinuity and the Boulware-Deser ghost were uncovered. We re-derive these results in a pedagogical manner, and develop the St¨ukelberg formalism to discuss them from the modern effective field theory viewpoint. We review recent progress of the last decade, including the dissolution of the vDVZ discontinuity via the Vainshtein screening mechanism, the existence of a consistent effective field theory with a stable hierarchy between the graviton mass and the cutoff, and the existence of particular interactions which raise the maximal effective field theory cutoff and remove the ghosts. In addition, we review some peculiarities of massive gravitons on curved space, novel theories in three dimensions, and examples arXiv:1105.3735v2 [hep-th] 2 Oct 2011 of the emergence of a massive graviton from extra-dimensions and brane worlds. 1E-mail address: [email protected] Contents 1 Introduction4 1.1 General relativity is massless spin 2 . .4 1.2 Modifying general relativity . .7 1.3 History and outline . 10 2 The free Fierz-Pauli action 13 2.1 Hamiltonian and degree of freedom count .
    [Show full text]
  • Massive Gravity Theories
    Colby College Digital Commons @ Colby Honors Theses Student Research 2013 Massive Gravity Theories Greg Seyfarth Colby College Follow this and additional works at: https://digitalcommons.colby.edu/honorstheses Part of the Elementary Particles and Fields and String Theory Commons Colby College theses are protected by copyright. They may be viewed or downloaded from this site for the purposes of research and scholarship. Reproduction or distribution for commercial purposes is prohibited without written permission of the author. Recommended Citation Seyfarth, Greg, "Massive Gravity Theories" (2013). Honors Theses. Paper 716. https://digitalcommons.colby.edu/honorstheses/716 This Honors Thesis (Open Access) is brought to you for free and open access by the Student Research at Digital Commons @ Colby. It has been accepted for inclusion in Honors Theses by an authorized administrator of Digital Commons @ Colby. Massive Gravity Theories Greg Seyfarth Department of Physics and Astronomy Colby College 2013 Abstract We examine Fierz and Pauli's work in 1939 of adding a mass term to the Lagrangian 1 2 µν 2 for linearized gravity with the form − 2 m (hµνh + λh ) where hµν is a small pertur- bation away from flat spacetime and showing that the only value for λ which gives a ghost-free theory with the correct five degrees of freedom expected for a massive spin- 2 particle such as the graviton is λ = −1 [1]. We start by rederiving the Lagrangian formulation for classical electrodynamics and giving examples of a ghost. Then we solve the linearized massless gravity equations to show their two degrees of freedom. We perform a Hamiltonian analysis for massless linearized gravity.
    [Show full text]