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Modifications to Gravitational Wave Equation from Canonical Quantum Louisiana State University LSU Digital Commons Faculty Publications Department of Physics & Astronomy 8-14-2020 Modifications ot gravitational wave equation from canonical quantum gravity Andrea Dapor Klaus Liegener Follow this and additional works at: https://digitalcommons.lsu.edu/physics_astronomy_pubs Part of the Astrophysics and Astronomy Commons, and the Physics Commons Eur. Phys. J. C (2020) 80:741 https://doi.org/10.1140/epjc/s10052-020-8333-8 Letter Modifications to gravitational wave equation from canonical quantum gravity Andrea Dapor1,a, Klaus Liegener2,b 1 Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA 2 II. Institute for Theoretical Physics, University of Hamburg, Luruper Chausee 149, 22761 Hamburg, Germany Received: 3 February 2020 / Accepted: 8 August 2020 / Published online: 14 August 2020 © The Author(s) 2020 Abstract It is expected that the quantum nature of space- detectable, and are expected to lead to a new understanding time leaves its imprint in all semiclassical gravitational sys- of the early universe. This includes the possibility to probe tems, at least in certain regimes, including gravitational the so far unknown frontier of quantum gravity. Indeed, first waves. In this paper we investigate such imprints on gravita- approaches exist which use GW to narrow down the possibil- tional waves within a specific framework: space is assumed ity of discrete spacetimes [8] featuring an intrinsic discrete- to be discrete (in the form of a regular cubic lattice), and ness scale typically associated to the Planck length. However this discrete geometry is quantised following Dirac’s canon- before these phenomenological tests can be carried out fully, ical quantisation scheme. The semiclassical behavior is then it is important to understand the theoretical predictions about extracted by promoting the expectation value of the Hamilto- the propagation of GW from a given theory of quantum grav- nian operator on a semiclassical state to an effective Hamil- ity. tonian. Considering a family of semiclassical states repre- The present work will elucidate on this endeavour. We senting small tensor perturbations to Minkowski background, will focus on a particular approach towards quantum gravity, we derive a quantum-corrected effective wave equation. The namely Loop Quantum Gravity (LQG) [9–12]. This candi- deviations from the classical gravitational wave equation are date for canonical quantum gravity has matured in the last found to be encoded in a modified dispersion relation and decades towards a mathematically well-defined theory. It controlled by the discreteness parameter of the underlying is a quantum field theory on a continuous manifold, how- lattice. For finite discretisations, several interesting effects ever the observables of geometrical quantities (such as areas appear: we investigate the thermodynamical properties of and volumes) come naturally with discrete spectra [13– these modified gravitons and, under certain assumptions, 16]. Whence, one could expect discrete features of quantum derive the tensor power spectrum of the cosmic microwave geometry to emerge at the level of observations. Including background. The latter is found to deviate from the classical this discreteness into reduced models led to many interesting prediction, in that an amplification of UV modes takes place. concepts: for example, Loop Quantum Cosmology (a quanti- We discuss under what circumstances such effect can be in sation of the cosmological sector of General Relativity using agreement with observations. techniques from LQG) resolved the initial singularity via a smooth bounce [17–20]. However, the evolution of such cos- mological models was prone to discretisation ambiguities. 1 Introduction Only recent studies enabled to draw a connection between the reduced dynamics and the Hamiltonian constraint of The observation of gravitational waves (GW) in recent years the full theory [21]. This was achieved via coherent states has opened a new window for insights into the cosmos sharply peaked on a discrete geometry [22,23]: interpreting [1–4]. This manifested itself in the emergence of the era the expectation value of the Hamiltonian constraint opera- of multi-messenger astronomy on the one hand [5,6] and tor thereon as an effective Hamiltonian extracts the quantum on the other hand with the further search for gravitational corrections due to the underlying discrete structure [24–28] waves of primordial origin [7]. Such waves might soon be in the form of a modified effective dynamics for the canonical variables. This paper applies this framework to the case of a e-mail: [email protected] GW propagating on flat Minkowski background. b e-mail: [email protected] (corresponding author) 123 741 Page 2 of 10 Eur. Phys. J. C (2020) 80 :741 Section 2 of this paper revisits the classical formulation where κ = 16πG and β>0 is the Immirzi parameter. of GW in the Hamiltonian description. Then, we introduce K is the extrinsic curvature and F the gauge curvature of in Sect. 3 the necessary formalism of LQG and how coher- connection A. The connection and its canonical conjugated ent states for GW on Minkowski spacetime are constructed. momentum, the triad E, can be computed for metric (1)(by A necessary condition is the perturbative treatment of the neglecting O(h2) contributions) and read modes, as their smallness is necessary to obey the linearised Einstein equations. We will outline how this simplifies the 1 = − h+ , 2 = + h+ , 1 = 2 =−h× E1 1 E2 1 E2 E1 formalism and what result is found for the expectation value 2 2 2 3 = , 1 = 2 =−β p+ , 2 = 1 = β p× of the Hamiltonian constraint, which is then thought of as an E3 1 A1 A2 A1 A2 (4) effective Hamiltonian on the phase space of linearised grav- 2 2 ity. Said effective constraint differs from its classical con- where p+/× is the canonical conjugated momentum to h+/× tinuous counterpart, the modifications being controlled by { ( ), ( )}=κδi δ( , )/ 2 respectively, i.e., hi z p j z j z z L . the discreteness parameter stemming from the discretised The Hamiltonian of General Relativity is quantum geometry. While the effective equations correspond to the limit h¯ → 0, the remaining discreteness implies that a H = dxdydz (NC + N Ca) (5) we have not taken any continuum limit. In Sect. 4, the modi- σ fied Hamilton’s equation for GW are derived and analytically a solved, leading to waves with a specific dispersion relation. with lapse function N and shift vector N . We will gauge fix = a = In Sect. 5, the modified dispersion relation for GW in LQG N 1 and N 0 such that for a spacetime given by (1) is studied. The propagating modes can be described as gravi- the Hamiltonian becomes (up to a boundary term): tons, for which a standard Fock quantisation is possible. We 2 1 ˙2 2 ˙2 2 will describe their thermodynamical properties. Section 6 H = L z [h + (∂ h+) + h + (∂ h×) ] κ d + z × z (6) presents a toy model to investigate whether the modified dis- 2 persion relation can lead to observable modifications to the This describes two decoupled, massless free scalar fields in Cosmic Microwave Background (CMB) tensor power spec- one dimension. The solutions are thus classical waves. trum. Finally, Sect. 7 concludes with a comparison to existing literature and an outlook of further research directions. 3 Loop quantisation 2 Canonical description Being interested in the consequences for GW from discrete spatial manifolds, we introduce an ad-hoc discretisation of σ M = σ × R σ ∼ [ , ]3 On manifold (with = 0 L a torus with in form of a cubic lattice with edges of coordinate length periodic boundaries) we consider for the spacetime metric a (and its dual cell complex); the number of vertices v in each perturbation around flat Minkowski background: direction is N = L/. Keeping finite and only considering finitely many degrees of freedom described by smearings = η + gμν μν hμν (1) along the edges and faces of the lattice allows to proceed with the canonical quantisation of General Relativity in a well- where h is sufficiently small so that its quadratic orders can defined manner, analogously to the quantisation procedure be neglected. By adopting the transverse traceless gauge, its of LQG. We emphasize that discretisation of space is not a only non vanishing elements are necessity in LQG, so we regard it as an additional postulate. The variables we are interested in are SU(2)-valued =− = , = = hxx h yy h+ hxy h yx h× (2) holonomies of A along the edges e of the lattice and gauge- 1 covariant fluxes of E across the dual faces Se for each edge which depend only on direction z and time t. e: In the Hamiltonian formulation of gravity, Einstein’s equations become equivalent to a totally constrained sys- † † P(e) = h(e[1/2]) h(ρx ) ∗ E(x)h(ρx ) h(e[1/2]) tem [29,30]. Expressed in the Ashtekar–Barbero variables Se ( a, I ) EI Aa [13,31], the constraints are = ∗E(x) + O(h2), h(e) = P exp − A S e Ea Eb e = IJK√ J K ( I − ( + β2) M N ) (7) C Fab 1 IMNKa K κ det E b 2 1 With the choice in [32,33], these coincide with conventional fluxes C = F J Eb , G = ∂ Ea + AK Ea (3) a κβ ab J J a J JKL a L up to O(h2). 123 Eur. Phys. J. C (2020) 80 :741 Page 3 of 10 741 I where A = A τI , E = EI τI and τI =−iσI /2, with σI dynamics generated by the discretised Hamiltonian of the ⊂ ( ) the Pauli matrices. e[1/2] e is starting at e 0 and ending system, that is, the leading order of the expectation value of ˆ ˆ at p = e ∩ Se and ρx some choice of of path contained in Se H = v C(v).
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