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This volume provides a self-contained introduction to applications of loop representations, and the related topic of knot theory, in particle physics and quantum gravity. These topics are of considerable interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. The book begins with a detailed review of loop representation theory and then describes loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed, with the following chapters considering knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research. First published in 1996, this title has been reissued as an Open Access publication on Cambridge Core.

It was recently pointed out that linear quantum fields $\hat \phi(x)$ can be meaningfully propagated across the big bang (and the big crunch) singularities of spatially flat Friedmann, Lema\^itre, Robertson, Walker (FLRW) universes \cite{ADLS2021}. Recall that $\hat \phi(x)$, as well as renormalized observables $\langle\hat \phi(x)^2 \rangle_{ren}$ and $\langle \hat T_{ab}(x)\rangle_{ren}$, are distribution-valued already in Minkowskian quantum field theories. It was shown that they can be extended as well-defined distributions even when these space-times are enlarged to include the big-bang (or the big crunch). We generalize these results to spatially closed and open FLRW models, showing that this `tameness' of cosmological singularities is not an artifact of the technical simplifications due to spatial flatness. Our analysis also provides explicit expressions of $\langle\hat \phi(x) \hat \phi(x') \rangle_{ren}$, $\langle\hat \phi(x)^2 \rangle_{ren}$ and $\langle \hat T_{ab}(x)\rangle_{ren}$ in closed and open universes for minimally coupled massless scalar fields and discuss the ambiguities in the definition of $\langle \hat T_{ab}(x)\rangle_{ren}$ at the big-bang. While the technical expressions are more complicated than in the spatially flat case, there is also an unexpected conceptual simplification: the infrared divergence \cite{fp} is now absent because, in effect, the spatial curvature provides a natural cutoff. Finally, we further clarify the sense in which quantum field theory can continue to be well defined even though the extended space-time is not globally hyperbolic because of the singularity, and suggest directions for further work., Comment: 22 pages + 2 appendices

The big bang and the Schwarzschild singularities are space-like. They are generally regarded as the "final frontiers" at which space-time ends and general relativity breaks down. We review the status of such space-like singularities from three increasingly more general perspectives. They are provided by (i) A reformulation of classical general relativity motivated by the Belinskii, Khalatnikov, Lifshitz conjecture on the behavior of the gravitational field near space-like singularities; (ii) The use of test quantum fields to probe the nature of these singularities; and, (iii) An analysis of the fate of these singularities in loop quantum gravity due to quantum geometry effects. At all three levels singularities turn out to be less menacing than one might a priori expect from classical general relativity. Our goal is to present an overview of the emerging conceptual picture and suggest lines for further work. In line with the \emph{Introduction to Current Research} theme, we have made an attempt to make it easily accessible to all researchers in gravitational physics., Invited article under "Introduction to Current Research", GRG (at press). Special issue in memory of Prof. T. Padmanabhan

The anticipated enhancements in detector sensitivity and the corresponding increase in the number of gravitational wave detections will make it possible to estimate parameters of compact binaries with greater accuracy assuming general relativity(GR), and also to carry out sharper tests of GR itself. Crucial to these procedures are accurate gravitational waveform models. The systematic errors of the models must stay below statistical errors to prevent biases in parameter estimation and to carry out meaningful tests of GR. Comparisons of the models against numerical relativity (NR) waveforms provide an excellent measure of systematic errors. A complementary approach is to use balance laws provided by Einstein's equations to measure faithfulness of a candidate waveform against exact GR. Each balance law focuses on a physical observable and measures the accuracy of the candidate waveform vis a vis that observable. Therefore, this analysis can provide new physical insights into sources of errors. In this paper we focus on the angular momentum balance law, using post-Newtonian theory to calculate the initial angular momentum, surrogate fits to obtain the remnant spin and waveforms from models to calculate the flux. The consistency check provided by the angular momentum balance law brings out the marked improvement in the passage from \texttt{IMRPhenomPv2} to \texttt{IMRPhenomXPHM} and from \texttt{SEOBNRv3} to \texttt{SEOBNRv4PHM} and shows that the most recent versions agree quite well with exact GR. For precessing systems, on the other hand, we find that there is room for further improvement, especially for the Phenom models., Comment: 12 pages, 6 figures. Appendix B added to provide heat maps showing the degree of violation of the angular momentum balance law in various non-precessing and precessing models, as functions of the binary parameters

Full, non-linear general relativity predicts a memory effect for gravitational waves. For compact binary coalescence, the total gravitational memory serves as an inferred observable, conceptually on the same footing as the mass and the spin of the final black hole. Given candidate waveforms for any LIGO event, then, one can calculate the posterior probability distribution functions for the total gravitational memory, and use them to compare and contrast the waveforms. In this paper we present these posterior distributions for the binary black hole merger events reported in the first Gravitational Wave Transient Catalog (GWTC-1), using the Phenomenological and Effective-One-Body waveforms. On the whole, the two sets of posterior distributions agree with each other quite well though we find larger discrepancies for the $\ell=2, m=1$ mode of the memory. This signals a possible source of systematic errors that was not captured by the posterior distributions of other inferred observables. Thus, the posterior distributions of various angular modes of total memory can serve as diagnostic tools to further improve the waveforms. Analyses such as this would be valuable especially for future events as the sensitivity of ground based detectors improves, and for LISA which could measure the total gravitational memory directly., Comment: 15 pages, 6 figures

By carrying out a systematic investigation of linear, test quantum fields $\hat{\phi}(x)$ in cosmological space-times, we show that $\hat{\phi}(x)$ remain well-defined across the big bang as operator valued distributions in a large class of Friedmann, Lema\^itre, Robertson, Walker space-times, including radiation and dust filled universes. In particular, the expectation values $\langle \hat{\phi}(x)\,\hat{\phi}(x')\rangle$ are well-defined bi-distributions in the extended space-time in spite of the big bang singularity. Interestingly, correlations between fields evaluated at spatially and temporally separated points exhibit an asymmetry that is reminiscent of the Belinskii, Khalatnikov, Lifshitz behavior. The renormalized products of fields $\langle \hat{\phi}^2(x)\rangle_{\rm ren}$ and $\langle \hat{T}_{ab}(x) \rangle_{\rm ren}$ also remain well-defined as distributions. Conformal coupling is not necessary for these considerations to hold. Thus, when probed with observables associated with quantum fields, the big bang (and the big crunch) singularities are quite harmless., Comment: 38 pages; Three small clarifications added and minor typos corrected. Version to appear in Advances in Mathematical and Theoretical Physics

An outstanding open issue in our quest for physics beyond Einstein is the unification of general relativity (GR) and quantum physics. Loop quantum gravity (LQG) is a leading approach toward this goal. At its heart is the central lesson of GR: Gravity is a manifestation of spacetime geometry. Thus, the approach emphasizes the quantum nature of geometry and focuses on its implications in extreme regimes -- near the big bang and inside black holes -- where Einstein's smooth continuum breaks down. We present a brief overview of the main ideas underlying LQG and highlight a few recent advances. This report is addressed to non-experts., 35 pages, 6 figures; Key Issues Review commissioned by the Journal

We investigate three issues that have been discussed in the context of inflation: Fading of the importance of quantum non-commutativity; the phenomenon of quantum squeezing; and the ability to approximate the quantum state by a distribution function on the classical phase space. In the standard treatments, these features arise from properties of mode functions of quantum fields in (near) de Sitter space-time. Therefore, the three notions are often assumed to be essentially equivalent, representing different facets of the same phenomenon. We analyze them in general Friedmann-Lemaitre- Robertson-Walker space-times, through the lens of geometric structures on the classical phase space. The analysis shows that: (i) inflation does not play an essential role; classical behavior can emerge much more generally; (ii) the three notions are conceptually distinct; classicality can emerge in one sense but not in another; and, (iii) the third notion is realized in a surprisingly strong sense; there is exact equality between completely general $n$-point functions in the classical theory and those in the quantum theory, provided the quantum operators are Weyl ordered. These features arise already for linear cosmological perturbations by themselves: considerations such as mode-mode coupling, decoherence, and measurement theory --although important in their own right-- are not needed for emergence of classical behavior in any of the three senses discussed. Generality of the results stems from the fact that they can be traced back to geometrical structures on the classical phase space, available in a wide class of systems. Therefore, this approach may also be useful in other contexts., Comment: 43 pages; 3 Figures. Note Added in response to arXiv:2009.09999 as Section VIII

In presence of gravitational radiation, the notion of angular momentum of an isolated system acquires an infinite dimensional supertranslation ambiguity. This fact has been emphasized in the mathematical general relativity literature over several decades. We analyze the issue in the restricted context of compact binary coalescence (CBC) where the initial total angular momentum of the binary and the final black hole spin generically refer to \emph{distinct} rotation subgroups of the Bondi-Metzner-Sachs group, related by \emph{supertranslations}. We show that this ambiguity can be quantified using gravitational memory and the `black hole kick'. Our results imply that, although the ambiguity is conceptually important, under assumptions normally made in the CBC literature, it can be ignored in practice for the current and foreseeable gravitational wave detectors., Comment: 28 pages, 1 Figure. Some clarifications and a figure added at referee's suggestion

Certain anomalies in the CMB bring out a tension between the six-parameter flat $\Lambda$CDM model and the CMB data. We revisit the PLANCK analysis with loop quantum cosmology (LQC) predictions and show that LQC alleviates both the large-scale power anomaly and the tension in the lensing amplitude. These differences arise because, in LQC, the primordial power spectrum is scale dependent for small $k$, with a specific power suppression. We conclude with a prediction of larger optical depth and power suppression in the $B$-mode polarization power spectrum on large scales., Comment: 6 pages, 5 figures, 1 table. The version that appeared in PRL. A few clarifications and several references added

Consider compact objects --such as neutron star or black hole binaries-- in \emph{full, non-linear} general relativity. In the case with zero cosmological constant $\Lambda$, the gravitational radiation emitted by such systems is described by the well established, 50+ year old framework due to Bondi, Sachs, Penrose and others. However, so far we do not have a satisfactory extension of this framework to include a \emph{positive} cosmological constant --or, more generally, the dark energy responsible for the accelerated expansion of the universe. In particular, we do not yet have an adequate gauge invariant characterization of gravitational waves in this context. As the next step in extending the Bondi et al framework to the $\Lambda >0$ case, in this paper we address the following questions: How do we impose the `no incoming radiation' condition for such isolated systems in a gauge invariant manner? What is the relevant past boundary where these conditions should be imposed, i.e., what is the \emph{physically relevant} analog of past null infinity $\mathcal{I}^{-}_{0}$ used in the $\Lambda=0$ case? What is the symmetry group at this boundary? How is it related to the Bondi-Metzner-Sachs (BMS) group? What are the associated conserved charges? What happens in the $\Lambda \to 0$ limit? Do we systematically recover the Bondi-Sachs-Penrose structure at $\mathcal{I}^{-}_{0}$ of the $\Lambda=0$ theory, or do some differences persist even in the limit? We will find that while there are many close similarities, there are also some subtle but important differences from the asymptotically flat case. Interestingly, to analyze these issues one has to combine conceptual structures and mathematical techniques introduced by Bondi et al with those associated with \emph{quasi-local horizons}., Comment: 56 pages, 5 figures; several clarifications and references added to make the material more easily accessible especially to geometric analysts and mathematical relativists interested in conceptual issues.Version to appear in PRD

Gravitational waveforms for compact binary coalescences (CBCs) have been invaluable for detections by the LIGO-Virgo collaboration. They are obtained by a combination of semi-analytical models and numerical simulations. So far systematic errors arising from these procedures appear to be less than statistical ones. However, the significantly enhanced sensitivity of the new detectors that will become operational in the near future will require waveforms to be much more accurate. This task would be facilitated if one has a variety of cross-checks to \emph{evaluate} accuracy, particularly in the regions of parameter space where numerical simulations are sparse. Currently errors are estimated by comparing the candidate waveforms with the numerical relativity (NR) ones, which are taken to be exact. The goal of this paper is to propose a qualitatively different tool. We show that full non-linear general relativity (GR) imposes an infinite number of sharp constraints on the CBC waveforms. These can provide clear-cut measures to evaluate the accuracy of candidate waveforms against exact GR, help find systematic errors, and also provide external checks on NR simulations themselves., 25 pages, 1 figure. The main results are the same as in V1. Focus of the article improved by expanding the material addressed to the waveform community and dropping the one primarily addressed to mathematical relativists. Discussion updated in view of new results of [46] - [48]

We present a new effective description of macroscopic Kruskal black holes that incorporates corrections due to quantum geometry effects of loop quantum gravity. It encompasses both the `interior' region that contains classical singularities and the `exterior' asymptotic region. Singularities are naturally resolved by the quantum geometry effects of loop quantum gravity, and the resulting quantum extension of the full Kruskal space-time is free of all the known limitations of previous investigations [1-11] of the Schwarzschild interior. We compare and contrast our results with these investigations and also with the expectations based on the AdS/CFT duality [12]., Comment: 5 pages, 1 figure, minor changes, version to appear in PRL

A new description of macroscopic Kruskal black holes that incorporates the quantum geometry corrections of loop quantum gravity is presented. It encompasses both the `interior' region that contains classical singularities and the `exterior' asymptotic region. Singularities are naturally resolved by the quantum geometry effects of loop quantum gravity. The resulting quantum extension of space-time has the following features: (i) It admits an infinite number of trapped, anti-trapped and asymptotic regions; (ii) All curvature scalars have uniform (i.e., mass independent) upper bounds; (iii) In the large mass limit, all asymptotic regions of the extension have the same ADM mass; (iv) In the low curvature region (e.g., near horizons) quantum effects are negligible, as one would physically expect; and (v) Final results are insensitive to the fiducial structures that have to be introduced to construct the classical phase space description (as they must be). Previous effective theories [1-10] shared some but not all of these features. We compare and contrast our results with those of these effective theories and also with expectations based on the AdS/CFT conjecture [11]. We conclude with a discussion of limitations of our framework, especially for the analysis of evaporating black holes., 47 pages, 5 figures, A short addendum is added after the last section to address misleading remarks in a recent pre-print [50] that may confuse the reader

There has been a recent resurgence of interest in the structure of the gravitational field at null infinity, sparked by new results on soft charges and infrared issues related to the S matrix theory in perturbative quantum gravity. We summarize these developments and put them in the broader context of research in the relativity community that dates back to several decades. In keeping with intent of this series, this overview is addressed to gravitational scientists who are not experts in this specific area., Comment: 19 pages, no figures. Prepared for the "Introduction to Current Research Series." Clarifications added following correspondence with expert, typos corrected, version to appear in the journal

It is surprising that the fundamental, microscopic laws of Nature are not invariant under time reversal. In his article, Three Merry Roads to T - Violation , Dr. Bryan Roberts provided a succinct summary of the theoretical frameworks normally used to interpret the results of the experiments that established this fact. They all rely on the detailed structure of quantum mechanics. In this ‘response’ to Dr. Robert׳s talk, I will show that these experiments can be interpreted using a much more general framework. Consequently, should quantum mechanics be eventually replaced by a new paradigm, e.g., because of quantum gravity, these experiments could still be used to argue that the microscopic laws violate T invariance.

Most of the literature on general relativity over the last century assumes that the cosmological constant $\Lambda$ is zero. However, by now independent observations have led to a consensus that the dynamics of the universe is best described by Einstein's equations with a small but positive $\Lambda$. Interestingly, this requires a drastic revision of conceptual frameworks commonly used in general relativity, \emph{no matter how small $\Lambda$ is.} We first explain why, and then summarize the current status of generalizations of these frameworks to include a positive $\Lambda$, focusing on gravitational waves., Comment: A Key Issues Review, Commissioned by Rep. Prog. Phys. 12 pages, 3 figures

Somewhat surprisingly, in many of the widely used monographs and review articles the term Transverse-Traceless modes of linearized gravitational waves is used to denote two entirely different notions. These treatments generally begin with a decomposition of the metric perturbation that is local in the momentum space (and hence non-local in physical space), and denote the resulting transverse traceless modes by $h_{ab}^{\TT}$. However, while discussing gravitational waves emitted by an isolated system --typically in a later section-- the relevant modes are extracted using a `projection operator' that is local in physical space. These modes are also called transverse-traceless and again labeled $h_{ab}^{\TT}$, implying that this is just a reformulation of the previous notion. But the two notions are conceptually distinct and the difference persists even in the asymptotic region. We show that this confusion arises already in Maxwell theory that is often discussed as a prelude to the gravitational case. Finally, we discuss why the distinction has nonetheless remained largely unnoticed, and also point out that there are some important physical effects where only one of the notions gives the correct answer., 36 pages. The detailed of the Letter arXiv:1707.07729; some references and minor clarifications added

The GRG journal’s Golden Oldies series has republished more than 80 important old papers in general relativity and gravitation. This announcement updates and replaces the one made in Gen. Relativ. Gravit. 39 (7), 1043 (2007) as modified by the editorial announcement in Gen. Relativ. Gravit. 44 (10), 2419 (2012).

Observational missions have provided us with a reliable model of the evolution of the universe starting from the last scattering surface all the way to future infinity. Furthermore given a specific model of inflation, using quantum field theory on curved space-times this history can be pushed \emph{back in time} to the epoch when space-time curvature was some $10^{62}$ times that at the horizon of a solar mass black hole! However, to extend the history further back to the Planck regime requires input from quantum gravity. An important aspect of this input is the choice of the background quantum geometry and of the Heisenberg state of cosmological perturbations thereon, motivated by Planck scale physics. This paper introduces first steps in that direction. Specifically we propose two principles that link quantum geometry and Heisenberg uncertainties in the Planck epoch with late time physics and explore in detail the observational consequences of the initial conditions they select. We find that the predicted temperature-temperature (T-T) correlations for scalar modes are indistinguishable from standard inflation at small angular scales even though the initial conditions are now set in the deep Planck regime. However, \emph{there is a specific power suppression at large angular scales}. As a result, the predicted spectrum provides a better fit to the PLANCK mission data than standard inflation, where the initial conditions are set in the general relativity regime. Thus, our proposal brings out a deep interplay between the ultraviolet and the infrared. Finally, the proposal also leads to specific predictions for power suppression at large angular scales also for the (T-E and E-E) correlations involving electric polarization. The PLANCK team is expected to release this data in the coming year., Invited article, to appear in CQG. This paper is addressed both to the quantum gravity and cosmology audiences. Cosmologists can focus just on sections I, IV.C, IV.D and V without loss of continuity. 43 pages, 13 figures. Version 2 contains a few clarifications and new references, especially to compare and contrast related results in the literature

It is with profound sadness that we report on the sudden passing away of Professor Sergio Dain, an Associate Editor of the journal General Relativity and Gravitation. He succumbed to cancer on February 24th, 2016. Although only 46, he had already been an internationally recognized relativist for quite some time. Sergio received his Ph.D. from the Universidad Nacional de Cordoba, Argentina in 1999 for his numerous contributions to asymptotically flat space-times and gravitational radiation. Part of his thesis work was carried out at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute—AEI) in Golm, Germany, under the auspices of a Fellowship of the German Academic Exchange Service. In 2000

This is a brief overview of the current status of symmetry reduced models in Loop Quantum Gravity (LQG), focussing on the very early universe. Over the last 3 years or so the subject has matured sufficiently to make direct contact with observations of the Cosmic Microwave Background (CMB). In particular, thanks to an unforeseen interplay between the ultraviolet features of quantum geometry and the infrared properties of quantum fields representing cosmological perturbations, \emph{Planck scale effects} of LQG can leave imprints on CMB \emph{at the largest angular scales}. In addition to a summary of these results, the article also contains a critical discussion of the symmetry reduction procedure used in discussions of quantum cosmology (and quantum black holes)., Comment: 25 pages, 2 figures. References updated; typos corrected. The abstract is expanded for readers who may not see the special volume of the journal in which this article will appear

Inflationary models of the early universe provide a natural mechanism for the formation of large scale structure. This success brings to forefront the question of naturalness: Does a sufficiently long slow roll inflation occur generically or does it require a careful fine tuning of initial parameters? In recent years there has been considerable controversy on this issue. In particular, for a quadratic potential, Kofman, Linde and Mukhanov have argued that the probability of inflation with at least 65 e-foldings is close to one, while Gibbons and Turok have argued that this probability is suppressed by a factor of ~ $\10^{-85}$. We first clarify that such dramatically different predictions can arise because the required measure on the space of solutions is intrinsically ambiguous in general relativity. We then show that this ambiguity can be naturally resolved in loop quantum cosmology (LQC) because the big bang is replaced by a big bounce and the bounce surface can be used to introduce the structure necessary to specify a satisfactory measure. The second goal of the paper is to present a detailed analysis of the inflationary dynamics of LQC using analytical and numerical methods. By combining this information with the measure on the space of solutions, we address a sharper question than those investigated in the literature: What is the probability of a sufficiently long slow roll inflation WHICH IS COMPATIBLE WITH THE SEVEN YEAR WMAP DATA? We show that the probability is very close to 1. The material is so organized that cosmologists who may be more interested in the inflationary dynamics in LQC than in the subtleties associated with measures can skip that material without loss of continuity., 34 pages, 3 figures

In loop quantum cosmology (LQC) the big bang is replaced by a quantum bounce which is followed by a robust phase of super-inflation. Rather than growing unboundedly in the past, the Hubble parameter \emph{vanishes} at the bounce and attains a \emph{finite universal maximum} at the end of super-inflation. These novel features lead to an unforeseen implication: in presence of suitable potentials all LQC dynamical trajectories are funneled to conditions which virtually guarantee slow roll inflation with more than 68 e-foldings, {without any input from the pre-big bang regime}. This is in striking contrast to certain results in general relativity, where it is argued that the a priori probability of obtaining a slow roll with 68 or more e-foldings is suppressed by a factor $e^{-204}$., Comment: 5 pages, 1 table

Loop quantum cosmology (LQC) is used to provide concrete evidence in support of the general paradigm underlying spin foam models (SFMs). Specifically, it is shown that: i) the physical inner product in the timeless framework equals the transition amplitude in the deparameterized theory; ii) this quantity admits a %convergent vertex expansion a la SFMs in which the $M$-th term refers just to $M$ volume transitions, without any reference to the time at which the transition takes place; iii) the exact physical inner product is obtained by summing over just the discrete geometries; no `continuum limit' is involved; and, iv) the vertex expansion can be interpreted as a perturbative expansion in the spirit of group field theory. This sum over histories reformulation of LQC also addresses certain other issues which are briefly summarized., 11 pages, no figures

There is a precise sense in which the requirement of background independence suffices to uniquely select the kinematics of loop quantum gravity (LQG). Specifically, the fundamental kinematic algebra of LQG admits a unique diffeomorphism invariant state. Although this result has been established rigorously, it comes as a surprise to researchers working with other approaches to quantum gravity. The goal of this article is to explain the underlying reasons in a pedagogical fashion using geometrodynamics, keeping the technicalities at their minimum. This discussion will bring out the surprisingly powerful role played by diffeomorphism invariance (and covariance) in non-perturbative, canonical quantum gravity., To appear in the special issue of General Relativity and Gravitation, dedicated to the memory of Professor Juergen Ehlers. 16 pages

A brief overview of loop quantum cosmology of homogeneous isotropic models is presented with emphasis on the origin of and subtleties associated with the resolution of big bang and big crunch singularities. These results bear out the remarkable intuition that John Wheeler had. Discussion is organized at two levels. The the main text provides a bird's eye view of the subject that should be accessible to non-experts. Appendices address conceptual and technical issues that are often raised by experts in loop quantum gravity and string theory., To appear in the Proceedings of the Bad Honef Workshop entitled \emph{Quantum Gravity: Challenges and Perspectives}, dedicated to the memory of John A. Wheeler

Basic features of dynamical black holes in full, non-linear general relativity are summarized in a pedagogical fashion. Qualitative properties of the evolution of various horizons follow directly from the celebrated Raychaudhuri equation.

The effective description of loop quantum cosmology (LQC) has proved to be a convenient platform to study phenomenological implications of the quantum bounce that resolves the classical big-bang singularity. Originally, this description was derived using Gaussian quantum states with small dispersions. In this paper we present a generalization to incorporate states with large dispersions. Specifically, we derive the \emph{generalized} effective Friedmann and Raychaudhuri equations and propose a generalized effective Hamiltonian which are being used in an ongoing study of the phenomenological consequences of a broad class of quantum geometries. We also discuss an interesting interplay between the physics of states with larger dispersions in standard LQC, and of sharply peaked states in (hypothetical) LQC theories with larger area gap., Comment: 21 pages, 4 figures; A paragraph discussing the relation to the "Chimera scheme" added in section V. Version to appear in PRD

Almost a century ago, Einstein used a weak field approximation around Minkowski space-time to calculate the energy carried away by gravitational waves emitted by a time changing mass-quadrupole. However, by now there is strong observational evidence for a positive cosmological constant, $\Lambda$. To incorporate this fact, Einstein's celebrated derivation is generalized by replacing Minkowski space-time with de Sitter space-time. The investigation is motivated by the fact that, because of the significant differences between the asymptotic structures of Minkowski and de Sitter space-times, many of the standard techniques, including the standard $1/r$ expansions, can not be used for $\Lambda >0$. Furthermore since, e.g., the energy carried by gravitational waves is always positive in Minkowski space-time but can be arbitrarily negative in de Sitter space-time \emph{irrespective of how small $\Lambda$ is}, the limit $\Lambda\to 0$ can fail to be continuous. Therefore, a priori it is not clear that a small $\Lambda$ would introduce only negligible corrections to Einstein's formula. We show that, while even a tiny cosmological constant does introduce qualitatively new features, in the end, corrections to Einstein's formula are negligible for astrophysical sources currently under consideration by gravitational wave observatories., Comment: 31 pages, 2 figures