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Linear cosmological perturbation theory is pivotal to a theoretical understanding of current cosmological experimental data provided e.g. by cosmic microwave anisotropy probes. A key issue in that theory is to extract the gauge invariant degrees of freedom which allow unambiguous comparison between theory and experiment. When one goes beyond first (linear) order, the task of writing the Einstein equations expanded to n'th order in terms of quantities that are gauge invariant up to terms of higher orders becomes highly non-trivial and cumbersome. This fact has prevented progress for instance on the issue of the stability of linear perturbation theory and is a subject of current debate in the literature. In this series of papers we circumvent these difficulties by passing to a manifestly gauge invariant framework. In other words, we only perturb gauge invariant, i.e. measurable quantities, rather than gauge variant ones. Thus, gauge invariance is preserved non perturbatively while we construct the perturbation theory for the equations of motion for the gauge invariant observables to all orders. In this first paper we develop the general framework which is based on a seminal paper due to Brown and Kuchar as well as the realtional formalism due to Rovelli. In the second, companion, paper we apply our general theory to FRW cosmologies and derive the deviations from the standard treatment in linear order. As it turns out, these deviations are negligible in the late universe, thus our theory is in agreement with the standard treatment. However, the real strength of our formalism is that it admits a straightforward and unambiguous, gauge invariant generalisation to higher orders. This will also allow us to settle the stability issue in a future publication., Comment: 77 pages, no figures

In our companion paper we identified a complete set of manifestly gauge-invariant observables for general relativity. This was possible by coupling the system of gravity and matter to pressureless dust which plays the role of a dynamically coupled observer. The evolution of those observables is governed by a physical Hamiltonian and we derived the corresponding equations of motion. Linear perturbation theory of those equations of motion around a general exact solution in terms of manifestly gauge invariant perturbations was then developed. In this paper we specialise our previous results to an FRW background which is also a solution of our modified equations of motion. We then compare the resulting equations with those derived in standard cosmological perturbation theory (SCPT). We exhibit the precise relation between our manifestly gauge-invariant perturbations and the linearly gauge-invariant variables in SCPT. We find that our equations of motion can be cast into SCPT form plus corrections. These corrections are the trace that the dust leaves on the system in terms of a conserved energy momentum current density. It turns out that these corrections decay, in fact, in the late universe they are negligible whatever the value of the conserved current. We conclude that the addition of dust which serves as a test observer medium, while implying modifications of Einstein's equations without dust, leads to acceptable agreement with known results, while having the advantage that one now talks about manifestly gauge-invariant, that is measurable, quantities, which can be used even in perturbation theory at higher orders., Comment: 51 pages, no figures

We give a Hamiltonian analysis of the asymptotically flat spherically symmetric system of gravity coupled to a scalar field. This 1+1 dimensional field theory may be viewed as the "standard model" for studying black hole physics. Our analysis is adapted to the flat slice Painleve-Gullstrand coordinates. We give a Hamiltonian action principle for this system, which yields an asymptotic mass formula. We then perform a time gauge fixing that gives a Hamiltonian as the integral of a local density. The Hamiltonian takes a relatively simple form compared to earlier work in Schwarzschild gauge, and therefore provides a setting amenable to full quantisation., Comment: 11 pages, refererences added, discussions clarified, version to appear in PRD

We calculate the power spectrum of vacuum fluctuations of a generic scalar field in a quantum cosmological setting that is manifestly singularity-free. The power spectrum is given in terms of the usual scale invariant spectrum plus scale dependent corrections. These are induced by well-defined quantum fluctuations of de Sitter spacetime, resulting in a modified dispersion relation for the scalar field. The leading correction turns out to be proportional to the ratio of Hubble scale and Planck Mass. The maximal relative change in the spectrum is on the ten percent level and might be observable with future CMB experiments., Comment: 4 pages, 2 figures, substantially revised version with improved results

We examine the singularity resolution issue in quantum gravity by studying a new quantization of standard Friedmann-Robertson-Walker geometrodynamics. The quantization procedure is inspired by the loop quantum gravity programme, and is based on an alternative to the Schr\"odinger representation normally used in metric variable quantum cosmology. We show that in this representation for quantum geometrodynamics there exists a densely defined inverse scale factor operator, and that the Hamiltonian constraint acts as a difference operator on the basis states. We find that the cosmological singularity is avoided in the quantum dynamics. We discuss these results with a view to identifying the criteria that constitute "singularity resolution" in quantum gravity., Comment: 12 pages

We summarize a recently proposed concrete programme for investigating the (semi)classical limit of canonical, Lorentzian, continuum quantum general relativity in four spacetime dimensions. The analysis is based on a novel set of coherent states labelled by graphs. These fit neatly together with an Infinite Tensor Product (ITP) extension of the currently used Hilbert space. The ITP construction enables us to give rigorous meaning to the infinite volume (thermodynamic) limit of the theory which has been out of reach so far., Comment: 37 p., latex2e, no figures

In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann. In this paper we establish the ``Ehrenfest Property'' of these states which are labelled by a point (A,E), a connection and an electric field, in the classical phase space. By this we mean that i) The expectation value of {\it all} elementary quantum operators $\hat{O}$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the corresponding classical function O evaluated at the phase space point (A,E) and ii) The expectation value of the commutator between two elementary quantum operators $[\hat{O}_1,\hat{O}_2]/(i\hbar)$ divided by $i\hbar$ with respect to the coherent state with label (A,E) is given to zeroth order in $\hbar$ by the value of the Poisson bracket between the corresponding classical functions $\{O_1,O_2\}$ evaluated at the phase space point (A,E). These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. It follows that the infinitesimal quantum dynamics of quantum general relativity is to zeroth order in $\hbar$ indeed given by classical general relativity., Comment: 49 pages, LATEX, no figures

In the canonical approach to Lorentzian Quantum General Relativity in four spacetime dimensions an important step forward has been made by Ashtekar, Isham and Lewandowski some eight years ago through the introduction of an appropriate Hilbert space structure. This Hilbert space, together with its generalization due to Baez and Sawin, is appropriate for semi-classical quantum general relativity if the spacetime is spatially compact. In the spatially non-compact case, however, an extension of the Hilbert space is needed in order to approximate metrics that are macroscopically nowhere degenerate. For this purpose, in this paper we apply von Neumann's theory of the Infinite Tensor Product (ITP) of Hilbert Spaces to Quantum General Relativity. The cardinality of the number of tensor product factors can take the value of any possible Cantor aleph as is needed for our problem, where a Hilbert space is attached to each edge of an arbitrarily complicated, generally infinite graph. The new framework opens a pandora's box full of techniques, appropriate to pose fascinating physical questions such as quantum topology change, semi-classical quantum gravity, effective low energy physics etc. from the universal point of view of the ITP. In particular, the study of photons and gravitons propagating on fluctuating quantum spacetimes is now in reach, the topic of the next paper in this series., Comment: 60 pages, LATEX, no figures

In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we resolve these issues, in particular, we prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation, ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical analysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity, enable error-controlled approximations and set a new starting point for {\it numerical canonical quantum general relativity and gauge theory}. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed., Comment: 70 pages, LATEX, 29 figures