Search

The Wilsonian renormalization group (RG) requires Euclidean signature. The conformal factor of the metric then has a wrong-sign kinetic term, which has a profound effect on its RG properties. Generically for the conformal sector, complete flows exist only in the reverse direction (i.e. from the infrared to the ultraviolet). The Gaussian fixed point supports infinite sequences of composite eigenoperators of increasing infrared relevancy (increasingly negative mass dimension), which are orthonormal and complete for bare interactions that are square integrable under the appropriate measure. These eigenoperators are non-perturbative in $\hbar$ and evanescent. For $\mathbb{R}^4$ spacetime, each renormalised physical operator exists but only has support at vanishing field amplitude. In the generic case of infinitely many non-vanishing couplings, if a complete RG flow exists, it is characterised in the infrared by a scale $\Lambda_\mathrm{p}>0$, beyond which the field amplitude is exponentially suppressed. On other spacetimes, of length scale $L$, the flow ceases to exist once a certain universal measure of inhomogeneity exceeds $O(1)+2\pi L^2\Lambda^2_\mathrm{p}$. Importantly for cosmology, the minimum size of the universe is thus tied to the degree of inhomogeneity, with spacetimes of vanishing size being required to be almost homogeneous. We initiate a study of this exotic quantum field theory at the interacting level, and discuss what the full theory of quantum gravity should look like, one which must thus be perturbatively renormalizable in Newton's constant but non-perturbative in $\hbar$., Comment: 52 pages, 4 figures; fixed typos; improved explanation of the sign of V, and the use of Sturm-Liouville theory. To be publ in JHEP

We show how the Weyl anomaly generated by gauge fields, can be computed from manifestly gauge invariant and diffeomorphism invariant exact renormalization group equations, without having to fix the gauge at any stage. Regularisation is provided by covariant higher derivatives and by embedding the Maxwell field into a spontaneously broken $U(1|1)$ supergauge theory. We first provide a realisation that leaves behind two versions of the original $U(1)$ gauge field, and then construct a manifestly $U(1|1)$ supergauge invariant flow equation which leaves behind only the original Maxwell field in the spontaneously broken regime., Comment: 24 pages

Simulation studies are computer experiments that involve creating data by pseudorandom sampling. The key strength of simulation studies is the ability to understand the behaviour of statistical methods because some 'truth' (usually some parameter/s of interest) is known from the process of generating the data. This allows us to consider properties of methods, such as bias. While widely used, simulation studies are often poorly designed, analysed and reported. This tutorial outlines the rationale for using simulation studies and offers guidance for design, execution, analysis, reporting and presentation. In particular, this tutorial provides: a structured approach for planning and reporting simulation studies, which involves defining aims, data-generating mechanisms, estimands, methods and performance measures ('ADEMP'); coherent terminology for simulation studies; guidance on coding simulation studies; a critical discussion of key performance measures and their estimation; guidance on structuring tabular and graphical presentation of results; and new graphical presentations. With a view to describing recent practice, we review 100 articles taken from Volume 34 of Statistics in Medicine that included at least one simulation study and identify areas for improvement., Comment: 31 pages, 9 figures (2 in appendix), 8 tables (1 in appendix)

In Cox regression it is sometimes of interest to study time-varying effects (TVE) of exposures and to test the proportional hazards assumption. TVEs can be investigated with log hazard ratios modelled as a function of time. Missing data on exposures are common and multiple imputation (MI) is a popular approach to handling this, to avoid the potential bias and loss of efficiency resulting from a 'complete-case' analysis. Two MI methods have been proposed for when the substantive model is a Cox proportional hazards regression: an approximate method (White and Royston, Statist. Med. 2009;28:1982-98) and a substantive-model-compatible method (Bartlett et al., SMMR 2015;24:462-87). At present, neither method accommodates TVEs of exposures. We extend them to do so for a general form for the TVEs and give specific details for TVEs modelled using restricted cubic splines. Simulation studies assess the performance of the methods under several underlying shapes for TVEs. Our proposed methods give approximately unbiased TVE estimates for binary exposures with missing data, but for continuous exposures the substantive-model-compatible method performs better. The methods also give approximately correct type I errors in the test for proportional hazards when there is no TVE, and gain power to detect TVEs relative to complete-case analysis. Ignoring TVEs at the imputation stage results in biased TVE estimates, incorrect type I errors and substantial loss of power in detecting TVEs. We also propose a multivariable TVE model selection algorithm. The methods are illustrated using data from the Rotterdam Breast Cancer Study. Example R code is provided.

We explain how to find the asymptotic form of fixed point solutions in functional truncations, in particular $f(R)$ approximations. We find that quantum fluctuations do not decouple at large $R$, typically leading to elaborate asymptotic solutions containing several free parameters. By a counting argument, these can be used to map out the dimension of the fixed point solution spaces. They are also necessary to validate the numerical solution, and provide the physical part in the limit that the cutoff is removed: the fixed point equation of state. As an example we apply the techniques to a recent $f(R)$ approximation by Demmel et al, finding asymptotic matches to their numerical solution. Depending on the value of the endomorphism parameter, we find many other asymptotic solutions and fixed point solution spaces of differing dimensions, yielding several alternative scenarios for the equation of state. Asymptotic studies of other $f(R)$ approximations are needed to clarify the picture., Comment: Minor corrections. Version to be published in PRD

In single-metric approximations to the exact renormalization group (RG) for quantum gravity, it has been not been clear how to treat the large curvature domain beyond the point where the effective cutoff scale $k$ is less than the lowest eigenvalue of the appropriate modified Laplacian. We explain why this puzzle arises from background dependence, resulting in Wilsonian RG concepts being inapplicable. We show that when properly formulated over an ensemble of backgrounds, the Wilsonian RG can be restored. This in turn implies that solutions should be smooth and well defined no matter how large the curvature is taken. Even for the standard single-metric type approximation schemes, this construction can be rigorously derived by imposing a modified Ward identity (mWI) corresponding to rescaling the background metric by a constant factor. However compatibility in this approximation requires the space-time dimension to be six. Solving the mWI and flow equation simultaneously, new variables are then derived that are independent of overall background scale., Comment: 34 pages, 2 figures; various small improvements

We present a discussion of the design issues and trade-offs that have been considered in putting together a new concept for MOSAIC, the multi-object spectrograph for the E-ELT. MOSAIC aims to address the combined science cases for E-ELT MOS that arose from the earlier studies of the multi-object and multi-adaptive optics instruments. MOSAIC combines the advantages of a highly-multiplexed instrument targeting single-point objects with one which has a more modest multiplex but can spatially resolve a source with high resolution (IFU). These will span across two wavebands: visible and near-infrared.

The performance of a wide-field adaptive optics system depends on input design parameters. Here we investigate the performance of a multi-object adaptive optics system design for the European Extremely Large Telescope, using an end-to-end Monte-Carlo adaptive optics simulation tool, DASP, with relevance for proposed instruments such as MOSAIC. We consider parameters such as the number of laser guide stars, sodium layer depth, wavefront sensor pixel scale, actuator pitch and natural guide star availability. We provide potential areas where costs savings can be made, and investigate trade-offs between performance and cost, and provide solutions that would enable such an instrument to be built with currently available technology. Our key recommendations include a trade-off for laser guide star wavefront sensor pixel scale of about 0.7 arcseconds per pixel, and a field of view of at least 7 arcseconds, that EMCCD technology should be used for natural guide star wavefront sensors even if reduced frame rate is necessary, and that sky coverage can be improved by a slight reduction in natural guide star sub-aperture count without significantly affecting tomographic performance. We find that adaptive optics correction can be maintained across a wide field of view, up to 7 arcminutes in diameter. We also recommend the use of at least 4 laser guide stars, and include ground-layer and multi-object adaptive optics performance estimates., Comment: Accepted for publication in MNRAS

We study the quantum behaviour of a particle moving in a one-dimensional double well potential. This double well is obtained by gluing together, at the origin, two shifted harmonic oscillator potentials. The Schr\"odinger equation is exactly solvable. The requirement that discontinuities, in the wavefunction and its first derivative, are absent at the origin, leads to the quantisation of the energy eigenvalues. We also show that oscillations in time take place between two nearby single harmonic oscillator ground states. Finally, the double well potential is augmented by a Dirac delta-function potentials at the origin and the corresponding Schr\"odinger equation is solved., Comment: This paper is withdrawn because the model is already treated in a substantially similar way in Merzbacher's text-book

The $\mathcal{O}(\partial^2)$ background independent flow equations for conformally reduced gravity are shown to be equivalent to flow equations naturally adapted to scalar field theory with a wrong sign kinetic term. This sign change is shown to have a profound effect on the renormalization group properties, broadly resulting in a continuum of fixed points supporting both a discrete and a continuous eigenoperator spectrum, the latter always including relevant directions. The properties at the Gaussian fixed point are understood in particular depth, but also detailed studies of the local potential approximation, and the full $\mathcal{O}(\partial^2)$ approximation are given. These results are related to evidence for asymptotic safety found by other authors., Comment: 58 pages, 15 figures. Added section that shows that for any sensible cutoff profile, space-time and field dimension, the LPA for standard scalar field theory can only have a discrete set of fixed points, while the sign change of the kinetic term results instead in a continuum of fixed points

We present an exact RG (renormalization group) analysis of $O(N)$-invariant scalar field theory about the Gaussian fixed point. We prove a series of statements that taken together show that the non-polynomial eigen-perturbations found in the LPA (local potential approximation) at the linearised level, do not lead to new interactions, \textit{i.e.} enlarge the universality class, neither in the LPA or treated exactly. Non-perturbatively, their RG flow does not emanate from the fixed point. For the equivalent Wilsonian effective action they can be re-expressed in terms of the usual couplings to polynomial interactions, which can furthermore be tuned to be as small as desired for all finite RG time. For the infrared cutoff Legendre effective action, this can also be done for the infrared evolution. We explain why this is nevertheless consistent with the fact that the large field behaviour is fixed by these perturbations., Comment: 32 pages, 1 figure; extended to exact treatment

Within the derivative expansion of conformally reduced gravity, the modified split Ward identities are shown to be compatible with the flow equations if and only if either the anomalous dimension vanishes or the cutoff profile is chosen to be power law. No solutions exist if the Ward identities are incompatible. In the compatible case, a clear reason is found for why Ward identities can still forbid the existence of fixed points; however, for any cutoff profile, a background independent (and parametrisation independent) flow equation is uncovered. Finally, expanding in vertices, the combined equations are shown generically to become either over-constrained or highly redundant beyond the six-point level., Comment: 31 pages, 4 figures

We construct a manifestly diffeomorphism invariant Wilsonian (Exact) Renormalization Group for classical gravity, and begin the construction for quantum gravity. We demonstrate that the effective action can be computed without gauge fixing the diffeomorphism invariance, and also without introducing a background space-time. We compute classical contributions both within a background-independent framework and by perturbing around a fixed background, and verify that the results are equivalent. We derive the exact Ward identities for actions and kernels and verify consistency. We formulate two forms of the flow equation corresponding to the two choices of classical fixed-point: the Gaussian fixed point, and the scale invariant interacting fixed point using curvature-squared terms. We suggest how this programme may completed to a fully quantum construction., Comment: 50 pages, 2 figures, minor typos fixed, minor additions to the Discussion, one reference added

Starting from a full renormalised trajectory for the effective average action (a.k.a. infrared cutoff Legendre effective action) $\Gamma_k$, we explicitly reconstruct corresponding bare actions, formulated in one of two ways. The first step is to construct the corresponding Wilsonian effective action $S^k$ through a tree-level expansion in terms of the vertices provided by $\Gamma_k$. It forms a perfect bare action giving the same renormalised trajectory. A bare action with some ultraviolet cutoff scale $\Lambda$ and infrared cutoff $k$ necessarily produces an effective average action $\Gamma^\Lambda_k$ that depends on both cutoffs, but if the already computed $S^\Lambda$ is used, we show how $\Gamma^\Lambda_k$ can also be computed from $\Gamma_k$ by a tree-level expansion, and that $\Gamma^\Lambda_k\to\Gamma_k$ as $\Lambda\to\infty$. Along the way we show that Legendre effective actions with different UV cutoff profiles, but which correspond to the same Wilsonian effective action, are related through tree-level expansions. All these expansions follow from Legendre transform relationships that can be derived from the original one between $\Gamma^\Lambda_k$ and $S^k$., Comment: 32 pages

We argue, at a very basic effective field theory level, that higher dimension operators in scalar theories that break symmetries at scales close to their ultraviolet completion cutoff, include terms that favour the breaking of translation (Lorentz) invariance, potentially resulting in striped, chequered board or general crystal-like phases. Such descriptions can be thought of as the effective low energy description of QCD-like gauge theories near their strong coupling scale where terms involving higher dimension operators are generated. Our low energy theory consists of scalar fields describing operators such as $\bar{q} q$ and $\bar{q} F^{(2n)} q$. Such scalars can have kinetic mixing terms that generate effective momentum dependent contributions to the mass matrix. We show that these can destabilize the translationally invariant vacuum. It is possible that in some real gauge theory such operators could become sufficiently dominant to realize such phases and it would be interesting to look for them in lattice simulations. We present a holographic model of the same phenomena which includes RG running. A key phenomenological motive to look at such states is recent work that shows that the non-linear response in $R^2$ gravity to such short range fluctuations can mimic a cosmological constant. Intriguingly in a cosmology with such a Starobinsky inflation term, to generate the observed value of the present day acceleration would require stripes at the electroweak scale. Unfortunately, low energy phenomenological constraints on Lorentz violation in the electron-photon system appear to strongly rule out any such possibility outside of a disconnected dark sector., Comment: 9 pages, 1 figure; minor changes. Version to be published in PRD

Within the conformally reduced gravity model, where the metric is parametrised by a function $f(\phi)$ of the conformal factor $\phi$, we keep dependence on both the background and fluctuation fields, to local potential approximation and $\mathcal{O}(\partial^2)$ respectively, making no other approximation. Explicit appearances of the background metric are then dictated by realising a remnant diffeomorphism invariance. The standard non-perturbative Renormalization Group (RG) scale $k$ is inherently background dependent, which we show in general forbids the existence of RG fixed points with respect to $k$. By utilising transformations that follow from combining the flow equations with the modified split Ward identity, we uncover a unique background independent notion of RG scale, $\hat k$. The corresponding RG flow equations are then not only explicitly background independent along the entire RG flow but also explicitly independent of the form of $f$. In general $f(\phi)$ is forced to be scale dependent and needs to be renormalised, but if this is avoided then $k$-fixed points are allowed and furthermore they coincide with $\hat k$-fixed points., Comment: 53 pages, broken reference corrected

A multi-object spectrograph on the forthcoming European Extremely Large Telescope will be required to operate with good sky coverage. Many of the interesting deep cosmological fields were deliberately chosen to be free of bright foreground stars, and therefore are potentially challenging for adaptive optics (AO) systems. Here we investigate multi-object AO performance using sub-fields chosen at random from within the Great Observatories Origins Deep Survey (GOODS)-S field, which is the worst case scenario for five deep fields used extensively in studies of high-redshift galaxies. Our AO system model is based on that of the proposed MOSAIC instrument but our findings are equally applicable to plans for multi-object spectroscopy on any of the planned Extremely Large Telescopes. Potential guide stars within these sub-fields are identified and used for simulations of AO correction. We achieve ensquared energies within 75~mas of between 25-35\% depending on the sub-field, which is sufficient to probe sub-kpc scales in high-redshift galaxies. We also investigate the effect of detector readout noise on AO system performance, and consider cases where natural guide stars are used for both high-order and tip-tilt-only AO correction. We also consider how performance scales with ensquared energy box size. In summary, the expected AO performance is sufficient for a MOSAIC-like instrument, even within deep fields characterised by a lack of bright foreground stars., Comment: Accepted for publication in MNRAS

We report results obtained during the characterization of a commercial front-illuminated progressive scan interline transfer CCD camera. We demonstrate that the unmodified camera operates successfully in temperature and pressure conditions (-40C, 4mBar) representative of a high altitude balloon mission. We further demonstrate that the centroid of a well-sampled star can be determined to better than 2% of a pixel, even though the CCD is equipped with a microlens array. This device has been selected for use in a closed-loop star-guiding and tip-tilt correction system in the BIT-STABLE balloon mission., Comment: 14 pages, 16 figures. Accepted in Journal of Astronomical Instrumentation, special issue on Scientific Ballooning

The first on-sky results obtained by CANARY, the Multi-Object Adaptive Optics (MOAO) demonstrator, are analysed. The data were recorded at the William Herschel Telescope, at the end of September 2010. We describe the command and calibrations algorithms used during the run and present the observing conditions. The processed data are MOAO-loop engaged or disengaged slope buffers, comprising the synchronised measurements of the four Natural Guide Stars (NGS) wavefront sensors running in parallel, and near Infra-Red (IR) images. We describe the method we use to establish the error budget of CANARY. We are able to evaluate the to- mographic and the open loop errors, having median values around 216 nm and 110 nm respectively. In addition, we identify an unexpected residual quasi-static field aberration term of mean value 110 nm. We present the detailed error budget analysed for three sets of data for three different asterisms. We compare the experimental budgets with the numerically simulated ones and demonstrate a good agreement. We find also a good agreement between the computed error budget from the slope buffers and the measured Strehl ratio on the IR images, ranging between 10% and 20% at 1 530 nm. These results make us confident in our ability to establish the error budget of future MOAO instruments., Comment: in press A&A (2014)

We study the effective stress-energy tensor induced by cosmological inhomogeneity in $f(R)=R+cR^2$ and equivalent scalar-tensor theories, motivated both by models of early universe inflation and by phenomenological alternative cosmologies to the standard $\Lambda$-CDM. We use Green and Wald's framework for averaging over classical fluctuations of short-wavelength $\lambda$. By ensuring that the leading non-linear terms from the fluctuations of the Einstein terms and the corrections both contribute in the formal limit as $\lambda\to0$, we derive a diffeomorphism invariant effective stress-energy tensor whose trace is non-vanishing and of the right sign to potentially account for the current acceleration of the universe. However a more phenomenologically acceptable dark energy model would be required if this effect were to fully account for the current acceleration., Comment: 23 pages, references updated, some arguments clarified

Dark Matter (DM) is an elusive form of matter which has been postulated to explain astronomical observations through its gravitational effects on stars and galaxies, gravitational lensing of light around these, and through its imprint on the Cosmic Microwave Background (CMB). This indirect evidence implies that DM accounts for as much as 84.5% of all matter in our Universe, yet it has so far evaded all attempts at direct detection, leaving such confirmation and the consequent discovery of its nature as one of the biggest challenges in modern physics. Here we present a novel form of low-mass DM $\chi$ that would have been missed by all experiments so far. While its large interaction strength might at first seem unlikely, neither constraints from particle physics nor cosmological/astronomical observations are sufficient to rule out this type of DM, and it motivates our proposal for direct detection by optomechanics technology which should soon be within reach, namely, through the precise position measurement of a levitated mesoscopic particle which will be perturbed by elastic collisions with $\chi$ particles. We show that a recently proposed nanoparticle matter-wave interferometer, originally conceived for tests of the quantum superposition principle, is sensitive to these collisions, too., Comment: 25 pages, 7 figures

The spectral resolution of a dispersive spectrograph is dependent on the width of the entrance slit. This means that astronomical spectrographs trade-off throughput with spectral resolving power. Recently, optical guided-wave transitions known as photonic lanterns have been proposed to circumvent this trade-off, by enabling the efficient reformatting of multimode light into a pseudo-slit which is highly multimode in one axis, but diffraction-limited in the other. Here, we demonstrate the successful reformatting of a telescope point spread function into such a slit using a three-dimensional integrated optical waveguide device, which we name the photonic dicer. Using the CANARY adaptive optics demonstrator on the William Herschel Telescope, and light centred at 1530 nm with a 160 nm FWHM, the device shows a transmission of between 10 and 20\% depending upon the type of AO correction applied. Most of the loss is due to the overfilling of the input aperture in poor and moderate seeing. Taking this into account, the photonic device itself has a transmission of 57 $\pm$ 4\%.We show how a fully-optimised device can be used with AO to provide efficient spectroscopy at high spectral resolution., Comment: 8 pages, 6 figures

Working within the familiar local potential approximation, and concentrating on the example of a single scalar field in three dimensions, we show that the commonly used approximation method of identifying the total and background fields, leads to pathologies in the resulting fixed point structure and the associated spaces of eigenoperators. We then show how a consistent treatment of the background field through the corresponding modified shift Ward identity, can cure these pathologies, restoring universality of physical quantities with respect to the choice of dependence on the background field, even within the local potential approximation. Along the way we point out similarities to what has been previously found in the f(R) approximation in asymptotic safety for gravity., Comment: 40 pages, version accepted by JHEP

In this paper we review the definition and properties of redundant operators in the exact renormalisation group. We explain why it is important to require them to be eigenoperators and why generically they appear only as a consequence of symmetries of the particular choice of renormalisation group equations. This clarifies when Newton's constant and or the cosmological constant can be considered inessential. We then apply these ideas to the Local Potential Approximation and approximations of a similar spirit such as the f(R) approximation in the asymptotic safety programme in quantum gravity. We show that these approximations can break down if the fixed point does not support a `vacuum' solution in the appropriate domain: all eigenoperators become redundant and the physical space of perturbations collapses to a point. We show that this is the case for the recently discovered lines of fixed points in the f(R) flow equations., Comment: 29 pages; improvements in presentation & correction of a subsidiary result, as reflected in the new abstract; typos corrected; references added

In the asymptotic safety programme for quantum gravity, it is important to go beyond polynomial truncations. Three such approximations have been derived where the restriction is only to a general function f(R) of the curvature R>0. We confront these with the requirement that a fixed point solution be smooth and exist for all non-negative R. Singularities induced by cutoff choices force the earlier versions to have no such solutions. However, we show that the most recent version has a number of lines of fixed points, each supporting a continuous spectrum of eigen-perturbations. We uncover and analyse the first five such lines. Sensible fixed point behaviour may be achieved if one consistently incorporates geometry/topology change. As an exploratory example, we analyse the equations analytically continued to R<0, however we now find only partial solutions.We show how these results are always consistent with, and to some extent can be predicted from, a straightforward analysis of the constraints inherent in the equations., Comment: Latex, 66 pages, published version, typos corrected

In the OPERA experiment, superluminal propagation of neutrinos can occur if one of the neutrino masses is extremely small. However the effect only has appreciable amplitude at energies of order this mass and thus has negligible overlap with the multi-GeV scale of the experiment., Comment: Typos corrected. Expanded to include discussion of boosted wave packets similar to Naumov & Naumov, and explanation for why causality is not threatened. References added

Recently it was suggested that the observation of superluminal neutrinos by the OPERA collaboration may be due to group velocity effects resulting from close-to-maximal oscillation between neutrino mass eigenstates, in analogy to known effects in optics. We show that superluminal propagation does occur through this effect for a series of very narrow energy ranges, but this phenomenum cannot explain the OPERA measurement., Comment: Reworking includes corrections due to finite width of the wave packet, further references, and comments on other corrections, causality, the new OPERA results, and why the size of neutrino wave packet provides important constraints

The canonical transformation that maps light-cone Yang-Mills theory to a Lagrangian description of the MHV rules is non-local, consequently the two sets of fields do not necessarily generate the same S-matrix. By deriving a new recursion relation for the canonical transformation expansion coefficients, we find a direct map between these coefficients and tree level light-cone diagrams. We use this to show that, at least up to one-loop with dimensionally regularised MHV vertices, the only difference is the omission of the one-loop amplitudes in which all gluons have positive helicity., Comment: 36 pages, JHEP3 style, published version, discussions on rational parts and supersymmetry added