Disentangling magnification in combined shear-clustering analyses

Monthly Notices of the Royal Astronomical Society Oxford University Press 491 (2019) 1756-1758

L Thiele, C Duncan, D Alonso

Tomographic measurement of the intergalactic gas pressure through galaxy–tSZ cross-correlations

Monthly Notices of the Royal Astronomical Society Oxford University Press 491 (2019) 5464-5480

N Koukoufilippas, D Alonso, M Bilicki, JA Peacock

We cross-correlate maps of the thermal Sunyaev–Zeldovich (tSZ) Compton-y parameter published by Planck with the projected distribution of galaxies in a set of low-redshift tomographic bins. We use the nearly full-sky 2MASS Photometric Redshift and WISE × SuperCOSMOS public catalogues, covering the redshift range z ≲ 0.4. Our measurements allow us to place constraints on the redshift dependence of the mass–observable relation for tSZ cluster count analyses in terms of the so-called hydrostatic mass bias parameter 1−bH⁠. These results can also be interpreted as measurements of the bias-weighted average gas pressure 〈bPe〉 as a function of redshift, a quantity that can be related to the thermodynamics of gas inside haloes and used to constrain energy injection processes. We measure 1−bH with ∼13 per cent precision in six equispaced redshift bins, and find no evidence for a redshift-dependent mass bias parameter, in agreement with previous analyses. Our mean value of 1−bH=0.59±0.03 is also in good agreement with the one estimated by the joint analysis of Planck cluster counts and cosmic microwave background anisotropies. Our measurements of 〈bPe〉, at the level of ∼10 per cent in each bin, are the most stringent constraints on the redshift dependence of this parameter to date, and agree well both with previous measurements and with theoretical expectations from shock-heating models.

Quantifying baryon effects on the matter power spectrum and the weak lensing shear correlation

Journal of Cosmology and Astroparticle Physics 2019 (2019)

A Schneider, R Teyssier, J Stadel, NE Chisari, AMCL Brun, A Amara, A Refregier

© 2019 IOP Publishing Ltd and Sissa Medialab. Feedback processes from baryons are expected to strongly affect weak-lensing observables of current and future cosmological surveys. In this paper we present a new parametrisation of halo profiles based on gas, stellar, and dark matter density components. This parametrisation is used to modify outputs of gravity-only N-body simulations (following the prescription of [1]) in order to mimic baryonic effects on the matter density field. The resulting baryonic correction model relies on a few well motivated physical parameters and is able to reproduce the redshift zero clustering signal of hydrodynamical simulations at two percent accuracy below k∼10 h/Mpc. A detailed study of the baryon suppression effects on the matter power spectrum and the weak lensing shear correlation reveals that the signal is dominated by two parameters describing the slope of the gas profile in haloes and the maximum radius of gas ejection. We show that these parameters can be constrained with the observed gas fraction of galaxy groups and clusters from X-ray data. Based on these observations we predict a beyond percent effect on the power spectrum above k=0.2-1.0 h/Mpc with a maximum suppression of 15-25 percent around k∼ 10 h/Mpc. As a result, the weak lensing angular shear power spectrum is suppressed by 15-25 percent at scales beyond ℓ∼ 100-600 and the shear correlations ξ + and ξ - are affected at the 10-25 percent level below 5 and 50 arc-minutes, respectively. The relatively large uncertainties of these predictions are a result of the poorly known hydrostatic mass bias of current X-ray observations as well as the generic difficulty to observe the low density gas outside of haloes.

The dependence of intrinsic alignment of galaxies on wavelength using KiDS and GAMA

Astronomy and Astrophysics 622 (2019)

C Georgiou, H Johnston, H Hoekstra, M Viola, K Kuijken, B Joachimi, NE Chisari, DJ Farrow, H Hildebrandt, BW Holwerda, A Kannawadi

© ESO 2019. The outer regions of galaxies are more susceptible to the tidal interactions that lead to intrinsic alignments of galaxies. The resulting alignment signal may therefore depend on the passband if the colours of galaxies vary spatially. To quantify this, we measured the shapes of galaxies with spectroscopic redshifts from the GAMA survey using deep gri imaging data from the KiloDegree Survey. The performance of the moment-based shape measurement algorithm DEIMOS was assessed using dedicated image simulations, which showed that the ellipticities could be determined with an accuracy better than 1% in all bands. Additional tests for potential systematic errors did not reveal any issues. We measure a significant difference of the alignment signal between the g,r and i-band observations. This difference exceeds the amplitude of the linear alignment model on scales below 2 Mpc h -1 . Separating the sample into central/satellite and red/blue galaxies, we find that the difference is dominated by red satellite galaxies.

Core Cosmology Library: Precision cosmological predictions for LSST

Astrophysical Journal Supplement American Astronomical Society 242 (2019) 2

J Neveu, A Villarreal, S Singh, T McClintock, J Ellison, Z Du, J Zuntz, A Mead, CS Lorenz, S Joudaki, T Tröster, J Sanchez, F Lanusse, M Ishak, R Hlozek, J Blazek, J-E Campagne, H Almoubayyed, T Eifler, M Kirby, D Kirkby, S Plaszczynski, A Slosar, M Vrastil, EL Wagoner

The Core Cosmology Library (CCL) provides routines to compute basic cosmological observables to a high degree of accuracy, which have been verified with an extensive suite of validation tests. Predictions are provided for many cosmological quantities, including distances, angular power spectra, correlation functions, halo bias, and the halo mass function through state-of-the-art modeling prescriptions available in the literature. Fiducial specifications for the expected galaxy distributions for the Large Synoptic Survey Telescope (LSST) are also included, together with the capability of computing redshift distributions for a user-defined photometric redshift model. A rigorous validation procedure, based on comparisons between CCL and independent software packages, allows us to establish a well-defined numerical accuracy for each predicted quantity. As a result, predictions for correlation functions of galaxy clustering, galaxy–galaxy lensing, and cosmic shear are demonstrated to be within a fraction of the expected statistical uncertainty of the observables for the models and in the range of scales of interest to LSST. CCL is an open source software package written in C, with a Python interface and publicly available at

Time-varying neutrino mass from a supercooled phase transition: Current cosmological constraints and impact on the Ωm-σ8 plane

Physical Review D 99 (2019)

CS Lorenz, L Funcke, E Calabrese, S Hannestad

© 2019 American Physical Society. In this paper we investigate a time-varying neutrino mass model, motivated by the mild tension between cosmic microwave background (CMB) measurements of the matter fluctuations and those obtained from low-redshift data. We modify the minimal case of the model proposed by [G. Dvali and L. Funcke, Phys. Rev. D 93, 113002 (2016)PRVDAQ2470-001010.1103/PhysRevD.93.113002] that predicts late neutrino mass generation in a postrecombination cosmic phase transition, by assuming that neutrino asymmetries allow for the presence of relic neutrinos in the late-time Universe. We show that, if the transition is supercooled, current cosmological data (including CMB temperature, polarization and lensing, baryon acoustic oscillations, and type Ia supernovae) prefer the scale factor as of the phase transition to be very large, peaking at as∼1, and therefore supporting a cosmological scenario in which neutrinos are almost massless until very recent times. We find that in this scenario the cosmological bound on the total sum of the neutrino masses today is significantly weakened compared to the standard case of constant-mass neutrinos, with mν<4.8 eV at 95% confidence, and in agreement with the model predictions. The main reason for this weaker bound is a large correlation arising between the dark energy and neutrino components in the presence of false vacuum energy that converts into the nonzero neutrino masses after the transition. This result provides new targets for the coming KATRIN and PTOLEMY experiments. We also show that the time-varying neutrino mass model considered here does not provide a clear explanation of the existing cosmological Ωm-σ8 discrepancies.

KiDS plus GAMA: Intrinsic alignment model constraints for current and future weak lensing cosmology


H Johnston, C Georgiou, B Joachimi, H Hoekstra, NE Chisari, D Farrow, MC Fortuna, C Heymans, S Joudaki, K Kuijken, A Wright

Towards emulating cosmic shear data: revisiting the calibration of the shear measurements for the Kilo-Degree Survey

Astronomy and Astrophysics EDP Sciences 624 (2019) A92

H Hoekstra, L Miller, A Kannawadi, T Erben, AH Wright, M Viola, IF Conti, R Herbonnet, H Hildebrandt, C Heymans, K Kuijken, M Vakili

Exploiting the full statistical power of future cosmic shear surveys will necessitate improvements to the accuracy with which the gravitational lensing signal is measured. We present a framework for calibrating shear with image simulations that demonstrates the importance of including realistic correlations between galaxy morphology, size and more importantly, photometric redshifts. This realism is essential so that selection and shape measurement biases can be calibrated accurately for a tomographic cosmic shear analysis. We emulate Kilo-Degree Survey (KiDS) observations of the COSMOS field using morphological information from {\it Hubble} Space Telescope imaging, faithfully reproducing the measured galaxy properties from KiDS observations of the same field. We calibrate our shear measurements from lensfit, and find through a range of sensitivity tests that lensfit is robust and unbiased within the allowed 2 per cent tolerance of our study. Our results show that the calibration has to be performed by selecting the tomographic samples in the simulations, consistent with the actual cosmic shear analysis, because the joint distributions of galaxy properties are found to vary with redshift. Ignoring this redshift variation could result in misestimating the shear bias by an amount that exceeds the allowed tolerance. To improve the calibration for future cosmic shear analyses, it will be essential to also correctly account for the measurement of photometric redshifts, which requires simulating multi-band observations.

Galaxy orientation with the cosmic web across cosmic time

Monthly Notices of the Royal Astronomical Society Oxford University Press 481 (2018) 4753–4774-

A Jindal, S Codis, NE Chisari, D Vibert, Y Dubois, C Pichon, J Devriendt

This work investigates the alignment of galactic spins with the cosmic web across cosmic time using the cosmological hydrodynamical simulation Horizon-AGN. The cosmic web structure is extracted via the persistent skeleton as implemented in the DISPERSE algorithm. It is found that the spin of low-mass galaxies is more likely to be aligned with the filaments of the cosmic web and to lie within the plane of the walls while more massive galaxies tend to have a spin perpendicular to the axis of the filaments and to the walls. The mass transition is detected with a significance of 9σ. This galactic alignment is consistent with the alignment of the spin of dark haloes found in pure dark matter simulations and with predictions from (anisotropic) tidal torque theory. However, unlike haloes, the alignment of low-mass galaxies is weak and disappears at low redshifts while the orthogonal spin orientation of massive galaxies is strong and increases with time, probably as a result of mergers. At fixed mass, alignments are correlated with galaxy morphology: the high-redshift alignment is dominated by spiral galaxies while elliptical centrals are mainly responsible for the perpendicular signal. The two regimes probed in this work induce competing galactic alignment signals for weak lensing, with opposite redshift and luminosity evolution. Understanding the details of these intrinsic alignments will be key to exploit future major cosmic shear surveys like Euclid or LSST.

The impact of relativistic effects on cosmological parameter estimation

Physical Review D American Physical Society 97 (2018) 1-14

C Lorenz, D Alonso, P Ferreira

Future surveys will access large volumes of space and hence very long wavelength fluctuations of the matter density and gravitational field. It has been argued that the set of secondary effects that affect the galaxy distribution, relativistic in nature, will bring new, complementary cosmological constraints. We study this claim in detail by focusing on a subset of wide-area future surveys: Stage-4 cosmic microwave background experiments and photometric redshift surveys. In particular, we look at the magnification lensing contribution to galaxy clustering and general relativistic corrections to all observables. We quantify the amount of information encoded in these effects in terms of the tightening of the final cosmological constraints as well as the potential bias in inferred parameters associated with neglecting them. We do so for a wide range of cosmological parameters, covering neutrino masses, standard dark-energy parametrizations and scalar-tensor gravity theories. Our results show that, while the effect of lensing magnification to number counts does not contain a significant amount of information when galaxy clustering is combined with cosmic shear measurements, this contribution does play a significant role in biasing estimates on a host of parameter families if unaccounted for. Since the amplitude of the magnification term is controlled by the slope of the source number counts with apparent magnitude, $s(z)$, we also estimate the accuracy to which this quantity must be known to avoid systematic parameter biases, finding that future surveys will need to determine $s(z)$ to the $\sim$5-10\% level. On the contrary, large-scale general-relativistic corrections are irrelevant both in terms of information content and parameter bias for most cosmological parameters, but significant for the level of primordial non-Gaussianity.

KiDS-i-800: Comparing weak gravitational lensing measurements from same-sky surveys

Monthly Notices of the Royal Astronomical Society Oxford University Press 477 (2018) 4285-4307

A Amon, C Heymans, D Klaes, T Erben, C Blake, H Hildebrandt, H Hoekstra, K Kuijken, L Miller, CB Morrison, A Choi, JTA De Jong, K Glazebrook, N Irisarri, B Joachimi, S Joudaki, A Kannawadi, C Lidman, N Napolitano, D Parkinson, P Schneider, E Van Uitert, M Viola, C Wolf

We present a weak gravitational lensing analysis of 815 deg2of i-band imaging from the Kilo-Degree Survey (KiDS-i-800). In contrast to the deep r-band observations, which take priority during excellent seeing conditions and form the primary KiDS data set (KiDS-r-450), the complementary yet shallower KiDS-i-800 spans a wide range of observing conditions. The overlapping KiDS-i-800 and KiDS-r-450 imaging therefore provides a unique opportunity to assess the robustness of weak lensing measurements. In our analysis we introduce two new 'null' tests. The 'nulled' two-point shear correlation function uses a matched catalogue to show that the calibrated KiDS-i-800 and KiDS-r-450 shear measurements agree at the level of 1 ± 4 per cent.We use five galaxy lens samples to determine a 'nulled' galaxy-galaxy lensing signal from the full KiDS-i-800 and KiDS-r-450 surveys and find that the measurements agree to 7 ± 5 per cent when the KiDS-i-800 source redshift distribution is calibrated using either spectroscopic redshifts, or the 30-band photometric redshifts from the COSMOS survey.

Weak-lensing study in VOICE survey - I. Shear measurement


L Fu, D Liu, M Radovich, X Liu, C Pan, Z Fan, G Covone, M Vaccari, V Amaro, M Brescia, M Capaccioli, D De Cicco, A Grado, L Limatola, L Miller, NR Napolitano, M Paolillo, G Pignata

Weak lensing study in VOICE survey - II. Shear bias calibrations


D Liu, L Fu, X Liu, M Radovich, C Wang, C Pan, Z Fan, G Covone, M Vaccari, MT Botticella, M Capaccioli, D De Cicco, A Grado, L Miller, N Napolitano, M Paolillo, G Pignata

Radio weak lensing shear measurement in the visibility domain - II. Source extraction

Monthly Notices of the Royal Astronomical Society Oxford University Press 476 (2018) 2053–2062-

M Rivi, L Miller

This paper extends the method introduced in Rivi et al. (2016b) to measure galaxy ellipticities in the visibility domain for radio weak lensing surveys. In that paper we focused on the development and testing of the method for the simple case of individual galaxies located at the phase centre, and proposed to extend it to the realistic case of many sources in the field of view by extracting visibilities of each source with a faceting technique, taking into account the contamination from the other sources. In this second paper we present a detailed algorithm for source extraction in the visibility domain and show its effectiveness as a function of the source number density by running simulations of SKA1-MID observations in the band 950-1150 MHz and comparing original and measured values of galaxies' ellipticities. Shear measurements from a realistic population of 10^4 galaxies randomly located in a field of view of 1 deg^2 (i.e. the source density expected for the current radio weak lensing survey proposal with SKA1) are also performed. At SNR &gt;= 10, the multiplicative bias is only a factor 1.5 worse than what found when analysing isolated sources, and is still comparable to the bias values reported for similar measurement methods at optical wavelengths. The additive bias is unchanged from the case of isolated sources, but is significantly larger than typically found in optical surveys. This bias depends on the shape of the Point Spread Function (PSF) and we suggest that a uv-plane weighting scheme to produce a more isotropic PSF could reduce and control additive bias.

KiDS-450 + 2dFLenS: Cosmological parameter constraints from weak gravitational lensing tomography and overlapping redshift-space galaxy clustering

Monthly Notices of the Royal Astronomical Society Oxford University Press 474 (2017) 4894–4924-

C Lidman, A Mead, A Amon, M Asgari, A Choi, M Viola, C Wolf, S Joudaki, C Blake, A Johnson, L Miller, D Parkinson, GB Poole, P Schneider, T Erben, K Glazebrook, J Harnois-Déraps, C Heymans, H Hildebrandt, H Hoekstra, D Klaes, K Kuijken

We perform a combined analysis of cosmic shear tomography, galaxy-galaxy lensing tomography, and redshift-space multipole power spectra (monopole and quadrupole) using 450 deg$^2$ of imaging data by the Kilo Degree Survey (KiDS) overlapping with two spectroscopic surveys: the 2-degree Field Lensing Survey (2dFLenS) and the Baryon Oscillation Spectroscopic Survey (BOSS). We restrict the galaxy-galaxy lensing and multipole power spectrum measurements to the overlapping regions with KiDS, and self-consistently compute the full covariance between the different observables using a large suite of $N$-body simulations. We methodically analyze different combinations of the observables, finding that galaxy-galaxy lensing measurements are particularly useful in improving the constraint on the intrinsic alignment amplitude (by 30%, positive at $3.5\sigma$ in the fiducial data analysis), while the multipole power spectra are useful in tightening the constraints along the lensing degeneracy direction (e.g. factor of two stronger matter density constraint in the fiducial analysis). The fully combined constraint on $S_8 \equiv \sigma_8 \sqrt{\Omega_{\rm m}/0.3} = 0.742 \pm 0.035$, which is an improvement by 20% compared to KiDS alone, corresponds to a $2.6\sigma$ discordance with Planck, and is not significantly affected by fitting to a more conservative set of scales. Given the tightening of the parameter space, we are unable to resolve the discordance with an extended cosmology that is simultaneously favored in a model selection sense, including the sum of neutrino masses, curvature, evolving dark energy, and modified gravity. The complementarity of our observables allows for constraints on modified gravity degrees of freedom that are not simultaneously bounded with either probe alone, and up to a factor of three improvement in the $S_8$ constraint in the extended cosmology compared to KiDS alone.

A rapid occultation event in NGC 3227

Monthly Notices of the Royal Astronomical Society 481 (2018) 2470-2478

TJ Turner, JN Reeves, V Braito, A Lobban, S Kraemer, L Miller

© 2018 The Author(s). NGC 3227 exhibits rapid flux and spectral variability in the X-ray band. To understand this behaviour, we conducted a coordinated observing campaign using 320 ks of XMM- Newton exposures together with 160 ks of overlapping NuSTAR observations, spanning a month. Here, we present a rapid variability event that occurs toward the end of the campaign. The spectral hardening event is accompanied by a change in the depth of an unresolved transition array (UTA), whose time-dependent behaviour is resolved using the RGS data. This UTA fingerprint allows us to identify this as a transit event, where a clump of gas having NH ~ 5 × 1022 atoms cm-2, log Ξ ~ 2 occults ~ 60 per cent of the continuum photons over the course of approximately a day. This occulting gas is likely associated with clouds in the inner broad-line region. An additional zone of gas with lower column and higher ionization, matches the outflow velocity of the variable zone, and may represent transmission through the cloud limb.

KiDS plus GAMA: Cosmology constraints from a joint analysis of cosmic shear, galaxy-galaxy lensing, and angular clustering

Monthly Notices of the Royal Astronomical Society Oxford University Press 476 (2018) 4662-4689

E Van Uitert, B Joachimi, S Joudaki, C Heymans, A Amon, F Koehlinger, M Asgari, C Blake, A Choi, T Erben, DJ Farrow, J Harnois-Deraps, H Hildebrandt, H Hoekstra, TD Kitching, D Klaes, L Miller, J Merten, K Kuijken, P Schneider, R Nakajima, E Valentijn, M Viola

We present cosmological parameter constraints from a joint analysis of three cosmological probes: the tomographic cosmic shear signal in~450 deg2of data from the Kilo Degree Survey (KiDS), the galaxy-matter cross-correlation signal of galaxies from the Galaxies And Mass Assembly (GAMA) survey determined with KiDS weak lensing, and the angular correlation function of the same GAMA galaxies. We use fast power spectrum estimators that are based on simple integrals over the real-space correlation functions, and show that they are practically unbiased over relevant angular frequency ranges. We test our full pipeline on numerical simulations that are tailored to KiDS and retrieve the input cosmology. By fitting different combinations of power spectra, we demonstrate that the three probes are internally consistent. For all probes combined, we obtain S8≡ σ8√ Ωm/0.3 = 0.800-0.027+0.029, consistent with Planck and the fiducial KiDS-450 cosmic shear correlation function results. Marginalizing over wide priors on the mean of the tomographic redshift distributions yields consistent results for S8with an increase of 28 per cent in the error. The combination of probes results in a 26 per cent reduction in uncertainties of S8over using the cosmic shear power spectra alone. The main gain from these additional probes comes through their constraining power on nuisance parameters, such as the galaxy intrinsic alignment amplitude or potential shifts in the redshift distributions, which are up to a factor of 2 better constrained compared to using cosmic shear alone, demonstrating the value of large-scale structure probe combination.

Agent-based modelling to predict policy outcomes: A food waste recycling example

Environmental Science & Policy Elsevier 87 (2018) 85-91

AC Skeldon, F Schiller, A Yang, T Balke-Visser, A Penn, N Gilbert

Optimising policy choices to steer social/economic systems efficiently towards desirable outcomes is challenging. The inter-dependent nature of many elements of society and the economy means that policies designed to promote one particular aspect often have secondary, unintended, effects. In order to make rational decisions, methodologies and tools to assist the development of intuition in this complex world are needed. One approach is the use of agent-based models. These have the ability to capture essential features and interactions and predict outcomes in a way that is not readily achievable through either equations or words alone. In this paper we illustrate how agent-based models can be used in a policy setting by using an example drawn from the biowaste industry. This example describes the growth of in-vessel composting and anaerobic digestion to reduce food waste going to landfill in response to policies in the form of taxes and financial incentives. The fundamentally dynamic nature of an agent-based modelling approach is used to demonstrate that policy outcomes depend not just on current policy levels but also on the historical path taken.

Distinguishing between neutrinos and time-varying dark energy through cosmic time

Physical Review D American Physical Society 96 (2017) 1-11

E Calabrese, C Lorenz, D Alonso

We study the correlations between parameters characterizing neutrino physics and the evolution of dark energy. Using a fluid approach, we show that time-varying dark energy models exhibit degeneracies with the cosmic neutrino background over extended periods of the cosmic history, leading to a degraded estimation of the total mass and number of species of neutrinos. We investigate how to break degeneracies and combine multiple probes across cosmic time to anchor the behavior of the two components. We use Planck cosmic microwave background data and baryonic acoustic oscillation measurements from the BOSS, SDSS, and 6dF surveys to present current limits on the model parameters, and then forecast the future reach from the CMB Stage-4 and DESI experiments. We show that a multiprobe analysis of current data provides only marginal improvement on the determination of the individual parameters and no reduction of the correlations. Future observations will better distinguish the neutrino mass and preserve the current sensitivity to the number of species even in case of a time-varying dark energy component

Next Generation Virgo Cluster Survey. XXI. The weak lensing masses of the CFHTLS and NGVS RedGOLD galaxy clusters and calibration of the optical richness

Astrophysical Journal American Astronomical Society 848 (2017) 114

A Raichoor, J Ford, L Ferrarese, SDJ Gwyn, TH Puzia, C Parroni, T Erben, S Mei, LV Waerbeke, R Licitra, L Miller, M Meneghetti, H Hildebrandt, P Côté, G Covone, J-C Cuillandre, P-A Duc

We measured stacked weak lensing cluster masses for a sample of 1323 galaxy clusters detected by the RedGOLD algorithm in the Canada–France–Hawaii Telescope Legacy Survey W1 and the Next Generation Virgo Cluster Survey at $0.2\lt z\lt 0.5$, in the optical richness range $10\lt \lambda \lt 70$. This is the most comprehensive lensing study of a $\sim 100 \% $ complete and $\sim 80 \% $ pure optical cluster catalog in this redshift range. We test different mass models, and our final model includes a basic halo model with a Navarro Frenk and White profile, as well as correction terms that take into account cluster miscentering, non-weak shear, the two-halo term, the contribution of the Brightest Cluster Galaxy, and an a posteriori correction for the intrinsic scatter in the mass–richness relation. With this model, we obtain a mass–richness relation of $\mathrm{log}{M}_{200}/{M}_{\odot }\,=(14.46\pm 0.02)+(1.04\pm 0.09)\mathrm{log}(\lambda /40)$ (statistical uncertainties). This result is consistent with other published lensing mass–richness relations. We give the coefficients of the scaling relations between the lensing mass and X-ray mass proxies, L X and T X, and compare them with previous results. When compared to X-ray masses and mass proxies, our results are in agreement with most previous results and simulations, and consistent with the expected deviations from self-similarity.