arXiv subjects

Mustapha Ishak

Publications and source records attributed to Mustapha Ishak.

At least 91 records · Page 5Linked to original sources

arXiv2019

Cosmological discordances III: more on measure properties, Large-Scale-Structure constraints, the Hubble constant and Planck

Consistency between cosmological data sets is essential for ongoing and future cosmological analyses. We first investigate the questions of stability and applicability of some moment-based inconsistency measures to multiple data sets. We show that the recently introduced index of inconsistency (IOI) is numerically stable while it can be applied to multiple data sets. We use an illustrative construction of constraints as well as an example with real data sets (i.e. WMAP versus Planck) to show some limitations of the application of the Karhunen-Loeve decomposition to discordance measures. Second, we perform various consistency analyzes using IOI between multiple current data sets while \textit{working with the entire common parameter spaces}. We find current Large-Scale-Structure (LSS) data sets (Planck CMB lensing, DES lensing-clustering and SDSS RSD) all to be consistent with one another. This is found to be not the case for Planck temperature (TT) versus polarization (TE,EE) data, where moderate inconsistencies are present. Noteworthy, we find a strong inconsistency between joint LSS probes and Planck with IOI=5.27, and a moderate tension between DES and Planck with IOI=3.14. Next, using the IOI metric, we compare the Hubble constant from five independent probes. We confirm previous strong tensions between local measurement (SH0ES) and Planck as well as between H0LiCOW and Planck, but also find new strong tensions between SH0ES measurement and the joint LSS probes with IOI=6.73 (i.e. 3.7-$σ$ in 1D) as well as between joint LSS and combined probes SH0ES+H0LiCOW with IOI=8.59 (i.e. 4.1-$σ$ in 1D). Whether due to systematic effects in the data sets or problems with the underlying model, sources of these old and new tensions need to be identified and dealt with.

astro-ph.CO↗

arXiv2019

Modified Gravity and Dark Energy models Beyond $w(z)$CDM Testable by LSST

One of the main science goals of the Large Synoptic Survey Telescope (LSST) is to uncover the nature of cosmic acceleration. In the base analysis, possible deviations from the Lambda-Cold-Dark-Matter ($Λ$CDM) background evolution will be probed by fitting a $w(z)$CDM model, which allows for a redshift-dependent dark energy equation of state with $w(z)$, within general relativity (GR). A rich array of other phenomena can arise due to deviations from the standard $Λ$CDM+GR model though, including modifications to the growth rate of structure and lensing, and novel screening effects on non-linear scales. Concrete physical models are needed to provide consistent predictions for these (potentially small) effects, to give us the best chance of detecting them and separating them from astrophysical systematics. A complex plethora of possible models has been constructed over the past few decades, with none emerging as a particular favorite. This document prioritizes a subset of these models along with rationales for further study and inclusion into the LSST Dark Energy Science Collaboration (DESC) data analysis pipelines, based on their observational viability, theoretical plausibility, and level of theoretical development. We provide references and theoretical expressions to aid the integration of these models into DESC software and simulations, and give justifications for why other models were not prioritized. While DESC efforts are free to pursue other models, we provide here guidelines on which theories appear to have higher priority for collaboration efforts due to their perceived promise and greater instructional value.

astro-ph.CO↗

arXiv2019

Core Cosmology Library: Precision Cosmological Predictions for LSST

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 https://github.com/LSSTDESC/CCL.

astro-ph.CO↗

arXiv2019

Inflation and Dark Energy from spectroscopy at $z > 2$

The expansion of the Universe is understood to have accelerated during two epochs: in its very first moments during a period of Inflation and much more recently, at $z < 1$, when Dark Energy is hypothesized to drive cosmic acceleration. The undiscovered mechanisms behind these two epochs represent some of the most important open problems in fundamental physics. The large cosmological volume at $2 < z < 5$, together with the ability to efficiently target high-$z$ galaxies with known techniques, enables large gains in the study of Inflation and Dark Energy. A future spectroscopic survey can test the Gaussianity of the initial conditions up to a factor of ~50 better than our current bounds, crossing the crucial theoretical threshold of $σ(f_{NL}^{\rm local})$ of order unity that separates single field and multi-field models. Simultaneously, it can measure the fraction of Dark Energy at the percent level up to $z = 5$, thus serving as an unprecedented test of the standard model and opening up a tremendous discovery space.

astro-ph.CO↗

arXiv2018

Testing General Relativity in Cosmology

We review recent developments and results in testing general relativity (GR) at cosmological scales. The subject has witnessed rapid growth during the last two decades with the aim of addressing the question of cosmic acceleration and the dark energy associated with it. However, with the advent of precision cosmology, it has also become a well-motivated endeavor by itself to test gravitational physics at cosmic scales. We overview cosmological probes of gravity, formalisms and parameterizations for testing deviations from GR at cosmological scales, selected modified gravity (MG) theories, gravitational screening mechanisms, and computer codes developed for these tests. We then provide summaries of recent cosmological constraints on MG parameters and selected MG models. We supplement these cosmological constraints with a summary of implications from the recent binary neutron star merger event. Next, we summarize some results on MG parameter forecasts with and without astrophysical systematics that will dominate the uncertainties. The review aims at providing an overall picture of the subject and an entry point to students and researchers interested in joining the field. It can also serve as a quick reference to recent results and constraints on testing gravity at cosmological scales.

astro-ph.CO↗

arXiv2018

Self-calibration method for II and GI types of intrinsic alignments of galaxies

We introduce a self-calibration method that can be applied to the intrinsic ellipticity--intrinsic ellipticity (II) and gravitational shear -- intrinsic ellipticity (GI) types of intrinsic alignment of galaxies. The method combines previous self-calibration techniques with modifications to one of them in order to use auto-spectra in addition to cross-spectra between redshift bins. This allows one to use the self-calibration while preserving all the constraining power of cosmic shear from surveys. We show that the new method provides more flexibility in using various redshift bin widths. We perform cosmological parameter constraint forecast when this method is applied to the Large Synoptic Survey Telescope (LSST). Compared to the original self-calibration, we find that the new method provides further significant reduction in any residual shift in the cosmological parameters (e.g. factors of $2-4$ for the dark energy equation of state) which is promising for accurate cosmology.

astro-ph.CO↗

arXiv2017

Effects of Self-Calibration of Intrinsic Alignment on Cosmological Parameter Constraints from Future Cosmic Shear Surveys

Intrinsic alignments (IA) of galaxies have been recognized as one of the most serious contaminants to weak lensing. These systematics need to be isolated and mitigated in order for ongoing and future lensing surveys to reach their full potential. The IA self-calibration (SC) method was shown in previous studies to be able to reduce the GI contamination by up to a factor of 10 for the 2-point and 3-point correlations. The SC method does not require to assume an IA model in its working and can extract the GI signal from the same photo-z survey offering the possibility to test and understand structure formation scenarios and their relationship to IA models. In this paper, we study the effects of the IA SC mitigation method on the precision and accuracy of cosmological parameter constraints from future cosmic shear surveys LSST, WFIRST and Euclid. We perform analytical and numerical calculations to estimate the loss of precision and the residual bias in the best fit cosmological parameters after the self-calibration is performed. We take into account uncertainties from photometric redshifts and the galaxy bias. We find that the confidence contours are slightly inflated from applying the SC method itself while a significant increase is due to the inclusion of the photo-z uncertainties. The bias of cosmological parameters is reduced from several-$σ$, when IA is not corrected for, to below 1-$σ$ after SC is applied. These numbers are comparable to those resulting from applying the method of marginalizing over IA model parameters despite the fact that the two methods operate very differently. We conclude that implementing the SC for these future cosmic-shear surveys will not only allow one to efficiently mitigate the GI contaminant but also help to understand their modeling and link to structure formation.

astro-ph.CO↗

arXiv2017

Cosmological discordances II: Hubble constant, Planck and large-scale-structure data sets

We examine systematically the (in)consistency between cosmological constraints as obtained from various current data sets of the expansion history, Large Scale Structure (LSS), and Cosmic Microwave Background (CMB) from Planck. We run (dis)concordance tests within each set and across the sets using a recently introduced index of inconsistency (IOI) capable of dissecting inconsistencies between two or more data sets. First, we compare the constraints on $H_0$ from five different methods and find that the IOI drops from 2.85 to 0.88 (on Jeffreys' scales) when the local $H_0$ measurements is removed. This seems to indicate that the local measurement is an outlier, thus favoring a systematics-based explanation. We find a moderate inconsistency (IOI=2.61) between Planck temperature and polarization. We find that current LSS data sets including WiggleZ, SDSS RSD, CFHTLenS, CMB lensing and SZ cluster count, are consistent one with another and when all combined. However, we find a persistent moderate inconsistency between Planck and individual or combined LSS probes. For Planck TT+lowTEB versus individual LSS probes, the IOI spans the range 2.92--3.72 and increases to 3.44--4.20 when the polarization data is added in. The joint LSS versus the combined Planck temperature and polarization has an IOI of 2.83 in the most conservative case. But if Planck lowTEB is added to the joint LSS to constrain $τ$ and break degeneracies, the inconsistency between Planck and joint LSS data increases to the high-end of the moderate range with IOI=4.81. Whether due to systematic effects in the data or to the underlying model, these inconsistencies need to be resolved. Finally, we perform forecast calculations using LSST and find that the discordance between Planck and future LSS data, if it persists as present, can rise up to a high IOI of 17, thus falling in the strong range of inconsistency. (Abridged).

astro-ph.CO↗

arXiv2017

Cosmological discordances: a new measure, marginalization effects, and application to geometry vs growth current data sets

The continuous progress toward more precise cosmological surveys and experiments has galvanized recent interest into consistency tests on cosmological parameters and models. At the heart of this effort is quantifying the degree of inconsistency between two or more cosmological data sets. We introduce an intuitive moment-based measure we call the \textit{index of inconsistency} (IOI) and show that it is sensitive to the separation of the means, the size of the constraint ellipsoids, and their orientations in the parameter space. We find that it tracks accurately the inconsistencies when present. Next, we show that parameter marginalization can cause a loss of information on the inconsistency between two experiments and we quantify such a loss using the drop in IOI. In order to zoom on a given parameter, we define the relative residual IOI and the relative drop in IOI. While these two quantities can provide insights on the parameters that are most responsible for inconsistencies, we find that the full IOI applied to the whole parameter spaces is what must be used to correctly reflect the degree of inconsistency between two experiments. We discuss various properties of IOI, provide its eigenmode decomposition, and compare it to other measures of discordance. Finally, we apply IOI to current geometry data sets (i.e. an improved Supernovae Type Ia compilation, baryon acoustic oscillations from 6dF, SDSS MGS and Lyman-$α$ forest, and high-$\ell$ CMB temperature data from Planck-2015) versus growth data sets (i.e. Redshift Space Distortions from WiggleZ and SDSS, Weak Lensing from CFHTLenS, CMB Lensing, Sunyav-Zeldovich effect, and low-$\ell$ CMB temperature and polarization data from Planck-2015). We find that a persistent inconsistency is present between the two data sets. This could reflect the presence of systematics in the data or inconsistencies in the underlying model.

astro-ph.CO↗

arXiv2016

Testing gravity theories using tensor perturbations

Primordial gravitational waves constitute a promising probe of the very early Universe and the laws of gravity. We study in this work changes to tensor-mode perturbations (TMPs) that can arise in various proposed modified gravity (MG) theories. These include additional friction effects, nonstandard dispersion relations involving a massive graviton, a modified speed, and a small-scale modification. We introduce a physically motivated parameterization of these effects and use current available data to obtain exclusion regions in the parameter spaces. Taking into account the foreground subtraction, we then perform a forecast analysis focusing on tensor-mode MG parameters as constrained by future experiments COrE, Stage-IV and PIXIE. For a fiducial value of the tensor-to-scalar ratio $r=0.01$, we find that an additional friction of $3.5\sim4.5\%$ compared to GR will be detected at $3$-$σ$ by these experiments, while a decrease in friction will be more difficult to detect. The speed of gravitational waves needs to be by $5\sim15\%$ different from the speed of light for detection. We find that the minimum detectable graviton mass is about $7.8\sim9.7\times 10^{-33}$ eV, which is of the same order of magnitude as the graviton mass that allows massive gravity theories to produce cosmic acceleration. Finally, we study TMPs in MG during inflation using our parameterization. We find that the tensor spectral index is related to $r$ and the friction parameter $ν_0$ by $n_T=-3ν_0-r/8$. Assuming that the friction parameter is unchanged throughout the history of the Universe and is much larger than $r$, future experiments will be able to distinguish this MG consistency relation from the standard inflation one, and thus can be used as a further test of MG. In summary, TMPs and cosmic-microwave-background B-mode polarization provide a complementary avenue to test gravity theories. (Abridged)

astro-ph.CO↗

arXiv2016

Ultra faint dwarf galaxies: an arena for testing dark matter versus modified gravity

The scenario consistent with a wealth of observations for the missing mass problem is that of weakly interacting dark matter particles. However, arguments or proposals for a Newtonian or relativistic modified gravity scenario continue to be made. A distinguishing characteristic between the two scenarios is that dark matter particles can produce a gravitational effect, in principle, without the need of baryons while this is not the case for the modified gravity scenario where such an effect must be correlated with the amount of baryonic matter. We consider here ultra-faint dwarf (UFD) galaxies as a promising arena to test the two scenarios based on the above assertion. We compare the correlation of the luminosity with the velocity dispersion between samples of UFD and non-UFD galaxies, finding a significant loss of correlation for UFD galaxies. For example, we find for 28 non-UFD galaxies a strong correlation coefficient of -0.688 which drops to -0.077 for the 23 UFD galaxies. Incoming and future data will determine whether the observed stochasticity for UFD galaxies is physical or due to systematics in the data. Such a loss of correlation (if it is to persist) is possible and consistent with the dark matter scenario for UFD galaxies but would constitute a new challenge for the modified gravity scenario.

astro-ph.CO↗

arXiv2016

Averaged universe confronted with cosmological observations: A fully covariant approach

One of the outstanding problems in general relativistic cosmology is that of the averaging. That is, how the lumpy universe that we observe at small scales averages out to a smooth Friedmann-Lemaitre-Robertson-Walker (FLRW) model. The root of the problem is that averaging does not commute with the Einstein equations that govern the dynamics of the model. This leads to the well-know question of backreaction in cosmology. In this work, we approach the problem using the covariant framework of Macroscopic Gravity (MG). We use its cosmological solution with a flat FLRW macroscopic background where the result of averaging cosmic inhomogeneities has been encapsulated into a backreaction density parameter denoted $Ω_\mathcal{A}$. We constrain this averaged universe using available cosmological data sets of expansion and growth including, for the first time, a full CMB analysis from Planck temperature anisotropy and polarization data, the supernovae data from Union 2.1, the galaxy power spectrum from WiggleZ, the weak lensing tomography shear-shear cross correlations from the CFHTLenS survey and the baryonic acoustic oscillation data from 6Df, SDSS DR7 and BOSS DR9. We find that $-0.0155 \le Ω_\mathcal{A} \le 0$ (at the 68\% CL) thus providing a tight upper-bound on the backreaction term. We also find that the term is strongly correlated with cosmological parameters such $Ω_Λ$, $σ_8$ and $H_0$. While small, a backreaction density parameter of a few percent should be kept in consideration along with other systematics for precision cosmology.

astro-ph.CO↗

arXiv2015

Constraints and tensions in testing general relativity from Planck and CFHTLenS including intrinsic alignment systematics

We present constraints on testing general relativity (GR) at cosmological scales using recent data sets and assess the impact of galaxy intrinsic alignment (IA) in the CFHTLenS lensing data on those constraints. We consider CMB temperature data from Planck, the galaxy power spectrum from WiggleZ, weak lensing tomography from the CFHTLenS, ISW-galaxy cross correlations, and BAO data from 6dF, SDSS DR7, and BOSS DR9. We use a parameterization of the modified gravity (MG) that is binned in redshift and scale, a parameterization that evolves monotonically in scale but is binned in redshift, and a functional parameterization that evolves only in redshift. We present the results in terms of the MG parameters $Q$ and $Σ$. We employ an IA model with an amplitude $A_{CFHTLenS}$ that is included in the parameter analysis. We find an improvement in the constraints on the MG parameters corresponding to $40-53\%$ increase on the figure of merit compared to previous studies, and GR is found consistent with the data at the $95\%$ CL. The bounds found on $A_{CFHTLenS}$ are sensitive to the MG parameterization used, and the correlations between $A_{CFHTLenS}$ and MG parameters are found to be weak to moderate. For all 3 MG parameterizations $A_{\rm CFHTLenS}$ is found consistent with zero when the whole lensing sample is used, however, when using the optimized early-type galaxy sample a significantly non-zero $A_{\rm CFHTLenS}$ is found for GR and the third MG parameterization. We find that the tensions observed in previous studies persist, and there is an indication that CMB data and lensing data prefer different values for MG parameters, particularly for the parameter $Σ$. The analysis of the confidence contours and probability distributions suggest that the bimodality found follows that of the known tension in the $σ_8$ parameter. (Abridged)

astro-ph.CO↗

arXiv2015

Expansion and Growth of Structure Observables in a Macroscopic Gravity Averaged Universe

We investigate the effect of averaging inhomogeneities on expansion and large-scale structure growth observables using the exact and covariant framework of Macroscopic Gravity (MG). It is well-known that applying the Einstein's equations and spatial averaging do not commute and lead to the averaging problem. For the MG formalism applied to the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric, this gives an extra dynamical term encapsulated as an averaging density parameter denoted $Ω_A$. An exact isotropic cosmological solution of MG for the flat FLRW metric is already known in the literature, we derive here an anisotropic exact solution. Using the isotropic solution, we compare the expansion history to current data of distances to supernovae, Baryon Acoustic Oscillations, CMB last scattering surface, and Hubble constant measurements, and find $-0.05 \le Ω_A \le 0.07$ (at the 95% CL). For the flat metric case this reduces to $-0.03 \le Ω_A \le 0.05$. We also find that the inclusion of this term in the fits can shift the values of the usual cosmological parameters by a few to several percents. Next, we derive an equation for the growth rate of large scale structure in MG that includes a term due to the averaging and compare it to that of the LCDM concordance model. We find that an $Ω_A$ of an amplitude range within the bounds above leads to a relative deviation of the growth from that of the LCDM of up to 2-4% at late times. Thus, the shift in the growth could be of comparable amplitude to that caused by similar changes in cosmological parameters like the dark energy density parameter or its equation of state. The effect could also be comparable in amplitude to some systematic effects considered for future surveys. This indicates that the averaging term and its possible effect need to be tightly constrained in future precision cosmological studies. (Abridged)

astro-ph.CO↗

arXiv2015

Effect of inhomogeneities on high precision measurements of cosmological distances

We study effects of inhomogeneities on distance measures in an exact relativistic Swiss-cheese model of the universe, focusing on the distance modulus. The model has LCDM background dynamics, and the `holes' are non-symmetric structures described by the Szekeres metric. The Szekeres exact solution of Einstein's equations, which is inhomogeneous and anisotropic, allows us to capture potentially relevant effects on light propagation due to nontrivial evolution of structures in an exact framework. Light beams traversing a single Szekeres structure in different ways can experience either magnification or demagnification, depending on the particular path, and we explore a small additional effect due time evolution of the structures. We study the probability distributions of $Δμ=μ_{LCDM}-μ_{SC}$ for sources at different redshifts in various Swiss-cheese constructions, where the light beams travel through a large number of randomly oriented holes with random impact parameters. We find for $Δμ$ the dispersions $0.004\le σ_{Δμ} \le 0.008$ mag for sources with redshifts $1.0\le z \le 1.5$, which are smaller than the intrinsic dispersion of, for example, magnitudes of type Ia supernovae. The shapes of the distributions we obtain for our Swiss-cheese constructions are peculiar in the sense that they are not consistently skewed toward the demagnification side, as they are in analyses of lensing in cosmological simulations, reflecting a limitation of these constructions. This could be the result of requiring the continuity of Einstein's equations throughout the overall spacetime patchwork, which imposes the condition that compensating overdense shells must accompany the underdense void regions in the holes. The possibility to explore other uses of these constructions that could circumvent this limitation and lead to different statistics remains open. (Abridged)

astro-ph.CO↗

arXiv2014

The Intrinsic Alignment of Galaxies and its Impact on Weak Gravitational Lensing in an Era of Precision Cosmology

The wealth of incoming and future cosmological observations will allow us to map out the structure and evolution of the observable universe to an unprecedented level of precision. Among these observations is the weak gravitational lensing of galaxies, e.g., cosmic shear that measures the minute distortions of background galaxy images by intervening cosmic structure. Weak lensing and cosmic shear promise to be a powerful probe of astrophysics and cosmology, constraining models of dark energy, measuring the evolution of structure in the universe, and testing theories of gravity on cosmic scales. However, the intrinsic alignment of galaxies -- their shape and orientation before being lensed -- may pose a great challenge to the use of weak gravitational lensing as an accurate cosmological probe, and has been identified as one of the primary physical systematic biases in cosmic shear studies. Correlations between this intrinsic alignment and the lensing signal can persist even for large physical separations, and isolating the effect of intrinsic alignment from weak lensing is not trivial. A great deal of work in the last two decades has been devoted to understanding and characterizing this intrinsic alignment, which is also a direct and complementary probe of structure formation and evolution in its own right. In this review, we report in a systematic way the state of our understanding of the intrinsic alignment of galaxies, with a particular emphasis on its large-scale impact on weak lensing measurements and methods for its isolation or mitigation. (Abridged)

astro-ph.CO↗

arXiv2014

The effects of structure anisotropy on lensing observables in an exact general relativistic setting for precision cosmology

The study of relativistic, higher order and nonlinear effects has become necessary in recent years in the pursuit of precision cosmology. We develop and apply here a framework to study gravitational lensing in exact models in general relativity that are not restricted to homogeneity and isotropy, and where full nonlinearity and relativistic effects are included. We apply the framework to a specific, anisotropic galaxy cluster model which is based on a modified NFW halo density profile and described by the Szekeres metric. We examine the effects of increasing levels of anisotropy in the galaxy cluster on lensing observables like the convergence and shear for various lensing geometries, finding a strong nonlinear response in both the convergence and shear for rays passing through anisotropic regions of the cluster. Deviation from the expected values in a spherically symmetric structure are asymmetric with respect to path direction and thus will persist as a statistical effect when averaged over some ensemble of such clusters. The resulting relative difference in various geometries can be as large as approximately 2%, 8%, and 24% in the measure of convergence for levels of anisotropy of 5%, 10%, and 15%, respectively, as a fraction of total cluster mass. For the total magnitude of shear, the relative difference can grow near the center of the structure to be as large as 15%, 32%, and 44% for the same levels of anisotropy, averaged over the two extreme geometries. The convergence is impacted most strongly for rays which pass in directions along the axis of maximum dipole anisotropy in the structure, while the shear is most strongly impacted for rays which pass in directions orthogonal to this axis, as expected. These effects due to anisotropic structures will affect lensing measurements and must be fully examined in an era of precision cosmology.

astro-ph.CO↗

arXiv2014

Cross-correlation between cosmic microwave background lensing and galaxy intrinsic alignment as a contaminant to gravitational lensing cross-correlated probes of the universe

We introduce here a cross-correlation term between CMB lensing and galaxy intrinsic alignment, noted hereby as $ϕI$. This effect acts as a contaminant to the cross-correlation between CMB lensing and galaxy lensing. The latter cross-correlation has recently been detected for the first time, and measurements will greatly improve as the area of overlap between galaxy and CMB surveys increases and measurements of the CMB polarization become more significant. This will constitute a powerful probe for studying the structure and evolution of the universe. The magnitude of the $ϕI$ term is found to be about 15% of the pure CMB lensing-galaxy lensing component and acts to reduce the magnitude of its measured spectrum. This offset in the spectrum will strongly impact its use for precision cosmological study if left unmitigated. We also propose here a method to calibrate this $ϕI$ contamination through use of a scaling relation that allows one to reduce the impact of $ϕI$ by a factor of 20 or more in all redshift bins, which would reduce its magnitude down to detection limits in almost all cases. This will allow the full use of this probe for precision cosmology.

astro-ph.CO↗