Fig 2 - uploaded by Pan Simon
Content may be subject to copyright.
Schematic illustration of the symmetric covariance matrix C for the case where there are N z = 2 source redshift bins and N ∆θ = 5 angular separation bins. Combinations of ξ L,U,LU identify covariance terms of the form given in Eq. (31).
Source publication
Cosmological weak lensing by the large scale structure of the Universe, cosmic shear, is coming of age as a powerful probe of the parameters describing the cosmological model and matter power spectrum. It complements CMB studies, by breaking degeneracies and providing a cross-check. An important measure of the cosmic shear signal are the shear corr...
Context in source publication
Context 1
... the outer average is performed over the N f = 795 surveys. If redshift binning is considered, there are N z (N z + 1) /2 combinations of correlation and cross- correlation functions and hence C is a matrix composed of [N z (N z + 1) /2] 2 blocks. Fig. 2 illustrates this for the simplified case where N z = 2 and N ∆θ = 5; for exam- ple the block in the upper left of the matrix ...
Similar publications
We propose a theoretical framework that can possibly constrain the initial vacuum by observation of the cosmic microwave background (CMB). With a generic vacuum without any particular choice a priori, thereby keeping both the Bogolyubov coefficients in the analysis, we compute observable parameters from two- and three-point correlation functions. W...
We extend our recent work on the effects of a time-varying fine-structure constant $\alpha$ in the cosmic microwave background, by providing a thorough analysis of the degeneracies between $\alpha$ and the other cosmological parameters, and discussing ways to break these with both existing and/or forthcoming data. In particular, we present the stat...
The E-type $\alpha$-attractor models of single-field inflation were generalized further in order to accommodate production of primordial black holes (PBH) via adding a near-inflection point to the inflaton scalar potential at smaller scales, in good agreement with measurements of the cosmic microwave background (CMB) radiation. A minimal number of...
This is the second paper in the series that confronts predictions of a model of the landscape with cosmological observations. We show here how the modifications of the Friedmann equation due to the decohering effects of long wavelength modes on the wave function of the Universe defined on the landscape leave unique signatures on the CMB spectra and...
Citations
... gal per arcmin 2 . As was shown by [64,65], tomography greatly enhances cosmological constraints with the power-spectrum. We see a similar result in figure 5. ...
We explore the effect of massive neutrinos on the weak lensing shear bispectrum using the Cosmological Massive Neutrino Simulations [1]. We find that the primary effect of massive neutrinos is to suppress the amplitude of the bispectrum with limited effect on the bispectrum shape. The suppression of the bispectrum amplitude is a factor of two greater than the suppression of the small scale power-spectrum. For an LSST-like weak lensing survey that observes half of the sky with five tomographic redshift bins, we explore the constraining power of the bispectrum on three cosmological parameters: the sum of the neutrino mass ∑ mν, the matter density Ωm and the amplitude of primordial fluctuations As. Bispectrum measurements alone provide similar constraints to the power spectrum measurements and combining the two probes leads to significant improvements than using the latter alone. We find that the joint constraints tighten the power spectrum 95% constraints by ∼ 32% for ∑ mν, 13% for Ωm and 57% for As.
... This is only to a little extent achieved in a 2-dimensional analysis: galaxy shapes are observed on the 2-dimensional celestial sphere and the shear components are line-of-sight projected quantities, with the projection causing loss of information on the redshift evolution (Jain & Seljak 1997;Takada & Jain 2003a,b;Munshi & Kilbinger 2006;Jee et al. 2013;Kilbinger et al. 2013). For this reason, as an alternative to a pure 2D projection, a tomographic analysis based on a binning in redshift of the sources has been first proposed in Hu (1999) and has since become the standard technique for cosmological weak lensing studies (Takada & White 2004;Simon et al. 2004;Takada & Jain 2004;Hollenstein et al. 2009;Kilbinger et al. 2009;Schäfer & Heisenberg 2012;Heymans et al. 2013). Galaxies are assigned to different bins according to their redshifts, so that intra-and interbin correlations of the binned shear field can be computed. ...
Cosmic shear, the weak gravitational lensing caused by the large-scale structure, is one of the primary probes to test gravity with current and future surveys. There are two main techniques to analyse a cosmic shear survey; a tomographic method, where correlations between the lensing signal in different redshift bins are used to recover redshift information, and a 3D approach, where the full redshift information is carried through the entire analysis. Here we compare the two methods, by forecasting cosmological constraints for future surveys like Euclid. We extend the 3D formalism for the first time to theories beyond the standard model, belonging to the Horndeski class. This includes the majority of universally coupled extensions to {\Lambda}CDM with one scalar degree of freedom in addition to the metric, which are still in agreement with current observations. Given a fixed background, the evolution of linear perturbations in Horndeski gravity is described by a set of four functions of time only. We model their time evolution assuming proportionality to the dark energy density fraction and place Fisher matrix constraints on the proportionality coefficients. We find that a 3D analysis can constrain Horndeski theories better than a tomographic one, in particular with a decrease in the errors on the Horndeski parameters of the order of 20 - 30$\%$. This paper shows for the first time a quantitative comparison on an equal footing between Fisher matrix forecasts for both a fully 3D and a tomographic analysis of cosmic shear surveys. The increased sensitivity of the 3D formalism comes from its ability to retain information on the source redshifts along the entire analysis.
... The integrated lensing signal is mostly sensitive to a combination of the cosmic mean matter density Ω m and the matter power spectrum amplitude σ 8 [5]. Tomographic methods can yield additional constraints on all parameters guiding the growth rate of structure [6,7], notably also on the properties of dark energy [8][9][10], but rely on low photometric redshift uncertainties [11]. Since lensing is a direct probe of the total matter, luminous and dark, it can be combined with measurements of the luminous matter distribution in order to learn about the relationship between baryons and dark matter. ...
We present a Bayesian reconstruction algorithm that infers the three-dimensional large-scale matter distribution from the weak gravitational lensing effects measured in the image shapes of galaxies. The algorithm assumes that the prior probability distribution of the matter density is lognormal, in contrast to many existing methods that assume normal (Gaussian) distributed density fields. We compare the reconstruction results for both priors in a suite of increasingly realistic tests on mock data. We find that in cases of high noise levels (i.e. for low source galaxy densities and/or high shape measurement uncertainties), both normal and lognormal priors lead to reconstructions of comparable quality. In the low-noise regime, however, the lognormal model produces significantly better reconstructions than the normal model: The lognormal model 1) enforces non-negative densities, while negative densities are present when a normal prior is employed, 2) better traces the extremal values and the skewness of the true underlying distribution, and 3) yields a higher correlation between the reconstruction and the true density distribution. Hence, the lognormal model is to be preferred over the normal model, in particular since these differences become relevant for data from current and futures surveys.
... The distances to the source galaxies can be estimated with sufficient accuracy by photometric (rather than spectroscopic) methods. Such estimates can be used to group the source-galaxy population by distance shells and thus to extract three-dimensional information on the lensing matter distribution [282,283,284,285,286,225,287]. ...
According to the theory of general relativity, masses deflect light in a way similar to convex glass lenses. This gravitational lensing effect is astigmatic, giving rise to image distortions. These distortions allow to quantify cosmic structures statistically on a broad range of scales, and to map the spatial distribution of dark and visible matter. We summarise the theory of weak gravitational lensing and review applications to galaxies, galaxy clusters and larger-scale structures in the Universe.
... In each bin a conventional two-dimensional analysis can be carried out; the range of the projection, however, is much smaller, and the correlations between signals in different redshift bins put constraints on the time evolution. This stacking approach has been termed 'tomography' (Hu 1999;Takada & White 2004;Simon et al. 2004;Takada & Jain 2004;Hollenstein et al. 2009;Kilbinger et al. 2009;Schäfer & Heisenberg 2012;Heymans et al. 2013). Several authors have focused in particular on the potential of weak lensing to constrain dark energy (Jain & Taylor 2003;Bernstein & Jain 2004;Hannestad et al. 2006;Amendola et al. 2008;Huterer 2010), leading to slighly weaker constraints compared to 3d-methods. ...
We present the first calculation of the cross-correlation between three-dimensional cosmic shear and the integrated Sachs-Wolfe
(iSW) effect. Both signals are combined in a single formalism, which permits the computation of the full covariance matrix.
In order to avoid the uncertainties presented by the non-linear evolution of the matter power spectrum and intrinsic alignments
of galaxies, our analysis is restricted to large scales, i.e. multipoles below ℓ = 1000. We demonstrate in a Fisher analysis
that this reduction compared to other studies of three-dimensional weak lensing extending to smaller scales is compensated
by the information that is gained if the additional iSW signal and in particular its cross-correlation with lensing data are
considered. Given the observational standards of upcoming weak lensing surveys like Euclid, marginal errors on cosmological parameters decrease by ten per cent compared to a cosmic shear experiment if both types
of information are combined without a CMB prior. Once the constraining power of CMB data is added, the improvement becomes
marginal.
... In addition to the auto-power spectra for each bin, one can also consider cross-power spectra or bispectra between tomographic bins. This not only adds more constraining information, and has been recognized as the optimal method for extracting cosmological information (e.g., Hu (1999); Huterer (2002); Simon et al. (2004); Takada and Jain (2004)), but can also be used in methods to deal with systematics in weak lensing such as the intrinsic alignment of galaxies (see Sec. 6). A tomographic galaxy sample is chosen such that each redshift bin i spans from some χ i (z i ) to χ i+1 (z i+1 ), with a meanχ i (z i ). ...
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 -- poses 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)
... Weak lensing tomography was first investigated by [126] who utilised power spectra to show that two or three bin tomography can recover most of the cosmological information in a data set. The optimisation of tomographic analysis of cosmic shear was investigated by [152] who found that parameter uncertainties are reduced by a factor of up to 10 by binning the data into 4 redshift bins, with the dark energy density parameter ⌦ ⇤ gaining the most. This is due to the fact that dark energy influences the expansion rate and the growth rate over time, so retaining redshift information allows us to constrain its behaviour at di↵erent epochs. ...
... However, these expressions assume gaussianity of the signal and underestimate the power on small scales. An alternative, easy-to-implement approach is to measure an estimate of the covariance from multiple realisations of simulated data [152,164]. This approach has the advantages of naturally accounting for cross-bin terms and is independent of survey configuration, and hence more easily applicable to real data. ...
... For this reason, we measure the area of the ⌦ m -8 68.3% contour with tomography as an optimisation metric (we denote this metric 1). [152] find that two-bin tomography partially breaks the degeneracy between the two parameters, indicating that tomography has the potential to put tight limits on the size of this contour. We also measure the width of the ⌦ m (metric 2) and 8 (metric 3) 68.3% likelihood contours around the fiducial values for di↵erent bin combinations, to see if the optimal binning for these parameters di↵ers. ...
Weak gravitational lensing is a powerful astronomical tool for constraining
cosmological parameters that is entering its prime. Lensing occurs because
gravitational fields deflect light rays and measuring this deflection through a
statistic known as cosmic shear allows us to directly measure the properties of
dark matter and dark energy on large scales. In principle, gravitational lensing
is a clean probe of the cosmology of the Universe, as it depends on gravity alone
and not on incomplete astrophysical models or approximations. In practice,
however, there are several factors that limit the accuracy and precision of lensing
measurements. These include accurate measurement of galaxy shapes, correctly
accounting for distortions to galaxy images due to the point spread function of
the telescope, the presence of intrinsic alignments (IAs) of galaxy shapes due
to physical processes, and inaccuracies in commonly-used galaxy photometric
redshift information. These effects may all introduce systematic errors in lensing
measurements which must be carefully accounted for to ensure that cosmological
constraints from lensing are unbiased and as precise as possible.
The Canada-France-Hawaii-Telescope Lensing Survey (CFHTLenS) is the
largest weak lensing survey completed to date, covering 154 square degrees of the
sky in 5 optical bands, with photometric redshift information for every survey
galaxy. With lensing measurements from more galaxies than ever before, the
statistical uncertainties on parameter estimates will be the lowest ever achieved
from weak lensing. If left unaccounted for, sources of systematic error would
dominate over the statistical uncertainty, potentially biasing parameter estimates
catastrophically. A technique known as tomography in which galaxies are sorted
into bins based on their redshift can help constrain cosmological parameters
more precisely. This is because utilising the redshifts of survey galaxies retains
cosmological information that would otherwise be lost, such as the behaviour of dark energy and the growth of structure over time. Tomography, however,
increases the demand for systematics-free galaxy catalogues as the technique is
strongly sensitive to the IA signal and photometric redshift errors. Therefore,
future lensing analyses will require a more sophisticated treatment of these
effects to extract maximal information from the lensing signal. A thorough
understanding of the error on lensing measurements is necessary in order to
produce meaningful cosmological constraints. One of the key features of cosmic
shear is that it is highly correlated over di erent angular scales, meaning that
error estimates must take into account the covariance of the data over different
angular scales, and in the case of tomography, between different redshift bins.
The behaviour and size of the (inverse) covariance matrix is one of the limiting
factors in such a cosmological likelihood analysis, so constructing an accurate,
unbiased estimate of the covariance matrix inverse is essential to cosmic shear
analysis.
This thesis presents work to optimise tomographic weak lensing analysis
and achieve the tightest parameter constraints possible for a CFHTLenS-like
survey. N-body simulations and Gaussian shear fields incorporating an IA model
(known as the `non-linear alignment' model) with a free parameter are used to
estimate fully tomographic covariance matrices of cosmic shear for CFHTLenS.
We simultaneously incorporate for the first time the error contribution expected
from the non-linear alignment model for IAs and realistic photometric redshift
uncertainties as measured from the CFHTLenS. We find that non-Gaussian
simulations that incorporate nonlinearity on small scales are needed to ensure
the covariance is not underestimated, and that the covariance matrix is shot-noise
dominated for almost all tomographic correlations. The number of realisations
of the simulations used to estimate the covariance places a hard limit on the
maximum number of tomographic bins that one can use in an analysis. Given
the available number of lines of sight generated from CFHTLenS-like simulations,
we find that up to ~ 15 tomographic bins may be utilised in a likelihood analysis.
The estimated tomographic covariance matrices are used in a least-squares
likelihood analysis in order to find the combination of both angular and
tomographic bins that gives the tightest constraints on some key cosmological
parameters. We find that the optimum binning is somewhat degenerate, with
around 6 tomographic and 8 angular bins being optimal, and limited by the available number of realisations of the simulations used to estimate the covariance.
We also investigate the bias on best- t parameter estimates that occurs if IAs
or photometric redshift errors are neglected. With our choice of IA model, the
effect of neglecting IAs on the best- t cosmological parameters is not significant
for a CFHTLenS-like survey, although this may not be true if the IA signal differs
substantially from the model, or for future wide-field surveys with much smaller
statistical uncertainties. Similarly, neglecting photometric redshift errors does
not result in significant bias, although we apply similar caveats.
Finally, we apply the results of this optimisation to the CFHTLenS cosmic
shear data, performing a preliminary analysis of the shear correlation function
to produce both 2D and optimal tomographic cosmological constraints. From
6-bin tomography, we constrain the matter density parameter
Ωm = 0:419+0:123-0:090,
the amplitude of the matter power spectrum σ8 = 0:623+0:101
-0:084 and the amplitude
parameter of the non-linear alignment model, A = -1:161+1:163
-0:597. We perform this
analysis to test the validity and limitations of the optimal binning on real data and
find that 6-bin tomography improves parameter constraints considerably, albeit
not as much as when performed on simulated data. This analysis represents
an important step in the development of techniques to optimise the recovery
of lensing information and hence cosmological constraints, while simultaneously
accounting for potential sources of bias in shear analysis.
... Sufficiently accurate information on the distance to the source galaxies is provided by photometric redshifts. [454] show that the accuracy of cosmic-shear parameter estimates can be improved by a factor of 5 . . . 10 by splitting the source galaxies into only four redshift bins. [175] developed an elegant formalism for extracting three-dimensional information from weak-lensing data (see also [385]), and [472] recovered the three-dimensional matter distribution in the field of the Combo-17 survey, using photometric redshifts as distance indicators for the source galaxies. ...
Gravitational lensing has developed into one of the most powerful tools for
the analysis of the dark universe. This review summarises the theory of
gravitational lensing, its main current applications and representative results
achieved so far. It has two parts. In the first, starting from the equation of
geodesic deviation, the equations of thin and extended gravitational lensing
are derived. In the second, gravitational lensing by stars and planets,
galaxies, galaxy clusters and large-scale structures is discussed and
summarised.
... It has been noted by several authors for the shear signal (GG) alone (e.g. Amara and Refregier 2006;Hu 1999;Jain et al 2007;Ma et al 2006;Simon et al 2004) that although a small amount of tomographic information greatly improves parameter constraints, continuing to subdivide into further redshift bins beyond ∼3 does not significantly reduce uncertainties further. This can be seen by the dotted line in figure 6. ...
Cosmic shear constrains cosmology by exploiting the apparent alignments of pairs of galaxies due to gravitational lensing by intervening mass clumps. However galaxies may become (intrinsically) aligned with each other, and with nearby mass clumps, during their formation. This effect needs to be disentangled from the cosmic shear signal to place constraints on cosmology. We use the linear intrinsic alignment model as a base and compare it to an alternative model and data. If intrinsic alignments are ignored then the dark energy equation of state is biased by ~50 per cent. We examine how the number of tomographic redshift bins affects uncertainties on cosmological parameters and find that when intrinsic alignments are included two or more times as many bins are required to obtain 80 per cent of the available information. We investigate how the degradation in the dark energy figure of merit depends on the photometric redshift scatter. Previous studies have shown that lensing does not place stringent requirements on the photometric redshift uncertainty, so long as the uncertainty is well known. However, if intrinsic alignments are included the requirements become a factor of three tighter. These results are quite insensitive to the fraction of catastrophic outliers, assuming that this fraction is well known. We show the effect of uncertainties in photometric redshift bias and scatter. Finally we quantify how priors on the intrinsic alignment model would improve dark energy constraints. Comment: 14 pages and 9 figures. Replaced with final version accepted in "Gravitational Lensing" Focus Issue of the New Journal of Physics at http://www.iop.org/EJ/abstract/1367-2630/9/12/E09
... three-dimensional distribution of dark structures. Although lensing observables measure the two-dimensional, projected density distribution, selecting sources at different distances allows structures along the line-of-sight to be resolved. Sufficiently accurate information on the distance to the source galaxies is provided by photometric redshifts. [262] show that the accuracy of cosmic-shear parameter estimates can be improved by a factor of 5 . . . 10 by splitting the source galaxies into only four redshift bins. [107] developed an elegant formalism for extracting three-dimensional information from weaklensing data (see also [229]), and [270] recovered the three-dimensional matter dis ...
Gravitational lensing originates from the deflection of light by masses, irrespective of their physical state or composition. Since it appears inescapable that most of the matter in the universe is dark, gravitational lensing has developed into one of the primary tools to learn about the amount, composition and distribution of masses in the universe.
The review will summarise the theory of gravitational lensing, starting from Fermat’s principle. This will first be applied to isolated lenses like compact objects, galaxies, and galaxy clusters. Cosmologically relevant applications will be described, such as searches for compact dark-matter objects in galactic halos, measurements of the Hubble constant in galaxy lenses, and methods for mapping the dark matter in galaxy clusters. Next, the theory of cosmological lensing will be introduced. The concepts of lensing by large-scale structures and its measurement will be discussed, concluding with an overview of results which have so far been obtained, and an outlook at what can be expected in the near future.
