2.2 Large scale structure
2.2.1 Weak gravitational lensing
In a similar way to using the temperature to calculate the CMB power spectrum, the lensing power spectrum involves extracting the lensing potential across the sky [218;219;236]. For weak lensing the Newtonian gravitational potential Φ can be defined by using the Newtonian gauge in cosmological perturbation theory, where δg00 = −2a2Ψ and δgij = 2a2Φδij in equation (I–2.11) [109, Chapter 4;
243]. There is no anisotropic stress when Ψ = Φ [109, Chapter 4;243]. Through equation (I–2.13) the lensing potential can be related to the matter in a ΛCDM universe, which in Fourier space is ([185;218;219;236])
k2Φ(k) = −3
2ΩmH02a−1δ(k) (III–2.4)
where Ωm is the matter density parameter, H0 is the Hubble constant, k is the wavevector with magnitude k in direction ˆn and a in the scale factor where a = 1 today. δ(k) is the Fourier transform of the density contrast at wavevector k. Integrating this along the line-of-sight gives the lensing potential, ψ, at an angle, (θ, ϕ) = ˆn, on the sky [219]. An amplification matrix A, describing the mapping of the coordinates of a lensed image to its unlensed source can be written in terms of this lensing potential ([218])
Aij = δij− ∂i∂jψ. (III–2.5)
Further, this amplification matrix can be decomposed into
κ = 1
2∇2ψ, (III–2.6)
where ∇2 is the Laplacian (∂θ∂θ+ ∂ϕ∂ϕ) and γ1 = 1
2(∂θ∂θ− ∂ϑ∂ϑ)ψ, (III–2.7)
γ2 = ∂θ∂ϕψ, (III–2.8)
where it is usual to combine γ1+ iγ2 = γ [218]. κ and γ are the convergence and shear which provide isotropic and anisotropic magnification of lensed im-ages [218; 219]. As well as anisotropic magnification, γ describes the shape distortion of the images of sources [218; 219]. When the anisotropic stress is zero (as when Ψ = Φ) then the only distortion of the shape of images must come from the gravitational tidal field and so the comic shear γ, totally encapsulates this [219].
As with the temperature fluctuations of the CMB, the lensing potential can be decomposed into functions on a sphere ([185;219;236;357])
ψ(ˆn) = X Cijψ(`) is proportional to the matter power spectrum Pmby integration along the line-of-sight, including a factor of 3ΩmH02/2 [219].
Galaxy lensing
Galaxy surveys measure the statistical shapes of galaxies. These can be related to the lensing potential through the shear and convergence power spectra [169].
The shear power spectrum can be related to the lensing power spectrum by Cijγ(`) = 1
where the details of the calculation are not important here, but arise due to the relation between the lensing potential to the shear via the Jacobi matrix [219].
It is most common to work with the shear correlation functions ξ+ and ξ−since they can be measured directly from the galaxy shape catalogues [204]. This means that the shape of galaxies viewed in galaxy surveys are directly related to the cosmic shear. The correlation functions are given by
ξ+(ˆn · ˆn0) = 1 where P`(ˆn· ˆn0) are the Legendre polynomials [219]. The flat sky power spectrum Pγ(`), analogue of equation (III–2.12) can be used when the correlations are over scales where the curvature of the sky is less important, which gives the correlation functions
where J0 and J4 are the Bessel functions of the first kind at order 0 and 4 [218;
219]. Measurements of the shear correlation functions therefore reveal informa-tion on Ωm directly through the relation to Cψ.
The convergence power spectrum is calculated in a similar way to the cosmic shear power spectrum and is related to the lensing power spectrum by ([219])
Cijκ = 1 4
(` + 1)!
(` − 1)!
4
Cijψ. (III–2.16)
From this, it can be seen that the convergence power spectrum is considerably larger than the shear power spectrum on large scales (low `), but comparable at large ` [219].
Although the correlation of ellipticity between galaxies reveals information on the shear field, it is contaminated by the intrinsic alignment of galaxies [357].
Galaxies which form near each other will be aligned due to their gravitational pull on each other. The galaxies which are in the same tidal field will also be aligned with each other along the line-of-sight [169]. If the observed shear field is expanded to
γobs = γ + γI, (III–2.17)
where γ is the true shear field and γIdescribes the correlated intrinsic alignment of galaxies the correlation functions which need to be considered are
hγiobsγjobsi = hγiγji + hγiγjIi + hγiIγji + hγiIγjIi, (III–2.18) where only hγiγji = ξ+ provides useful constraints on cosmological parame-ters [170]. The other objects are expected to be small and can be encapsulated by modelling of galactic physics and controlled via uncertainty biases during analysis. Modelling the gravitational lensing signature is difficult since it in-volves knowing, to a high precision, galaxy dynamics [220].
The galaxy lensing measurements used in this Thesis are:
Lensing2013 : When using Planck 2013+WP+BAO or WMAP+highL+BAO the CFHTLenS tomographic blue galaxy sample is used as the galaxy lensing data. This was shown in [170] to have an intrinsic alignment signal that was con-sistent with zero. This eliminates the need to marginalise over any additional nuisance parameters. The cosmic shear correlation functions are estimated in six redshift bins, each with an angular range 1.5 < θ < 35 arcmin. The power
spectrum on non-linear scales can be corrected using the Halofit fitting formu-lae [337;350], which has been shown to be accurate enough to use with massive neutrinos [62]. This data set is always combined with the Planck 2013 lensing data.
CFHTLenS (Strong): This relates directly to the Min case in [199] fig-ure 12 which has the strongest assumptions made about astrophysical uncer-tainties. There are seven angular bins and seven tomographic redshift bins which each have their own uncertainties related to them. These redshift un-certainties are Gaussians about ∆z1 = −0.045 ± 0.014, ∆z2 = −0.013 ± 0.010,
∆z3 = 0.008 ± 0.008, ∆z4 = 0.042 ± 0.017 and ∆z5 = 0.042 ± 0.034 leaving the last two bins with flat priors of ∆z6,7 = [−0.1, 0.1], keeping all angular scales.
There are also tight priors on the amplitude of intrinsic alignments and the in-trinsic alignment luminosity and redshift dependence are zero.
CFHTLenS (Weak): As for the CFHTLenS (Strong) case, this also comes from [199] where it is denoted Max. The astrophysical assumptions are greatly reduced with wide flat priors on intrinsic alignment measurements and ∆z = [−0.1, 0.1] for each of the seven tomographic bins, while non-linear scales are cut in the matter power spectrum. The cut to the non-linear scales is the main cause for measurements from CFHTLenS (Weak) being much less constraining than CFHTLenS (Strong).
DES Science Verification: The results from the Dark Energy Survey (DES) follow the prescription in [1] where the range of angular scales included is less than in either of the CFHTLenS analyses for each of its three redshift bins.
Here uncertainties in the redshift bins are not taken into account and intrin-sic alignments are set to zero. As such the constraints are not as tight as the CFHTLenS (Strong) but provide a stronger constraint than CFHTLenS (Weak).
Kilo-Degree Survey: During the preparation of this Thesis the Kilo-Degree Survey (KIDS) [226] has produced results which are similar in many ways to those produced by CFHTLenS. Given this, a value for the discordance has not been quoted for this data in the next section, presuming it to be close to that for CFHTLenS.
CMB lensing
Measuring the gravitational lensing of CMB photons can also provide informa-tion about cosmological shear correlainforma-tions related to the matter power spectrum,
hence revealing information about Ωm and σ8 [236]. As lensing maps the tem-perature Θ(ˆn) → Θ(ˆn + ∇ψ) throughout space, the effects of lensing on the CMB power spectrum can be calculated [236]. The correlation of the lensed temperature is given by [185]
where C` is the CMB temperature power spectrum from equation (I–2.22) and C` is the lensing power spectrum from equation (III–2.11). This represents a convolution of the unlensed temperature power spectrum with the lensing power spectrum to flatten peaks and shift power from large scales to smaller scales in the temperature power spectrum [236]. Equation (III–2.19) shows that the mea-surements of the temperature anisotropies of the CMB are closely linked to the lensing potential the photons travel through to be observed. By disentangling the lensed spectrum from the unlensed temperature therefore allows information to be learned about the matter power spectrum and the density of matter. The CMB lensing data used is:
Planck2013 lensing: Lensing2013 constraints of cosmic shear are always com-bined with the gravitational lensing of the CMB using reconstructions from Planck 2013 [10] and SPT [129]. The combination of Planck 2013 lensing and Lensing2013 will simply denoted Lensing for convenience.
Planck2015 lensing: The updated Planck 2015 lensing contains measurements of the lensing power spectrum between 40 < ` < 400 as in [15] where other scales are cut due to spurious features.