integrated Sachs-Wolfe angular power spectrum

Fig.13.7shows the angular power spectra Cτ(`) of the iSW-effectτ(θ) and Cχ(`) of the iSW temperature gradient

χ(θ) which have been obtained by applying the projection formula (13.26) to the spectrumhqk(k)q∗

k(k)iwith the weighing function WiSW(w)= 3H2 0Ω0 c2 fK(w) a(w) , (13.71)

which can be read offfrom eqn. (13.64). The redshift-weightings and the time-evolution of the density and velocity fields can be combined, which yields the function (13.72) after substitutingy=kw,

ψ`(k)4pt= "Z ymax 0 dyWiSW y k dJl(y) dy D(y)G(y) #2 (13.72) which mediates between the 3-dimensional power spectrumhqk(k)q∗<sub>k</sub>(k)iand the angular power spectrum Cτ(`) by convolution:

Cτ(`)=2π

Z

dkhqk(k)q∗k(k)i ×ψ`(k). (13.73) Again, the 3-fold integration in eqn. (13.26) is reduced to a 2-fold integration. The shape of the functionψ`(k) is

depicted in Fig.13.6for various values of`. In contrast to the functionϕ`(k) used in the projection of the lensing

power spectra, the functionψ`(k) is symmetric about its peak, which is caused by the replacement of J`(y)/ywith

the derivative dJ`(y)/dy. The fast variability is again due to the strong influence of the velocity time evolution

G(y).Details concerning numerics of the integration in eqn.13.72which involves a rapidly oscillating function are discussed in AppendixD.

The angular power spectrum Cτ(`) of the iSW temperature fluctuationsτ(θ) along with the primary CMB fluctu- ations and the limiting PLANCK-sensitivity is depicted in Fig.13.7. The angular power spectrum has an amplitude of'3×10−11K2 at small`and shows but little variation with the multipole order`. The amplitude agrees well with the result fromSeljak (1996b), but the decline of the power spectrum on large angular scales could not be confirmed, which is due to the fact that for large angles, the Bessel functions Jl(x) are a poor approximation to the Legendre polynomials P`(x). The position of the peak in the projection kernelψ`(k) suggests that on the largest

scales considered here, the angular spectrum Cτ(`) is dominated by fluctuations at the maximum of P(k) on scales

at k−1 _{'10 Mpc. With increasing multipole order`, the peak inψ}

`(k) shifts only slowly towards higher values of k,

which explains the small variation of Cχ(`)=`(`+1)Cτ(`).

The channel averaged PLANCK-sensitivy is described by (Knox 1995,Tegmark & Efstathiou 1996): Cnoise(`)=

4πσ2

Npix

exphθ2_b`(`+1)i, (13.74)

where Npix'5.03×107is the number of pixels andθbthe FWHM extension of the PLANCK-beam. For the average

amplitude of the noiseσeffper solid angle subtended by a single pixel I use the quadratic harmonic mean over all

six HFI-channels: 1 σ2 eff = 6 X i=1 1 σ2 i −→σ_eff=13.42µK. (13.75)

13.5 Summary 100−2 10−1 100 101 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PSfrag replacements contrib ution ψ` ( k )

comoving wave vectorkMpc/h−1

Figure 13.6.: Contributionψ`(k) of the 4-point term to the angular power spectrum Cτ(`) of the iSW temperature fluctu-

ationsτas a function of wave vector k, for`=100 (solid line),`=300 (dashed line),`=1000 (dash-dotted line) and

`=3000 (dotted line). The curves have been normalised to a peak value of unity.

The sensitiviy considerations suggest that the iSW-effect is well above the noise level of the combined PLANCK HFI-channels, so that the power spectrum of Cτ(`) should be observable for angular scales`∼<200 as a contribution to the primary CMB fluctuations CCMB(`), which in Fig.13.7have been computed using the CMBfastcode by Seljak & Zaldarriaga(1996)).

13.5. Summary

The scope of this paper is to derive the corrections to the power spectrum of weak gravitational lensing due to gravitomagnetic terms in the metric by perturbation theory. Within the same formalism, the power spectrum of the iSW-effect can be determined as well.

• The iSW-effect and gravitomagnetic lensing measure the evolution of velocities and densities in the large- scale structure and are sensitive to the cosmological parametersΩM andσ8. Applied to single objects like

clusters, where the above described formalism equally applies, the iSW-effect would allow to measure the cosmological evolution of merger rates and dark matter accretion strengths (van den Bosch 2002,Wechsler et al. 2002,Zhao et al. 2003).

• Gravitomagnetic lensing would test general relativity on the largest scales (Mpc - Gpc) to second order, and could help decide in favour of or against other metric theories of gravity. It should be emphasised that in the current theoretical description of structure formation or in current numerical simulations the motion of bodies is described by classical mechanics, i.e. instantaneous propagation of potentials and no relativistic increase of inertial mass with velocity, but the interaction of light with matter should be treated in the framework of the post-Newtonian limits of general relativity.

• Gravitomagnetic terms influence the weak lensing power spectrum most notably on large spatial and angular scales, which are difficult to access experimentally. Furthermore, cosmic variance and galactic foregrounds prevent accurate measurements on the scales in question, i.e. ∼>Gpc/h and above. The small gravitomagnetic corrections could be amplified by cross correlation with the kinetic Sunyaev-Zel’dovich effect (Sunyaev & Zel’dovich 1972), once future CMB telescopes will provide accurate measurements of line-of-sight velocities

101 ₁₀2 ₁₀3 ₁₀4 10−4 10−2 100 102 104 106

PSfrag replacements angular

po wer spectrum ` ( ` + 1) C∆T ( ` )[ µ K] 2 multipole order`

Figure 13.7.: Angular power spectrum C∆T(`)=T2

CMBCτ(`) of the iSW temperature fluctuationsτ(θ) (dashed line). The CMB power spectrum CCMB(`) for theΛCDM cosmology (solid line) and the limiting PLANCK-sensitivies Cnoise(`) for angular resolutions∆θ=5.0

0 (dash-dotted line) and∆θ=9.0

1 (dotted line) are depicted for comparison.

or with the velocity information from optical galaxy surveys. For current weak lensing surveys, gravitomag- netic corrections to cosmic shear do not play a significant role.

• The iSW-effect is described by a line-of-sight integration over the divergence of the gravitomagnetic poten- tials. By this argument, the iSW-effect is reduced to a second order lensing effect. Every iSW quantity has a correspondence in weak gravitational lensing and the derivation of the power spectrum Cτ(`) proceeds in

complete analogy to that of any weak lensing quantity, for instance that of the convergence Cκ(`). The most

important difference of the derivation presented here to the ones carried out bySeljak(1996b) orCooray

(2002) is that my derivation explicily pays tribute to the lensing nature of the iSW-effect.

• Gravitomagnetic lensing and the iSW-effect are complementary in measuring the matter flows parallel and perpendicular to the line-of-sight. The picture emerging is that (subject to the approximations made) in gravitational light deflection (including the gravitomagnetic term Az), the photon’s k-vector is rotated but

its normalisation is conserved. Contrarily, the components of A transverse to the line-of-sight change the normalisation of the k-vector, i.e. the photon’s energy, but leave the direction of k invariant.

• Both effects, gravitomagnetic lensing and the iSW-effect, are achromatic which makes them only accessible by their n-point statistics. Furthermore, the iSW-effect needs to be separated from other achromatic CMB structures such as the kinetic Sunyaev-Zel’dovich effect and the Ostriker-Vishniac effect. The derivation predicts iSW temperature fluctuations of∆T = τTCMB ' 5.4 µK on large angular scales, which is within

reach of future CMB experiments like the European PLANCK-mission.

• The gradientχ(θ) of the iSW temperature fluctuation fieldτ(θ) should directly map regions of large matter flows, e.g. filaments and clusters with high peculiar velocities, but it can be expected to be very susceptible to noise due to the differentiation required in obtainingχ(θ) fromτ(θ), which is reflected by the fact that ratio of the angular power spectra Cχ(`)/Cτ(`) is proportional to`(`+1).

The verification of the theoretical approach by a ray-tracing simulation of photons through a cosmological n- body simulation will be the subject of future research. The non-Gaussian features the iSW-effect and gravitational lensing exhibit and the mode-coupling in nonlinear structure growth are unaccessible to perturbation theory and are important on small scales. The novel approach to the iSW-effect presented here should allow a much improved

13.5 Summary

precision in the numerical treatment, because inaccuracies in interpolating the scalar potential’s time derivative ∂Φ/∂ηfor each integration time step and in integrating a rapidly oscillating function inherent the direct approach (e.g.Tuluie & Laguna 1995a,b) are alleviated.

The results of this chapter were derived in collaboration with M. Bartelmann (ITA, Heidelberg). A resulting paper entitled Gravitational lensing in the second post-Newtonian approximation: Gravitomagnetic potentials and the integrated Sachs-Wolfe effect will be submitted to the journal Montly Notices of the Royal Astronomical Society.

14. Summary and outlook

The main subject of this thesis is the simulation of observations of cluster of galaxies for the European PLANCK- satellite. PLANCK will be able to detect clusters of galaxies by their thermal Sunyaev-Zel’dovich signature in sub- millimetric data and will open a new observational window for investigating structure growth and baryonic physics inside clusters of galaxies. In Chapter 5, I present all-sky maps of the thermal and kinetic Sunyaev-Zel’dovich effects which was constructed from numerical data by combining template clusters extracted from a high-resolution hydrodynamical simulation and a cluster catalogue following from a large-volume dark matter simulation. By con- struction, the map correctly shows the clustering properties on large spatial scales, deviation from canonical scaling relations and asymmetric and non-analytic temperature and density profiles of the individual clusters of galaxies. In the kinetic Sunyaev-Zel’dovich map, the peculiar velocities correspond to the local density field. The comparison to estimates of the Sunyaev-Zel’dovich signal strengths following from virial arguments showed that the number of clusters detectable with PLANCK is likely to be overestimated.

These maps were combined with various Galactic and ecliptic foregrounds. Specifically, I considered synchrotron radiation, free-free emission, infrared emission by thermal dust, line transition produced in rotational transitions of carbon monoxide molecules and the thermal radiation of planets and asteroids of the Solar system. I combined the Sunyaev-Zel’dovich maps with these foreground maps and with a realisation of the fluctuating CMB while taking care of the different spectral properties of the respective emission components and convolved the individual spectra with PLANCK’s frequency response functions. The maps were successively convolved with PLANCK’s respective point-spread functions, yielding antenna temperature maps for all nine PLANCK channels. In order to simulate the finite sensitivity of PLANCK’s receivers, I generated noise maps that incorporate the spatial highly-non uniform exposure pattern due to PLANCK’s scanning strategy, which were successively added. In Chapter6 I describe the simulation in detail and investigate the complicated cross- and autocorrelation properties which have special relevance to filtering and component separation.

In Chapter7, I describe an approach how the weak Sunyaev-Zel’dovich signal can be amplified and extracted by matched and scale-adaptive filtering. These filter schemes are particularly appealing because they are based on a variational principle. The algorithms require filter kernels to minimise the variance of a data set with the condition that the amplitude of the filtered field is an unbiased estimator of the underlying signal and that the amplitude of the filtered field is maximal if the spatial scale of the filter corresponds to the spatial extension of the signal. These filtering schemes were extended to multifrequency observations and to spherical topologies. In collaboration with C. Pfrommer I could derive formulae that yield filter kernels for a given signal profile, for a specific spectral behaviour of the signal, and for the angular cross- and autocorrelation function of the spurious components. I derived filter kernels numerically for the simulated PLANCK antenna temperature maps and verified their functionality.

The characterisation of the PLANCK Sunyaev-Zel’dovich cluster sample is the subject of Chapter 8. It was shown that the SZ-cluster sample derived in this work, which contains 6×103 _{entries above 3σ does not live}

up to the high expectations claimed by analytic estimates. But the SZ-sample was shown to be clean and not to contain spurious detections on a significant level. The redshift range probed by PLANCK is restricted to redshifts smaller than z < 0.8, which is due to the highly structured noise on small scales. The sample was analysed in detail and the distributions of mass, redshift and detection significance are given. The spatial distribution was demonstrated to be spatially non-uniform on a significant level, irrespective of the filtering scheme, which is due to the improper removal of long-wavelength modes. The linearity of the filters was proved and position accuracies were demonstrated to be too coarse for direct follow-up studies in the X-ray band or in optical wavelengths.

The SZ-cluster catalogue of PLANCK will exceed classic X-ray catalogues with respect to number of detec- tions and will complement our view of the baryonic processes inside clusters of galaxies. Furthermore, aspects of structure formation ranging from dark energy parameters, especially the dark energy densityΩQ, its equation of state parameterwQ, the shape of the power spectrum on Mpc scales and its normalisationσ8 will be a high-

SZ observations with PLANCK carried out so far and covers all important aspects of cluster physics, foreground contamination, observation and instrumental imperfections, filtering and peak extraction.

In a supplementary project aiming at PLANCK data analysis algorithms I worked out a new pixel numbering scheme for the HEALPix tesselation commonly used in analyses of CMB data. A core quantity in many CMB data analysis tasks is the pairwise pixel covariance matrix. Common pixel numbering schemes face the difficulty that the covariance matrix does not have a simple shape and is difficult to access algorithmically and numerically. Basic matrix manipulations like inversion and computation of the determinant are very difficult to carry out, keeping the vast number of pixels of current and future CMB experiments in mind. In Chapter9, I propose to use a pixel numbering based on a fractal, self-similar Peano-Hilbert curve that runs through all pixels on the sphere. If pixels were numbered successively along this curve, the pairwise pixel covariance matrix would assume a band-diagonal shape if correlations on large angular separations are neglected. For band diagonal matrices, fast and efficient algorithms for computing e.g. determinants and inverses exist. I tested the locality of the spherical Peano-Hilbert curve and investigated the shape of the covariance matrix for typical shapes of the correlation function and found its properties with respect to locality to be superior to the two existing pixel numbering schemes. It is planned to add the Peano-numbering to the HEALPix software package.

Aiming at future high-resolution CMB observations I analysed the morphology of simulated SZ-maps of clusters of galaxies with wavelets. It was found that the spectrum of wavelet coefficients can be described with elementary functions that have certain characteristics which are non-degenerate indicators of redshift. These morphological redshifts will be particularly useful for future SZ surveys that are expected to detect thousands of clusters in order to select targets for e.g. X-ray follow up observations. A detailed analysis in Chapter10examined the redshift estimation based on wavelet decomposition and found the relative accuracy in the distance estimation to be accurate to a few percent out to redshifts of unity. Adding noise contributions such as instrumental noise at reasonable levels and CMB fluctuations in order to simulate monochromatic observations proved the method to be very robust. Other complications like finite instrumental resolution, cool cores of clusters and systematic deviations from the universal baryon fraction that significantly alter the SZ-morphology of a cluster or impact on the SZ-scaling relations were shown to be controllable. Morphological redshift estimators will be of particular use for dedicated high-yield SZ observatories in order to select targets for optical or X-ray follow-up observations.

The statistical description of the CMB based on Gaussian random fields leaded to an unexpected application of this cosmological key concept to X-ray andγ-ray imaging in high energy astronomy. Imaging of highly energetic radiation by refractive or reflective optics is far from easy. Imaging at these high photon energies is commonly achieved by coded mask imaging, where the shadow cast of a mask consisting of randomly placed open elements is registered by a position sensitive detector. By using correlation techniques, it is possible to reconstruct the distribution of sources inside the field-of-view from the shadowgram, which is a superposition of the intensity distributions imaged by each individual pinhole. In Chapter11, I propose to use Gaussian random fields as coded mask patterns, because they can be constructed to encode a specific functional shape of the point-spread function. I investigated the properties of Gaussian random fields in coded mask imagers in extensive photon ray-tracing studies and found the Gaussian random fields to perform well in the observation of extended sources which are unaccessible to traditional coded mask instruments and to yield a moderate performance in the observation of point sources.

Apart from the interaction of photons with the electrons of the intra-cluster medium I studied their gravitational interaction with clusters of galaxies and with the cosmic large-scale structure in Chapter13. I was able to explain the integrated Sachs-Wolfe/Rees-Sciama effect, which predicts a frequency shift of photons transversing time-variable gravitational wells to be a second-order gravitational lensing effect emerging in the post-Newtonian expansion of general relativity. In this approximation, the Rees-Sciama effect measures the divergence of the gravitomagnetic vector potential integrated along the line-of-sight. By using this access, I could show interesting analogies between gravitational lensing quantities and Rees-Sciama quantities and point out many analogies in the respective formulae. I derived the angular autocorrelation function of the Rees-Sciama temperature fluctuations in the quasilinear regime in perturbation theory by using the gravitomagnetic formalism. The angular power spectrum was found to be detectable by PLANCK as a correction to the primordial CMB power spectrum at low multipoles. The Rees- Sciama effect on these scales will be an important diagnostic for dark matter clustering as it probes the transition from the linear into the nonlinear regime of structure formation.

By using the same tools, I addressed gravitomagnetic corrections to weak gravitational lensing of the large scale