The geometry of the universe and the nature of dark energy can also be constrained by galactic lensing (Jullo et al., 2010). This work will be included in a future paper on CMB lensing and the HI intensity map (Prince et al., 2016b).
Standard Cosmological Model
- Cosmological Principle
- General Relativity
- Equation of State and Density Parameters
- Evolution of the Scale Factor and Hubble Parameter
- Redshift
- Distances
We can use the cosmological principle to determine an appropriate criterion for the standard model of the universe (Robertson, 1935). The critical density of the universe is the total energy density required for the universe to be spatially flat.
Perturbation Theory and Large-Scale Structure Formation
- Perturbed Einstein Equations
- Gauge Transformations
- Equations Governing the Evolution of Perturbations
- Evolution of the Power Spectrum
For scalar perturbations in the conformal Newtonian gauge, v = −v,i andB = 0, and the perturbation to the stress energy tensor is thus. Einstein's field equations for scalar perturbations of a perfect fluid (Σ = 0 soΦ = Ψ) in the conformal Newtonian gauge are given by.
Inflation
The early period of exponential expansion results in a decreasing Hubble radius, meaning that regions that were in early causal contact became causally disconnected as the Hubble radius contracted. Thus, inflation results in a particle horizon much larger than the Hubble radius: regions that were not causally connected when the CMB formed were able to interact early on, which explains the near-uniformity of the CMB.
The Cosmic Microwave Background
Statistics of CMB Temperature Anisotropies
If the fluctuations are Gaussian, as we expect, then the angular power spectrum contains all available statistical information about the temperature fluctuations. Inhomogeneities of the temperature field during recombination are seen by us as anisotropies, because when we look at the sky in different directions, we see photons from different parts of the final scattering surface.
Physics of CMB Temperature Anisotropies
The next section describes the physics of these anisotropies. clumps of cold dark matter), while the pressure of photons opposed the compression of the photon-baryonic fluid caused by gravity. This makes intuitive sense because if the length scale of the potential fluctuations were smaller (i.e., k was larger), it would take less time for the photon-baryon fluid to compress into the potential (i.e., ω would be larger).
CMB Polarisation
The rich structure of the power spectrum allows us to detect the effects of gravitational lensing, which will be discussed in the next section. The acoustic peaks and small scale damping can be seen in the temperature power spectrum plotted in figure where we used the flat air approximation.
Weak Gravitational Lensing of the Cosmic Microwave Background
Deflection Angle
The deflection angle predicted by Einstein's general theory of relativity is twice that predicted by Newtonian theory of gravity, due to the effects of spacetime curvature. The top panel shows a map of the CMB without a lens on the left and the CMB with a lens on the right.
Lensing Potential
The difference between the two maps is correlated with the modulus of the deflection angle, in the lower left and right panels, respectively. It is the rich structure of the anisotropies, described statistically by the CMB power spectrum, that allows us to detect the effects of lensing and reconstruct the lensing potential. We write the angular power spectrum of the lens potential Ψ in terms of the three-dimensional power spectrum of the gravitational potential Ψ as follows.
Magnification Matrix
Theils are spherical Bessel functions, which are used to project three-dimensional potentials onto the two-dimensional sky. We can simplify this expression to the Limber approximation, for which we assume that PΨ(k) varies slowly compared to the Bessel functions. 1∂x2 determine the distortion of the source shape along different axes due to the tidal gravitational field.
Lensed CMB Power Spectra
In this chapter, we derive real-space estimators that reconstruct convergence and lensing directly from CMB temperature and polarization maps. This limit corresponds to the study of lens fields (κ0, γ+andγ×) on large scales (small L) and focusing on. This is a fairly good approximation, because small-scale anisotropies contribute most of the statistical information about the lens, and the lens potential peaks at quite low L (see Figure (2.4)).
Derivation from Lensed Power Spectra
The Fourier transform of the unlensed and lensed temperature maps in the flat-sky approximation is found by integrating the areas A and A0 to give . It is now straightforward to find the lensed angular power spectra and the cross-spectra of the CMB temperature and E- and B-mode polarizations in terms of the primordial spectra. These terms indicate how real-space estimators are implemented to obtain lens convergence and shear maps from temperature and polarization maps.
Real Space Estimators as a limit of Harmonic Space Estimators
We can see that the harmonic space estimator for L2ψˆXY for XY = T T, EE and T E is the inverse variance-weighted combination of the real space estimators for the convergence κˆ0 and the shift E-mode ˆγE. The B-mode component of the shear is forbidden in weak lensing, and is therefore not present. Thus, our real space estimators for the T T, EE, and T E spectra are simply the small L limits of the corresponding harmonic space estimators.
Derivation from Lensed Correlation Functions
The lensed correlation functions can thus be expressed in terms of the unlensed correlation functions defined in equation (3.62) and the lens fields that make up the deformation tensor, in a region where κ0,γ+ogγ× are constant and small, e.g. The estimator can be translated to different points on the map, giving ˆ. 3.73) Again, we can translate this estimator to different pixels on the map, which gives 3.74). We can make similar comparisons for the other estimators by deriving their kernels from those defined in Section 3.1.1.
Features of Real Space Estimators
Cumulative Information
For example, most of the information for the AdvACT TT estimator is found between l ~1800 and l ~3000, while for the EE estimator the corresponding scales are l ~ 1000 and l ~ 2000, and for the EB estimator most of the information comes from the scales between l ~400 and l ~1200. The improved resolution of AdvACT (solid curves) with θbeam = 1.4 arcmin as opposed to Planck's ~7 arcmin beam means that more information comes from smaller scales or larger rl for AdvACT than for Planck. For example, for the TT estimator most of the information for Planck comes from angular wavenumbers between l ~800 and l ~1600, while for AdvACT the corresponding wavenumbers are between l ~1800 and l ~3000.
Reconstruction Kernels
Most of the information in the lens kernels in real space (in the column on the left of the figures). One of the dominant CMB foregrounds comes from clustered cosmic infrared background sources (Dunkley et al., 2013), which begin to dominate the CMB signal. Thus, for higher resolution experiments such as AdvACT, much of the apparent small-scale signal will actually come from the foreground, which will act as an additional source of noise in our reconstruction.
Multiplicative Bias
We obtain the multiplicative bias for the convergence and shift estimators using their respective kernels. The AdvACT and Planck form factors are shown for the shift (upper panel) and convergence (lower panel) estimators in Figure (3.5) (the shift plus and shift cross estimators have the same . form factor). The temperature and EB estimators have the most promising form factors, as the multiplicative bias falls off much less sharply with L than for the other estimators.
Simulated CMB Lensing Map Reconstructions
Lensing Reconstruction Applications
The cosmic variance of the reconstructed lensing map due to the unlensed CMB maps is seen as an excess power in the power spectrum of the lensing reconstruction (Kesden et al., 2003). The CMB lensing power spectrum was used for this purpose, for example, in Sherwin et al. CMB lenses probe the power spectrum of matter and thus can constrain the neutrino mass (Allison et al., 2015a).
Neutral Hydrogen
- HI Lines
- Intensity Mapping
- HI Intensity Mapping Signal
- HI Power Spectrum
- Thermal Noise from HI Intensity Mapping
- Foregrounds
- HI Signal to Noise
Thus the Fourier transform of the HI signal in a frequency or redshift bin in the flat-sky approximation is given by. The multi-frequency angular power spectrum (taken between frequencies ν1 and ν2 divided by ∆ν) of HI temperature fluctuations is given by (Bharadwaj and Ali, 2005; Datta et al., 2007). The square of the observed visibilities is related to the angular power spectrum by (Zaldarriaga et al., 2004).
CMB Lensing Convergence
Lensing Signal
Lensing Reconstruction Noise
Down to the zeroth order of the lens potential, the noise of the reconstruction is given by the inverse of the normalization factor NLψ = 1. The reconstruction noise for an AdvACT-like experiment with specifications given in Table (3.3) is shown in Figure (4.6). ), together with the angular power spectrum of the lens convergence. A survey of the southern sky is planned for AdvACT, which will overlap with the part of the sky surveyed by HIRAX.
Cross-Correlation of CMB Lensing with HI Intensity Mapping
Cross-Correlation Angular Power Spectrum
The cross-correlation between the convergence of CMB lenses and the temperature fluctuations of HI intensity mapping is obtained from equations (4.19) and (4.35). Under these assumptions, the cross-correlation becomes hδT(~l)κ∗(~l0)i= 1. 4.49) The amplitude of the cross-correlation increases as the bandwidth increases, as shown in the left panel of Figure (4.7). , because there is a larger overlap between the HI bin and the lens kernel. The cross-correlation angular power spectra in this and later graphs were calculated using a modified version of the code developed by A P'enin.
Cross-Correlation Noise
4.49) The cross-correlation amplitude increases with increasing bandwidth, as shown in the left panel of figure (4.7), because there is a greater overlap between the HI bin and the lens core. We include noise in the total angular power since κandδT are noisy experimental quantities. Figure (4.8) shows the effect of reducing kk,min in a 100 MHz bin centered onz = 1, resulting in the crosstalk amplitude increasing by almost an order of magnitude.
Cross-Correlation Signal to Noise
The signal to noise for individual modes is not highly significant, but we can add the contributions from different annuli inlspace in quadrature to obtain the cumulative signal to noise, shown in the bottom panel of figure (4.10). We find that a significant statistical detection of the cross-correlation will be possible, with a total signal to noise of between 20 and 50 for the four 100 MHz bins, shown in the lower right panel of Figure (4.10). We expect that the overall signal to noise will be somewhat reduced once foreground residuals are included in our treatment, which will be considered in our upcoming article.
Constraints from the Lensing-HI Cross-Correlation
A measurement of the gravitational lensing of the cosmic microwave background through galaxy clusters using data from the South Pole Telescope. The Atacama Cosmology Telescope: a measurement of the cosmic microwave background power spectrum at 148 and 218 GHz from the 2008 Southern Survey. Detection of the cosmic microwave background lensing power spectrum by the Atacama Cosmology Telescope.
