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  • CMB Review
    at matter radiation equality and the damping scale as well as the value of the baryon photon momentum density ratio sets the spacing between of the peaks and compete to determine their amplitude through radiation driving and diffusion damping sets the baryon loading and along with the potential well depths set by fixes the modulation of the even and odd peak heights The initial conditions set the phase or equivalently the location of the first peak in units of and an overall tilt in the power spectrum In the model of Plate 1 these numbers are and and in this family of models the parameter sensitivity is approximately Hu et al 2001 24 and Current observations indicate that and Knox et al 2001 see also Wang et al 2001 Pryke et al 2001 de Bernardis et al 2001 if gravitational waves contributions are subdominant and the reionization redshift is low as assumed in the working cosmological model see 2 1 The acoustic peaks therefore contain three rulers for the angular diameter distance test for curvature i e deviations from However contrary to popular belief any one of these alone is not a standard ruler whose absolute scale is known even in the working cosmological model This is reflected in the sensitivity of these scales to other cosmological parameters For example the dependence of on and hence the Hubble constant is quite strong But in combination with a measurement of the matter radiation ratio from this degeneracy is broken The weaker degeneracy of on the baryons can likewise be broken from a measurement of the baryon photon ratio The damping scale provides an additional consistency check on the implicit assumptions in the working model e g recombination and the energy contents of the Universe during this epoch What makes the peaks

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  • CMB Review
    Preparation 307 GR Perturbation Theory 408 Advanced CMB 448 University of Chicago Astronomy Department KICP Thunch astro ph CO ADS InSpire Review Home Introduction Observables Cosmological Paradigm Temperature Field Polarization Field Acoustic Peaks Introduction Basics Initial Conditions Gravitational Forcing Baryon Loading Radiation Driving Damping Polarization Integral Approach Parameter Sensitivity Matter Power Spectrum Introduction Physical Description Cosmological Implications Gravitational Secondaries Introduction ISW Effect RS Moving Halo Gravitational Waves Gravitational Lensing Scattering

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  • CMB Review
    fall into the Newtonian potential wells with the cold dark matter an event usually referred to as the end of the Compton drag epoch We claimed in 3 5 that above the horizon at matter radiation equality the potentials are nearly constant This follows from the dynamics where pressure gradients are negligible infall into some initial potential causes a potential flow of see Equation 19 and causes density enhancements by continuity of The Poisson equation says that the potential at this later time so that this rate of growth is exactly right to keep the potential constant Formally this Newtonian argument only applies in general relativity for a particular choice of coordinates Bardeen 1980 but the rule of thumb is that if what is driving the expansion including spatial curvature can also cluster unimpeded by pressure the gravitational potential will remain constant Because the potential is constant in the matter dominated epoch the large scale observations of COBE set the overall amplitude of the potential power spectrum today Translated into density this is the well known COBE normalization It is usually expressed in terms of the matter density perturbation at the Hubble scale today Since the observed temperature fluctuation is approximately 25 where the second equality follows from the Poisson equation in a fully matter dominated universe with The observed COBE fluctuation of K Smoot et al 1992 implies For corrections for where the potential decays because the dominant driver of the expansion cannot cluster see Bunn White 1997 On scales below the horizon at matter radiation equality we have seen in 3 5 that pressure gradients from the acoustic oscillations themselves impede the clustering of the dominant component i e the photons and lead to decay in the potential Dark matter density perturbations remain but grow only logarithmically from

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  • CMB Review
    the near scale invariant initial spectrum of fluctuations tells us that by the present fluctuations in the cold dark matter or baryon density fields will have gone non linear for all scales Mpc It is a great triumph of the standard cosmological paradigm that there is just enough growth between and to explain structures in the Universe across a wide range of scales In particular since this non linear scale also corresponds to galaxy clusters and measurements of their abundance yields a robust measure of the power near this scale for a given matter density The agreement between the COBE normalization and the cluster abundance at low and the observed Hubble constant Freedman et al 2001 was pointed out immediately following the COBE result e g White et al 1993 Bartlett Silk 1993 and is one of the strongest pieces of evidence for the parameters in the working cosmological model Ostriker Steinhardt 1995 Krauss Turner 1995 More generally the comparison between large scale structure and the CMB is important in that it breaks degeneracies between effects due to deviations from power law initial conditions and the dynamics of the matter and energy contents of the Universe Any dynamical effect that reduces the amplitude of the matter power spectrum corresponds to a decay in the Newtonian potential that boosts the level of anisotropy see 3 5 and 4 2 1 Massive neutrinos are a good example of physics that drives the matter power spectrum down and the CMB spectrum up The combination is even more fruitful in the relationship between the acoustic peaks and the baryon wiggles in the matter power spectrum Our knowledge of the physical distance between adjacent wiggles provides the ultimate standard ruler for cosmology Eisenstein et al 1998 For example at very low the radial distance out

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  • CMB Review
    Function WMAP Likelihood Reionization PPF for CAMB Halo Mass Conversion Cluster Abundance Cosmology 321 Current Topics 282 Galaxies and Universe 242 Radiative Processes 305 Research Preparation 307 GR Perturbation Theory 408 Advanced CMB 448 University of Chicago Astronomy Department KICP Thunch astro ph CO ADS InSpire Review Home Introduction Observables Cosmological Paradigm Temperature Field Polarization Field Acoustic Peaks Introduction Basics Initial Conditions Gravitational Forcing Baryon Loading Radiation Driving Damping Polarization Integral Approach Parameter Sensitivity Matter Power Spectrum Introduction Physical Description Cosmological Implications Gravitational Secondaries Introduction ISW Effect RS Moving Halo Gravitational Waves Gravitational Lensing Scattering Secondaries Introduction Peak Suppression Large Angle Polarization Doppler Effect Modulated Doppler Effect SZ Effect Non Gaussianity Data Analysis Introduction Mapmaking Bandpower Estimation Parameter Estimation Discussion Discussion Bibliography Gravitational Secondaries Gravitational secondaries arise from two sources the differential redshift from time variable metric perturbations Sachs Wolfe 1967 and gravitational lensing There are many examples of the former one of which we have already encountered in 3 8 in the context of potential decay in the radiation dominated era Such gravitational potential effects are usually called the integrated Sachs Wolfe ISW effect in linear perturbation theory 4 2 1 the Rees Sciama 4 2 2 effect in

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  • CMB Review
    domination and the onset dark energy or spatial curvature domination If the potential decays between the time a photon falls into a potential well and when it climbs out it gets a boost in temperature of due to the differential gravitational redshift and due to an accompanying contraction of the wavelength see 3 3 Potential decay due to residual radiation was introduced in 3 8 but that due to dark energy or curvature at late times induces much different changes in the anisotropy spectrum What makes the dark energy or curvature contributions different from those due to radiation is the longer length of time over which the potentials decay on order the Hubble time today Residual radiation produces its effect quickly so the distance over which photons feel the effect is much smaller than the wavelength of the potential fluctuation Recall that this meant that in the integral in Equation 23 could be set to and removed from the integral The final effect then is proportional to and adds in phase with the monopole The ISW projection indeed the projection of all secondaries is much different see Plate 3 Since the duration of the potential change is much longer photons typically travel through many peaks and troughs of the perturbation This cancellation implies that many modes have virtually no impact on the photon temperature The only modes which do have an impact are those with wavevectors perpendicular to the line of sight so that along the line of sight the photon does not pass through crests and troughs What fraction of the modes contribute to the effect then For a given wavenumber and line of sight instead of the full spherical shell at radius only the ring with participate Thus the anisotropy induced is suppressed by a factor of or

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  • CMB Review
    Clusters Transfer Function WMAP Likelihood Reionization PPF for CAMB Halo Mass Conversion Cluster Abundance Cosmology 321 Current Topics 282 Galaxies and Universe 242 Radiative Processes 305 Research Preparation 307 GR Perturbation Theory 408 Advanced CMB 448 University of Chicago Astronomy Department KICP Thunch astro ph CO ADS InSpire Review Home Introduction Observables Cosmological Paradigm Temperature Field Polarization Field Acoustic Peaks Introduction Basics Initial Conditions Gravitational Forcing Baryon Loading Radiation Driving Damping Polarization Integral Approach Parameter Sensitivity Matter Power Spectrum Introduction Physical Description Cosmological Implications Gravitational Secondaries Introduction ISW Effect RS Moving Halo Gravitational Waves Gravitational Lensing Scattering Secondaries Introduction Peak Suppression Large Angle Polarization Doppler Effect Modulated Doppler Effect SZ Effect Non Gaussianity Data Analysis Introduction Mapmaking Bandpower Estimation Parameter Estimation Discussion Discussion Bibliography Rees Sciama and Moving Halo Effects The ISW effect is linear in the perturbations Cancellation of the ISW effect on small scales leaves second order and non linear analogues in its wake Rees Sciama 1968 From a single isolated structure the potential along the line of sight can change not only from evolution in the density profile but more importantly from its bulk motion across the line of sight In the context of clusters of galaxies

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  • CMB Review
    Analysis Introduction Mapmaking Bandpower Estimation Parameter Estimation Discussion Discussion Bibliography Gravitational Waves A time variable tensor metric perturbation similarly leaves an imprint in the temperature anisotropy Sachs Wolfe 1967 A tensor metric perturbation can be viewed as a standing gravitational wave and produces a quadrupolar distortion in the spatial metric If its amplitude changes it leaves a quadrupolar distortion in the CMB temperature distribution Polnarev 1985 Inflation predicts a nearly scale invariant spectrum of gravitational waves Their amplitude depends strongly on the energy scale of inflation power Rubakov et al 1982 Fabbri Pollock 1983 and its relationship to the curvature fluctuations discriminates between particular models for inflation Detection of gravitational waves in the CMB therefore provides our best hope to study the particle physics of inflation Figure Gravitational waves and the energy scale of inflation Left temperature and polarization spectra from an initial scale invariant gravitational wave spectrum with power Right 95 confidence upper limits statistically achievable on and the scalar tilt by the MAP and Planck satellites as well as an ideal experiment out to in the presence of gravitational lensing modes Gravitational waves like scalar fields obey the Klein Gordon equation in a flat universe and their amplitudes begin oscillating and decaying once the perturbation crosses the horizon While this process occurs even before recombination rapid Thomson scattering destroys any quadrupole anisotropy that develops see 3 6 This fact dicates the general structure of the contributions to the power spectrum see Figure 4 left panel they are enhanced at the present quadrupole and sharply suppressed at multipole larger than that of the first peak Abbott Wise 1984 Starobinskii 1985 Crittenden et al 1993 As is the case for the ISW effect confinement to the low multipoles means that the isolation of gravitational waves is severely limited by cosmic

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