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  • Polarization Primer
    Polarization Scalar Perturbations Vector Perturbations Tensor Perturbations Polarization Patterns E and B Modes E and B Spectra Temperature Correlation Small Angle Correlation Large Angle Correlation Model Reconstruction Last Scattering Reionization Scalar Vector Tensor Adiabatic Isocurvature Inflation Defects Phenomenology Observations Foregrounds Data Analysis Future References Reionization Since polarization directly probes the last scattering epoch the first thing we learn is when that occurred i e what fraction of photons last scattered at when the universe recombined and what fraction rescattered when the intergalactic medium reionized at Since rescattering erases fluctuations below the horizon scale and regenerates them only weakly Efstathiou 1988 we already know from the reported excess over COBE of sub degree scale anisotropy that the optical depth during the reionized epoch was and hence It is thus likely that our universe has the interesting property that both the recombination and reionization epoch are observable in the temperature and polarization spectrum Unfortunately for the temperature spectrum at these low optical depths the main effect of reionization is an erasure of the primary anisotropies from recombination as This occurs below the horizon at last scattering since only on these scales has there been sufficient time to convert the originally isotropic temperature fluctuations into anisotropies The uniform reduction of power at small scales has the same effect as a change in the overall normalization For 20 the difference in the power spectrum is confined to large angles Here the observations are limited by cosmic variance the fact that we only have one sample of the sky and hence only samples of any given multipole Cosmic variance is the dominant source of uncertainty on the low temperature spectrum in Fig 14 The same is not true for the polarization As we have seen the polarization spectrum is very sensitive to the epoch of

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node15.html (2015-06-26)
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  • Polarization Primer
    Temperature Correlation Small Angle Correlation Large Angle Correlation Model Reconstruction Last Scattering Reionization Scalar Vector Tensor Adiabatic Isocurvature Inflation Defects Phenomenology Observations Foregrounds Data Analysis Future References Scalars Vectors Tensors There are three types of fluctuations scalars vectors and tensors and four observables the temperature E mode B mode and temperature cross polarization power spectra The CMB thus provides sufficient information to separate these contributions which in turn can tell us about the generation mechanism for fluctuations in the early universe see 4 5 Ignoring for the moment the question of foregrounds to which we turn in 5 2 if the E mode polarization greatly exceeds the B mode then scalar fluctuations dominate the anisotropy Conversely if the B mode is greater than the E mode then vectors dominate If tensors dominate then the E and B are comparable see Fig 15 These statements are independent of the dynamics and underlying spectrum of the perturbations themselves The causal constraint on the generation of a quadrupole moment and hence the polarization introduces further distinctions It tells us that the polarization peaks around the scale the horizon subtends at last scattering This is about a degree in a flat universe and scales with the angular diameter distance to last scattering Geometric projection tells us that the low tails of the polarization can fall no faster than and for scalars vectors and tensors see 3 2 The cross spectrum falls no more rapidly than for each We shall see below that these are the predicted slopes of an isocurvature model Furthermore causality sets the scale that separates the large and small angle temperature polarization correlation pattern Well above this scale scalar and vector fluctuations should show anticorrelation tangential around hot spots whereas tensor perturbations should show correlations radial around hot spots Of course

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node16.html (2015-06-26)
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  • Polarization Primer
    Angle Correlation Model Reconstruction Last Scattering Reionization Scalar Vector Tensor Adiabatic Isocurvature Inflation Defects Phenomenology Observations Foregrounds Data Analysis Future References Adiabatic vs Isocurvature Perturbations The scalar component is interesting to isolate since it alone is responsible for large scale structure formation There remain however two possibilities Density fluctuations could be present initially This represents the adiabatic mode Alternately they can be generated from stresses in the matter which causally push matter around This represents the isocurvature mode The presence or absence of density perturbations above the horizon at last scattering is crucial for the features in both the temperature and polarization power spectrum As we have seen in 3 3 1 it has as strong effect on the phase of the acoustic oscillation In a typical isocurvature model the phase is delayed by moving structure in the temperature spectrum to smaller angles see Fig 13 Consistency checks exist in the E polarization and cross spectrum which should be out of phase with the temperature spectrum and oscillating at twice the frequency of the temperature respectively However if the stresses are set up sufficiently carefully this acoustic phase test can be evaded by an isocurvature model Turok 1996 Perhaps more importantly acoustic features can be washed out in isocurvature models with complicated small scale dynamics to force the acoustic oscillation as in many defect models Albrecht et al 1996 Even in these cases the polarization carries a robust signature of isocurvature fluctuations The polarization isolates the last scattering surface and eliminates any source of confusion from the epoch between last scattering and the present In particular the delayed generation of density perturbations in these models implies a steep decline in the polarization above the angle subtended by the horizon at last scattering The polarization power thus hits the asymptotic limits

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node17.html (2015-06-26)
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  • Polarization Primer
    of the fluctuations but also the means by which they are generated We assume of course that they are not merely placed by fiat in the initial conditions Let us first divide the possibilities into broad classes In fact the distinction between isocurvature and adiabatic fluctuations is operationally the same as the distinction between conventional causal sources e g defects and those generated by a period of superluminal expansion in the early universe i e inflation It can be shown that inflation is the only causal mechanism for generating superhorizon size density curvature fluctuations Liddle 1995 Since the slope of the power spectrum in E can be traced directly to the presence of superhorizon size temperature and hence curvature fluctuations at last scattering it represents a test of inflation Hu White 1997 Spergel Zaldarriaga 1997 The acoustic phase test either in the temperature or polarization represents a marginally less robust test that should be easily observable if the former fails to be These tests while interesting do not tell us anything about the detailed physics that generates the fluctuations Once a distinction is made between the two possibilities one would like to learn about the mechanism for generating the fluctuations in more detail For example in the inflationary case there is a well known test of single field slow roll inflation which can be improved by using polarization information In principle inflation generates both scalar and tensor anisotropies If we assume that the two spectra come from a single underlying inflationary potential their amplitudes and slopes are not independent This leads to an algebraic consistency relation between the ratio of the tensor and scalar perturbation spectra and the tensor spectral index However information on the tensor contribution to the spectrum is limited by cosmic variance and is easily confused with

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node18.html (2015-06-26)
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  • Polarization Primer
    nbsp Power Prehistory nbsp Legacy Material 96 nbsp PhD Thesis 95 Baryon Acoustic Oscillations Cosmic Shear 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 Thomson Scattering Anistropy

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node19.html (2015-06-26)
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  • Polarization Primer
    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 Thomson Scattering Anistropy to Polarization Scalar Perturbations Vector Perturbations Tensor Perturbations Polarization Patterns E and B Modes E and B Spectra Temperature Correlation Small Angle Correlation Large Angle Correlation Model Reconstruction Last Scattering Reionization Scalar Vector Tensor Adiabatic Isocurvature Inflation Defects Phenomenology Observations Foregrounds Data Analysis Future References Observations While the theoretical case for observing polarization is strong it is a difficult experimental task to observe signals of the low level of several K and below Nonetheless polarization experiments have one potential advantage over temperature anisotropy experiments They can reduce atmospheric emission effects by differencing the polarization states on the same patch of sky instead of physically chopping between different angles on the sky since atmospheric emission is thought to be nearly unpolarized see 5 2 However to be successful an experiment must overcome a number of systematic effects many of which are discussed in Keating et al 1997 It must at least balance the sensitivity of the instrument to the orthogonal polarization channels including the far side lobes to nearly 8 orders of magnitude

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node20.html (2015-06-26)
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  • Polarization Primer
    effect is not expected to polarize the emission by more than 10 Keating et al 1997 The emission is larger at low frequencies but is not expected to dominate the polarization at any frequency The polarization of dust is not well known In principle emission from dust particles could be highly polarized however Hildebrand Dragovan 1995 find that in their observations the majority of dust is polarized at the level at m with a small fraction of regions approaching polarization Moreover Keating et al 1997 show that even at 100 polarization extrapolation of the IRAS 100 m map with the COBE FIRAS index shows that dust emission is negligible below 80GHz At higher frequencies it will become the dominant foreground Radio point sources are polarized due to synchrotron emission at level For large angle experiments the random contribution from point sources will contribute negligibly but may be of more concern for the upcoming satellite missions Galactic synchrotron emission is the major concern It is potentially highly polarized with the fraction dependent on the spectral index and depolarization from Faraday rotation and non uniform magnetic fields The level of polarization is expected to lie between 10 75 of a total intensity which itself is approximately K at 30GHz This estimate follows from extrapolating the Brouw Spoelstra 1976 measurements at 1411 MHz with an index of Due to their different spectral indices the minimum in the foreground polarization like the temperature lies near 100GHz For full sky measurements since synchrotron emission is more highly polarized than dust the optimum frequency at which to measure intrinsic CMB polarization is slightly higher than for the anisotropy Over small regions of the sky where one or the other of the foregrounds is known a priori to be absent the optimum frequency would clearly be different

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node21.html (2015-06-26)
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  • Polarization Primer
    simplest and most powerful way to obtain the power spectrum is to perform a likelihood analysis of the data The likelihood function encodes all of the information in the measurement and can be modified to correctly account for non uniform noise sky coverage foreground subtraction and correlations between measurements Operationally one computes the probability of obtaining the measured points and assuming a given theory including a model for foregrounds and detector noise and maximizes the likelihood over the theories For our purposes the theories could be given simply by the polarization bandpowers in E and B for example or could be a more realistic model such as CDM with a given reionization history The confidence levels on the parameters are obtained as moments of the likelihood function in the usual way Such an approach also allows one to generalize the analysis to include temperature information for the cross correlation if it becomes available Assuming that the fluctuations are gaussian the likelihood function is given in terms of the data and the correlation function of Q and U for any pair of the n data points The calculation of this correlation function is straightforward and Kamionkowski et al 1997 discuss the problem extensively Let us assume that we are fitting only one component or have only one frequency channel The generalization to multiple frequencies with a model for the foreground is also straightforward We shall also assume for notational simplicity that we are fitting only to polarization data though again the generalization to include temperature data is straightforward The construction is as follows We define a data vector which contains the Q and U information referenced to a particular coordinate system in principle this coordinate system could change between different subsets of the data Call this data vector which has N

    Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node22.html (2015-06-26)
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