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- Polarization Primer

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 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 Polarization Patterns The considerations of 2 imply that scalars vectors and tensors generate distinct patterns in the polarization of the CMB However although they separate cleanly into polarization patterns for a single plane wave perturbation in the coordinate system referenced to in general there will exist a spectrum of fluctuations each with a different Therefore the polarization pattern on the sky does not separate into modes In fact assuming statistical isotropy one expects the ensemble averaged power for each multipole to be independent of m Nonetheless certain properties of the polarization patterns discussed in

Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node7.html (2015-06-26)

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E and magnetic B components This decomposition is useful both observationally and theoretically as we will discuss below There are two equivalent ways of viewing the modes that reflect their global and local properties respectively The nomenclature reflects the global property Like multipole radiation the harmonics of an E mode have parity on the sphere whereas those of a B mode have parity Under the E mode thus remains unchanged for even whereas the B mode changes sign as illustrated for the simplest case in Fig 9 recall that a rotation by 90 represents a change in sign Note that the E and B multipole patterns are rotations of each other i e and Since this parity property is obviously rotationally invariant it will survive integration over Fig 9 The electric E and magnetic B modes are distinguished by their behavior under a parity transformation n n E modes have 1 l parity and B modes have 1 l 1 here l 2 m 0 even and odd respectively The local distinction between the two is that the polarization direction is aligned with the principal axes of the polarization amplitude for E and crossed 45 degrees for B Dotted lines represent a sign reversal in the polarization The local view of E and B modes involves the second derivatives of the polarization amplitude second derivatives because polarization is a tensor or spin 2 object In much the same way that the distinction between electric and magnetic fields in electromagnetism involves vanishing of gradients or curls i e first derivatives for the polarization there are conditions on the second covariant derivatives of Q and U For an E mode the difference in second covariant derivatives of U along and vanishes as does that for Q along and For a B mode

Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node8.html (2015-06-26)

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modulated over the last scattering surface by the plane wave spatial dependence of the perturbation compare Figs 3 and 10 The modulation changes the amplitude sign and angular structure of the polarization but not its nature e g a Q polarization remains Q Nonetheless this modulation generates a B mode from the local E mode pattern Fig 10 Modulation of the local scalar pattern in Fig 3 by plane wave fluctuations on the last scattering surface Yellow points represent polarization out of the plane of the page with magnitude proportional to sign The plane wave odulation changes the amplitude and the sign of the polarization but does not mix Q and U Modulation can mix E and B however if U is also present The reason why this occurs is best seen from the local distinction between E and B modes Recall that E modes have polarization amplitudes that change parallel or perpendicular to and B modes in directions away from the polarization direction On the other hand plane wave modulation always changes the polarization amplitude in the direction or N S on the sphere Whether the resultant pattern possesses E or B contributions depends on whether the local polarization has Q or U contributions For scalars the modulation is of a pure Q field and thus its E mode nature is preserved Kamionkowski et al 1997 Zaldarriaga Seljak 1997 For the vectors the U mode dominates the pattern and the modulation is crossed with the polarization direction Thus vectors generate mainly B modes for short wavelength fluctuations Hu White 1997 For the tensors the comparable Q and U components of the local pattern imply a more comparable distribution of E and B modes at short wavelengths see Fig 11 a These qualitative considerations can be quantified by noting that plane wave modulation simply represent the addition of angular momentum from the plane wave with the local spin angular dependence The result is that plane wave modulation takes the local angular dependence to higher smaller angles and splits the signal into E and B components with ratios which are related to Clebsch Gordan coefficients At short wavelengths these ratios are B E 0 6 8 13 in power for scalars vectors and tensors see Fig 11 b and Hu White 1997 The distribution of power in multipole space is also important Due to projection a single plane wave contributes to a range of angular scales where r is the comoving distance to the last scattering surface From Fig 10 we see that the smallest angular largest variations occur on lines of sight or though a small amount of power projects to as The distribution of power in multipole space of Fig 11 b can be read directly off the local polarization pattern In particular the region near shown in Fig 11 a determines the behavior of the main contribution to the polarization power spectrum Fig 11 The E and B components of a planewave perturbation for scalars vectors and tensors

Original URL path: http://background.uchicago.edu/~whu/polar/webversion/node9.html (2015-06-26)

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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 Temperature Polarization Correlation As we have seen in 2 the polarization pattern reflects the local quadrupole anisotropy at last scattering Hence the temperature and polarization anisotropy patterns are correlated in a way that can distinguish between the scalar vector and tensor sources There are two subtleties involved in establishing the correlation First the quadrupole moment of the temperature anisotropy at last scattering is not generally the dominant source of anisotropies on the sky so the correlation is neither 100 nor necessarily directly visible as patterns in the map The second subtlety is that the correlation occurs through the E mode unless the polarization has been Faraday or otherwise rotated between the last scattering surface and the present As we have seen an E mode is modulated in the direction of or perpendicular to its polarization axis To be correlated with the temperature this modulation must also correspond to the modulation of the temperature perturbation The two options are that E is parallel or perpendicular to crests in the temperature perturbation As modes of different direction are superimposed this translates into a radial or tangential polarization pattern around hot spots see Fig 12 a On the other hand B modes do not correlate with the temperature In other words the rotation of the pattern in Fig 12 a by 45 into those of Fig 12 b solid and dashed lines cannot be generated by Thomson scattering The temperature field that generates the polarization has no way to distinguish between points reflected across the symmetric hot spot and so has no way to choose between

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Consider first the large angle scalar perturbations Here the dominant source of correlated anisotropies is the temperature perturbation on the last scattering surface itself The Doppler contributions can be up to half of the total contribution but as we have seen in 2 3 do not correlate with the quadrupole moment Contributions after last scattering while potentially strong in isocurvature models for example also rapidly lose their correlation with the quadrupole at last scattering As we have seen the temperature gradient associated with the scalar fluctuation makes the photon fluid flow from hot regions to cold initially Around a point on a crest therefore the intensity peaks in the directions along the crest and falls off to the neighboring troughs This corresponds to a polarization perpendicular to the crest see Fig 12 Around a point on a trough the polarization is parallel to the trough As we superpose waves with different we find the pattern is tangential around hot spots and radial around cold spots Crittenden et al 1995 It is important to stress that the hot and cold spots refer only to the temperature component which is correlated with the polarization The correlation increases at scales approaching the horizon at last scattering since the quadrupole anisotropy that generates polarization is caused by flows Fig 12 Temperature polarization cross correlation E parity polarization perpendicular parallel to crests generates a tangential radial polarization field around hot spots B polarization does not correlate with temperature since the 45 degree rotated contributions from oppositely directed modes cancel For the vectors no temperature perturbations exist on the last scattering surface and again Doppler contributions do not correlate with the quadrupole Thus the main correlations with the temperature will come from the quadrupole moment itself The correlated signal is reduced since the strong B contributions

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Angle Correlation Large Angle Correlation Model Reconstruction Last Scattering Reionization Scalar Vector Tensor Adiabatic Isocurvature Inflation Defects Phenomenology Observations Foregrounds Data Analysis Future References Small Angle Correlation Pattern Until now we have implicitly assumed that the evolution of the perturbations plays a small role as is generally true for scales larger than the horizon at last scattering Evolution plays an important role for small scale scalar perturbations where there is enough time for sound to cross the perturbation before last scattering The infall of the photon fluid into troughs compresses the fluid increasing its density and temperature For adiabatic fluctuations this compression reverses the sign of the effective temperature perturbation when the sound horizon s grows to be see Fig 13 a This reverses the sign of the correlation with the quadrupole moment Infall continues until the compression is so great that photon pressure reverses the flow when Again the correlation reverses sign This pattern of correlations and anticorrelations continues at twice the frequency of the acoustic oscillations themselves see Fig 13 a Of course the polarization is only generated at last scattering so the correlations and anticorrelations are a function of scale with sign changes at multiples of where is the sound horizon at last scattering As discussed in 3 2 these fluctuations project onto anisotropies as Fig 13 Time evolution of acoustic oscillations The polarization is related to the flows v red which form quadrupole anisotropies such that its product green with the effective temperature blue reflects the temperature polarization cross correlation As described in the text the adiabatic a and isocurvature b modes differ in the phase of the oscillation in all three quantities Temperature and polarization are anticorrelated in both cases at early times or large scales ks much less than unity Any scalar fluctuation will

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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 Model Reconstruction While it is clear how to compare theoretical predictions of a given model with observations the reconstruction of a phenomenological model from the data is a more subtle issue The basic problem is that in the CMB we see the whole history of the evolution in redshift projected onto the two dimensional sky The reconstruction of the evolutionary history of the universe might thus seem an ill posed problem Fortunately one needs only to assume the very basic properties of the cosmological model and the gravitational instability picture before useful information may be extracted The simplest example is the combination of the amplitude of the temperature fluctuations which reflect the conditions at horizon crossing and large scale structure today Another example is the acoustic peaks in the temperature which form a snapshot of conditions at last scattering on scales below the horizon at that time In most models the acoustic signature provides a wealth of information on cosmological parameters

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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 Last Scattering The main reason why polarization is so useful to the reconstruction process is that cosmologically it can only be generated by Thomson scattering The polarization spectrum of the CMB is thus a direct snapshot of conditions on the last scattering surface Contrast this with temperature fluctuations which can be generated by changes in the metric fluctuations between last scattering and the present such as those created by gravitational potential evolution This is the reason why the mere detection of large angle anisotropies by COBE did not rule out wide classes of models such as cosmological defects To use the temperature anisotropies for the reconstruction problem one must isolate features in the spectrum which can be associated with last scattering or more generally the universe at a known redshift Furthermore the polarization spectrum has potential advantages even for extracting information from features that are also present in the temperature spectrum e g the acoustic peaks The polarization spectrum is generated by local quadrupole anisotropies alone whereas the temperature spectrum has comparable contributions from the local monopole and dipole as well as possible contributions between last scattering and the present This property enhances the prominence of features see Fig 14 as does the fact that the scalar polarization has a relatively sharp projection due to the geometry see 3 2 Unfortunately these considerations are mitigated by the fact that the polarization amplitude is so much weaker than the temperature With present day detectors one needs to measure it in broad bands to increase the signal to noise see Fig 14 and Table 2 Fig 14 Temperature polarization

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