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  • CMB Review
    of parameters 26 That is the noise the difference between the data and the modulated signal is assumed to be Gaussian with covariance There are two important theorems useful in the construction of a map and more generally in each step of the data pipeline Tegmark et al 1997 The first is Bayes Theorem In this context it says that the probability that the temperatures are equal to given the data is proportional to the likelihood function times a prior Thus with a uniform prior 27 with the normalization constant determined by requiring the integral of the probability over all to be equal to one The probability on the left is the one of interest The most likely values of therefore are those which maximize the likelihood function Since the log of the likelihood function in question Equation 26 is quadratic in the parameters it is straightforward to find this maximum point Differentiating the argument of the exponential with respect to and setting to zero leads immediately to the estimator 28 where As the notation suggests the mean of the estimator is equal to the actual and the variance is equal to The second theorem states that this maximum likelihood estimator is also the minimum variance estimator The Cramer Rao inequality says no estimator can measure the with errors smaller than the diagonal elements of where the Fisher matrix is defined as 29 Inspection of Equation 26 shows that in this case the Fisher matrix is precisely equal to Therefore the Cramer Rao theorem implies that the estimator of Equation 28 is optimal it has the smallest possible variance Tegmark 1997a No information is lost if the map is used in subsequent analysis instead of the timestream data but huge factors of compression have been gained For example in the recent Boomerang experiment Netterfield et al 2001 the timestream contained numbers while the map had only pixels The map resulted in compression by a factor of There are numerous complications that must be dealt with in realistic applications of Equation 28 Perhaps the most difficult is estimation of the timestream noise covariance This typically must be done from the data itself Ferreira Jaffe 2000 Stompor et al 2001 Even if were known perfectly evaluation of the map involves inverting a process which scales as the number of raw data points cubed For both of these problems the assumed stationarity of it depends only on is of considerable utility Iterative techniques to approximate matrix inversion can also assist in this process Wright et al 1996 Another issue which has received much attention is the choice of pixelization The community has converged on the Healpix pixelization scheme now freely available Perhaps the most dangerous complication arises from astrophysical foregrounds both within and from outside the Galaxy the main ones being synchrotron bremmsstrahlung dust and point source emission All of the main foregrounds have different spectral shapes than the blackbody shape of the CMB Modern experiments typically observe at several different frequencies so

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  • CMB Review
    Equation 2 for the ensemble average leads to 30 where depends on the theoretical parameters through see Equation 3 Here the window function is proportional to the Legendre polynomial and a beam and pixel smearing factor For example a Gaussian beam of width dictates that the observed map is actually a smoothed picture of true signal insensitive to structure on scales smaller than If the pixel scale is much smaller than the beam scale Techniques for handling asymmetric beams have also recently been developed Wu et al 2001 Wandelt Gorski 2001 Souradeep Ratra 2001 Using bandpowers corresponds to assuming that is constant over a finite range or band of equal to for Plate 1 gives a sense of the width and number of bands probed by existing experiments For Gaussian theories then the likelihood function is 31 where and is the number of pixels in the map As before is Gaussian in the anisotropies but in this case are not the parameters to be determined the theoretical parameters are the upon which the covariance matrix depends Therefore the likelihood function is not Gaussian in the parameters and there is no simple analytic way to find the point in parameter space which is multi dimensional depending on the number of bands being fit at which is a maximum An alternative is to evaluate numerically at many points in a grid in parameter space The maximum of the on this grid then determines the best fit values of the parameters Confidence levels on say can be determined by finding the region within which say for limits This possibility is no longer viable due to the sheer volume of data Consider the Boomerang experiment with A single evaluation of involves computation of the inverse and determinant of the matrix both of which scale as While this single evaluation might be possible with a powerful computer a single evaluation does not suffice The parameter space consists of bandpowers equally spaced from up to A blindly placed grid on this space would require at least ten evaluations in each dimension so the time required to adequately evaluate the bandpowers would scale as No computer can do this The situation is rapidly getting worse better since Planck will have of order pixels and be sensitive to of order a bands It is clear that a smart sampling of the likelihood in parameter space is necessary The numerical problem searching for the local maximum of a function is well posed and a number of search algorithms might be used tends to be sufficiently structureless that these techniques suffice Bond et al 1998 proposed the Newton Raphson method which has become widely used One expands the derivative of the log of the likelihood function which vanishes at the true maximum of around a trial point in parameter space Keeping terms second order in leads to 32 where the curvature matrix is the second derivative of with respect to and Note the subtle distinction between the curvature matrix and

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  • CMB Review
    Non Gaussianity Data Analysis Introduction Mapmaking Bandpower Estimation Parameter Estimation Discussion Discussion Bibliography Cosmological Parameter Estimation The huge advantage of bandpowers is that they represent the natural meeting ground of theory and experiment The above two sections outline some of the steps involved in extracting them from the observations Once they are extracted any theory can be compared with the observations without knowledge of experimental details The simplest way to estimate the cosmological parameters in a set is to approximate the likelihood as 35 and evaluate it at many points in parameter space the bandpowers depend on the cosmological parameters Since the number of cosmological parameters in the working model is this represents a final radical compression of information in the original timestream which recall has up to data points In the approximation that the band power covariance is independent of the parameters maximizing the likelihood is the same as minimizing This has been done by dozens of groups over the last few years especially since the release of CMBFAST Seljak Zaldarriaga 1996 which allows fast computation of theoretical spectra Even after all the compression summarized in Figure 5 these analyses are still computationally cumbersome due to the large numbers of parameters varied Various methods of speeding up spectra computation have been proposed Tegmark Zaldarriaga 2000 based on the understanding of the physics of peaks outlined in 3 and Monte Carlo explorations of the likelihood function Christensen et al 2001 Again the inverse Fisher matrix gives a quick and dirty estimate of the errors Here the analogue of Equation 29 for the cosmological parameters becomes 36 In fact this estimate has been widely used to forecast the optimal errors on cosmological parameters given a proposed experiment and a band covariance matrix which includes diagonal sample and instrumental noise variance The

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  • CMB Review
    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 DISCUSSION Measurements of the acoustic peaks in the CMB temperature spectrum have already shown that the Universe is nearly spatially flat and began with a nearly scale invariant spectrum of curvature fluctuations consistent with the simplest of inflationary models In a remarkable confirmation of a longstanding prediction of Big Bang Nucleosynthesis the CMB measurements have now verified that baryons account for about four percent of the critical density Further they suggest that the matter density is some ten times higher than this implying the existence of non baryonic dark matter and dark energy Future measurements of the morphology of the peaks in the temperature and polarization should determine the baryonic and dark matter content of the Universe with exquisite precision Beyond the peaks gravitational wave imprint on the polarization the gravitational lensing of the CMB and gravitational and scattering secondary anisotropies hold the promise of understanding the physics of inflation and the impact of

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  • CMB Review
    403 L1 L3 Ferreira Jaffe 2000 Ferreira PG Jaffe AH 2000 MNRAS 312 89 102 Ferreira et al 1998 Ferreira PG Magueijo J Gorski KM 1998 Ap J Lett 503 L1 L4 Fixsen et al 1996 Fixsen DJ Cheng ES Gales JM Mather JC Shafer RA et al 1996 Ap J 473 576 587 Freedman et al 2001 Freedman WL Madore BF Gibson BK Ferrarese L Kelson DD et al 2001 Ap J 553 47 72 Goldberg Spergel 1999 Goldberg DM Spergel DN 1999 Phys Rev D59 103002 Gruzinov Hu 1998 Gruzinov A Hu W 1998 Ap J 508 435 439 Gunn Peterson 1965 Gunn JE Peterson BA 1965 Ap J 142 1633 1636 Guth Pi 1985 Guth AH Pi SY 1985 Phys Rev D 32 1899 1920 Halverson et al 2001 Halverson NW et al 2001 Ap J In Press astro ph 0104489 Hanany et al 2000 Hanany S Ade P Balbi A Bock J Borrill J et al 2000 Ap J Lett 545 L5 L9 Hawking 1982 Hawking SW 1982 Phys Lett B115 295 297 Heavens 1998 Heavens AF 1998 MNRAS 299 805 808 Hedman et al 2001 Hedman MM Barkats D Gundersen JO Staggs ST Winstein B 2001 Ap J Lett 548 L111 L114 Hogan et al 1982 Hogan CJ Kaiser N Rees MJ 1982 Royal Soc London Phil Trans Series 307 97 109 Holder Carlstrom 2001 Holder G Carlstrom J 2001 Ap J In Press astro ph 0105229 Hu 1998 Hu W 1998 Ap J 506 485 494 Hu 2000a Hu W 2000a Ap J 529 12 25 Hu 2000b Hu W 2000b Phys Rev D 62 043007 Hu 2001a Hu W 2001a Phys Rev D 64 083005 Hu 2001b Hu W 2001b Phys Rev D In press astro ph 0108090 Hu 2001c Hu W 2001c Ap J Lett 557 L79 L83 Hu et al 2001 Hu W Fukugita M Zaldarriaga M Tegmark M 2001 Ap J 549 669 680 Hu et al 1998 Hu W Seljak U White MJ Zaldarriaga M 1998 Phys Rev D57 3290 3301 Hu Sugiyama 1995 Hu W Sugiyama N 1995 Ap J 444 489 506 Hu Sugiyama 1996 Hu W Sugiyama N 1996 Ap J 471 542 570 Hu et al 1997 Hu W Sugiyama N Silk J 1997 Nature 386 37 Hu White 1996 Hu W White M 1996 Ap J 471 30 51 Hu White 1997a Hu W White M 1997a New Astronomy 2 323 344 Hu White 1997b Hu W White M 1997b Phys Rev D 56 596 615 Hu White 1997c Hu W White M 1997c Ap J 479 568 579 Hu White 2001 Hu W White M 2001 Ap J 554 67 73 Hui et al 2001 Hui L Burles S Seljak U Rutledge RE Magnier E et al 2001 Ap J 552 15 35 Huterer et al 2001 Huterer D Knox L Nichol RC 2001 Ap J 555 547 557 Jungman et al 1996 Jungman G Kamionkowski M Kosowsky A Spergel DN 1996 Phys Rev D 54 1332 1344 Kaiser 1983 Kaiser N 1983 MNRAS 202 1169 1180 Kaiser 1984 Kaiser N 1984 Ap J 282 374 381 Kamionkowski et al 1997 Kamionkowski M Kosowsky A Stebbins A 1997 Phys Rev D55 7368 7388 Kamionkowski et al 1994 Kamionkowski M Spergel DN Sugiyama N 1994 Ap J Lett 426 L57 L60 Keating et al 2001 Keating B O Dell C de Oliveira Costa A Klawikowski S Stebor N et al 2001 Ap J Lett In Press astro ph 0107013 Knox 1995 Knox L 1995 Phys Rev D 52 4307 4318 Knox 1999 Knox L 1999 MNRAS 307 977 983 Knox et al 2001 Knox L Christensen N Skordis C 2001 Ap J Lett In Press astro ph 0109232 Knox et al 1998 Knox L Scoccimarro R Dodelson S 1998 Phys Rev Lett 81 2004 2007 Kofman Starobinskii 1985 Kofman LA Starobinskii AA 1985 Soviet Astronomy Letters 11 271 Kogut et al 1996 Kogut A Banday AJ Bennett CL Gorski KM Hinshaw G et al 1996 Ap J Lett 464 L29 L33 Komatsu Kitayama 1999 Komatsu E Kitayama T 1999 Ap J Lett 526 L1 L4 Komatsu Seljak 2001 Komatsu E Seljak U 2001 MNRAS In Press astro ph 0106151 Krauss Turner 1995 Krauss LM Turner MS 1995 Gen Rel Grav 27 1137 1144 Lee et al 2001 Lee A et al 2001 Ap J In Press astro ph 0104459 Lewis et al 2001 Lewis A Challinor A Turok N 2001 Phys Rev D In Press astro ph 0106536 Lewis et al 2000 Lewis GF Babul A Katz N Quinn T Hernquist L et al 2000 Ap J 536 623 644 Liddle Lyth 1993 Liddle AR Lyth DH 1993 Phys Rept 231 1 105 Limber 1954 Limber DN 1954 Ap J 119 655 681 Loeb Barkana 2001 Loeb A Barkana R 2001 Annu Rev Astron Astrophys 39 19 66 Luo 1994 Luo X 1994 Ap J Lett 427 L71 L74 Ma Bertschinger 1995 Ma C Bertschinger E 1995 Ap J 455 7 25 Miller et al 1999 Miller AD Caldwell R Devlin MJ Dorwart WB Herbig T et al 1999 Ap J Lett 524 L1 L4 Narkilar Padmanabhan 2001 Narkilar J Padmanabhan T 2001 Annu Rev Astron Astrophys 39 211 248 Netterfield et al 2001 Netterfield C Ade P Bock J Bond J Borrill J et al 2001 Ap J In Press astro ph 0104460 Oh et al 1999 Oh SP Spergel DN Hinshaw G 1999 Ap J 510 551 563 Ostriker Steinhardt 1995 Ostriker JP Steinhardt PJ 1995 Nature 377 600 602 Ostriker Vishniac 1986 Ostriker JP Vishniac ET 1986 Ap J Lett 306 L51 L54 Padin et al 2001 Padin S Cartwright JK Mason BS Pearson TJ Readhead ACS et al 2001 Ap J Lett 549 L1 L5 Peacock 1991 Peacock JA 1991 MNRAS 253 1P 5P Pearce et al 2001 Pearce FR Jenkins A Frenk CS White SDM Thomas PA et al 2001 MNRAS 326 649 666 Peebles 1968 Peebles PJE 1968

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  • Animations
    nbsp Power Animations nbsp Lensing 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 Power

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  • Animations
    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 Power Spectra Baryons Matter Curvature DarkEnergy Reionization Tensors Damping Tail Baryon

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  • Animations
    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 Power Spectra Baryons Matter Curvature DarkEnergy Reionization Tensors Damping Tail

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