2MRS Auto Power • ~40,000 galaxies over full sky, but a lot of power on all scales, indicating extremely strong clustering • Far from Gaussian. We need to include non- Gaussian error bars [connected trispectrum] • Many “nuisance” parameters [nuisance for cosmologists but not necessarily for others] 44
Nuisance parameters, or: How galaxies populate halos? Projecting 3-d galaxy power spectrum onto 2-d: • Halo model (Seljak 2000) 45
5 nuisance parameters (just for galaxies) • Halo model (Seljak 2000) 46
Nasty degeneracy among nuisance parameters… But who cares, we just marginalise 47
Ando, Benoit-Lévy & EK (2018) Dominated by 1-halo term in most of the angular scales => Good for cross-correlation with SZ clusters 48
Why should we believe this? • Nuisance parameters - too phenomenological? And horrible posterior… How do we know that these numbers make any sense? • Uniqueness of a low-z survey like 2MASS: we actually see these parameters in the sky, because we resolve all galaxy clusters/groups! 49
Satellite galaxy radial profiles Ando, Benoit-Lévy & EK (2018)
Number of satellites per halo Ando, Benoit-Lévy & EK (2018) 51
Planck Collaboration (2016) Bolliet, Comis, EK, Macias-Pérez (2018) SZ Auto Power B. Bolliet • Far from Gaussian. We need to include non- Gaussian error bars with non-Gaussian error [connected trispectrum] without • When fitting, the Planck team used Gaussian covariance ignoring the non-Gaussian term Foregrounds = Nuisance Parameters • We also have a bunch of nuisance parameters 52
Interpreting Planck’s SZ Power Spectrum • Planck Collaboration (2015) • Ignored trispectrum; Nuisance parameters marginalised over • Horowitz & Seljak (2017); Salvati et al. (2018) • Included trispectrum; Nuisance parameters not marginalised over • Hurier & Lacasa (2017) • Included trispectrum; Nuisance parameters not marginalised over but performed cleaning in C l space 53
Interpreting Planck’s SZ Power Spectrum We vary/marginalise over everything: • Planck Collaboration (2015) • Ignored trispectrum; Nuisance parameters • SZ model parameters marginalised over • All relevant cosmological parameters • Horowitz & Seljak (2017); Salvati et al. (2018) • Nuisance parameters • Included trispectrum; Nuisance parameters not marginalised over with the trispectrum that depends also • Hurier & Lacasa (2017) on the SZ+cosmological parameters • Included trispectrum; Nuisance parameters not marginalised over but performed cleaning in C l space 54
Bolliet, Comis, EK, Macias-Pérez (2018) SZ power is lower than Planck without with trispectrum 55
EK, Kitayama (1999); EK, Seljak (2002) Simple Interpretation C l [not “l 2 C l ”] multipole • Randomly-distributed point sources = Poisson spectrum = ∑ i (flux i ) 2 / 4 π 56
EK, Kitayama (1999); EK, Seljak (2002) Simple Interpretation C l [not “l 2 C l ”] multipole • Extended sources = the power spectrum reflects intensity profiles 57
l(l+1)C l /2 π [ μ K 2 ] >5x10 13 M sun >5x10 14 M sun Adding smaller clusters >10 15 M sun >2x10 15 M sun Multipole 58
<latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> <latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> <latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> <latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> Planck Mass Bias dz dV dM dn Z Z dM | y ` ( M, z ) | 2 C ` = dz • The key ingredient of the power spectrum is a profile of thermal pressure of electrons ˜ M 500 c = M 500 c, true /B 59
Bolliet, Comis, EK, Macias-Pérez (2018) Inferred SZ Amplitude 2.6% measurement! Essentially cosmological model-independent 60
<latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> <latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> <latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> <latexit sha1_base64="jG4q3CLuBbOdwxz2VYtq2xgJvfo=">ACD3icbZC7SgNBFIZn4y3G26qlzWAQLSTuiqIWQoiNjRDBmEASwuzkJBkye2HmrBCWfQbX8XGQsXW1s63cXIRNPrDwMd/zuHM+b1ICo2O82lZmbn5heyi7ml5ZXVNXt941aHseJQ4aEMVc1jGqQIoICJdQiBcz3JFS9/sWwXr0DpUY3OAgqbPuoHoCM7QWC17t4FCtoFetZJjx+EpPf/G/aShfIoqhjQ9KLXsvFNwRqJ/wZ1AnkxUbtkfjXbIYx8C5JpXedCJsJUyi4hDTXiDVEjPdZF+oGA+aDbiajg1K6Y5w27YTKvADpyP05kTBf64HvmU6fYU9P14bmf7V6jJ3TZiKCKEYI+HhRJ5YUQzpMh7aFAo5yYIBxJcxfKe8xTiaDHMmBHf65L9QOSycFdzro3yxNEkjS7bINtkjLjkhRXJyqRCOLknj+SZvFgP1pP1ar2NWzPWZGaT/JL1/gUgTpr4</latexit> Bolliet, Comis, EK, Macias-Pérez (2018) Inferred SZ Amplitude Mass Bias [B=(1–b) –1 ] ˜ M 500 c = M 500 c, true /B 61
Makiya, Ando & EK (2018) Joint Analysis 2MASS Auto 62
Makiya, Ando & EK (2018) Joint Analysis SZ Auto
Makiya, Ando & EK (2018) Joint Analysis Cross!
Full covariance including the trispectrum term 65
Marignalise over EVERYTHING! 66
Makiya, Ando & EK (2018) Mass-bias Consistency SZ Auto [for Planck TT+lowP+lensing] No obvious systematics for B from the SZ auto power spectrum alone SZ-2MASS Cross + 2MASS Auto • B = 1.54 ± 0.098 67
Makiya, Ando & EK (2018) Mass-bias Consistency SZ Auto [for Planck TT+lowP+lensing] • B = 1.54 ± 0.098 • 1–b =B –1 = 0.649 ± 0.041 Similar value was inferred from the SZ cluster number counts: additional consistency SZ-2MASS Cross + 2MASS Auto 68
Makiya, Ando & EK (2018) Mass Tomography Mass Dependence of SZ Auto 69
Makiya, Ando & EK (2018) Mass Tomography Mass Dependence of Cross Cross is sensitive to less massive halos: We can use this to explore the mass bias as a function of mass! 70
Makiya, Ando & EK (2018) Mass Tomography Pressure ~ (M/B) 2/3+ α p 71
Summary, Part I • If you have two data sets, do 3x2pt • Not just auto alone, or cross alone. Many people do just autos or cross • 3x2pt analysis requires good knowledge of both data sets. Challenging but rewarding! Can your machine learn to do this too? • Novelty in our 3x2pt of SZ and galaxies • Parameter-dependent trispectrum in the full covariance • Marginalised over everything; deeper understanding of 2MASS auto and SZ auto spectra • Joint analysis revealed mass-(in)dependent of mass bias; and consistency of mass bias inferred from auto and cross • Useful (and rewarding)! 72
Do I have 5 more minutes? • To be submitted on November 9 73
“Internal Template” Cleaning • CMB = [ Map(140GHz) – α x Map(other freq) ] / [1– α ] Jean-François said this method does not assume anything about foregrounds, but it does: It assumes that the spectral property of foreground does not depend on sky directions 74
Example: Synchrotron • Power-law index: δ T synch ~ ν β from WMAP 23 GHz and Haslam 408 MHz 75
Katayama, EK (2011) “Tensor-to-scalar Ratio” Bias of r~2x10 –3 if β is assumed constant Parameter, r 76
Katayama, EK (2011) “Tensor-to-scalar Ratio” Bias of r~2x10 –3 if β is assumed constant Parameter, r My first reaction: Ah, not bad at al! 77
JAXA ESA + possible participations from USA, Canada, Europe 2025– [proposed] LiteBIRD 2027– [proposed] Target: δ r<0.001 (68%CL)
JAXA ESA + possible participations from USA, Canada, Europe 2025– [proposed] LiteBIRD 2027– [proposed] Target: δ r<0.001 (68%CL) Gosh, we need to do better
Ichiki, Kanai, Katayama, EK (to be submitted) Delta-map Method observed foreground foreground polarisation spectrum emission at [ p I : I th parameter] frequency ν * • Foreground spectrum varies spatially due to spatially varying spectral parameters p (e.g., β ) • Inference on p (n)? Time consuming/non-linear… • Spectral variations are actually not so large. (Bias in the tensor-to- scalar ratio parameter is only of order r~O(10 –3 )) • Let’s Taylor-expand!
Ichiki, Kanai, Katayama, EK (to be submitted) Delta-map Method foreground mean spectrum fluctuation contribution • Key : the fluctuation part simply acts as an additional foreground component with spatially-uniform coefficients, D ν ,I • We can then form a linear combination of frequencies to remove both [Q f ,U f ] and δ p [Q f ,U f ]. Easy! • Long story short: this eliminates bias in r to negligible level. But… 81
What am I doing? • This sounds a bit ad hoc • Sure, maybe it is unbiased, but is it minimum- variance? • What is the Bayesian foundation for this method? • I can answer that 82
Bottom-up CMB Solutions • Simplest: Uniform spectral index. We need at least two frequencies to remove it. The CMB solution is This is nothing but the simplest template cleaning 83
Bottom-up CMB Solutions • Spatially-varying power law. We need at least three frequencies to remove it. The CMB solution is 84
Top-down Now, likelihood • m : Data (maps) • D : “mixing matrix” (frequency dependence of stuff in the sky) This depends on foreground parameters (we expand this to first order in perturbation) • s : Signal (CMB and foregrounds) • N : Noise covariance of data 85
Top-down Maximum Likelihood When we have a uniform spectral index and two frequencies, this gives the internal template solution: 86
Top-down Expanding D to 1st order 87
Top-down Expanding D to 1st order When we have a spatially-varying power law and three frequencies, this gives what we saw already: 88
Posterior • OK, these cleaned maps are ML solutions when the number of frequencies is equal to the number of components. Sweet. • What about the posterior? • Let’s derive it in a heuristic way 89
Bottom-up Posterior [Simplest] • Square and average to get covariance and plug it into a Gaussian… –2ln(posterior[r, β |m]) = 90
Top-down. We do it better now. Likelihood: Prior: [Flat on the mixing matrix] Marginalise over s CMB because all we care is its signal covariance S CMB 91
Top-down Now what? What you do from now on will decide which foreground-removal method you are using. (Almost?) all foreground removal methods in the literature can be categorised by the way you go 92 from here. (Flavien Vansyngel’s thesis, 2014)
Top-down Now what? We take the ML estimate of s f : 93
Top-down Now what? We take the ML estimate of s f : And plug it into the top: 94
Top-down Looks terrible. However… actually… when we have, e.g., a uniform spectral index and two frequencies… We take the ML estimate of s f : And plug it into the top: 95
Top-down Looks terrible. However… actually… when we have, e.g., a uniform spectral index and two frequencies… where 96
Ichiki, Kanai, Katayama, EK (to be submitted) That’s it! Summary: • We can account for spatially-varying foreground spectra and estimate the tensor-to-scalar ratio by • I talked only about a power-law synchrotron, but you can use this for any number of other components. All you need is the derivative of emission law as a function of spectral parameters. • For example… 97
• And, we figured out how to account for polarised anomalous microwave emission (AME; spinning dust) and frequency decorrelation due to a superposition of varying spectra. If you are interested, wait for November!
Back-up Slides
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