1. Non-parametric Determination of Luminosity Evolutions, Correlations, and Distributions in AGN from Flux Limited Survey Data Jack Singal University of Richmond Stat. Chall. In 21 st Century Cosmology May 25, 2016 J. Singal, U. Richmond V. Petrosian, Stanford

2. Orientation What we have: Flux-limited survey catalogs What we want: The true population distributions and correlations for some class of objects (directly from the data) Specifically: - Luminosity-redshift correlations (“Luminosity Evolution”) - Luminosity distributions (“Luminosity Functions”) - Redshift distribution (“Density Evolution”) - (actual) Correlations between luminosities in different wavebands (e.g. optical and infrared)

3. Example: Optical Quasar Data For quasars, over 100,000 have been identified by SDSS (Schneider et al., 2010, AJ, 139, 2360) Quasars are distinguished from ordinary galaxies by their “colors” (the ratio of brightness at different wavelengths – red, blue, green, etc…) – Need an optical light observation if identifying quasars this way and/or using spectroscopic redshifts

4. Quasar Data: SDSS x FIRST Radio Optical Luminosity Radio Luminosity 5445 quasars seen in optical and radio FIRST has a flux limit of 1 mJy We are ‘missing’ most of the data because of the flux limits! Missing these and these

5. Quasar Data: SDSS x WISE infrared Missing these Individually different flux limits Optical Luminosity Infrared Luminosity Missing these “Truncated” data!

6. Another situation: Fermi-LAT Blazars Gamma-ray observations of blazars with the Fermi Gamma-ray Space Telescope Flux Missing these How flux varies with frequency

7. Challenges: Data truncated in possibly complicated ways Data truncated in multiple dimensions Logarithmic distributions in luminosities – nowhere near normal e.g. Optical Luminosity Missing these e.g. Radio Luminosity and these (If we’d like to get the true population distributions directly from this type of data) We’d like to get the population characteristics in a direct way

8. Methods We have been using a custom variant of a statistical rank test with “associated sets” to access the true intrinsic distributions of populations from flux-limited and otherwise truncated surveys Techniques explored in : - Singal et al., 2011, ApJ, 743, 104 - Singal et al., 2012, ApJ, 753, 45 - Singal et al., 2013, ApJ, 764, 43 - Singal, Petrosian, & Ko, 2014, ApJ, 786, 109 - Singal, 2015, MNRAS, 115, 122 - Petrosian & Singal, 2015, Proc IAU S313

9. Step 1: Evolving Luminosity Functions If a class of object changes in average luminosity over time, that is called “Luminosity Evolution” Let’s parameterize the luminosity evolution in waveband a - as a function of redshift (z) - like this: with If k a is positive, the objects get brighter (on average) in waveband a with increasing redshift (back in time). or g a (z) is the best-fit correlation function between band a luminosity and redshift

10. Step 1: Evolving Luminosity Functions Parameterize luminosity function in a band : density evolution ‘local’ luminosity function luminosity evolution with redshift Ψ a (L a,z) gives # of objects per luminosity per comoving volume Integrate dL dz to get total number

11. Step 2: Associated Sets We determine the best-fit correlations in truncated data by a customized statistical rank test with the method of ‘associated sets’ (first proposed by B. Efron, & V. Petrosian, 1992, ApJ, 399, 345 & 1999, JASA, 94, 447) Example of associated set: Say I wanted to determine the luminosity rank of the red point among all points of a lower redshift excluded – would not be seen if at redshift of point in question The associated set is an unbiased set for comparison Will be more complicated to form associated sets if both luminosities matter, truncation is not simple etc… (Because of the truncation, the raw rank would be seriously biased)

12. Step 3: Rank Test for Correlations Example : find best fit redshift evolution of L opt For SDSS quasars full optical set 1 σ range of best-fit values is where -1 < τ < 1 How to determine distributions a bit later…

13. Example: SDSS x FIRST quasars - J. Singal, V. Petrosian, A. Lawrence, & L. Stawarz, 2011, ApJ, 743, 104 - J. Singal, V. Petrosian, A. Lawrence, & L. Stawarz, 2012, ApJ, 753, 45 Let’s start with the luminosity evolutions…

14. SDSS x FIRST: Luminosity Evolutions Limits for inclusion in associated sets have to scale by g a (z) also! Objects can move in and out of associated sets when k a (z)s are adjusted.

15. SDSS x FIRST: Luminosity Evolutions Have to do a simultaneous minimization of k opt =3.4±0.2, k rad =5.5±0.2 Quasars have undergone significant evolution with redshift in both radio and optical, with greater evolution in radio. Quasars were more radio loud in the past! For combined radio and optical dataset τ comb = 1 and 2 σ contours

16. SDSS x FIRST: Luminosity Functions (With best-fit redshift evolution taken out “Local” Luminosity function) Cumulative distribution function determined by Lynden-Bell method (1971, MNRAS, 155, 95) modified with associated sets # of objects with L’ greater than object j which are in object j’s associated set Differential lum. fn. is derivative Luminosity function is a distribution

17. SDSS x FIRST: Luminosity Functions (With best-fit redshift evolution taken out) stars: joint rad opt data squares: opt. only data Other colors: diff rad flux limits big stars: assuming induced corr little stars: assuming intrinsic corr other colors: diff rad flux limits crosses: deep Kimball et al. (2011) optical radio

18. ‘Naïve’ result with our data (not treating missing data properly) has a ‘dip’ SDSS x FIRST: Radio loudness distribution The reconstructed intrinsic distribution of radio loudness is very different from the raw observed distribution With the local optical and radio luminosity functions, the local radio loudness distribution G(R’) can be constructed (now unbiased with truncations removed) Circles - intrinsic Triangles – raw observed Orange squares – Ivezic et al. (2004) Red squares – Cirasuolo et al. (2006) No sign of bi-modality

19. Example: FSRQ Blazars: Gamma-Ray Luminosity and Index Evolutions Singal, Petrosian, & Ko, 2014, ApJ, 786, 109 Here the associated set for an object j consists of those objects that would still be present in the survey if they were at object j’s redshift, given their flux and index and the truncation in the flux-index plane.

20. Example: FSRQ Blazars: Gamma-Ray Luminosity and Index Evolutions k Lγ =6.5±0.3, k Γ =0±0.1 FSRQ blazars have undergone significant gamma-ray luminosity evolution with redshift, but not photon index evolution Requires simultaneous determination of best-fit evolutions 1, 2, and 3 σ contours Form for best-fit redshift evolution of quantities Singal, Petrosian, & Ko, 2014, ApJ, 786, 109

21. SDSS x WISE: Luminosity Evolutions Singal, George, & Gerber, 2016, submitted Optical Luminosity Infrared Luminosity

22. SDSS x WISE: Luminosity Evolutions Have to do a simultaneous minimization of k opt =3.35±0.1, k inf =2.75±0.05 Quasars have undergone less evolution with redshift in 22 μm infrared than optical or radio For combined radio and optical dataset τ comb = 1 and 2 contours Singal, George, & Gerber, 2016, submitted

23. SDSS x WISE: Luminosity Functions Singal, George, & Gerber, 2016, submitted optical stars: joint inf opt data little stars: previous rad opt result stars: assuming intrinsic corr triangles: assuming induced corr mid-infrared

24. Luminosity Correlations For luminosities in two wavebands, the total luminosity function may or may not be separable (people always seem to assume it is…) It is only separable if L b and L d are not intrinsically correlated. L d and L b are (of course) observationally correlated, but the observed correlation may or may not be induced by similar redshift evolution and all of the truncations. Determining degree of intrinsic vs. induced correlation is not straightforward and is quite involved (Petrosian et al., in prep). These results so far have considered both cases.

25. Luminosity Correlations Can we use partial correlations to determine the extent of the intrinsic correlations?

26. Extra Slides

27. Example: Fermi-LAT Blazars - Correlation Between Gamma-Ray Flux and Photon Index Singal, 2015, MNRAS, 786, 109

28. SDSS x FIRST: Density Evolution Similar to previous results from e.g. Hopkins et al., 2007, AJ, 654, 731 Cumulative density evolution σ(z) determined with Lynden-Bell method (1971, MNRAS, 155, 95) modified with associated sets # of objects with redshift less than object j which are in object j’s associated set cumulative differential

29. Implications of these Quasar results Quasars were much brighter in the past, and there were more of them Quasars were more radio loud in the past There are not two distinct populations of quasars determined by radio loudness Supermassive Black Holes used to be spinning faster Going back in the past, jets get bigger relative to disks There’s fundamentally only one kind of supermassive black hole system Further discussion / info: J. Singal et al., 2011, ApJ, 743, 104 and J. Singal et al., 2012, ApJ, 753, 45

30. But Jiang et al. claim the opposite for radio loudness evolution… Jiang et al. (2007, ApJ, 656, 680) famously used SDSS DR3 quasars x FIRST and claim that the radio loud fraction decreases with increasing redshift We believe the difference is because Jiang et al. used both radio detected and non-radio detected SDSS optical quasars Since every SDSS quasar (by definition) needs an optical detection to flag it for spectroscopic followup, this is a highly asymmetric and biased way to treat the radio and optical flux limits We believe it is appropriate to use a sample that is both optically and radio limited for this analysis, and take this limits into account as we have Figure 7b from Jiang et al 2007

31. Other results on bi-modality in R question Mahoney et al. (2012, ApJ, 754, 12) claim no bimodality and single population in 20 GHz radio (R 20 ) with X-ray selected sample and radio flux down to 20 μJy. Broderick & Fender (2011, MNRAS, 417,184) claim no bimodality in X-ray loudness R X Kimball et al. (2011, ApJL, 739, L29) detect in radio almost every SDSS quasar in a field in a volume limited sample, down to a 6 GHz flux limit of 20 μJy. Using V/V max method, they construct radio luminosity function and claim it is best fit by a two population model separated at R~10. However they do not claim a bi- modality. Also they disfavor a population with R<0.1, as they have very few objects there.

32. Important Topics not discussed in this talk: Why we chose the criteria we did to match sources in radio and optical catalogs? (angular separation, what to do about multiple matches…) Limits on a bifurcation of the quasar population in radio loudness at R values below those considered here (but see Kimball et al. paper just mentioned) Determinations of extent of intrinsic nature of radio-optical luminosity correlations (work in prep) Above topics addressed in: J. Singal, V. Petrosian, A. Lawrence, & L. Stawarz, ApJ, submitted, arXiv:1207.3396 Can apply similar methods to blazars (see e.g. J. Singal, V. Petrosian, & M. Ajello, 2012, ApJ, 753, 45) Implications for source modeling of cosmic radio background (see J. Singal et al., 2010, MNRAS, 409, 1172)

33. Results: Density Evolution Cumulative density evolution σ(z) determined with Lynden- Bell method (1971, MNRAS, 155, 95) modified with associated sets # of objects with redshift less than object j which are in object j’s associated set Raw data True density ev.

34. FSRQ Blazars: Gamma-Ray Luminosity Function z=0 “local” lum. fn. (stars) with Ajello et al. (2012, ApJ, 751, 108) (lines) z=1 lum. fn. (stars) with Ajello et al. (2012, ApJ, 751, 108) (lines) and Inoue et al. (2010, PASJ, 62, 1005) (dash-dot)