Predicting Herschel & SCUBA2 Confusion Measuring “Fluctuations” & “Counts” Mattia Vaccari &...

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Confusion Confusion Measuring “Fluctuations” & “Counts” Measuring “Fluctuations” & “Counts” Mattia Vaccari & Alberto Franceschini & Giulia Rodighiero & Stefano Mattia Vaccari & Alberto Franceschini & Giulia Rodighiero & Stefano Berta Berta Department of Astronomy - University of Padova Department of Astronomy - University of Padova A) Summary A) Summary Confusion in Herschel & SCUBA2 bands is estimated using two Confusion in Herschel & SCUBA2 bands is estimated using two complementary criteria complementary criteria ISO, Spitzer & SCUBA counts are used to constrain models of ISO, Spitzer & SCUBA counts are used to constrain models of extragalactic populations extragalactic populations Based on such models the two approaches respectively measure Based on such models the two approaches respectively measure fluctuations due to unresolved sources and set a maximum number density fluctuations due to unresolved sources and set a maximum number density for resolved sources for resolved sources These two methods provide results which are in good agreement for all These two methods provide results which are in good agreement for all bands bands Recommendations for planning Herschel & SCUBA2 extragalactic surveys Recommendations for planning Herschel & SCUBA2 extragalactic surveys are outlined are outlined Mattia Vaccari Mattia Vaccari mattia@mattiavaccari .net www.mattiavaccari.net E) “Fluctuations” vs. “Counts” Confusion Limits E) “Fluctuations” vs. “Counts” Confusion Limits 3 3 “Fluctuations” are very similar to “Counts” 10 bps for PACS but nearer to “Fluctuations” are very similar to “Counts” 10 bps for PACS but nearer to 20 bps for SPIRE 20 bps for SPIRE This is due to the different slopes of PACS and SPIRE counts at and fainter This is due to the different slopes of PACS and SPIRE counts at and fainter than their 10 bps levels than their 10 bps levels The availability of multi- The availability of multi- (notably MIPS 24/70 (notably MIPS 24/70 m and/or PACS) data will be m and/or PACS) data will be invaluable in accurately pin-pointing SPIRE & SCUBA2 sources in order to reach invaluable in accurately pin-pointing SPIRE & SCUBA2 sources in order to reach down to 10 bps fluxes in all Herschel bands down to 10 bps fluxes in all Herschel bands G) Ongoing Work G) Ongoing Work Fitting additional observables (e.g. redshift distributions) into Fitting additional observables (e.g. redshift distributions) into refined modelling picture refined modelling picture Comparing our estimates with results obtained using other confusion Comparing our estimates with results obtained using other confusion estimators estimators Providing a “consensus“ confusion model suitable for survey planning Providing a “consensus“ confusion model suitable for survey planning B) Confusion at Long Wavelengths with Herschel & B) Confusion at Long Wavelengths with Herschel & SCUBA2 SCUBA2 Confusion sets a major limit to the sensitivity of long- Confusion sets a major limit to the sensitivity of long- extragalactic surveys extragalactic surveys As IRAS first dramatically showed, the FIR/sub-mm sky is very As IRAS first dramatically showed, the FIR/sub-mm sky is very densely populated densely populated ISO & Spitzer & SCUBA confirmed this view providing a wealth of ISO & Spitzer & SCUBA confirmed this view providing a wealth of additional data additional data Great care must be taken in taking stock of current knowledge in Great care must be taken in taking stock of current knowledge in order to predict Herschel & SCUBA2 confusion limits and thus allow a order to predict Herschel & SCUBA2 confusion limits and thus allow a timely planning of observations timely planning of observations D) Measuring Confusion with Fluctuations & D) Measuring Confusion with Fluctuations & Counts Counts Fluctuations” measure fluctuations in the background due to Fluctuations” measure fluctuations in the background due to unresolved sources [Franceschini et al. 1989 (ApJ, 344) ] and set unresolved sources [Franceschini et al. 1989 (ApJ, 344) ] and set “confusion” at the “confusion” at the 3 3 fluctuation level fluctuation level Counts” measure source counts and set “confusion” where a maximum Counts” measure source counts and set “confusion” where a maximum number of sources per beam, or rather a minimum number of beams per number of sources per beam, or rather a minimum number of beams per source (bps), is reached [Franceschini et al. 2001 (A&A, 378)] source (bps), is reached [Franceschini et al. 2001 (A&A, 378)] Roughly speaking, while “Fluctuations” follow the trend of counts Roughly speaking, while “Fluctuations” follow the trend of counts fainter than the confusion limit, “Counts” follow the trend at fainter than the confusion limit, “Counts” follow the trend at brighter fluxes than that. The degree of consistency between the two brighter fluxes than that. The degree of consistency between the two depends on the slope of counts at and near this flux limit depends on the slope of counts at and near this flux limit F) Recommendations for Herschel & SCUBA2 Survey F) Recommendations for Herschel & SCUBA2 Survey Planning Planning Target sensitivities envisaged in Herschel & SCUBA2 surveys appear to Target sensitivities envisaged in Herschel & SCUBA2 surveys appear to be realistic from a confusion stand-point when modelling ISO & Spitzer & be realistic from a confusion stand-point when modelling ISO & Spitzer & SCUBA results SCUBA results In particular, reaching the ~10 mJy level in all SPIRE bands appears to In particular, reaching the ~10 mJy level in all SPIRE bands appears to be feasible and thus somehow mandatory in order to fulfill Herschel’s be feasible and thus somehow mandatory in order to fulfill Herschel’s potential for extragalactic surveys. Such deep SPIRE surveys should be potential for extragalactic surveys. Such deep SPIRE surveys should be undertaken over sky areas where deep Spitzer MIPS 24/70 undertaken over sky areas where deep Spitzer MIPS 24/70 m and/or PACS m and/or PACS coverage is or will be available, e.g. within the most popular “Cosmic coverage is or will be available, e.g. within the most popular “Cosmic Windows” Windows” Survey-like exploratory efforts should be undertaken at a relatively Survey-like exploratory efforts should be undertaken at a relatively early time during Herschel & SCUBA2 operations in order to measure actual early time during Herschel & SCUBA2 operations in order to measure actual confusion levels and thus ensure to make the most of Herschel’s 3-year confusion levels and thus ensure to make the most of Herschel’s 3-year life span and of SCUBA2’s 2-year & 5-year plans. life span and of SCUBA2’s 2-year & 5-year plans. Channel Channel PACS1 PACS1 PACS2 PACS2 PACS3 PACS3 SPIRE1 SPIRE1 SPIRE2 SPIRE2 SPIRE3 SPIRE3 SC2-S SC2-S SC2-B SC2-B m m 70 70 110 110 170 170 250 250 350 350 500 500 450 450 850 850 Beam FWHM Beam FWHM 4 .74” 4 .74” 6.96” 6.96” 10.76” 10.76” 17.1” 17.1” 24.4” 24.4” 34.6” 34.6” 7.4” 7.4” 13.7” 13.7” 3 3 [mJy] [mJy] 0.07047 0.07047 1.002 1.002 7.484 7.484 18.12 18.12 22.55 22.55 20.19 20.19 1.967 1.967 1.383 1.383 10 bps 10 bps [mJy] [mJy] 0.1100 0.1100 1.263 1.263 7.090 7.090 14.00 14.00 15.23 15.23 13.23 13.23 1.917 1.917 1.302 1.302 20 bps 20 bps [mJy] [mJy] 0.3029 0.3029 2.746 2.746 11.82 11.82 20.49 20.49 21.65 21.65 18.31 18.31 3.426 3.426 2.125 2.125 30 bps 30 bps [mJy] [mJy] 0.4887 0.4887 4.034 4.034 15.23 15.23 25.40 25.40 26.34 26.34 21.49 21.49 4.604 4.604 2.755 2.755 40 bps 40 bps [mJy] [mJy] 0.6656 0.6656 5.110 5.110 18.12 18.12 29.18 29.18 29.93 29.93 24.09 24.09 5.630 5.630 3.253 3.253 50 bps 50 bps [mJy] [mJy] 0.8350 0.8350 6.107 6.107 20.45 20.45 32.47 32.47 33.05 33.05 26.18 26.18 6.467 6.467 3.701 3.701 C) Modelling ISO, Spitzer & SCUBA Observations C) Modelling ISO, Spitzer & SCUBA Observations Models by Franceschini et al. 2001 (A&A, 378) were updated to account Models by Franceschini et al. 2001 (A&A, 378) were updated to account for intervened reanalyses of ISO 15 for intervened reanalyses of ISO 15 m data and results from Spitzer m data and results from Spitzer MIPS and SCUBA observations MIPS and SCUBA observations Models describe available observables in terms of four populations Models describe available observables in terms of four populations slowly or non-evolving disk galaxies [ slowly or non-evolving disk galaxies [ blue blue dotted lines] dotted lines] type-1 AGNs evolving as shown by UV and X-ray selected quasars & type-1 AGNs evolving as shown by UV and X-ray selected quasars & Seyferts Seyferts [ [ green green long-short dashed lines] long-short dashed lines] moderate-luminosity starbursts with peak emission at z ~ 1 [ moderate-luminosity starbursts with peak emission at z ~ 1 [ cyan cyan dot-dashed lines] dot-dashed lines] ultra-luminous starbursts with peak evolution between z = 2 and z = ultra-luminous starbursts with peak evolution between z = 2 and z = 4 4 [ [ red red long dashed lines] long dashed lines] Confusion sets in at a flux level determined by the Beam Confusion sets in at a flux level determined by the Beam FWHM line (left below) and the 10 beams per source line FWHM line (left below) and the 10 beams per source line (right below) respectively (right below) respectively MIPS 24 MIPS 24 m m MIPS 70 MIPS 70 m m (See C) for (See C) for Legenda) Legenda) SCUBA2 850 SCUBA2 850 SCUBA2 450 SCUBA2 450 SPIRE 350 SPIRE 350 PACS 170 PACS 170 PACS 70 PACS 70 Name Name Area Area Field Field PACST PACST ime ime SPIRE SPIRE Time Time 70 70 110 110 170 170 250 250 350 350 500 500 - - deg^2 deg^2 - - hr hr hr hr mJy mJy mJy mJy mJy mJy mJy mJy mJy mJy mJy mJy Cluster Cluster s s - - - - 80 80 100 100 - - - - - - - - - - - - Level 1 Level 1 0.04 0.04 GOODS-S GOODS-S 230 230 10 10 + + 30 30 1.0 1.0 1.0 1.0 1.0 1.0 3.3 3.3 4.0 4.0 4.6 4.6 Level 2 Level 2 0.04 0.04 GOODS-N GOODS-N 27 27 10 10 2.0 2.0 2.8 2.8 3.0 3.0 6.7 6.7 8.1 8.1 9.2 9.2 Level 3 Level 3 0.25 0.25 GOODS-S GOODS-S 34 34 25 25 2.2 2.2 6.2 6.2 6.7 6.7 10.5 10.5 12.7 12.7 14.5 14.5 0.25 0.25 Groth Groth Strip Strip 34 34 25 25 2.2 2.2 6.2 6.2 6.7 6.7 10.5 10.5 12.7 12.7 14.5 14.5 0.25 0.25 Lockman Lockman 34 34 25 25 2.2 2.2 6.2 6.2 6.7 6.7 10.5 10.5 12.7 12.7 14.5 14.5 Level 4 Level 4 2 2 COSMOS COSMOS 110 110 50 50 6.0 6.0 9.8 9.8 10.5 10.5 21.1 21.1 25.5 25.5 29.1 29.1 2 2 XMM-LSS XMM-LSS 110 110 50 50 18 18 9.8 9.8 10.5 10.5 21.1 21.1 25.5 25.5 29.1 29.1 Level 5 Level 5 10 10 Spitzer Spitzer 185 185 200 200 18 18 16.9 16.9 18.0 18.0 23.6 23.6 28.5 28.5 32.5 32.5 Level 6 Level 6 50 50 Spitzer Spitzer - - 150 150 18 18 - - 120 120 61 61 74 74 84 84 e (hr) : e (hr) : PACS (659) PACS (659) SPIRE (850) SPIRE (850) Harwit (10) Harwit (10) (Spitzer Depths) (Spitzer Depths) Herschel Extragalactic GT Survey Wedding Herschel Extragalactic GT Survey Wedding Cake Cake Differential Counts Differential Counts Integral Counts Integral Counts (See C) for (See C) for Legenda) Legenda)

Transcript of Predicting Herschel & SCUBA2 Confusion Measuring “Fluctuations” & “Counts” Mattia Vaccari &...

Page 1: Predicting Herschel & SCUBA2 Confusion Measuring “Fluctuations” & “Counts” Mattia Vaccari & Alberto Franceschini & Giulia Rodighiero & Stefano Berta Department.

Predicting Herschel & SCUBA2 ConfusionPredicting Herschel & SCUBA2 ConfusionMeasuring “Fluctuations” & “Counts”Measuring “Fluctuations” & “Counts”

Mattia Vaccari & Alberto Franceschini & Giulia Rodighiero & Stefano BertaMattia Vaccari & Alberto Franceschini & Giulia Rodighiero & Stefano BertaDepartment of Astronomy - University of PadovaDepartment of Astronomy - University of Padova

Predicting Herschel & SCUBA2 ConfusionPredicting Herschel & SCUBA2 ConfusionMeasuring “Fluctuations” & “Counts”Measuring “Fluctuations” & “Counts”

Mattia Vaccari & Alberto Franceschini & Giulia Rodighiero & Stefano BertaMattia Vaccari & Alberto Franceschini & Giulia Rodighiero & Stefano BertaDepartment of Astronomy - University of PadovaDepartment of Astronomy - University of Padova

A) SummaryA) Summary Confusion in Herschel & SCUBA2 bands is estimated using two complementary criteriaConfusion in Herschel & SCUBA2 bands is estimated using two complementary criteria ISO, Spitzer & SCUBA counts are used to constrain models of extragalactic populationsISO, Spitzer & SCUBA counts are used to constrain models of extragalactic populations Based on such models the two approaches respectively measure fluctuations due to Based on such models the two approaches respectively measure fluctuations due to

unresolved sources and set a maximum number density for resolved sourcesunresolved sources and set a maximum number density for resolved sources These two methods provide results which are in good agreement for all bandsThese two methods provide results which are in good agreement for all bands Recommendations for planning Herschel & SCUBA2 extragalactic surveys are outlinedRecommendations for planning Herschel & SCUBA2 extragalactic surveys are outlined

Mattia VaccariMattia [email protected]

E) “Fluctuations” vs. “Counts” Confusion LimitsE) “Fluctuations” vs. “Counts” Confusion Limits 3 3 “Fluctuations” are very similar to “Counts” 10 bps for PACS but nearer to 20 bps for SPIRE “Fluctuations” are very similar to “Counts” 10 bps for PACS but nearer to 20 bps for SPIRE This is due to the different slopes of PACS and SPIRE counts at and fainter than their 10 bps levelsThis is due to the different slopes of PACS and SPIRE counts at and fainter than their 10 bps levels The availability of multi-The availability of multi- (notably MIPS 24/70 (notably MIPS 24/70 m and/or PACS) data will be invaluable in accurately m and/or PACS) data will be invaluable in accurately

pin-pointing SPIRE & SCUBA2 sources in order to reach down to 10 bps fluxes in all Herschel bandspin-pointing SPIRE & SCUBA2 sources in order to reach down to 10 bps fluxes in all Herschel bands

G) Ongoing WorkG) Ongoing Work Fitting additional observables (e.g. redshift distributions) into refined modelling pictureFitting additional observables (e.g. redshift distributions) into refined modelling picture Comparing our estimates with results obtained using other confusion estimatorsComparing our estimates with results obtained using other confusion estimators Providing a “consensus“ confusion model suitable for survey planningProviding a “consensus“ confusion model suitable for survey planning Applying techniques to other projects, e.g. Akari & SPICA & FIRMApplying techniques to other projects, e.g. Akari & SPICA & FIRM

B) Confusion at Long Wavelengths with Herschel & SCUBA2B) Confusion at Long Wavelengths with Herschel & SCUBA2 Confusion sets a major limit to the sensitivity of long-Confusion sets a major limit to the sensitivity of long- extragalactic surveys extragalactic surveys As IRAS first dramatically showed, the FIR/sub-mm sky is very densely populatedAs IRAS first dramatically showed, the FIR/sub-mm sky is very densely populated ISO & Spitzer & SCUBA confirmed this view providing a wealth of additional dataISO & Spitzer & SCUBA confirmed this view providing a wealth of additional data Great care must be taken in taking stock of current knowledge in order to predict Great care must be taken in taking stock of current knowledge in order to predict

Herschel & SCUBA2 confusion limits and thus allow a timely planning of observationsHerschel & SCUBA2 confusion limits and thus allow a timely planning of observations

D) Measuring Confusion with Fluctuations & CountsD) Measuring Confusion with Fluctuations & Counts ““Fluctuations” measure fluctuations in the background due to unresolved sources Fluctuations” measure fluctuations in the background due to unresolved sources

[Franceschini et al. 1989 (ApJ, 344) ] and set “confusion” at the [Franceschini et al. 1989 (ApJ, 344) ] and set “confusion” at the 3 3 fluctuation level fluctuation level “ “Counts” measure source counts and set “confusion” where a maximum number of Counts” measure source counts and set “confusion” where a maximum number of

sources per beam, or rather a minimum number of beams per source (bps), is reached sources per beam, or rather a minimum number of beams per source (bps), is reached

[Franceschini et al. 2001 (A&A, 378)][Franceschini et al. 2001 (A&A, 378)] Roughly speaking, while “Fluctuations” follow the trend of counts fainter than the Roughly speaking, while “Fluctuations” follow the trend of counts fainter than the

confusion limit, “Counts” follow the trend at brighter fluxes than that. The degree of confusion limit, “Counts” follow the trend at brighter fluxes than that. The degree of

consistency between the two depends on the slope of counts at and near this flux limitconsistency between the two depends on the slope of counts at and near this flux limit

F) Recommendations for Herschel & SCUBA2 Survey PlanningF) Recommendations for Herschel & SCUBA2 Survey Planning Target sensitivities envisaged in Herschel & SCUBA2 surveys appear to be realistic from a Target sensitivities envisaged in Herschel & SCUBA2 surveys appear to be realistic from a

confusion stand-point when modelling ISO & Spitzer & SCUBA resultsconfusion stand-point when modelling ISO & Spitzer & SCUBA results In particular, reaching the ~10 mJy level in all SPIRE bands appears to be feasible and thus In particular, reaching the ~10 mJy level in all SPIRE bands appears to be feasible and thus

somehow mandatory in order to fulfill Herschel’s potential for extragalactic surveys. Such somehow mandatory in order to fulfill Herschel’s potential for extragalactic surveys. Such

deep SPIRE surveys should be undertaken over sky areas where deep Spitzer MIPS 24/70 deep SPIRE surveys should be undertaken over sky areas where deep Spitzer MIPS 24/70 m m

and/or PACS coverage is or will be available, e.g. within the most popular “Cosmic Windows”and/or PACS coverage is or will be available, e.g. within the most popular “Cosmic Windows” Survey-like exploratory efforts should be undertaken at a relatively early time during Survey-like exploratory efforts should be undertaken at a relatively early time during

Herschel & SCUBA2 operations in order to measure actual confusion levels and thus ensure Herschel & SCUBA2 operations in order to measure actual confusion levels and thus ensure

to make the most of Herschel’s 3-year life span and of SCUBA2’s 2-year & 5-year plans.to make the most of Herschel’s 3-year life span and of SCUBA2’s 2-year & 5-year plans.

ChannelChannel PACS1PACS1 PACS2PACS2 PACS3PACS3 SPIRE1SPIRE1 SPIRE2SPIRE2 SPIRE3 SPIRE3 SC2-S SC2-S SC2-BSC2-B

mm 7070 110110 170170 250250 350350 500500 450450 850850

Beam FWHMBeam FWHM 4 .74”4 .74” 6.96”6.96” 10.76”10.76” 17.1”17.1” 24.4”24.4” 34.6”34.6” 7.4”7.4” 13.7”13.7”

3 3 [mJy] [mJy] 0.070470.07047 1.0021.002 7.4847.484 18.1218.12 22.5522.55 20.1920.19 1.9671.967 1.3831.383

10 bps [mJy]10 bps [mJy] 0.11000.1100 1.2631.263 7.0907.090 14.0014.00 15.2315.23 13.2313.23 1.9171.917 1.3021.302

20 bps [mJy]20 bps [mJy] 0.30290.3029 2.7462.746 11.8211.82 20.4920.49 21.6521.65 18.3118.31 3.4263.426 2.1252.125

30 bps [mJy]30 bps [mJy] 0.48870.4887 4.0344.034 15.2315.23 25.4025.40 26.3426.34 21.4921.49 4.6044.604 2.7552.755

40 bps [mJy]40 bps [mJy] 0.66560.6656 5.1105.110 18.1218.12 29.1829.18 29.9329.93 24.0924.09 5.6305.630 3.2533.253

50 bps [mJy]50 bps [mJy] 0.83500.8350 6.1076.107 20.4520.45 32.4732.47 33.0533.05 26.1826.18 6.4676.467 3.7013.701

C) Modelling ISO, Spitzer & SCUBA ObservationsC) Modelling ISO, Spitzer & SCUBA Observations Models by Franceschini et al. 2001 (A&A, 378) were updated to account for intervened Models by Franceschini et al. 2001 (A&A, 378) were updated to account for intervened

reanalyses of ISO 15 reanalyses of ISO 15 m data and results from Spitzer MIPS and SCUBA observationsm data and results from Spitzer MIPS and SCUBA observations Models describe available observables in terms of four populationsModels describe available observables in terms of four populations

slowly or non-evolving disk galaxies [slowly or non-evolving disk galaxies [blueblue dotted lines] dotted lines] type-1 AGNs evolving as shown by UV and X-ray selected quasars & Seyfertstype-1 AGNs evolving as shown by UV and X-ray selected quasars & Seyferts

[[greengreen long-short dashed lines] long-short dashed lines] moderate-luminosity starbursts with peak emission at z ~ 1 [moderate-luminosity starbursts with peak emission at z ~ 1 [cyancyan dot-dashed lines] dot-dashed lines] ultra-luminous starbursts with peak evolution between z = 2 and z = 4 ultra-luminous starbursts with peak evolution between z = 2 and z = 4 [[redred

long dashed lines]long dashed lines]

Confusion sets in at a flux level determined by the Beam FWHM line Confusion sets in at a flux level determined by the Beam FWHM line

(left below) and the 10 beams per source line (right below) respectively(left below) and the 10 beams per source line (right below) respectively

MIPS 24 MIPS 24 mm MIPS 70 MIPS 70 mm

(See C) for Legenda)(See C) for Legenda)

SCUBA2 850SCUBA2 850SCUBA2 450SCUBA2 450SPIRE 350SPIRE 350PACS 170PACS 170PACS 70PACS 70

NameName AreaArea FieldField PACSTPACSTimeime

SPIRE SPIRE TimeTime 7070 110110 170170 250250 350350 500500

-- deg^2deg^2 -- hrhr hrhr mJymJy mJymJy mJymJy mJymJy mJymJy mJymJy

ClustersClusters -- -- 8080 100100 -- -- -- -- -- --

Level 1Level 1 0.040.04 GOODS-SGOODS-S 230230 1010++3030 1.01.0 1.01.0 1.01.0 3.33.3 4.04.0 4.64.6

Level 2Level 2 0.040.04 GOODS-NGOODS-N 2727 1010 2.02.0 2.82.8 3.03.0 6.76.7 8.18.1 9.29.2

Level 3Level 3 0.250.25 GOODS-SGOODS-S 3434 2525 2.22.2 6.26.2 6.76.7 10.510.5 12.712.7 14.514.5

0.250.25 Groth StripGroth Strip 3434 2525 2.22.2 6.26.2 6.76.7 10.510.5 12.712.7 14.514.5

0.250.25 LockmanLockman 3434 2525 2.22.2 6.26.2 6.76.7 10.510.5 12.712.7 14.514.5

Level 4Level 4 22 COSMOSCOSMOS 110110 5050 6.06.0 9.89.8 10.510.5 21.121.1 25.525.5 29.129.1

22 XMM-LSSXMM-LSS 110110 5050 1818 9.89.8 10.510.5 21.121.1 25.525.5 29.129.1

Level 5Level 5 1010 SpitzerSpitzer 185185 200200 1818 16.916.9 18.018.0 23.623.6 28.528.5 32.532.5

Level 6Level 6 5050 SpitzerSpitzer -- 150150 1818 -- 120120 6161 7474 8484

Time (hr) : Time (hr) : PACS (659)PACS (659) SPIRE (850) SPIRE (850) Harwit (10)Harwit (10) (Spitzer Depths)(Spitzer Depths)

Herschel Extragalactic GT Survey Wedding CakeHerschel Extragalactic GT Survey Wedding Cake

Differential CountsDifferential Counts

Integral CountsIntegral Counts

(See C) for Legenda)(See C) for Legenda)