Neutrinos from GRBs, and the connection to gamma- and cosmic-rays

Transcription

1 Neutrinos from GRBs, and the connection to gamma- and cosmic-rays Philipp Baerwald Institut für Theoretische Physik und Astrophysik Universität Würzburg in collaboration with Mauricio Bustamante, Svenja Hümmer, and Walter Winter KIAA Workshop in Beijing May 7, 2013 P. Baerwald GK / 27

2 Introduction GRB models (What we assume) Several seconds long bursts of γ-rays from one point source Most luminous events in the known universe Connected to ultra-relativistic expanding fireball, candidates for UHE cosmic ray sources Interactions of protons with photons can lead to the production of neutrinos from GRB fireball model by [Rees and Meszaros, MNRAS 258 (2), 41P (1992)] currently most popular model for neutrino calculations. P. Baerwald GK / 27

3 Introduction Modeling neutrinos from GRB Photohadronic production of neutrinos Basic approach by Waxman and Bahcall: approximation of pγ interaction cross section using -resonance { p + γ + n + π + 1/3 of all cases p + π 0 2/3 of all cases The π + decay producing ν µ, ν µ and ν e in the process Standard conclusion π + µ + + ν µ µ + e + + ν e + ν µ ν result from interaction of p and γ in GRBs, with a ratio (ν e : ν µ : ν τ ) of (1 : 2 : 0), or (1 : 1 : 1) after flavor mixing. See e.g. [Waxman and Bahcall, Phys. Rev. Lett. 78 (12), 2292 (1997)] P. Baerwald GK / 27

4 Introduction Modeling neutrinos from GRB The spectral shape (IceCube approach) F γ ( ) Eγ αγ ɛ γ,b ( Eγ ɛ γ,b ) βγ for E γ ɛ γ,b for E γ > ɛ γ,b from [Waxman and Bahcall, Phys. Rev. D59, (1998)] P. Baerwald GK / 27

5 Introduction Modeling neutrinos from GRB The normalization (IceCube approach) 0 de ν E νf ν(e ν) = x π ν = {}}{ 1 8 Fraction of energy f π energy in protons {}}{ ) 1 10 MeV (1 (1 x p π ) R/λpγ de γ E γf γ(e γ) }{{} f e 1 kev f π with number of interactions ( ) ( R L iso ) ( ) γ 0.01 s ( ) MeV = λ pγ erg s 1 t var and average energy lost to pions (per interaction) x p π 0.2 Γ jet ɛ γ from [Abbasi et al., Astrophys. J. 710, 346 (2010)], based on [Guetta et al., Astropart. Phys. 20, 429 (2004)] P. Baerwald GK / 27

6 Introduction Modeling neutrinos from GRB GRB fireballs in trouble? LETTER doi: /nature11068 An absence of neutrinos associated with cosmic-ray acceleration in c-ray bursts IceCube Collaboration* Very energetic astrophysical events are required to accelerate cosmic rays to above electronvolts. GRBs (c-ray bursts) have been proposed as possible candidate sources 1 3. In the GRB fireball model, cosmic-ray acceleration should be accompanied by neutrinos producedinthe decayofcharged pions created ininteractionsbetween the high-energy cosmic-ray protons and c-rays 4. Previous searches for such neutrinos found none, but the constraints were weak because the sensitivity was at best approximately equal to the predicted flux 5 7. Here we report an upper limit on the flux of energetic neutrinos associated with GRBs that is at least a factor of 3.7 below the predictions 4,8 10. This implies either that GRBs are not the only sources of cosmic rays with energies exceeding electronvolts or that the efficiency of neutrino production is much lower than has been predicted. Neutrinos from GRBs are produced in the decay of charged pions produced in interactions between high-energy protons and the intense c-ray background within the GRB fireball, for example in the D-resonance process p 1 c R D 1 R n 1 p 1 (p, proton; c, photon (here c-ray); D 1, delta baryon; n, neutron; p 1, pion). When these pions decay via p 1 R m 1 nm and m z?e z vevm, they produce a flux of highenergy muon neutrinos (nm) and electron neutrinos (ne), coincident As in our previous study 7, we conducted two analyses of the IceCube data. In a model-dependent search, we examine data during the period of c-ray emission reported by any satellite for neutrinos with the energy spectrum predicted from the c-ray spectra of individual GRBs 6,9. The model-independent analysis searches more generically for neutrinos on wider timescales, up to the limit of sensitivity to small numbers of events at 61 day, or with different spectra. Both analyses follow the methods used in our previous work 7, with the exception of slightly changed event selection and the addition of the Southern Hemisphere to the model-independent search. Owing to the large background of downgoing muons from the southern sky, the Southern Hemisphere analysis is sensitive mainly to higher-energy events (Supplementary Fig. 3). Systematic uncertainties fromdetector effects have been included in the reported limits from both analyses, and were estimated by varying the simulated detector response and recomputing the limit, with the dominant factor being the efficiency of the detector s optical sensors. In the 59-string portion of the model-dependent analysis, no events were found to be both on-source and on time (within 10u of a GRB and between Tstart and Tstop). From the individual burst spectra 6,9 with an assumed ratio of energy in protons to energy in electrons ep/ee 5 10 P. with Baerwald the c-rays, and peaking at energies of several hundredgk1147 tera- (ref. 6), 8.4 signal events were predicted from the combined 2-year data 6 / 27

7 Refined particle physics calculations Detailed cross section The measured cross section Σ Ε Μbarn Ε r GeV from [Hümmer, Rüger, Spanier, and Winter, Astrophys. J. 721, 630 (2010)] Up to ɛ γ = 18 GeV measurements conducted at DESY, SLAC, and Cornell University Synchrotron in late 60s and early 70s Energy range from GeV reached at Fermilab in 1978 Higher energies indirectly accessible through simulations using the proton structure functions from DESY (ZEUS) at the end of last century and confirmed by measurements of cascades at Baksan neutrino observatory in 2003 P. Baerwald GK / 27

8 Refined particle physics calculations Detailed cross section Contributions to the full photohadronic cross section 10 7 NeuCosmA 2010 E 2 Φ ΝΜ ΝΜ GeV sr 1 s 1 cm WB approx. WB flux P. Baerwald GK / 27

9 Refined particle physics calculations Detailed cross section Contributions to the full photohadronic cross section Contributions to (ν µ + ν µ) flux from π ± decay divided in: 10 7 NeuCosmA 2010 (1232)-resonance E 2 Φ ΝΜ ΝΜ GeV sr 1 s 1 cm WB approx. WB flux 1232 only from [PB, S. Hümmer, and W. Winter, Phys. Rev. D83, (2011)] P. Baerwald GK / 27

10 Refined particle physics calculations Detailed cross section Contributions to the full photohadronic cross section Contributions to (ν µ + ν µ) flux from π ± decay divided in: 10 7 NeuCosmA 2010 (1232)-resonance Higher resonances E 2 Φ ΝΜ ΝΜ GeV sr 1 s 1 cm WB approx. WB flux Higher resonances 1232 only from [PB, S. Hümmer, and W. Winter, Phys. Rev. D83, (2011)] P. Baerwald GK / 27

11 Refined particle physics calculations Detailed cross section Contributions to the full photohadronic cross section Contributions to (ν µ + ν µ) flux from π ± decay divided in: 10 7 NeuCosmA 2010 (1232)-resonance Higher resonances t-channel (direct production) E 2 Φ ΝΜ ΝΜ GeV sr 1 s 1 cm t channel WB approx. Higher resonances WB flux 1232 only from [PB, S. Hümmer, and W. Winter, Phys. Rev. D83, (2011)] P. Baerwald GK / 27

12 Refined particle physics calculations Detailed cross section Contributions to the full photohadronic cross section Contributions to (ν µ + ν µ) flux from π ± decay divided in: 10 7 NeuCosmA 2010 Total flux (1232)-resonance Higher resonances t-channel (direct production) High energy processes (multiple π) E 2 Φ ΝΜ ΝΜ GeV sr 1 s 1 cm t channel Multi Π WB approx. Higher resonances WB flux 1232 only from [PB, S. Hümmer, and W. Winter, Phys. Rev. D83, (2011)] Especially Multi π contribution leads to change of flux shape; neutrino flux higher by up to a factor of 3 compared to WB treatment P. Baerwald GK / 27

13 Refined particle physics calculations Beyond the Delta-resonance Further particle decays π + µ + + ν µ, µ + e + + ν e + ν µ, π µ + ν µ, µ e + ν e + ν µ, K + µ + + ν µ, n p + e + ν e. P. Baerwald GK / 27

14 Refined particle physics calculations Beyond the Delta-resonance Further particle decays Resulting ν e flux (at the observer) π + µ + + ν µ, µ + e + + ν e + ν µ, 10 7 NeuCosmA 2010 Total flux π µ + ν µ, µ e + ν e + ν µ, K + µ + + ν µ, E 2 Φ Νe Νe GeV sr 1 s 1 cm from Μ WB flux from n n p + e + ν e from [PB, S. Hümmer, and W. Winter, Phys. Rev. D83, (2011)] P. Baerwald GK / 27

15 Refined particle physics calculations Beyond the Delta-resonance Further particle decays Resulting ν µ flux (at the observer) π + µ + + ν µ, µ + e + + ν e + ν µ, π µ + ν µ, µ e + ν e + ν µ, K + µ + + ν µ, n p + e + ν e. E 2 Φ ΝΜ ΝΜ GeV sr 1 s 1 cm NeuCosmA 2010 Total flux from Μ 10 8 WB flux from Π from K from [PB, S. Hümmer, and W. Winter, Phys. Rev. D83, (2011)] P. Baerwald GK / 27

16 Refined particle physics calculations Beyond the Delta-resonance Neutrino spectra incl. flavor mixing Electron neutrino spectrum 10 7 NeuCosmA Muon neutrino spectrum NeuCosmA 2010 E 2 Φ Νe Νe GeV sr 1 s 1 cm Total flux ΝΜ WB flux Νe E 2 Φ ΝΜ ΝΜ GeV sr 1 s 1 cm Total flux rescaled Total flux Νe ΝΜ WB flux E GeV from [PB, S. Hümmer, and W. Winter, Phys. Rev. D83, (2011)] Characteristic double peak structure from µ and π decay in both flavors, additional peak from K + decay at 10 8 to 10 9 GeV P. Baerwald GK / 27

17 Refined particle physics calculations The new prediction How the spectrum changes... At example of GRB080603A: 1. Correction to analytical model (IC-FC RFC) 2. Change due to full numerical calculation IC-FC: RFC: NFC: IceCube-Fireball Calculation Revised Fireball Calculation Numerical Fireball Calculation from [Hümmer, PB, and Winter, Phys. Rev. Lett. 108, (2012)] P. Baerwald GK / 27

18 Refined particle physics calculations The new prediction Revised fireball model Corrections to shape (c s): Revised shape Correct energy losses of secondaries Full energy dependencies See also [Li, Phys. Rev. D85, (2012)]. Corrections to f π (c fπ ): Integral normalization of photon spectrum Rounding errors Width of -resonance Compare to [Guetta et al., Astropart. Phys. 20, 429 (2004)]. E Ν 2 F Ν EΝ GeV cm NeuCosmA 2011 IC FC RFC c fπ c s shape revised f CΓ f f Σ from [Hümmer, PB, and Winter, Phys. Rev. Lett. 108, (2012)] P. Baerwald GK / 27

19 Refined particle physics calculations The new prediction Result of re-computation E Ν 2 Φ Ν E GeV cm 2 s 1 sr NeuCosmA 2012 IC40 IC40 59 IC86, 10y extrapolated NFC prediction GRB, all GRB, z known stat. error astrophysical uncertainties f e 10 5 Significantly reduced prediction Uncertainties of astrophysical parameters; tested ranges: t v = s, Γ = , α p = , and ɛ B /ɛ e = Conservative bounds only with known parameters, here: known redshifts Additional uncertainty from statistics in stacking analysis from [Hümmer, PB, and Winter, Phys. Rev. Lett. 108, (2012)]; see also [He, Liu, Wang, Nagataki, Murase, and Dai, Astrophys.J. 752, 29 (2012)] P. Baerwald GK / 27

20 Refined particle physics calculations Comparison different computations Uncertainties for different computations E Ν 2 ΦΝ E GeV cm 2 s 1 sr NeuCosmA 2011 IC40 IC40 59 IC86, 10y extrapolated NFC prediction IC FC NFC uncertainties IC FC uncertainties He et al. uncertainties IceCube: based on [Guetta, Spada, and Waxman, Astrophys. J. 559, 101 (2001)]: origin of target photons fixed (synchrotron, IC scattering) Hümmer et al.: uncertainties of parameters in basic fireball model, target photon spectrum from observation He et al.: most general assumption with emission radius as free parameter; includes possibilities of other models (dissipative photosphere, magnetic reconnection...) Data from [IceCube Collaboration, Nature 484, 351 (2012)], [Hümmer, PB, and Winter, Phys. Rev. Lett. 108, (2012)], and [He, Liu, Wang, Nagataki, Murase, and Dai, Astrophys.J. 752, 29 (2012)]. P. Baerwald GK / 27

21 Refined particle physics calculations Comparison different computations Model-(in)dependent predictions taken from [Zhang and Kumar, Phys. Rev. Lett. 110(12), (2013)] P. Baerwald GK / 27

22 Refined particle physics calculations Comparison different computations NextGen analyses -2 dn/de [GeV cm E 2 ] E ν [GeV] ANTARES GRB analysis presented at Moriond 2013 on March 13 by J. Schmid Analytical model and NeuCosmA prediction depicted P. Baerwald GK / 27 9

23 The cosmic ray connection Existing constraints One neutrino per cosmic ray p + γ + n + π + n : not subject of magnetic confinement and can contribute to cosmic rays n p e ν e π + : contributes to neutrino flux π + e + ν e ν µ ν µ = one ν µ per CR proton Constraints, see e.g. [Ahlers, Gonzalez-Garcia, and Halzen, Astropart. Phys. 35 (2), (2011)], [IceCube Collaboration, Nature 484, (2012)]. P. Baerwald GK / 27

24 The cosmic ray connection And circumventing them Particle escape from a shell On particle scale: particles can travel certain distance, λ mfp, without interacting, even if magnetically confined P. Baerwald GK / 27

25 The cosmic ray connection And circumventing them Particle escape from a shell On particle scale: particles can travel certain distance, λ mfp, without interacting, even if magnetically confined Idea A shell is an open system, so particles should be able to escape from within λ mfp to the edge ( leaky-box-model ). Fraction of directly escaping particles: V direct λ mfp V iso r with λ mfp(e ) = min [ r, R L(E ), c t ] pγ r Size of region, ensures fraction is normalized to 1 R L Larmor radius, estimate from where charged particles can escape even when magnetically confined; E B c t pγ interaction length for photohadronic interactions, depends mostly on photon spectrum and photon density P. Baerwald GK / 27

26 The cosmic ray connection And circumventing them Optically thin case E 2 sh GeV cm 2 t' 1 s L Γ,iso erg s L Γ,iso erg s 1 1 t' syn 1 t' acc 1 t' pγ 1 t' dyn 1 t' eff, dir E GeV Τn E GeV Τn NeuCosmA 2013 E 2 sh GeV cm 2 t' 1 s L Γ,iso erg s 1 initial p CR from n direct escaping p ΝΜ ΝΜ L Γ,iso erg s 1 E GeV Τn 0.34 Ep,max E GeV Τn 0.34 Direct escape dominates (at highest energies) Optical thickness of neutrons τ n (at E p,max) is smaller than unity optically thin Only (total) luminosity L iso changed Assumed burst parameters: Γ = 300 t v = 0.01 s T 90 = 10 s η = 1 ɛ e/ɛ B = 1 f e = 0.1 α γ = 1 β γ = 2 ε γ,b = 1 kev [PB, Bustamante, and Winter, Astrophys. J. 768, 186 (2013)] P. Baerwald GK / 27

27 The cosmic ray connection And circumventing them Optically thick case E 2 sh GeV cm 2 t' 1 s L Γ,iso erg s 1 initial p CR from n direct escaping p ΝΜ ΝΜ L Γ,iso erg s 1 1 t' syn 1 t' acc 1 t' pγ 1 t' dyn 1 t' eff, dir E GeV Τn 3.37 Ep,max E GeV Τn 3.37 NeuCosmA 2013 E 2 sh GeV cm 2 t' 1 s L Γ,iso erg s L Γ,iso erg s 1 E GeV Τn E GeV Τn 35.6 Neutron escape dominates Optical thickness of neutrons τ n (at E p,max) is larger than unity optically thick Only (total) luminosity L iso changed Assumed burst parameters: Γ = 300 t v = 0.01 s T 90 = 10 s η = 1 ɛ e/ɛ B = 1 f e = 0.1 α γ = 1 β γ = 2 ε γ,b = 1 kev [PB, Bustamante, and Winter, Astrophys. J. 768, 186 (2013)] P. Baerwald GK / 27

28 The cosmic ray connection Parameter space study Possible escape regimes 4 Η 0.1 LAT invisible different escape possibilities: tv s optically thin neutron escape optically thick neutron escape direct escape Η 0.1 full ΣpΓ WB resonance NeuCosmA 2013 direct escape: protons at highest energies escape directly optically thin neutron escape: n from pγ interactions give usual contribution to CR, usually considered case optically thick neutron escape: amount of ν enhanced due to higher number of interactions Standard escape may be constrained to limited region in parameter space. [PB, Bustamante, and Winter, Astrophys. J. 768, 186 (2013)] P. Baerwald GK / 27

29 The cosmic ray connection Parameter space study Possible escape regimes 4 Η 1.0 LAT invisible different escape possibilities: tv s optically thick neutron escape direct escape Η 1.0 optically thin neutron escape full ΣpΓ WB resonance NeuCosmA 2013 direct escape: protons at highest energies escape directly optically thin neutron escape: n from pγ interactions give usual contribution to CR, usually considered case optically thick neutron escape: amount of ν enhanced due to higher number of interactions Standard escape may be constrained to limited region in parameter space. [PB, Bustamante, and Winter, Astrophys. J. 768, 186 (2013)] P. Baerwald GK / 27

30 The cosmic ray connection Application and advantages Effect of additional component to CR spectrum HiRes I HiRes II E 3 J E GeV 2 cm 2 s 1 sr Α p 2.5, two comp. model Α p 2.5, neutron escape only dip model Α p 2.3, neutron escape only dip model Α p 2.0, neutron escape only transition model GeV [PB, Bustamante, and Winter, Astrophys. J. 768, 186 (2013)] P. Baerwald GK / 27

31 The cosmic ray connection The ultimate goal A self-consistent multi-messenger picture GRB models Number of bursts per year, baryonic loading f e, f π Fluence of burst Ν Γ strong constraints Diffuse vs. point source data Number of bursts per year, baryonic loading f e Contradictory claims in literature [Eichler et al., arxiv: ] vs. [Waxman, arxiv: ] CR [Ahlers, Gonzalez-Garcia, and Halzen, Astropart. Phys. 35 (2), (2011)]: direct connection (Y n Y π + ) P. Baerwald GK / 27

32 The cosmic ray connection The ultimate goal...including GW Γ GW Ν CR see talks by Kunihito Ioka, Szabolcs Marka, Shiho Kobayashi, Julian Osborne, Luciano Rezzolla, Kenta Hotokezaka, Imre Bartos, Zigao Dai, Yizhong Fan, Xue-Feng Wu, and He Gao P. Baerwald GK / 27

33 New Physics Neutrino decay Theoretical framework Very long lifetimes κ 1 i of neutrino mass eigenstates ν i (i = 1, 2, 3) may be tested by cosmological νs. [ κ 1 s ] i τ i [s] L [Mpc] 102 ev m i [ev] E [TeV] Proper differential decay equation including redshift dependence: dn i (E 0, z) dz = κ i dl N i (E 0, z) E 0 dz 1 + z L(z) [Gpc] light-travel distance luminosity distance L H =3.89 Gpc Correct distance measure for decay given by 10-2 light-travel (or look-back) distance z [PB, Bustamante, and Winter, JCAP 1210, 020 (2012)] Observational bounds Detection of neutrinos from SN1987A gives strong bound on lifetime of ν 1 (κ s ev 1), and effectively ν e. P. Baerwald GK /

34 New Physics Neutrino decay Consequences of decay E 2 ΦΝ GeV s 1 cm 2 E 2 ΦΝ GeV s 1 cm 2 NeuCosmA 2012 stable 10 5 Νe Νe ΝΜ ΝΜ 10 6 ΝΤ ΝΤ NeuCosmA 2012 Κ 1 1 s ev E 2 ΦΝ GeV s 1 cm Κ s ev NeuCosmA 2012 Possible decay scenario: ν 2 and ν 3 could be unstable; ν 1 bounded by SN1987A (stable) leads to suppression of cosmical ν µ and ν τ ν e nearly unaffected = currently only ν e cascades give reliable bounds E 0 GeV E 0 GeV [PB, Bustamante, and Winter, JCAP 1210, 020 (2012)] P. Baerwald GK / 27

35 Conclusion Summary Standard GRB neutrino flux shape is modified by full particle physics treatment. Simulation of diffuse neutrino flux is model-dependent. Numerical re-analysis of IC40 GRB neutrino prediction is significantly lower than original prediction. Cosmic rays can be obtained from same calculation and constrain neutrino predictions. But there are possibilities to circumvent these. Standard CR escape via neutrons only applies to limited range in parameter space. Ultimate goal: Self-consistent particle physics model of astrophysical sources. P. Baerwald GK / 27

36 Back-up slides P. Baerwald GK / 10

37 Used inputs Proton spectrum N p(e ) E 2 Photon spectrum resembling (simplified) Band function with parameters α, β, and E γ,break Contributions to σ pγ from (1232)-resonance, higher resonances, t-channel (direct production), and high energy processes (multiple π) Decays of π ±, µ ±, n, and K + considered, in case of µ ± including helicity dependence [P. Lipari, M. Lusignoli, and D. Meloni, Phys. Rev. D75, (2007)] Simple neutrino mixing including three mass eigenstates and θ 13 = 0 Normalization at the end matched to WB GRB ν µ flux bound P. Baerwald GK / 10

38 GRB rate from corrected SFR by Hopkins and Beacom SFR by Hopkins and Beacom: ρ (1 + z) 3.44 for 0 < z (1 + z) 0.26 for 0.97 < z (1 + z) 7.8 for 4.48 < z Correction factor taken from [Kistler et al., Astrophys. J. 705, L104-L108 (2009)] from [A. Hopkins, and J. Beacom, Astrophys. J. 651, (2006)] E(z) (1 + z) 1.2 P. Baerwald GK / 10

39 Observational bounds Recent calculations exclude this (arbitrary) one-to-one correspondence for generic ν-spectra. See e.g. [Ahlers, Gonzalez-Garcia, and Halzen, Astropart. Phys. 35 (2), (2011)] (right), [Ice- Cube Collaboration, Nature 484, (2012)] (below). P. Baerwald GK / 10

40 Single burst detection probability estimate Assumption: Question: Diffuse flux from bursts is at the level of the WB flux (1/10 of WB), leading to a certain number of events in the full IceCube detector during 10 years of operation. How near would a single burst have to be to lead to at least 3 events? Answer: Burst has to be at most at z max = 0.14 (z max = 0.05). The probability to have at least one such a close burst is about 40% (2%). from [PB, Hümmer, and Winter, Astropart. Phys. 35, 508 (2012)] P. Baerwald GK / 10

41 Aggregation of fluxes 300 NeuCosmA 2011 Diffuse flux = result of large number of (unresolved) individual sources GRB rate follows SFR z decoupled from other parameters General scaling: F d 2 L Number of GRBs from [PB, Hümmer, and Winter, Astropart. Phys. 35, 508 (2012)] P. Baerwald GK / 10

42 IceCube-40 GRB sample Experimental realization: 117 observed bursts, between April 2008 and May 2009 Individual neutrino spectra stacked Parameters measured or standard value Diffuse limit for 667 bursts per year from [Abbasi et al., Phys. Rev. Lett. 106, (2011)] P. Baerwald GK / 10

43 Resulting aggregated flux EΝ 2 ΦΜ GeV sr 1 s 1 cm NeuCosmA 2011 Standard spectrum WB spectrum All z Peak contribution from bursts at z 1, few close bursts dominate Leads to shift to higher energies compared to single burst (with assumed z = 2) Characteristic features of single burst flux shape are preserved Analysis of other source parameters showed that only the magnetic field can wash out these features P. Baerwald GK / 10

44 Result of statistical analysis Simulated samples of n bursts with different redshift, and extrapolated flux limits from stacked flux. Analysis of resulting limits gave the results: n Rel. error 90% Rel. error 3σ Variation of extrapolated bound for 100 bursts (comparable with the 117 bursts from IceCube) still too high to rule out the (most basic) fireball model due to statistics. For variation of multiple parameters effect gets even stronger. However, additional effects, such as bias through minimal observable flux when considering simultaneous variation of redshift and luminosity, reduce uncertainty compared to independent variation case. P. Baerwald GK / 10

45 Picked samples with different redshifts according to redshift distribution shown earlier for each sample size n 100 ±50 NeuCosmA NeuCosmA 2011 n Σ Probability C.L n ±10% ±20% ±50% n Rel. error 90% Rel. error 3σ P. Baerwald GK / 10