Predicting neutrino flavor evolution in astrophysics: an - - PowerPoint PPT Presentation

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Predicting neutrino flavor evolution in astrophysics: an unconventional application of data assimilation

Eve Armstrong

Computational Neuroscience Initiative University of Pennsylvania

International Symposium on Data Assimilation RIKEN Center for Computational Science, Kobe, Japan

2019 January 21

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What information must an Earth-based detector receive in

• rder to infer the evolutionary

history of cosmological events?

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What information must an Earth-based detector receive in

• rder to infer the evolutionary

history of cosmological events?

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The Super-Kamiokande neutrino observatory, Institute for Cosmic Ray Research, University of Tokyo (Physics Today, 2018 October 4, DOI:10.1063/PT.6.2.20181004a)

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The trouble with numerical integration

We seek to determine the flavor evolution history of neutrinos arriving at a detector. In a forward-scattering-only scenario, the steady-state evolution can be evolved in one direction. i dρE (r)

dr

= [HE (r), ρE (r)]

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The trouble with numerical integration

We seek to determine the flavor evolution history of neutrinos arriving at a detector. In a forward-scattering-only scenario, the steady-state evolution can be evolved in one direction. i dρE (r)

dr

= [HE (r), ρE (r)] + i CE (r). But with back-scattering (the neutrino “halo”), flavor states at later distances can influence states at earlier distances. Or: phase information propagates outward and inward.

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The trouble with numerical integration

We seek to determine the flavor evolution history of neutrinos arriving at a detector. In a forward-scattering-only scenario, the steady-state evolution can be evolved in one direction. i dρE (r)

dr

= [HE (r), ρE (r)] + i CE (r). But with back-scattering (the neutrino “halo”), flavor states at later distances can influence states at earlier distances. Or: phase information propagates outward and inward. This effect can significantly impact flavor evolution through the envelope (Cherry et al. 2012).

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The trouble with numerical integration

We seek to determine the flavor evolution history of neutrinos arriving at a detector. In a forward-scattering-only scenario, the steady-state evolution can be evolved in one direction. i dρE (r)

dr

= [HE (r), ρE (r)] + i CE (r). But with back-scattering (the neutrino “halo”), flavor states at later distances can influence states at earlier distances. Or: phase information propagates outward and inward. This effect can significantly impact flavor evolution through the envelope (Cherry et al. 2012). How to solve for flavor evolution in this scenario?

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Can data assimilation (DA) be formulated in an integration-blind manner to solve this problem? ◮ less computationally expensive; ◮ may offer a test for whether failure of the procedure reflects lack of convergence

• r numerical issues.

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A simple test problem for DA

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check.

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx).

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx).Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere.

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx).Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere.

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx). Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere.

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx). Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere. At r = 0, neutrinos are in flavor state νe, or: Pz = 1 (the z-component of polarization vector P).

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx). Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere. At r = 0, neutrinos are in flavor state νe, or: Pz = 1 (the z-component of polarization vector P). In the adiabatic limit (oscillation length <<∇Vmatter, or: neutrino energy remains constant): beginning in νe, efficient flavor conversion to νx (Mikheyev et al. 1985; Wolfenstein 1978).

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx). Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere. At r = 0, neutrinos are in flavor state νe, or: Pz = 1 (the z-component of polarization vector P). In the adiabatic limit (oscillation length <<∇Vmatter, or: neutrino energy remains constant): beginning in νe, efficient flavor conversion to νx (Mikheyev et al. 1985; Wolfenstein 1978). At what location r does the transition occur?

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx). Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere. The task for DA Given “measurements” of Pz,1 and Pz,2 at r = 0 and r = R: ◮ predict flavor evolution between r = 0 and r = R; ◮ estimate ν-ν and ν-matter coupling strengths.

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A simple test problem for DA

Consider forward-scattering only, to retain a consistency check. A two-flavor model: electron-flavored (νe) and “x-flavored” neutrinos (νx). Two mono-energetic beams with different energies interact with each other (Vνν ∼ Cνν/r3) and with a background of free nucleons and electrons (Vmatter ∼ Cm/r3); r is distance from the neutrino sphere. The task for DA Given “measurements” of Pz,1 and Pz,2 at r = 0 and r = R: ◮ predict flavor evolution between r = 0 and r = R; ◮ estimate ν-ν and ν-matter coupling strengths. Two sets of experiments: Eν1/Eν2 = 2.5 and 0.01. For each set: νν coupling strength Cνν = 0, 1, 100, 1000.

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Result

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Results for Eν1/Eν2 = 2.5 (Armstrong et al. 2017)

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Results for Eν1/Eν2 = 0.01 (Armstrong et al. 2017)

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Keys

◮ For six out of eight experiments, the measurements contained sufficient information to capture overall flavor evolution history. Two exceptions: when ν2 is strongly coupled to neither matter nor ν1.

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Keys

◮ For six out of eight experiments, the measurements contained sufficient information to capture overall flavor evolution history. Two exceptions: when ν2 is strongly coupled to neither matter nor ν1. ◮ The measurements contain insufficient information to break degeneracy in parameter estimates (Cνν and Cm).

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Keys

◮ For six out of eight experiments, the measurements contained sufficient information to capture overall flavor evolution history. Two exceptions: when ν2 is strongly coupled to neither matter nor ν1. ◮ The measurements contain insufficient information to break degeneracy in parameter estimates (Cνν and Cm). Evidently, the flavor evolution is insensitive to the parameter values within the permitted search ranges.

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Specifics of the DA procedure

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The cost function: A0 =

errormeasurement

• j

L

• l

Rl

m

2 (yl(n) − xl(n))2 +

errormodel

• N−1
• n

D

• a

Ra

f

2 (xa(n + 1) − fa(①(n), ♣))2 +

errorunitarity

• k

N

• n

|g1(①(n)) − 1|2 + k

N

• n

|g2(①(n)) − 1|2, where ◮ rn ≡ n ◮ yl are measurements (Pz,1, Pz,2) ◮ xa are state variables (Px,1, Py,1, Pz,1, Px,2, Py,2, Pz,2) ◮ ❢ is the discretized model ◮ p are unknown parameters.

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The cost function: A0 =

errormeasurement

• j

L

• l

Rl

m

2 (yl(n) − xl(n))2 +

errormodel

• N−1
• n

D

• a

Ra

f

2 (xa(n + 1) − fa(①(n), ♣))2 +

errorunitarity

• k

N

• n

|g1(①(n)) − 1|2 + k

N

• n

|g2(①(n)) − 1|2, Details ◮ Optimization at all points simultaneously, to eliminate the problem inherent in forward (or reverse) integration. ◮ Global minimum? → Annealing in coefficient Rf (Ye et al. 2015). ◮ Variational approach to minimization: Seek the path ❳ 0 = {①(0), ①(1), . . . , ①(n), ♣} in state space on which A0 attains a minimum value:

◮ ∂A0(❳) ∂❳

• ❳=❳ 0 = 0

◮ ∂2A0(❳) ∂❳ 2

• ❳=❳ 0 > 0.

◮ Open-source Interior-point Optimizer (Ipopt) (W¨ achter 2009).

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Next

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First, while retaining an independent consistency check: ◮ Determine whether a measurement of one neutrino contains information about past interactions with unmeasured neutrinos. ◮ Add more complicated potentials inside the envelope, and impose constraints on values of variables within the envelope at particular locations. Then let optimization lead: add a reflecting surface to include back-scattering.

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Thank you

Co-authors: Henry Abarbanel, George Fuller, Lucas Johns, Chad Kishimoto, Amol Patwardhan Collaborators & consultants: Shashank Shalgar, Baha Balentekin, Nirag Kadakia, Paul Rozdeba, Sasha Shirman, Jingxin Ye Thank you to Henry Abarbanel and the organizers of ISDA 2019! Henry Abarbanel George Fuller Amol Patwardhan Luke Johns Chad Kishimoto

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References ◮

Abarbanel, H.D.I., Predicting the Future: Completing models of Observed Complex Systems. Springer Science & Business Media (2013)

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Armstrong, E., Patwardhan, A. V., Johns, L., Kishimoto, C. T., Abarbanel, H. D., Fuller, G. M. An

• ptimization-based approach to calculating neutrino flavor evolution. Physical Review D, 96(8), 083008

(2017)

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Cherry, J. F., Carlson, J., Friedland, A., Fuller, G. M., Vlasenko, A. Neutrino scattering and flavor transformation in supernovae. Physical review letters, 108(26), 261104 (2012)

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Kalnay, E. Atmospheric modeling, data assimilation and predictability. Cambridge university press (2003)

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Mikheyev, S. P. and Smirnov, A. Y. , Yad. Fiz. 42 (1985)

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Quinn, J.C. and Abarbanel, H.D.I., Quarterly Journal of the Royal Meteorological Society, 136(652), 2010

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W¨ achter, A. Short tutorial: getting started with ipopt in 90 minutes. In Dagstuhl Seminar Proceedings. Schloss Dagstuhl-Leibniz-Zentrum f¨ ur Informatik (2009)

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Wolfenstein, L., Phys. Rev. D 17, 2369 (1978)

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Ye, J., Kadakia, N., Rozdeba, P. J., Abarbanel, H. D. I., Quinn, J. C. Improved variational methods in statistical data assimilation. Nonlinear Processes in Geophysics, 22(2), 205-213 (2015)

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Additional slides

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Final equations of motion in terms of polarization vectors P, which incorporate vacuum, matter, and neutrino-neutrino coupling Hamiltonians: dPx,1 dr = (∆ cos 2θ − V (r))Py,1 + µ(r)(Py,2Pz,1 − Pz,2Py,1), dPy,1 dr = −(∆ cos 2θ − V (r))Px,1 − ∆ sin 2θPz,1 + µ(r)(Pz,2Px,1 − Px,2Pz,1), dPz,1 dr = ∆ sin 2θ Py,1 + µ(r)(Px,2Py,1 − Py,2Px,1), dPx,2 dr = (∆ cos 2θ − V (r))Py,2 + µ(r)(Py,1Pz,2 − Pz,1Py,2), dPy,2 dr = −(∆ cos 2θ − V (r))Px,2 − ∆ sin 2θPz,2 + µ(r)(Pz,1Px,2 − Px,1Pz,2), dPz,2 dr = ∆ sin 2θ Py,2 + µ(r)(Px,1Py,2 − Py,1Px,2), where: V (r) = C r3 , µ(r) = Q r3

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Data assimilation (DA)

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◮ is the use of information contained in experimental data to estimate unknown model parameters, where the test of success is the prediction;

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Data assimilation (DA)

:

◮ is the use of information contained in experimental data to estimate unknown model parameters, where the test of success is the prediction; ◮ can inform experimental design, by determining which measurements must be made to complete a model;

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Data assimilation (DA)

:

◮ is the use of information contained in experimental data to estimate unknown model parameters, where the test of success is the prediction; ◮ can inform experimental design, by determining which measurements must be made to complete a model; ◮ is used widely in the geosciences and weather prediction, where the available data are sparse and where the corresponding model consists of degrees of freedom coupled in a nonlinear manner (e.g. Kalnay 2003);

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Data assimilation (DA)

:

◮ is the use of information contained in experimental data to estimate unknown model parameters, where the test of success is the prediction; ◮ can inform experimental design, by determining which measurements must be made to complete a model; ◮ is used widely in the geosciences and weather prediction, where the available data are sparse and where the corresponding model consists of degrees of freedom coupled in a nonlinear manner (e.g. Kalnay 2003); ◮ performs best with strongly-coupled nonlinear sets of model equations;

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Data assimilation (DA)

:

◮ is the use of information contained in experimental data to estimate unknown model parameters, where the test of success is the prediction; ◮ can inform experimental design, by determining which measurements must be made to complete a model; ◮ is used widely in the geosciences and weather prediction, where the available data are sparse and where the corresponding model consists of degrees of freedom coupled in a nonlinear manner (e.g. Kalnay 2003); ◮ performs best with strongly-coupled nonlinear sets of model equations; ◮ can be formulated as an integration-blind optimization procedure.

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The model The model F is defined by the evolution of D state variables xa (a = 1, 2, . . . , D): dxa(r) dr = Fa(①(r), ♣), where ♣ are unknown model parameters.

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The model The model F is defined by the evolution of D state variables xa (a = 1, 2, . . . , D): dxa(r) dr = Fa(①(r), ♣), where ♣ are unknown model parameters.

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The model The model F is defined by the evolution of D state variables xa (a = 1, 2, . . . , D): dxa(r) dr = Fa(①(r), ♣), where ♣ are unknown model parameters. The corresponding physical system We have L measurements yl (l = 1, 2, . . . , L), each corresponding to one of the D model state variables (e.g. polarization vector component Pz).

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The model The model F is defined by the evolution of D state variables xa (a = 1, 2, . . . , D): dxa(r) dr = Fa(①(r), ♣), where ♣ are unknown model parameters. The corresponding physical system We have L measurements yl (l = 1, 2, . . . , L), each corresponding to one of the D model state variables (e.g. polarization vector component Pz). Typically, L ≪ D.

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The model The model F is defined by the evolution of D state variables xa (a = 1, 2, . . . , D): dxa(r) dr = Fa(①(r), ♣), where ♣ are unknown model parameters. The corresponding physical system We have L measurements yl (l = 1, 2, . . . , L), each corresponding to one of the D model state variables (e.g. polarization vector component Pz). Typically, L ≪ D. For us: D = 6 (Px,1, Py,1, Pz,1, Px,2, Py,2, Pz,2); L = 2 (Pz,1, Pz,2)