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Variances and covariances of geoneutrino signals

Eligio Lisi, INFN, Bari, Italy

Neutrino Geoscience, Paris, 2015

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Report on work in progress with

Marica Baldoncini, Fabio Mantovani Bill McDonough, Virginia Strati

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Introduction

In the absence of direct mantle geoν data (from Hanohano-like or directional detectors), the only way to get the mantle signal is:

DATA – CRUST = MANTLE

Observed Estimated Inferred

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KamLAND & Borexino data + Crustal model:

2012*: MANTLE = 23 ± 10 TNU (~2.3σ)

2014**: MANTLE = 8 ± 6 TNU (~1.3σ)

Expect C.L. of ~1σ-2σ from each (future) experiment. Hope to reach >2-3σ by combining future data Since every tiny bit of information is precious, careful statistics & crust modeling are desirable!

*arXiv:1204.1932 (Fiorentini, Fogli, Lisi, Mantovani & Rotunno); close to the value reported yesterday from Borexino 2015 alone! **arXiv:1310.1961 (Ludhova & Zavatarelli)

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Ways to improve statistics & modeling (I)

Always deal with (Th, U) signals, not summed Th+U [There is more info in the joint (Th,U) distributions] Must take into account (Th,U) covariances. E.g., KamLAND:

(Th,U) data errors: negative correlation. If one !, the other "

(Th,U) model errors: positive correlation. Both !! or ""

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‣Analysis - Rate+Shape+Time Analysis (1)NU vs NTh

0 50 100 150 2000

50

100

150

200

UN

ThN

68.3%

95.4%

99.7%earth model predictionEPSL 258, 147 (2007)

ratio fixed

best-fit (NU, NTh) = (116,8)

0 50 100 1500

2

4

6

8

10

UN

2r

6

m1

m2

m3

0 50 100 1500

2

4

6

8

10

ThN

2r

6

m1

m2

m3

NU

NTh

NU 0 signal : rejected at 2.6σ (99.0%)

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Reminder about bivariate (gaussian) errors

var(x)=(σx)2

var(y)=(σy)2

cov(x,y)= ρ σx σy

E.g., error correlation ρ =

,

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Ways to improve statistics & modeling (II)

Combine information from more experiments:

D1-C1=M1 D2-C2=M2

D3-C3=M3 ...

For isotropic mantle (M=M1=M2=M3=...), the M error is then smaller than individual Mi errors But: combined subtraction is tricky, since the Ci errors are correlated via common crust #

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$ $

$

E.g., KamLAND, JUNO, JinPing, RENO-50 probe a common chunk of not-so-far-field crust

Crust signal errors for (i,j) sites

i,j “very near”

i,j “very far”

σi"

σj"

σj"

σi"

How far is “far”? In the context of geoneutrinos, the length scale dividing “near” and “far” is presumably a few hundred km, but...

$

ρ∼1 "

ρ<<1 "

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... in principle, it would be nice to anchor this length scale to point-to-point correlation functions for the world crust, i.e., global “variograms”

A local variogram for U outcrops over 2100 km2 in Sardinia (Ferrara et al group)

[In this data sample, the local correlation length scale is a few kilometers.]

Need to distinguish in some way “local” and “far-field” crust reservoirs

But, we have no “variogram analog” for big regions of the (deep) crust.

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Note: for local crust reservoirs, oscillations are not completely averaged out, and their δm2L/E dependence must be taken into account in estimating signals. Not completely negligible!

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Ways to improve statistics & modeling (III)

Try to better characterize the error distributions, e.g., normal vs log-normal.

Sudbury, local crust. Pink = tonalite

a(U) distribution is not Gaussian...

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Ways to improve the statistical approach (III)

Try to better characterize the error distributions, e.g., normal vs log-normal.

Sudbury, local crust. Pink = tonalite

... but log a(U) is reasonably Gaussian...

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Ways to improve the statistical approach (III)

Try to better characterize the error distributions, e.g., normal vs log-normal.

Sudbury, local crust. Pink = tonalite

... and log(Th,U) is a bivariate Gaussian

ρ=0.74

Correlation is often significant

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(Bivariate) lognormals have two good features... and a bad one: 1) empirically favored w.r.t. normal distributions in Earth sciences (whenever there are enough data to distinguish between them) 2) never lead to unphysical values of the argument <0 (no negative abundances or crustal fluxes <0) 3) but ... the combination of lognormals is not lognormal! (a significant complication in error propagation)*

(1) & (2) enough to prefer lognormal to normal distributions, despite the issue (3)

*multivariate Gaussian propagation was worked out in arXiv:physics/0608025

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Ways to improve statistics and modeling (IV)

Good practices...

$Rely on (large) data samples as a much as possible $If data are filtered/processed, describe procedure

$Declare explicitly all guess-estimates and assumptions

$If data are replaced by proxies, describe extrapolation

$Provide full, joint error distributions of predicted signals

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Good practices...

$Rely on (large) data samples as a much as possible $If data are filtered/processed, describe procedure

$Declare explicitly all guess-estimates and assumptions

$If data are replaced by proxies, describe extrapolation

$Provide full, joint error distributions of predicted signals

There is already a literature on the previous improvements (I)-(IV), but a more systematic approach is desirable, in order to refine the crustal model for subtraction from data:

DATA – CRUST = MANTLE

Ways to improve statistics and modeling (IV)

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Methodology (no numbers yet – work in progress)

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Methodology (no numbers yet – work in progress) Take the best current model for the global crust geometry (vertical and horiz.)

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Methodology (no numbers yet – work in progress) Take the best current model for the global crust geometry (vertical and horiz.) Identify local reservoirs around expt. sites (currently: Gran Sasso, Kamioka, Sudbury)

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Methodology (no numbers yet – work in progress) Take the best current model for the global crust geometry (vertical and horiz.) Identify local reservoirs around expt. sites (currently: Gran Sasso, Kamioka, Sudbury) Identify large sub-reservoirs in the “rest of the crust”

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Methodology (no numbers yet – work in progress) Take the best current model for the global crust geometry (vertical and horiz.) Identify local reservoirs around expt. sites (currently: Gran Sasso, Kamioka, Sudbury) Identify large sub-reservoirs in the “rest of the crust” In each of the above reservoirs, assign central values, errors and correlat. to log(Th) and log(U) abundances

(LogTh,LogU)

In each one:

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Methodology (no numbers yet – work in progress) Take the best current model for the global crust geometry (vertical and horiz.) Identify local reservoirs around expt. sites (currently: Gran Sasso, Kamioka, Sudbury) Identify large sub-reservoirs in the “rest of the crust” In each of the above reservoirs, assign central values, errors and correlat. to log(Th) and log(U) abundances [If needed, add volumetric uncertainties]

(LogTh,LogU)

In each one:

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Methodology (no numbers yet – work in progress) Take the best current model for the global crust geometry (vertical and horiz.) Identify local reservoirs around expt. sites (currently: Gran Sasso, Kamioka, Sudbury) Identify large sub-reservoirs in the “rest of the crust” In each of the above reservoirs, assign central values, errors and correlat. to log(Th) and log(U) abundances [If needed, add volumetric uncertainties] Declare all the assumptions underlying the above choices, for book-keeping and for future improvements

(LogTh,LogU)

In each one:

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Methodology (no numbers yet – work in progress) Take the best current model for the global crust geometry (vertical and horiz.) Identify local reservoirs around expt. sites (currently: Gran Sasso, Kamioka, Sudbury) Identify large sub-reservoirs in the “rest of the crust” In each of the above reservoirs, assign central values, errors and correlat. to log(Th) and log(U) abundances [If needed, add volumetric uncertainties] Declare all the assumptions underlying the above choices, for book-keeping and for future improvements Based on this model, calculate the Th and U oscillated crust rates and errors at each site. Provide full, joint distributions of their errors

Th (j)

U (i)

For both i=j (same site) and i≠j (different sites)

C.L. of crustal rates

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Methodology (no numbers yet – work in progress)

Th (j)

U (i)

For both i=j (same site) and i≠j (different sites)

C.L. of crustal rates The idea is to get not only a “reference crust model” but also an associated set of “reference error distributions” to be used in the geo-ν context. At the very least, get a “template” for model+errors. [Historically, this kind of improvements has characterized all “mature” fields of particle physics, including ν physics] Although the path to follow is clear, its practical implementation is not obvious, but it offers a chance to revisit and learn more on geo-chemical/physical features of both local and global crust. A few highlights and issues follow#

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!ID!! !Reservoir!

Central"3le"

1" Cenozoic"terrigenous"units"

2" Meso<Cenozoic"Basinal"Carbonate"units"

3" Mesozoic"Carbonate"units"

4" Upper"crust"

5" Lower"crust"

Rest"of"the"region"

6" Sediments"

7" Upper"crust"

8" Lower"crust"

Borexino local reservoirs (N=8)

In part, need to rely on “proxies” (crust outcrops taken from other regions of Italy). Extrapolation errors/biases?

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ID!! !Reservoir!

Local"region"

9" Upper"crust"

10" Middle"crust"

11" Lower"crust"

KamLAND local reservoirs (N=3)

Not enough data to really characterize more than three independent reserv.

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SNO+ local reservoirs (N=7)

Paleozoic sediments (Great Lakes) (12)

Hurionan Supergroup: sedimentary rocks (13)

Granite or granodioritic intrusions (14)

Sudbury Igneous Complex (15)

Volcanics and metavolcanics rocks (16)

Tonalite and tonalite gneiss (17)

Central Gneiss Belt (18)

Middle crust (19)

Lower crust (20)

+ detailed vertical structure

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Volcanics and metavolcanis rocks of SNO area.

Bimodal distribution in bilogarithmic plot (% mafic vs felsic #)

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Volcanics and metavolcanis rocks of SNO area.

Need to account for the effect of two populations, for their relative weights, for outliers... (also in some other reservoirs)

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Rest of the crust or “far field crust”

common to all expts!

(N=7)

!ID!! !Reservoir!

Con3nental"crust"

21" Sediments"SED"

22" Upper"crust"UC"

23" Middle"crust"MC"

24" Lower"crust"LC"

25" Con3nental"lithospheric"mantle"LM"

Oceanic"crust"26" Sediments"SED"

27" Crust"C"

Currently, our crust model is partitioned into a total of twenty-seven reservoirs (assumed to be indepedent). [Working on assignment of central values, errors and correlations of log(Th), log(U) in each of them.]

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Revisiting the (far-field) upper crust

Is it “homogeneous” for practical geo-ν purposes? Apparently yes. E.g., no statistically significant difference between: - U & Th abundances in archean and non-archean UC (not shown) - U & Th abundances in different kinds of UC “mixtures” #

Natural!mixers!of!upper!crust!

Ice!

Wind!

Water!

Tillites!

Loess!

Shales!

Consistent!results!from!=llites,!loess!and!shales!

Graph from R. Rudnick at Goldschdmit 2013 - Florence (in collaboration with Richard Gaschnig, Bill McDonough, Yu Huang)

(Th,U)# (logTh, logU)

Data suggest a bivariate lognormal for the overall UC (apart from shales outliers with very low abundances)

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Final estimates for the joint (Th,U) crustal signals... ...will come out from sums of lognormals – which are not lognormal! [but will always have upper tails longer than lower tails]

Brute-force approach is needed. Two ways: (work in progress) (1) MonteCarlo simulations. However: highly inefficient for sampling the most interesting part (=tails). Tricky to get joint MC distributions of signals in different sites. (2) Maximum-likelihood approach (χ2 minimization). Tricky to minimize over ~O(50) correlated & nonlinear variables, but maybe better than (1). Moreover, it allows further contraints [e.g., from heat-flow or chondrite data]

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Conclusions

Th

U

C.L. of crustal rates Hope to get plots like this, with (x,y) for the same site or for different sites...

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Conclusions

Th

U

C.L. of crustal rates Hope to get plots like this, with (x,y) for the same site or for different sites...

...with applications to real BX+KL data and to future data...

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Conclusions

Th

U

C.L. of crustal rates Hope to get plots like this, with (x,y) for the same site & for different sites...

...with applications to real BX+KL data and to future data...

...for the next Neutrino Geoscience meeting!

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Thank you for your attention.