1. N EUTRINO O SCILLATION W ORKSHOP Conca Specchiulla 9-16 Sept. 2006 Anna Maria Rotunno Dip. Di Fisica & Sez. INFN di Bari Geo-Neutrino: Theoretical Aspects Based on: G.L. Fogli, E. Lisi, A. Palazzo, A.M. Rotunno, GeoNeutrinos: an approach to their uncertainties and correlations, to appear in Earth, Moon and Planets; & long preprint in preparation (2006)

2. Contents - Introduction to Geo-Neutrinos - The Geo-Neutrino Source Model (GNSM) - Covariance and Correlation - Forward Analysis - Backward Analysis - Conclusions and Prospects Purpose of this Work - We show predictions about several experiments (“forward approach”) and how future data can constrain the error matrix of the model (“backward approach”). Geo-Neutrinos emitted by heat producing elements (U, Th, K) can probe Earth interior. Their fluxes present large and correlated uncertainties. Handling them is difficult but necessary, if we want to quantify how future data can reduce errors. - We propose an approach in terms of covariance matrices. - We briefly discuss the construction of a tentative Geo-Neutrino Source Model (GNSM) describing U, Th, K abundances in Earth reservoirs.

3. Introduction to Geo-Neutrinos

4. T = 1500 ºC (Mg, Fe, Al) (Al,Si)O 2 pervoskite CaSiO 2 pervoskite, (Fe,Mg)O Fe-Ni, Si, S, O, H, etc. T = 4300 ºC T = 3700 ºC T = 4000 ºC What do we know about Earth Interior? -Seismology: based on sound velocity measurement from seismic data - reconstructs density profile throughout the Earth - infers crust-mantle-core layer structure - does not reach deep Earth -GeoChemistry: based on direct sampling - gives direct information on chemical composition of crust and upper part of mantle

5. -The mantle convects even though it is “solid”. - Main issue today: Whole or layered mantle convection?

6. - Earth’s Global Heat: - 30 – 45 TW: not well constrained due to scarce oceanic sampling and model dependence - probably 40 – 60% has radiogenic origin: mainly from decays of 238 U, 232 Th, 40 K (trace elements) inside crust and mantle Geo-Neutrinos from radioactive decays of 238 U, 232 Th, 40 K trace elements in crust and mantle of Earth bring to surface information about: - the whole planet - its radioactive contents - energetics and thermal history Where are radioactive elements located? anti-neutrino energy E (MeV) 238 U series 232 Th series 40 K counts/MeV/parent

7. “Standard Model” of Earth Global Composition in Trace Elements Original Earth global composition similar to Carbonaceous Chondrites (CI) - Escape of volatile elements (e.g. K) - Crust/Mantle(Upper/Lower Mantle) Differentiation - Refractory Lithophile elements (e.g. U, Th) differently distributed in crust and mantle Oldest meteorites ≡ undifferentiated rock and metal mixture Today’s Earth composition is not CI ! Planetary Evolution: Low (<1200 K) condensation temperature High (>1400 K) condensation temperature Preferentially embedded in rocks rather than iron Bulk Silicate Earth (BSE) Model Th/U abundance ratio is 3.9 i.e. before crust/mantle differentiation - “primitive mantle” - present crust+mantle system describes - Earth Refractory Elements in chondritic proportions - U, Th, K absent in the Earth core assumes Constraints: Direct sampling (crust & upper mantle) & Neutrino Geophysics (in the future) 4.5 GY ago

8. A recent new field in Neutrino Physics: Geo-Neutrino detection by Liquid Scintillator 2005: first Geo– e observation at KamLAND KamLAND Coll., Nature 436,499 (2005) Some Important Facts: - Observable Geo- e events: from U, Th decay only - e from K decay below threshold for detection -  Th ( e ) &  U ( e ) in KamLAND weighted by 1/L 2 - U, Th, K more abundant in the crust than in the mantle - Assumptions on the relative Th, U (and K) abundances need to be explicitly reported - Earth science constraints - Uncertainty evaluation Sea of Japan Japan Trench KamLAND our reanalysis of Kamland data In this context we illustrate our approach to uncertainties and correlations Th/U = 3.8

9. Question 1: What do we really know about U, Th, K abundances? Question 2: What do we expect to know from geo- data? Usually advertised Goal: measure the Earth Radiogenic Heat But…. … based on future U and Th geo- flux measurements, we might say something more (e.g., about mantle convection) We report about a systematic approach to include U, Th and K abundance uncertainties and correlations in reservoirs (“Geo-Neutrino Source Model”)

10. The Geo-Neutrino Source Model (GNSM)

11. Purpose: to incorporate the best available knowledge of U, Th and K distributions inside Earth. Our GNSM geometry is based on: - PREM model (Dziewonsky & Anderson, 1981): spherical symmetry of Earth below crust - CRUST 2.0 model (G. Laske et al., 2001): crustal characterization on a 2°  2° grid. Global reservoirs: around detector sites that are: - Japan (KamLAND) 13 crustal tiles - Hawaii - BOREXINO - SNO - LENA 9 crustal tiles GNSM Structural Details - core - lower mantle - upper mantle - continental crust - oceanic crust Local reservoirs: lower/middle/upper crust Local composition may be ≠ from global composition (in terms of U, Th, K)

12. {a i S } i=1,…N (S=U,Th,K) a = {a i } i=1,…3N, N = number of reservoirs set of abundances (i.e. abundance vector) of reservoirs, a i : a i = a i ±  i and [  2 ] ij =  ij  i  j where a i = central value,  2 = covariance matrix,  = error correlation matrix. For abundance values and references, we refer to G.L. Fogli, E. Lisi, A. Palazzo, GeoNeutrinos: an approach to their uncertainties and correlations, to appear in Earth, Moon and Planets Entries for the above equations: -BSE Model: gives global constraints on elemental abundances (“mass balance constraints”) -Vertical crust structure: relevant within local reservoirs -Missing information is supplied by educated guesses, whenever possible, or arbitrary but explicit assumptions, when unavoidable -“local” abundance fluctuations assumed to be decoupled from “global” abundance uncertainties GNSM geochemical details:

13. An example: U, Th, K uncertainties and correlations in BSE For Uranium, Thorium: - a BSE /a CI expected to be the same for all Refractory Lithophile Elements not volatilized during Earth formation (e.g. U, Th, Al) - Benchmark: Alluminiummore abundant than trace elements U, Th We obtain: a Th BSE = a Th CI (a Al BSE /a Al CI ) a U BSE = a U CI (a Al BSE /a Al CI )  U,Th BSE = 0.936 (U,Th) correlation Sources: - CI meteoritic data (1988-2003) - recent BSE models: McDonough & Sun (1995) Allegre et al. (2001) Palme & O’Neill (2003) - relative U & Th abundances in CI from Ref. Rochall & Jochum (1993), Goreva & Burnett (2001) For Potassium: - K not constrained by meteorites, because moderately volatile - we conservatively increase the K/U ratio error usually quoted in the geochemical literature (Ref. Jochum et al, 1983) because unrealistic We obtain a K BSE,  K,Th BSE = 0.648 &  K,U BSE = 0.701 Similarly, we survey all the available literature for upper mantle (UM), continental crust (CC) and oceanic crust (OC) to estimate abundances (central values), errors and correlation

14. - LM abundance obtained by subtraction: LM = BSE–UM–CC–OC - Derivation of errors (by propagation) and correlations Qualifying result of our work Global Reservoirs (correlated) OC CC UM LM OC CC UM LM CC = continental crust OC = oceanic crust UM = upper mantle LM = lower mantle (core is excluded) Local Reservoirs (uncorrelated) i-th reservoir U Th K U Th K ≡ LM abundances anti-correlated with the other reservoirs because of subtraction Structure of correlation matrix of abundance Lower Mantle (LM) not accessible! Derived by mass balance constraints “local” fluctuations have nothing to do with “global” estimates

15. Geo-Neutrino Source Model for Global Reservoirs BSE CC OC UM LM U 21.9×10 -9 ± 14 %1+.936+.701000000000+.908+.893+.690 Th 82.1×10 -9 ± 14 %1+.648000000000+.850+.954+.638 K 26.3×10 -5 ± 21 %1000000000+.636+.618+.985 U 1.46×10 -6 ± 17 %1+.906 000000-.409-.263-.146 Th 6.29×10 -6 ± 10 %1+.595000000-.371-.291-.096 K 1.62×10 -2 ± 10 %1000000-.371-.173-.161 U 1.00×10 -7 ± 30 %1+.906+.868000-. 012-.007 Th 2.20×10 -7 ± 30 %1+.764000-.011-.001-.006 K 1.25×10 -3 ± 28 %1000-.010-.006-.008 U 3.95×10 -9 ± 30 %1+.906+.868-.093-.065-.058 Th 10.8×10 -9 ± 30 %1+.764-.084-.071-.051 K 5.02×10 -5 ± 28 %1-.081-.054-.066 U 17.3×10 -9 ± 30 %1+.924+.692 Th 60.4×10 -9 ± 30 %1+.640 K 21.7×10 -5 ± 28 %1 Reser. Elem. Abund. ± 1  U Th K U Th K U Th K U Th K U Th K BSE CC OC UM LM Geo-Neutrino Source Model (GNSM): Abundances, errors and correlations of radiogenic elements (U, Th, K) in global reservoirs Qualifying result of our work NEW Similar to previous work by Enomoto et al., Fiorentini et al. Numerical Results allows well-defined statistical analyses

16. - All relevant observables and constraints can be expressed as linear functions of such abundances (with known coefficients) - (U,Th,K) abundances within a given reservoir are typically positively correlated - (U,Th,K) correlations among different reservoirs can take any value > 0 local abundances  ij < 0 complementary reservoirs ~ 0 decoupled reservoirs - Measured Geo-Neutrino event rates (R U, R Th ) are anticorrelated R Th (TNU) Solid line: KamLAND data fit Dashed line: Adapted Gaussian R U = 12.5 ± 48.9 TNU R Th = 34.7 ± 28.5 TNU  (U,Th) = - 0.645 1 TNU = 1 event/year/10 32 protons R U (TNU) our reanalysis of Kamland data Negative correlation due to experimental sensitivity to R U +R Th rather than R U and R Th separately Covariance approach relevant for GeoNeutrino physics because:

17. Forward Analysis: Event Rates at KamLAND - GNSM compatible with data at 1 . - Data do not constrain model yet. - Background reduction and much higher statistics required. Dashed Line: KamLAND data R U = 12.5±48.9 TNU R Th = 34.7±28.5 TNU  (U,Th) = -0.645 Solid Line: GNSM R U = 24.9±2.0 TNU R Th = 6.7±0.5 TNU  (U,Th) = 0.901 Th/U = 3.8

18. KamiokaGran SassoSudburyPhyasalmiBaksanHawaii Kamioka31.6 ± 2.51.000.720.650.630.620.83 Gran Sasso40.6 ± 2.91.000.710.730.700.64 Sudbury47.9 ± 3.21.000.690.650.55 Pyhasalmi49.9 ± 3.51.000.690.48 Baksan50.7 ± 3.41.000.51 Hawaii13.4 ± 2.21.00 Site Rate ± 1  Correlation Matrix of GNSM predictions (TNU) Forward Analysis: Total Event Rates (including oscillations) with errors and correlations at various detector sites all positively correlated (they measure in part the same flux)

19. Forward Analysis: Total Radiogenic Heat vs Total Event Rate at KamLAND GNSM R U+Th = 31.6 ± 2.5 TNU H U+Th+K = 21.1 ± 3.0 TW  (R,H) = +0.858 The ellipse selects the allowed band of total radiogenic heat around GNSM prediction

20. Two extremes: 1) homogeneous mantle: whole mantle convection, i.e. a LM = a UM 2) two-layered model: geochemically decoupled UM and LM LM with primitive abundances a LM = a BSE Within 3  : - a LM a UM (left panel) whole mantle convection - a LM a BSE (right panel) two-layered mantle model The two extreme cases are recovered at ± 3  in our GNSM GNSM central values: a UM < a LM < a BSE partial mantle convection Mantle Convection Problem : still debated today 1, 2, 3  homogeneous two-layered GNSM

21. In an optimistic future scenario with: - 6 detectors operative - U, Th event separate collection for 20 kton years - no background - no systematics + DATA In principle, it might allow to reject at >> 3  the case a LM = a UM (global mantle convection). Really relevant result in geophysics and geochemistry More realistic (or less optimistic) simulations of prospective data need to be performed. Backward Analysis: Hypothetical future Results about Mantle Convection partial convection preferred

22. We expect that a network of detectors in different points of the Earth’s continental and oceanic crust would be useful to: - REDUCE THE EXPERIMENTAL ERROR; - CONSTRAIN THE GNSM PARADIGM NEW EXPERIMENTS in sites with both LOWER & HIGHER FLUX - BOREXINO - LENA - Sudbury - Hawaii - Baksan We are currently studying the synergy of a world detector network from a quantitative viewpoint.

23. - We have presented a tentative Geo-Neutrino Source Model (GNSM) embedding a full error matrix for the (U, Th, K) abundances in relevant local and global reservoirs. It is based on published data (when available) and on supplementary assumptions (when needed). - Covariance analysis may provide a useful template for current and future studies. Applications of our approach have been given in terms of predictions for future experiments (forward propagations of errors) and of GNSM error reduction through prospective data (backward update). - We are still far from a satisfactory approach of this kind in (U, Th, K) geochemistry, due to intrinsic difficulties (large uncertainties, incomplete data, sometimes conflicting estimates, ecc.) - Interdisciplinary studies of more refined geochemical and geophysical Earth models and of future possible observations of Geo-Neutrino signals will be beneficial to Earth sciences. Conclusions and Prospects