Maïa Kuga

Bernard Marty

Yves Marrocchi

Laurent Tissandier

Synthesis of refractory organic matter in the ionized gas phase of the Solar Nebula

Refractory organic compounds are ubiquitous in primitive chondrites and cometary samples, though their origin is poorly understood. Those organic compounds are the main host of primordial noble gases, known as Q-gases, and nitrogen, which isotopic fractionations recorded physicochemical conditions of the solar system formation. Here, we report the characterization of organic compounds synthesized under ionizing conditions in a plasma setup from gas mixtures (H 2 (O)-CO-N 2 -Noble gases) reminiscent of the protosolar nebula composition. Ionization of the gas phase was achieved at temperature up to 1000 K. Synthesized solid compounds share chemical and structural features with chondritic organics, and trapped noble gases reproduce the elemental and isotopic Q-gases patterns. These results suggest that both the formation of chondritic refractory organics and the trapping of Q-gases took place simultaneously in ionized areas of the protoplanetary disk, via photon-and/or electron-driven reactions and processing. Thus synthesis of primitive organics might not have required a cold environment as often assumed, and could have occurred anywhere it is ionized in the disk, including in its warm regions. This scenario also supports N 2 photodissociation as the cause of the large nitrogen isotopic range in the solar system. solar nebula

Introduction

The solar system formed from the gravitational collapse of a dense core within a molecular cloud that led to the ignition of a central star surrounded by a protoplanetary disk. Protoplanetary disks are evolving and dynamic systems, in which complex chemistry occurs and results in the formation and agglomeration of solids. Among them, organic molecules are of great astrobiological interest because they may represent the very first building blocks of prebiotic molecules. Moreover, they are the main carriers of volatile elements (i.e., H, C, N, and noble gases) that were directly involved in shaping the atmospheres of rocky planets. Organic matter is ubiquitous in primitive solar system bodies (e.g., chondrites especially the carbonaceous ones, interplanetary dust particles -IDPs, cometary dust -Comet 81P/Wild 2 and Ultra-Carbonaceous Antarctic Micro-Meteorites -UCAMMs (1-3)). Most of meteoritic organics (up to 99%) are in the form of a refractory and insoluble macromolecular solid, commonly referred to as insoluble organic matter (IOM) [START_REF] Derenne | Model of molecular structure of the insoluble organic matter isolated from Murchison meteorite[END_REF]. Notably, IOM displays bulk D and 15 N isotope excesses relative to solar composition, which can reach extreme values at a microscopic scale [START_REF] Remusat | Enrichment of deuterium in insoluble organic matter from primitive meteorites: A solar system origin?[END_REF][START_REF] Busemann | Interstellar Chemistry Recorded in Organic Matter from Primitive Meteorites[END_REF]. These compositional and isotopic characteristics bear a unique record of the processes and environmental conditions of IOM synthesis, and, by extension, of processing of solid materials and volatile elements in the early solar nebula, but the message is still cryptic.

IOM is also the main carrier of the heavy noble gases (Ar, Kr and Xe) as well as a small fraction of He and Ne trapped in chondrites. The exact host phase of these elements -nicknamed Phase Q (7) -is not yet characterized but appears to be a part of, or at least chemically associated with refractory organic compounds [START_REF] Lewis | Host phase of a strange xenon component in Allende[END_REF][START_REF] Ott | Noble-gas-rich separates from the Allende meteorite[END_REF]. This Q component is ubiquitous in primitive chondrites [START_REF] Huss | The "normal planetary" noble gas component in primitive chondrites: Compositions, carrier, and metamorphic history[END_REF][START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF], IDPs and in Antarctic micrometeorites [START_REF] Spring | paper presented at the 45th Lunar and Planetary Science Conference[END_REF]. Thus, Q-gases may represent the most important noble gas reservoir outside the Sun at the time of solid accretion in the protosolar solar nebula (PSN). Q-gases are elementally and isotopically fractionated relative to the solar composition (as inferred by the analysis of solar wind composition, [START_REF] Asplund | The Chemical Composition of the Sun[END_REF][START_REF] Heber | Isotopic Mass Fractionation of Solar Wind: Evidence from Fast and Slow Solar Wind Collected by the Genesis mission[END_REF][START_REF] Pepin | Xenon and krypton isotopes in extraterrestrial regolith soils and in the solar wind[END_REF][START_REF] Meshik | Heavy noble gases in solar wind delivered by Genesis mission[END_REF]), in favor of heavy elements and isotopes, by about 1%/amu for Xe isotopes [START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF]. So far, only plasma experiments involving noble gas ionization were able to fractionate them elementally and isotopically to extents comparable to those of Q [START_REF] Frick | the 10th LPSC[END_REF][START_REF] Dzizckniec | paper presented at the 12th Lunar and Planetary Science Conference[END_REF][START_REF] Fukunaga | Vapor-growth carbon and the origin of carbonaceous material in ureilites[END_REF][START_REF] Marrocchi | Adsorption of xenon ions onto defects in organic surfaces: Implications for the origin and the nature of organics in primitive meteorites[END_REF][START_REF] Hohenberg | Active capture and anomalous adsorption: New mechanisms for the incorporation of heavy noble gases[END_REF]. Both Q-gases and IOM are intimately related and ubiquitous in primitive solar system bodies, likely pointing to a common and pre-accretion origin.

Several processes for IOM synthesis have been proposed and/or studied experimentally, reflecting the variety of astrophysical regions where organosynthesis may occur. Those include: (i) cold scenarios (e.g., below 40 K) via UV-induced grain surface polymerization of organic molecules in the icy mantles of dust grains, either in the interstellar medium (ISM) or in the outer solar nebula [START_REF] Bernstein | Organic compounds produced by photolysis of realistic interstellar and cometary ice analogs containing methanol[END_REF][START_REF] Ciesla | Organic Synthesis via Irradiation and Warming of Ice Grains in the Solar Nebula[END_REF]; (ii) warm or high temperature scenarios (e.g., 300 K or above) such as Fischer-Tropsch-like reactions onto metal grains in the inner part of the solar nebula [START_REF] Nuth | A Self-Perpetuating Catalyst for the Production of Complex Organic Molecules in Protostellar Nebulae[END_REF][START_REF] Llorca | Reaction between H2, CO, and H2S over Fe, Ni metal in the solar nebula: Experimental evidence for the formation of sulfur-bearing organic molecules and sulfides[END_REF]; or (iii) organosynthesis onto the parent body of chondrites via polymerization of interstellar formaldehyde [START_REF] Cody | Establishing a molecular relationship between chondritic and cometary organic solids[END_REF][START_REF] Kebukawa | Exploring the Potential Formation of Organic Solids in Chondrites and Comets through Polymerization of Interstellar Formaldehyde[END_REF]. Laboratory analogues reproduce partially the compositional features of chondritic/cometary refractory organics, but generally fail to reproduce the large D and 15 N excesses observed in inner planets, meteorites and comets (relative to the PSN composition), and do not address the elemental and isotopic fractionations of noble gases.

In this work, we present a plasma experiment designed to produce refractory solid organics from mixtures of C-and Nbearing gases representative of the composition of the solar nebula (i.e, CO, N 2 with addition of noble gases). This is the

Significance

In this paper, we report an experimental investigation of the origin of refractory organics as well as of primordial noble gases, known as Q-gases, in primitive chondrites. We designed and performed a plasma experiment to examine for the first time organosynthesis and noble gas issues simultaneously. The synthetic compounds share chemical and structural features with chondritic organics, and trapped noble gases reproduce the elemental and isotopic characteristics of Q-gases. Our results demonstrate that Q-gases formed via ionization of noble gases within the early solar nebula and points to the most ionized areas of the nebula as the place were organosynthesis occurred, consistently with N 2 photodissociation at the origin of 15 N enrichments in chondritic organics. first integrative study that investigates organosynthesis and noble gas issues simultaneously. Refractory and insoluble organic compounds are produced at temperatures >500 K, and share compositional and structural similarities with IOM. These solids also contain trapped noble gases with Q-like elemental/isotopic fractionation. Our results suggest that noble gas-rich complex organic compounds could have been formed in the ionized regions of the protoplanetary disk via gas phase reactions and grain coagulation, even in the warm regions of the disk.

Reserved for Publication Footnotes

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Methods

Production of carbonaceous solids was undertaken in a microwave (2.45 GHz) plasma reactor, called Nebulotron ((27), Fig. S1), from gas mixtures composed of CO ± N 2 ± noble gases (+ traces of H 2 /H 2 O). Typical experiments were run with a total gas flow of 6-10 sccm at ∼ 1 mbar for two to six hours. Carbonaceous solid particles synthesis is initiated by the ionization and dissociation of CO/CO-N 2 by electronic and heavy particle processes in the gas phase at ∼ 800-1000 K, a temperature typical of microwave discharges [START_REF] Kuga | Nitrogen isotopic fractionation during abiotic synthesis of organic solid particles[END_REF][START_REF] Fridman | Plasma chemistry[END_REF]. The synthesized carbonaceous solids from CO ± noble gases and CO-N 2 ± noble gases experiments were recovered and analyzed by a combination of analytical techniques in order to address their nature and structure and their noble gas content (see SI for experimental and analytical details).

Results

Complex CHON compounds

Elemental analyses of the Nebulotron-synthesized solids reveal their organic nature, with carbon, hydrogen, oxygen and nitrogen representing 30-70 mol%, 23-35 mol%, 6-20 mol% and 1-22 mol%, respectively (Table S1). Solids produced from CO and CO-Noble gas mixtures are N-poor samples (i.e., N/C < 0.1), whereas those prepared from a CO-N 2 ± noble gases mixture contain significant amounts of nitrogen (i.e., N/C > 0.5), demonstrating the very efficient incorporation of nitrogen into the skeleton of the organic solids when N 2 is present in the flowing gas mixture. H 2 was not used in this study but H 2 /H 2 O impurities from the gas tanks and/or from reactor's surfaces contributed to the gas budget during the experiments. Interestingly, hydrogen is efficiently incorporated into molecular precursors of solid compounds and certainly helps to build the organic skeleton of the macromolecules recovered after the experiments.

The organic and complex nature of the Nebulotronsynthesized compounds is also revealed by their FTIR spectra (Fig. 1), which display characteristic bands of aliphatic and aromatic/oleifinic groups, and O-rich functional groups (hydroxyl, carbonyl and ether groups). The N-rich Nebulotron samples synthesized from CO-N 2 experiments present additional bands identifying N-rich functional groups (amine/amide, nitrile and imine groups). The C=O and C=C bands display the largest relative intensities, suggesting an unsaturated molecular structure for the Nebulotron-synthetized organics. The absence of the aromatic C-H stretching band (∼ 3050 cm -1 ) is consistent with high cross-linking and substitution of aromatics in those solids. The addition of noble gases in the gas mixture, as well as the increase of the electric discharge power, is expected to modify the electric equilibrium of the plasma and the gas temperature, thus changing the chemical reactions and/or their rates in the gas phase [START_REF] Fridman | Plasma chemistry[END_REF]. Such effects can account for differences in the relative intensities of the bands in the FTIR spectra of samples Neb-CO28, Neb-COGR29 and Neb-COGR30 (Table S1).

A low degree of carbon structural organization: HRTEM and Raman

High resolution transmission electron microscopy (HRTEM) images (Fig. 2) illustrate a gradual increase degree of organization of the carbon structure of the Nebulotron samples with the temperature of the discharge, as well as some heterogeneities at the nanoscale. The degree of structural organization ranges from highly disordered or "fluffly" (29) with very short fringes staked by two or three, to lamellar and porous nanostructure with fringes staked by more than five and longer than 10 nm. Consistently, qualitative Raman observations show that most of the Nebulotron samples display large D and G carbon bands in 514 and 244 nm Raman spectra (Figs. S2 andS3), confirming a low degree of carbon structural organization [START_REF] Bonal | Determination of the petrologic type of CV3 chondrites by Raman spectroscopy of included organic matter[END_REF][START_REF] Busemann | Characterization of insoluble organic matter in primitive meteorites by microRaman spectroscopy[END_REF]. Quantitative extraction of 244 nm Raman spectral parameters illustrate a limited heterogeneity among Nebulotron samples (Fig. 3), except for the I D /I G ratio (Fig. 3a), and for the FWHM-G, much larger for N-rich Nebulotron samples (Fig. 3b). The Nebulotron CO-9 sample falls out the trend, clearly reflecting a higher degree of structural organization consistent with HRTEM imaging (Fig. 2). This particular sample was produced with a higher electric discharge power compared to other samples, implying a higher gas temperature in the plasma setup (estimated > 800 K, [START_REF] Fridman | Plasma chemistry[END_REF]). This illustrates a gradual organization of the carbon structure with the plasma temperature.

Trapping, elemental and isotopic fractionation of heavy noble gases

Both Ar and Kr trapped in the synthesized solids are depleted by one order of magnitude relative to Xe and to the flowing noble gas mixture composition (Fig. 4a: the Ar depletion factor is an upper limit as the blank contribution for Ar was larger than the trapped Ar content). All the Nebulotron samples produced from CO-Noble gases ± N 2 mixtures present large Kr and Xe mass-dependent isotopic fractionations, where heavy isotopes are enriched relative to the lighter ones by 1.3±0.7%/amu and 1.0±0.4%/amu, respectively (Figs. 4b and4c). Both the elemental and isotopic fractionation factors are also comparable with those obtained in previous plasma noble gas - High resolution transmission electron microscopy images of two Nebulotron solids (CO-28 and CO-9) presenting different degrees of structural organization and of chondritic IOMs (adapted from ( 29)). Nebulotron samples CO-28 and CO-9 were produced at discharge power of 30 and 60 W, respectively. [START_REF] Frick | the 10th LPSC[END_REF][START_REF] Dzizckniec | paper presented at the 12th Lunar and Planetary Science Conference[END_REF][START_REF] Marrocchi | Adsorption of xenon ions onto defects in organic surfaces: Implications for the origin and the nature of organics in primitive meteorites[END_REF], and in Q (average of chondrites measured by [START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF]). The delta notation is used and the ratios are normalized to 132 Xe and to air composition for plasma experiments and to solar composition for Q. Typical errors (1σ) are shown for Q only for visibility. Xe-Q isotopes do not follow a linear trend with the mass contrary to Xe trapped in plasma-synthesized organics. This has been interpreted as the result of addition of other noble gas components found in meteorites such as Xe-HL, carried by nanodiamonds, s-process Xe and radiogenic 129 Xe to account for the 129 Xe excess displayed by Xe-Q [START_REF] Meshik | Heavy noble gases in solar wind delivered by Genesis mission[END_REF]. (c) Isotopic composition of Kr in the Nebulotron solids (grey range corresponding to 7 samples), in plasma synthesized solids by [START_REF] Frick | the 10th LPSC[END_REF] and in Q (average of chondrites measured by [START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF]). The delta notation is used and the ratios are normalized to 84 Kr and to air composition for plasma experiments and to solar composition for Q. Errors (1σ) are smaller than the symbol size for Kr-Q and are ± 5-40‰ for synthetic samples, depending on the isotopic ratio.

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Fig. 5. Schematic diagram of the solar nebula (0.1-100 AU), irradiated by stellar and interstellar UVs and X-rays. White lines represent electron fraction isolines. Both the electron fractions and the gas temperature scale are adapted from [START_REF] Walsh | Chemical Processes in Protoplanetary Disks. II. On the Importance of Photochemistry and X-Ray Ionization[END_REF]. Synthesis of 15 N-and noble gas-rich organics is possible in the most ionized areas of the disk, via photon and/or electrongas interactions. The source of 15 N-enrichment is the UV photodissociation of N 2 in the PDR region only. Dispersion of organics within the disk is possible thanks to turbulence and settlement. Organics may interact with ices in the cold and shielded middle part of the disk.

performed in ionizing conditions [START_REF] Frick | the 10th LPSC[END_REF][START_REF] Dzizckniec | paper presented at the 12th Lunar and Planetary Science Conference[END_REF][START_REF] Fukunaga | Vapor-growth carbon and the origin of carbonaceous material in ureilites[END_REF][START_REF] Marrocchi | Adsorption of xenon ions onto defects in organic surfaces: Implications for the origin and the nature of organics in primitive meteorites[END_REF]. The concentration of trapped Xe in the Nebulotron-synthesized solids is very high (from 2.0×10 -12 to 2.4×10 -9 mol/g), in agreement with results from [START_REF] Marrocchi | Adsorption of xenon ions onto defects in organic surfaces: Implications for the origin and the nature of organics in primitive meteorites[END_REF]. It is worth noting that without ionization, the trapping efficiency of Xe atoms in organics is very low and does not induces detectable isotopic fractionation [START_REF] Marrocchi | Experimental determination of the xenon isotopic fractionation during adsorption[END_REF]. The present data confirm that ionization of noble gases is the sine qua non condition for their efficient trapping and for the generation of elemental and isotopic fractionations.

Comparison of laboratory synthesized organics with refractory organic solids from primitive chondrites

The elemental analyses and FTIR data show that the N-poor Nebulotron samples (produced from CO + traces of H 2 /H 2 O ± noble gases) display a chemical composition very close to that of IOM from unheated chondrites (Table S1 and Fig. 1a). N-poor Nebulotron samples display elemental ratios in the range of those of IOM from primitive chondrites [START_REF] Alexander | The origin and evolution of chondrites recorded in the elemental and isotopic compositions of their macromolecular organic matter[END_REF] and their FTIR spectra are comparable to those of IOM. The main differences are: (i) the larger O/C ratio for N-poor synthesized samples compared to chondritic IOM, also illustrated in the FTIR spectra by the larger relative intensities of C=O and OH bands in the Nebulotron samples relative to IOM FTIR spectra, and (ii) the lower aliphatic content in the Nebulotron synthesized solids than in IOM, as shown by the overall weaker aliphatic stretching modes in FTIR spectra of the Nebulotron samples relative to IOM (Fig. 1a).

Raman and HRTEM studies also support common structural features between the Nebulotron-synthesized organics and IOM or other extraterrestrial organics (Figs. 2 and3). Both the carbon nanotextures (Fig. 2) and the Raman spectral parameters (Fig. 3) are comparable (although not exactly identical as illustrated by the D band parameters (Fig. 3a)). Interestingly, the Nebulotron samples plot close to the extraterrestrial primitive organics in the Raman spectral parameters space (Fig. 3). Altogether, these observations show that organosynthesis from very simple gas molecules at temperatures > 500 K can account for the production of complex and unsaturated organic compounds with carbon structural disorder fairly similar to the one of chondritic IOMs.

The elemental and isotopic fractionations of trapped Xe-Q and Kr-Q in meteorites relative to the solar composition [START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF] are well reproduced in the Nebulotron plasma experiment (Fig. 4). Ionization, which increases dramatically the trapping efficiency of noble gases in growing solids [START_REF] Marrocchi | Adsorption of xenon ions onto defects in organic surfaces: Implications for the origin and the nature of organics in primitive meteorites[END_REF] may also account for the extremely high noble gas content of Phase Q [START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF].

FTIR and 244 nm Raman spectra show that organics produced in the Nebulotron setup from a mixture of CO+N 2 (+ traces of H) are richer in nitrogen (up to 22 % mol) than chondritic IOM ((34, 35), max. 4 %mol. in CR, [START_REF] Alexander | The origin and evolution of chondrites recorded in the elemental and isotopic compositions of their macromolecular organic matter[END_REF]). This difference was already observed for laboratory analogues produced in a N 2 -CH 4 plasma (tholins from PAMPRE setup, [START_REF] Bonnet | Formation of analogs of cometary nitrogen-rich refractory organics from thermal degradation of tholin and HCN polymer[END_REF], Figs. 1 andS3). Interestingly, N-rich refractory organics have been identified in some IDPs [START_REF] Aléon | Nitrogen isotopic composition of macromolecular organic matter in interplanetary dust particles[END_REF] and in UCAMMs [START_REF] Dartois | UltraCarbonaceous Antarctic micrometeorites, probing the Solar System beyond the nitrogen snow-line[END_REF], which are believed to be of cometary origin. Both N-rich Nebulotron samples and PAMPREtholins display fairly similar FTIR spectra to the one of UCAMM (but with either too much oxygen or aliphatic abundance, Fig. 1).

The low nitrogen abundance in chondritic IOM may result from different processes. Organic precursors of IOM might have been initially N-rich, such as "cometary" organics. Processing in the disk or onto the parent-body might have resulted in selective loss of nitrogen [START_REF] Bonnet | Formation of analogs of cometary nitrogen-rich refractory organics from thermal degradation of tholin and HCN polymer[END_REF]. The fact that IOM still contains trapped noble gases with a common composition among different classes of meteorites may not support this possibility. However, one could argue that Phase Q is only a small and refractory fraction of IOM, so that nitrogen would have been lost from more labile sites than Phase Q. Alternatively, meteoritic IOM might have originated from synthesis of organics in a gas of non canonical composition, that is, in a gaseous environment depleted in nitrogen relative to C-bearing species, or from precursors having trapped preferentially C and H relative to N. Both scenarios have to be investigated in details and experiments are currently under way at CRPG.

Discussion

Noble gases and nitrogen isotopes: tracers of organic synthesis in ionized areas of the solar nebula

The origin of the refractory organic compounds found in chondrites is actively debated. Cold scenarios propose UVinduced photo-polymerization of organic molecules in the icy mantles of dust grains, either in the interstellar medium [START_REF] Bernstein | Organic compounds produced by photolysis of realistic interstellar and cometary ice analogs containing methanol[END_REF] or in the solar nebula [START_REF] Ciesla | Organic Synthesis via Irradiation and Warming of Ice Grains in the Solar Nebula[END_REF]. Low temperature conditions are consistent with models calling for ion-molecule reactions as the origin of D and 15 N enrichments (relative to solar values) of chondritic IOMs [START_REF] Yang | Interstellar organic matter in meteorites[END_REF][START_REF] Terzieva | The possibility of nitrogen isotopic fractionation in interstellar clouds[END_REF]. However, theoretical studies suggest that ion-molecule reactions are unable to produce the extreme 15 N-enrichments of several thousands of permils observed in some IOM hotspots, or under specific conditions only [START_REF] Terzieva | The possibility of nitrogen isotopic fractionation in interstellar clouds[END_REF][START_REF] Rodgers | Nitrogen superfractionation in dense cloud cores[END_REF]. Other scenarios favor an origin within the inner solar system via high temperature condensation reactions, including Fischer-Tropsch-like reactions [START_REF] Nuth | A Self-Perpetuating Catalyst for the Production of Complex Organic Molecules in Protostellar Nebulae[END_REF][START_REF] Llorca | Reaction between H2, CO, and H2S over Fe, Ni metal in the solar nebula: Experimental evidence for the formation of sulfur-bearing organic molecules and sulfides[END_REF] and condensation reactions assisted by electric discharge, such as in the present experiment [START_REF] Sagan | Tholins-organic chemistry of interstellar grains and gas[END_REF][START_REF] Morgan | A new mechanism for the formation of meteoritic kerogen-like material[END_REF][START_REF] Saito | Origin of organic globules in meteorites: Laboratory simulation using aromatic hydrocarbons[END_REF]. More recently, a postaccretion scenario was proposed, via interstellar formaldehyde polymerization onto the parent body [START_REF] Cody | Establishing a molecular relationship between chondritic and cometary organic solids[END_REF][START_REF] Kebukawa | Exploring the Potential Formation of Organic Solids in Chondrites and Comets through Polymerization of Interstellar Formaldehyde[END_REF]. However, none of them satisfy all the existing constraints. More specifically, the trapping of volatile elements into meteorite organics and their associated isotopic fractionation have received little attention even though these elements are essential tracers for deciphering the origin of IOM.

The similarity between the organic solids synthesized in the Nebulotron and chondritic IOM suggests that gas phase chemistry under ionizing conditions was able to form organic compounds in the disk of gas and dust surrounding the T-Tauri Sun. Plasma-synthesized organic dust also reproduces the noble gas elemental and isotopic fractionations of chondrites, contrary to the experiments simulating neutral or extremely weakly ionized cold environments via noble gas trapping into ices (e.

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dark clouds or mid-plane regions in the outer solar nebula (45)). However, our experiments do not reproduce the 15 N-enrichments observed in IOM, of ∼ 500-750 ‰ in for bulk IOM (up to 1,100 ‰ for CR chondrites, [START_REF] Alexander | The origin and evolution of chondrites recorded in the elemental and isotopic compositions of their macromolecular organic matter[END_REF]) and up to 5,800 ‰ in IOM hotspots (Bells CM2 chondrites, ( 6)), all data relative to the PSN nitrogen isotopic composition [START_REF] Marty | A 15N-poor isotopic composition for the solar system as shown by Genesis solar wind samples[END_REF]. We have shown in a previous study that nitrogen trapped in organics produced during plasma organosynthesis is moderately depleted in 15 N by about 15-25‰ [START_REF] Kuga | Nitrogen isotopic fractionation during abiotic synthesis of organic solid particles[END_REF]. Therefore, electron-driven reactions cannot account for solar system 15 N-enrichments. The only experiment done so far leading to strong 15 N enrichments involves the illumination of N 2 +H 2 gas mixtures by synchrotron-generated UV photons in the range 80-98 nm. Extreme 15 N enrichments of ∼ 2,200‰ in average, up to 13,400 ‰ at 90 nm, were obtained in forming the photoproduct NH 3 [START_REF] Chakraborty | Massive isotopic effect in vacuum UV photodissociation of N2 and implications for meteorite data[END_REF]. Such enrichments were interpreted as the result of selfand mutual-shielding of N 2 during photodissociation, associated with quantum perturbations that may explain the extreme 15 N excursion at 90 nm [START_REF] Chakraborty | Massive isotopic effect in vacuum UV photodissociation of N2 and implications for meteorite data[END_REF][START_REF] Clayton | Solar System: Self-shielding in the solar nebula[END_REF]. None of these processes can take place in an electron dominated environment because (i) ionization by electrons is volume-correlated thus precluding self or mutual shielding, and (ii) contrary to photons, plasma electrons have a continuum spectrum of energy that prohibits quantum effects [START_REF] Fridman | Plasma chemistry[END_REF]. The noble gases are not concerned by this limitation because their elemental and isotopic fractionations are not related to molecular dissociation as N 2 is.

Noble gas fractionation requires that their host phase formed in an ionizing environment. The 15 N enrichments of solar system objects and reservoirs permit to identify the mode of ionization. Indeed, experimental results point to UV-irradiated regions for the generation of 15 N-rich compositions [START_REF] Chakraborty | Massive isotopic effect in vacuum UV photodissociation of N2 and implications for meteorite data[END_REF]. Other modes of ionization (e.g., electrons, radioactivity) could have contributed to noble gas fractionation, while not affecting nitrogen isotopes. Thus variations of the N isotope composition in the solar system, together with near-constant noble gas fractionation, could be the result of heterogeneities in the respective modes of ionization.

Ionization in the solar nebula and organosynthesis

In protoplanetary disks, ionization sources are multiple, resulting in the common occurrence of plasma environments, i.e., partially ionized regions [START_REF] Fromang | The ionization fraction in α models of protoplanetary discs[END_REF]. Beyond 1 AU, ionization is mainly controlled by irradiation sources, such as X-ray and UV photons from the central star and/or from the interstellar radiation field [START_REF] Glassgold | X-ray ionization of protoplanetary disks[END_REF]. In the very inner part of the disk, temperatures are high enough (> 10 3 K) for thermal ionization to occur. Radioactive decay may also contribute but to a lesser extent [START_REF] Fromang | The ionization fraction in α models of protoplanetary discs[END_REF]. The electron fraction, depending on disk parameters, is expected to be in the range 10 -12 (in dead zones) -0.1 (in the most irradiated surface region close to the star), generating a stratified ionization structure of the disk [START_REF] Walsh | Chemical Processes in Protoplanetary Disks. II. On the Importance of Photochemistry and X-Ray Ionization[END_REF]. Hence, weakly to almost fully ionized plasmas constitute the main fraction of the PSN. In these ionized regions, electrons generated by direct cosmic rays and by Xray and UV photons are likely suprathermal. The exact energy spectrum of electrons is not known but it has been suggested that the mean energy of the secondary electrons may be around 20 eV or higher [START_REF] Gredel | Cosmic-ray-induced photodissociation and photoionization rates of interstellar molecules[END_REF][START_REF] S.-I. Inutsuka | Self-sustained ionization and vanishing dead zones in protoplanetary disks[END_REF], well above the dissociation and ionization thresholds of H 2 , CO, N 2 and noble gases. Therefore, these electrons may drive chemical reactions such as the ones occurring in laboratory plasmas and be responsible for the synthesis of organic compounds.

In the PSN, organosynthesis and isotopic fractionation of volatiles may have occurred either via electron-driven reactions, provided an electron fraction of at least the ones of laboratory plasmas (10 -6 or above), or via photon-driven reactions, the latter specifically for the generation of 15 N excesses by self-shielding. Such modes of ionized regions are consistent with the so-called photon-dominated region (PDR) and the warm molecular layer (WML) in disk models. The PDR, at the very surface of the disk, is necessarily the place were N 2 self-shielding occurs in disk because extreme UV photons may rapidly be absorbed by gas molecules and dust in the underlying layers. This region is almost fully ionized because of strong irradiation [START_REF] Walsh | Chemical Processes in Protoplanetary Disks. II. On the Importance of Photochemistry and X-Ray Ionization[END_REF], and is expected to be a place of synthesis by photochemistry, but also of destruction of organic molecules [START_REF] Henning | Chemistry in Protoplanetary Disks[END_REF]. As a consequence, the organic molecules that form in the PDR may have a short lifetime but, in any case, the carrier of 15 N-enrichments must survive to a certain extent. The WML, although partly shielded from photon irradiation, is largely ionized too and is likely to be a place where organosynthesis and noble gas fractionation occur. Vertical transport, predicted to occur in turbulent disk, may also help to conserve a part of the newly formed organics and the noble gas and nitrogen isotopic signatures by moving to deeper, colder and more shielded parts of the disk [START_REF] Ciesla | Organic Synthesis via Irradiation and Warming of Ice Grains in the Solar Nebula[END_REF]. As a matter of fact, the PDR an the WML temperatures are expected to be > 500 K and 100 -500 K, respectively [START_REF] Walsh | Chemical Processes in Protoplanetary Disks. II. On the Importance of Photochemistry and X-Ray Ionization[END_REF]. Such "high" temperatures are generally not considered for organosynthesis in disk models, but our experiment demonstrates that even higher temperatures promote, with satisfactory yields, the formation of compounds that present organic complexity that is required to account for IOM precursors. In numerical models of protoplanetary disk chemistry, gas phase reactions are generally efficient only in producing primary organic molecules such as HCO + , HCN, N 2 H + or H 2 CO, but not for producing more complex organic compounds [START_REF] Walsh | Chemical Processes in Protoplanetary Disks. II. On the Importance of Photochemistry and X-Ray Ionization[END_REF]. However, our experimental results challenge this view, and suggest that suprathermal electrons as a source of ionization and chemical reactivity may be underestimated. In the hot and thermally ionized part of the disk within 1 AU, organic synthesis via gas-phase reactions and fractionation of noble gases may occur as well. Simple organic molecules such as HCN, C 2 H 2 , CO 2 , H 2 O and OH have been observed within 3 AU in the protoplanetary disk of AA Tauri [START_REF] Carr | Organic molecules and water in the planet formation region of young circumstellar disks[END_REF], supporting the hypothesis of an active organic chemistry in the hot and ionized inner part of the disk. The densities invoked for protoplanetary disks in such active regions (∼ 10 8 cm -3 ) are orders of magnitude lower than those used in laboratory plasmas at ∼ 1 mbar (10 15 -10 16 cm -3 ), likely resulting in much less efficient synthesis of organic compounds. Nonetheless, the time during which those processes operate would be much longer in the disk than in a laboratory, likely counterbalancing the lower gas density.

The present experimental study provides a challenging but nonetheless plausible scenario predicting that precursors of chondritic IOM, Q-gases and the 15 N enrichments originate in a common environment from the interaction of photons and/or electrons with gas species (Fig. 5). Heterogeneities in the composition of refractory organics and radial variations of 15 N/ 14 N ratios among solar system objects and reservoirs with heliocentric distance could reflect heterogeneities in ionization processes. UV photon ionization and processing may have dominated in the outer solar system whereas electron-dominated reactions may have been prevalent in hotter regions of the inner solar system. Please review all the figures in this paginated PDF and check if the figure size is appropriate to allow reading of the text in the figure.

Fig. 1 .

 1 Fig. 1. FTIR baseline-corrected spectra of the carbonaceous solids synthesized from CO ± Noble gas (top) or from CO + N 2 ± Noble gas (bottom) mixtures in the Nebulotron. Shown for comparison: IOMs isolated from Orgueil, Cold Bokkeveld and PCA91008 ((34), top), tholins from the PAMPRE setup (PAMPRE-5%, produced from a mixture composed of 5% CH 4 in N 2 , (56), bottom), and organics from the UCAMM particle DC65 ((38), bottom). The main absorption bands are highlighted. All spectra are normalized to the intensity at 1600 cm -1 (C=C), in order to compare the relative intensities of the other bands.

Fig. 2 .

 2 Fig. 2.High resolution transmission electron microscopy images of two Nebulotron solids (CO-28 and CO-9) presenting different degrees of structural organization and of chondritic IOMs (adapted from (29)). Nebulotron samples CO-28 and CO-9 were produced at discharge power of 30 and 60 W, respectively.

Fig. 3 .

 3 Fig. 3. Quantitative extraction of spectral parameters from 244 nm (a and b) and 514 nm (c) Raman spectra. (a) D band and I D /I G ratio extracted from 244 nm Raman spectra. Chondritic IOM and UCAMM data are from (57) and (35), respectively. (b) G band Raman spectral parameters extracted from 244 nm Raman spectra. Chondritic IOM data are from (57). The N-rich synthetic samples (Nebulotron/PAMPRE) present a much larger G band as pointed in Fig. S3. N-poor Nebulotron solids present G band parameters that are in the area covered by chondritic IOMs within errors. (c) G band spectral parameters extracted from the 514 nm Raman spectra for N-poor Nebulotron samples (empty circles). Data for UCAMMs (35), IDP and Stardust grains (58-60), carbonaceous chondrites (30, 31, 57, 61) and ion-irradiated organics (62) are shown for comparison. Arrows illustrate the evolution of Raman spectral parameters with thermal alteration and irradiation with ions in the keV range. Nebulotron samples plot close to primitive extraterrestrial organics, with one sample (CO-9) showing thermal alteration.

Fig. 4 .

 4 Fig. 4. Noble gases. (a) Elemental fractionation of noble gases in the Nebulotron solids (black dots), in synthetic solids produced in plasma setups (grey area, (16-18)) and in chondritic IOM for Q-gases (red dots, (10)) relative to 132 Xe and to the gas reference (air composition for laboratory compounds and solar composition for Q-gases). Arrows show upper limits. (b) Isotopic composition of Xe in the Nebulotron solids (grey range corresponding to 15 samples), in other plasma synthesized solids[START_REF] Frick | the 10th LPSC[END_REF][START_REF] Dzizckniec | paper presented at the 12th Lunar and Planetary Science Conference[END_REF][START_REF] Marrocchi | Adsorption of xenon ions onto defects in organic surfaces: Implications for the origin and the nature of organics in primitive meteorites[END_REF], and in Q (average of chondrites measured by[START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF]). The delta notation is used and the ratios are normalized to132 Xe and to air composition for plasma experiments and to solar composition for Q. Typical errors (1σ) are shown for Q only for visibility. Xe-Q isotopes do not follow a linear trend with the mass contrary to Xe trapped in plasma-synthesized organics. This has been interpreted as the result of addition of other noble gas components found in meteorites such as Xe-HL, carried by nanodiamonds, s-process Xe and radiogenic 129 Xe to account for the 129 Xe excess displayed by Xe-Q[START_REF] Meshik | Heavy noble gases in solar wind delivered by Genesis mission[END_REF]. (c) Isotopic composition of Kr in the Nebulotron solids (grey range corresponding to 7 samples), in plasma synthesized solids by[START_REF] Frick | the 10th LPSC[END_REF] and in Q (average of chondrites measured by[START_REF] Busemann | Primordial noble gases in "phase Q" in carbonaceous and ordinary chondrites studied by closed-system stepped etching[END_REF]). The delta notation is used and the ratios are normalized to 84 Kr and to air composition for plasma experiments and to solar composition for Q. Errors (1σ) are smaller than the symbol size for Kr-Q and are ± 5-40‰ for synthetic samples, depending on the isotopic ratio.

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www.pnas.org ------Footline Author

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This study was funded by the European Research Council (FP/7 2007-2013, grant agreement 267255 to B.M.) and by the Programme National de Planétologie (PNP) through grants to Y. M. and to L. T. We warmly thank L. Zimmermann for help during noble gas analysis and G. Cernogora, E. Quirico, F.R. Orthous-Daunay for helpful comments, discussions and IR measurements..


Soledad Torres
email: soledad.torres@uv.cl

Ciprian A Tudor
email: tudor@univ-paris1.fr

Donsker type theorem for the Rosenblatt process and a binary market model

Keywords: 2000 AMS Classification Numbers: 60F17, 91B70 Rosenblatt process, random walk, stock price model, binary market model

In this paper, we prove a Donsker type approximation theorem for the Rosenblatt process, which is a selfsimilar stochastic process exhibiting long range dependence. By using numerical results and simulated data, we show that this approximation performs very well. We use this result to construct a binary market model driven by this process and we show that the model admits arbitrage opportunities.

Introduction

Long range dependence stochastic processes have been intensively used as models for different physical phenomena. First, these properties appeared in empirical studies in areas like hydrology and geophysics; more recently, they appeared to play an important role in network traffic analysis and telecommunications. As a consequence, efficient mathematical models based on long range dependence (or long memory) processes have been proposed in these directions.

The notions of long range dependence and selfsimilarity have also been considered in mathematical finance. An excellent survey on the different aspects of the appearance of the long range dependence in practice is the paper [START_REF] Cont | Long range dependence in financial markets[END_REF]. The debate on the presence of long memory in stock prices is actually not new. The idea that asset returns could exhibit long range dependence comes from Mandelbrot ([11]) and then observed in several empirical studies. We refer, among others, to [START_REF] Willinger | Long range dependence and stock returns[END_REF] for concrete examples and for interesting comments on this question. We also mention that some other authors rejected the idea of the presence of long memory in asset returns (see e.g. [START_REF] Lo | Long memory in stock market prices[END_REF]).

A rather general opinion is that long range dependence in financial models is strongly related to the presence of arbitrage. For example, in the case of market models driven by the fractional Brownian motion, this has been explicitly shown by Rogers ([17]) or Sottinen ([22]). In special situations, for example under transaction costs, arbitrage could be eliminated (see [START_REF] Guasoni | No arbitrage under transaction costs, with fractional Brownian motion and beyond[END_REF] or [START_REF] Salopek | Tolerance to Arbitrage[END_REF]). Different approaches, based on Wick-Itô calculus, have been developed in e.g. [START_REF] Biagini | Minimal variance hedging for fractional Brownian motion[END_REF], [START_REF] Hu | Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion[END_REF]. Models driven by long range dependence processes others than the fractional Brownian motion have been, from the stochastic calculus point of view, less considered. This is actually one of the motivations of our work: we propose a binary market model driven by a non-Gaussian selfsimilar process (called the Rosenblatt process) which exhibits long range dependence and we show that this type of model admits arbitrage opportunities. It is actually known that in general arbitrage-free models imply the fact that the price process is a semimartingale ( [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF]) but it has been recently proved in [START_REF] Coviello | Modeling financial assets without semimartingales[END_REF] that by modifying the class of admissible strategies, one can also consider arbitrage-free models driven by non-martingales.

To construct our binary market model based on the Rosenblatt process we need a Donsker type theorem to approximate in law this process by some disturbed two-dimensional random walks; this results could be useful by itself. In fact, this theorem extends a result by Sottinen [START_REF] Sottinen | Fractional Brownian m!otion, random walks and binary market models[END_REF] and represents a variant of the so-called Non Central Limit Theorem proved in [START_REF] Dobrushin | Non-central limit theorems for non-linear functionals of Gaussian fields[END_REF] and [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]. We mention that, since we are now in a non-Gaussian context, the proof of this result demands different techniques.

Our paper is organized as follows. In Section 2 we describe the basic properties of the Rosenblatt process. Section 3 contains the proof of the Donsker theorem to approximate weakly, in the Skorohod topology, the Rosenblatt process by walks. In Section 4, we introduce our binary market model which is showed to converge to the Black and Scholes model with Rosenblatt noise. We show that the model admits arbitrage opportunities and we construct a such opportunity.

Finally, in Appendix we present numerical results based on simulated data which show that the approximation method performs very well. Related numerical results can be found in [START_REF] Abry | Wavelet-based synthesis of the Rosenblatt process[END_REF] or [START_REF] Pipiras | Wavelet type expansion of the Rosenblatt process[END_REF].

Preliminaries

The Rosenblatt process appears as a limit in the so called Non Central Limit Theorem (see [START_REF] Dobrushin | Non-central limit theorems for non-linear functionals of Gaussian fields[END_REF] or [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]). We recall the general context. Consider (ξ n ) n∈Z a stationary Gaussian sequence with mean zero and covariance 1 such that its correlation function satisfies

r(n) := E (ξ 0 ξ n ) = n 2H-2 k L(n), (1)
with k ≥ 1 integer, H ∈ ( 1 2 , 1) and L is a slowly varying function at infinity . Denote by H m (x) the Hermite polynomial of degree m given by H

m (x) = (-1) m e x 2 2 d m dx m e -x 2
2 . Let g be a function such that E(g(ξ 0 )) = 0 and E(g(ξ 0 ) 2 ) < ∞. Suppose that g has Hermite rank equal to k; that is, if g admits the following expansion in Hermite polynomials

g(x) = j≥0 c j H j (x), c j = 1 j! E (g(ξ 0 H j (ξ 0 ))) , then k = min{j; c j = 0}.
Since E [g(ξ 0 )] = 0, we have k ≥ 1. The Non Central Limit Theorem ([6], [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]) says that the sequence of stochastic processes

1 n H [nt] j=1 g(ξ j )
converges as n → ∞, in the sense of finite dimensional distributions, to the process (called the Hermite process)

Z k H (t) = c(H, k) R k t 0   k j=1 (s -y i ) -( 1 2 + 1-H k ) +   dsdB(y 1 ) . . . dB(y k ), (2)
where x + = max(x, 0) and the above integral is a multiple Wiener-Itô stochastic integral with respect to a Brownian motion B(y)) y∈R (see [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF]) .

Let us list some basic properties of the Hermite processes.

• it exhibits long-range dependence (the covariance function decay at a power function at zero -"Joseph effect")

• it is H-selfsimilar in the sense that for any c > 0, (Z k H (ct)) = (d) (c H Z k H (t))
, where " = (d) " means equivalence of all finite dimensional distributions

• it has stationary increments, that is, the joint distribution of (Z k

H (t + h) -Z k H (h), t ∈ [0, T ]) is independent of h > 0.
• the covariance function is

E(Z k H (t)Z k H (s)) = 1 2 t 2H + s 2H -|t -s| 2H , s, t ∈ [0, T ]
and consequently, for every s, t

∈ [0, T ] E Z k H (t) -Z k H (s) 2 = |t -s| 2H (3)
• the Hermite process is Holdër continuous of order δ < H

• if k ≥ 2, then Z k H is non-Gaussian. When k = 1 the
process given by ( 2) is nothing else that the fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1 2 ). For k ≥ 2 the process is not Gaussian. If k = 2 then the process ( 2) is known as the Rosenblatt process (it has been actually named by M. Taqqu).

We focus here our attention on the case k = 2. We will work with the representation of this processes as integral with respect to a Wiener process on a finite interval. Recall that the fBm B H with Hurst parameter H > 1 2 can be written as

B H t = t 0 K H (t, s)dW s , t ∈ [0, T ] (4)
with (W t , t ∈ [0, T ]) a standard Wiener process and

K H (t, s) = c H s 1 2 -H t s (u -s) H-3 2 u H-1 2 du ( 5
)
where t > s and c H =

H(2H-1) β(2-2H,H-1 2 ) 1 2
. From ( 5) we obtain that for t > s,

∂K ∂t (t, s) = c H s t 1 2 -H (t -s) H-3 2 . (6)
Aa analogous representation for the Rosenblatt process has been given in [START_REF] Tudor | Analysis of the Rosenblatt process[END_REF]. We have

Z 2 H (t) := Z t = (d) d(H) t 0 t 0 t y 1 ∨y 2 ∂K H ′ ∂u (u, y 1 ) ∂K H ′ ∂u (u, y 2 )du dW y 1 dW y 2 (7)
where (W t , t ∈ [0, T ]) is a Brownian motion,

H ′ = H + 1 2 (8)
and

d(H) = 1 H+1 H 2(2H-1) -1 2 . Note that H > 1 2 implies H ′ > 3 4 .

Convergence in law to the Rosenblatt process

This part in consecrated to a Donsker invariance principle for the Rosenblatt process. From now on, we will consider the Rosenblatt process to be given by the formula [START_REF] Guasoni | No arbitrage under transaction costs, with fractional Brownian motion and beyond[END_REF]. We will denote, for every t ∈ [0, T ]

F (t, y 1 , y 2 ) = d(H)1 [0,t] (y 1 )1 [0,t] (y 2 ) t y 1 ∨y 2 ∂K H ′ ∂u (u, y 1 ) ∂K H ′ ∂u (u, y 2 )du
and then

Z t = T 0 T 0 F (t, y 1 , y 2 )dW (y 1 )dW (y 2 ), t ∈ [0, T ]. (9)
The kernel K H ′ (denoted simply by K in the sequel) is the standard kernel of the fractional Brownian motion [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF].

Let us first recall some known facts. Consider (ξ i ) i≥1 a sequence of i.i.d random variables with E(ξ i ) = 0 and E(ξ 2 i ) = 1. The Donsker Invariance Principle says that the sequence of processes

W n t = 1 √ n [nt] i=1 ξ i (10)
converges weakly, in the Skorohod topology, to a standard Brownian motion. Here [x] denotes the biggest integer smaller than x. This result has been extended in [START_REF] Sottinen | Fractional Brownian m!otion, random walks and binary market models[END_REF] to the fractional Brownian motion (see also [START_REF] Nieminem | Fractional Brownian motion and Martingale differences[END_REF]). Define

K n (t, s) := n s s-1 n K( [nt] n , u)du, n ≥ 1
and put

B n t = t 0 K n (t, s)dW n s = [nt] i=1 n i n i-1 n K( [nt] n , s)ds ξ i √ n , n ≥ 1.
Then it has been proved in [START_REF] Sottinen | Fractional Brownian m!otion, random walks and binary market models[END_REF] that the disturbed random walk B n converges weakly to the fractional Brownian motion.

From the above results and the representation [START_REF] Hu | Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion[END_REF] it is quite natural to define the following approximation for the Rosenblatt process

Z n t = [nt] i,j=1;i =j n 2 i n i-1 n j n j-1 n F [nt] n , u, v dvdu ξ i √ n ξ j √ n , t ∈ [0, T ]. ( 11
)
Remark 1 We eliminate the diagonal "i=j" because the Rosenblatt process is defined as a double Wiener-Itô integral and as a consequence it has zero mean. When the diagonal i = j is included in the sum (11) then the limit is in general a double Stratonovich integral (see [START_REF] Hu | Sur les intégrales multiples de Stratonovich[END_REF] or [START_REF] Sole | Stratonovich integral and trace[END_REF]).

Proposition 1 The family of stochastic processes (Z n t ) t∈[0,T ] converges in the sense of finite dimensional distributions to the process (Z t ) t∈[0,T ] (9).

Proof: We will proof this result in several steps.

Step 1: Let us consider an arbitrary sequence of partitions of the interval [0, T ] of the form

π m : 0 = t m 0 < t m 1 < . . . < t m m = T with |π m | → 0 as m → ∞. Define Z π m t = m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, u, v)dvdu W (∆ m i )W (∆ m j ) (12)
where we denoted by ∆ m i = [t m i-1 , t m i ) and by

W (∆ m i ) = W t m i -W t m i-1 .
Then it follows from [START_REF] Sole | Stratonovich integral and trace[END_REF], Theorem 3.4, or [START_REF] Hu | Sur les intégrales multiples de Stratonovich[END_REF] that for fixed t the sequence Z π m t converges in L 2 (Ω) as |π m | → 0 to the multiple Wiener-Itô integral of F (t, •) with respect to the Brownian motion

W T 0 T 0 F (t, u, v)dW u dW v = Z t .
Step 2: Secondly, define the process

Z π m ,n t = m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, u, v)dvdu W n (∆ m i )W n (∆ m j ) (13)
where W n is the random walk given by [START_REF] Lo | Long memory in stock market prices[END_REF]. Then clearly, for fixed m, as n goes to ∞, the finite dimensional distributions of Z π m ,n converges to the finite dimensional distributions of Z π m (this comes from the weak convergence of W n to the Wiener process W ).

Step 3: We prove now that for every t ∈ [0, T ], the sequence Z π m ,n t converges in L 2 (Ω) to Z ′ ,n t as m → ∞, where

Z ′ ,n t = [nt] i,j=1;i =j n 2 i n i-1 n j n j-1 n F (t, u, v) dvdu ξ i √ n ξ j √ n , t ∈ [0, T ]. (14)
Consider the sequence

F π m (t, u, v) = m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, u, v)dvdu 1 ∆ m i (u)1 ∆ m j (v). Then F π m (t, •) converges to F (t, •) in L 2 ([0, T ] 2 ) as m → ∞ (see [21], [8]). First note that Z ′ ,n t can be approximated in L 2 (Ω) as m → ∞ by Z ′ ,π m ,n t = [nt] k,l=1;k =l n 2 k n k-1 n l n l-1 n F π m (t, u, v)dvdu ξ k √ n ξ l √ n .
Indeed,

E Z ′ ,π m ,n t -Z ′ ,n t 2 = [nt] k,l=1;k =l n 2 k n k-1 n l n l-1 n F π m (t, u, v) -F (t, u, v) dvdu 2 E(ξ 2 k )E(ξ 2 l ) ≤ [nt] k,l=1;k =l k n k-1 n l n l-1 n (F π m (t, u, v) -F (t, u, v)) 2 dvdu ≤ T 0 T 0 (F π m (t, u, v) -F (t, u, v)) 2 dvdu
and this clearly goes to zero as m → ∞.

It remains to observe that Z ′ ,π m ,n t is equal to Z π m t for every t, m, n. We can write, if λ denotes the Lebesque measure,

Z ′ ,π m ,n t = [nt] k,l=1;k =l n 2 k n k-1 n l n l-1 n dvdu ×   m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, x, y)dydx 1 ∆ m i (u)1 ∆ m j (v)   ξ k √ n ξ l √ n = m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, x, y)dydx × [nt] k,l=1;k =l n 2 ξ k √ n ξ l √ n k n k-1 n l n l-1 n 1 ∆ m i (u)1 ∆ m j (v)dvdu = m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, x, y)dydx × [nt] k,l=1;k =l n 2 ξ k √ n ξ l √ n λ [ k -1 n , k n ) ∆ m i λ [ l -1 n , l n ) ∆ m j = m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, x, y)dydx × k;[ k-1 n , k n )⊂∆ m i l =k;[ l-1 n , l n )⊂∆ m j ξ k √ n ξ l √ n
and on the other hand by using ( 12) and ( 10), one has

Z π m ,n t = m i,j=1;i =j 1 |∆ m i ||∆ m j | ∆ m i ∆ m j F (t, x, y)dydx [nt m i ] k=[nt m i-1 ]+1 [nt m j ] l=[nt m j-1 ]+1 ξ k √ n ξ l √ n
and it is not difficult to see that Z π m ,n t and Z

′ ,π m ,n t coincide.

Step 4: At this point we conclude that the family of processes Z ′ ,n converges in the sense of finite dimensional distributions to the Rosenblatt process Z t . Let h be a function defined on R p and consider s 1 , . . . , s p ∈ [0, T ]. We will show that

E h(Z ′ ,n s 1 , . . . , Z ′ ,n sp ) -E h(Z s 1 , . . . , Z sp )
converges to zero as n → ∞. This can be bounded by A + B + C where

A = E h(Z s 1 , . . . , Z sp ) -E h(Z π m s 1 , . . . , Z π m sp ) B = E h(Z π m ,n s 1 , . . . , Z π m ,n sp ) -E h(Z π m s 1 , . . . , Z π m sp ) and C = E h(Z π m ,n s 1 , . . . , Z π m ,n sp ) -E h(Z ′ ,n s 1 , . . . , Z ′ ,n sp ) .
By Step 1 and Step 3 we have that for fixed c > 0 the terms A and C are bounded (uniformly in n) by c when m is large enough. the term B tends to zero as n → ∞ from Step 2.

Step 5: Clearly the family Z ′ ,n can be replaced by the family Z n (11) because their difference goes to zero in L 2 (Ω) as n → ∞.

Next, we show the tightness.

Proposition 2

The family Z n given by ( 11) is tight.

Proof: Let s < t, s, t ∈ [0, T ]. It holds, since the kernel F (s, u, v) vanishes when u or v are bigger that s, E |Z n t -Z n s | 2 = E   [nt] i,j=1;i =j n 2 i n i-1 n j n j-1 n F [nt] n , u, v -F [ns] n , u, v dvdu ξ i √ n ξ j √ n   2 = [nt] i,j=1;i =j n 2 i n i-1 n j n j-1 n F [nt] n , u, v -F [ns] n , u, v dvdu 2 ≤ [nt] i,j=1;i =j i n i-1 n j n j-1 n F [nt] n , u, v -F [ns] n , u, v 2 dvdu ≤ T 0 T 0 F [nt] n , u, v -F [ns] n , u, v 2 dvdu = [nt] n - [ns] n 2 .
Now the conclusion follows by using exactly the same arguments as in [START_REF] Sottinen | Fractional Brownian m!otion, random walks and binary market models[END_REF], end of the proof of Theorem 1.

The main result of this section is a consequence of Proposition 1 and Proposition 2.

Theorem 1 The family of stochastic processes [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF] converges weakly, in the Skorohod topology, to the Rosenblatt process.

Binary market model based on the Rosenblatt process

The binary market constitutes a financial model where the asset are traded at discrete times.

In classical cases (for example when the driven process is the Wiener process) the binary model approximates the Black and Scholes model. Let us start by introducing the Black and Scholes model driven by the Rosenblatt process. As usually, we will consider two assets: a safe investment satisfying

B t = t 0 r s B s ds, (15)
where r is a differentiable deterministic function and a risky asset with price dynamic following the stochastic equation

S t = S 0 + t 0 a s S s ds + σ t 0 S s dZ s , ( 16
)
where Z is a Rosenblatt process, σ > 0 and a is a differentiable deterministic function. The integral with respect to Z is understood here in a pathwise sense. Since the trajectories of the Rosenblatt process are enough regular (in fact, they are Hölder continuous of order δ < H and H > 1 2 ) it is possible to consider pathwise integrals with respect to it and to solve some stochastic equations in the pathwise sense. We refer, among others, to [START_REF] Nourdin | Schémas d'approximation associés à une équation différentielle dirigée par une fonction höldérienne; cas du mouvement brownien fractionnaire[END_REF], [START_REF] Zaehle | Integration with respect to fractal functions and stochastic calculus[END_REF] or [START_REF] Russo | Forward backward and symmetric stochastic integration[END_REF]. In particular, the solution of ( 16) is given by (see [START_REF] Nourdin | Some linear fractional stochastic equations[END_REF], [START_REF] Zaehle | Integration with respect to fractal functions and stochastic calculus[END_REF], [START_REF] Nourdin | Schémas d'approximation associés à une équation différentielle dirigée par une fonction höldérienne; cas du mouvement brownien fractionnaire[END_REF]

S t = S 0 e t 0 asds+σZt , t ∈ [0, T ]. ( 17
)
Clearly the solution of ( 15) is

B t = B 0 e t 0 rsds , t ∈ [0, T ]. (18)
Moreover, we will assume in the sequel that the interest rates a and r are deterministic bounded functions.

Let us describe now the binary market model with Rosenblatt influence. The two assets are traded now at successive times periods t 1 < t 2 < . . . < ... and their dynamics are given by

B n = (1 + r n )B n-1 and S n = (a n + (1 + X n ))S n-1 ;
That means that B n and S n represent the prices of the bond and of the stock in the period between t n and t n+1 and r n and a n are the interest rates valuable in this period. The stochastic process X is binary, that is, given X n-1 it can takes at time n two possible values denoted by d n ("down") and u n ("up"). The binary market excludes arbitrage opportunities if for every n it holds that (see [START_REF] Sottinen | Fractional Brownian m!otion, random walks and binary market models[END_REF])

d n < r n -a n < u n . (19)
In the following we will choose our binary model to be determined by

X n = ∆Z N n N (20)
where Z N is defined by [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF] and

r n = 1 N r n N and a n = 1 N a n N ( 21
)
where a and r are the interest rates appearing in ( 15) and ( 16).

We have

Proposition 3

The binary market model with X a and r given by ( 20) and ( 21) converges as N → ∞ to the Black and Scholes model given by ( 15) and ( 16).

and

g n-1 (x 1 , . . . , x n-1 ) = 2σN n-1 i=1 i N i-1 N n N n-1 N F n N , u, v dvdu x i .
Then obviously

u n = f n-1 (ξ 1 , . . . , ξ n-1 ) + g n-1 (ξ 1 , . . . , ξ n-1 ) (22)
and

d n = f n-1 (ξ 1 , . . . , ξ n-1 ) -g n-1 (ξ 1 , . . . , ξ n-1 ). ( 23
)
The last result of our paper is the following.

Proposition 4

The binary market model with ( 21) and ( 20) admits arbitrage.

Proof: Throughout this proof, we will denote by c(H) a generic constant depending only on H. Let us show now that the condition ( 19) fails for some n ≥ 2. We will actually prove that the sequence

f n-1 (1, 1, . . . , 1) -g n-1 (1, 1, . . . 1) → n→∞ ∞ (24)
and then clearly [START_REF] Russo | Stochastic calculus with respect to a finite quadratic variation process[END_REF] does not hold because r n and a n are assumed to be bounded. We have

f n-1 (1, 1, . . . , 1) = σN n-1 i,j=1;i =j i N i-1 N j N j-1 N F n N , u, v -F n -1 N , u, v dvdu = σN n-1 i,j=1;i i N i-1 N j N j-1 N F n N , u, v -F n -1 N , u, v dvdu - n-1 i=1 i N i-1 N i N i-1 N F n N , u, v -F n -1 N , u, v dvdu := σN (A -B).

Appendix: Monte Carlo Simulation

This simulation study is intended to show our proposed model using simulated data. We simulate data using different values for H. Below, we describe the procedure used in generating the data to be used in the simulation study.

We have implemented this simulation on a standard personal computing platform (PC), and have observed that it performs very well using simulated data as can be seen from the simulated data in the figures 1 and 2 below. Despite the apparent algebraic complexity of the equations [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF], the problem poses no difficulty for standard symbolic algebra packages. Using Matlab's simulations and algebra capabilities yielded the best computing times. In our implementation, which performs an iteration of the algorithm from i = 0 to i = n. Figure 1 shows the histogram for a fixed time t for the marginal density of the Rosenblatt process. We can see the skew structure of the distribution. Figure 2 shows the some paths of the discretization for the Rosenblatt process. We use the values for the parameter H (H = 0.8 and H = 0.9).

Figure 1 :

 1 Figure 1: Marginal distributions for H = 0.5; H = 0.8 and H = 0.9.

Figure 2 :

 2 Figure 2: Simulations of the Trajectory for the Rosenblatt process with H = 0.8 and H = 0.9. Acknowledgments: This work was partially supported by the research project Nucleus Millenium P04-069-F "Information and Randomness: Fundamentals and Applications;

Proof: Let us consider the jump

and the quadratic variation

We will show that the process [Z N ] converges in L 1 ([0, T ] × Ω) to zero. Then the conclusion will follow exactly as in [START_REF] Sottinen | Fractional Brownian m!otion, random walks and binary market models[END_REF], proof of Lemma 1.

We have, since the jumps are at times k N , k integer,

and then

which goes to 0 as N goes to ∞. The next step is to show that the market admits arbitrage opportunities. Clearly, we have

and we will take the random variables ξ to be binary, that is

for every i ≥ 1. We can write, by isolating the part involving ξ n ,

where for every n ≥ 2

Using the expression of the kernel F and ( 6), the term A can be minorized as follows

We majorize now the term B. We can write

From the above computations we obtain that

because the integral

). We then obtain [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF].

Comments: i) A concrete arbitrage opportunity can be easily described. For example, suppose that a > r. Ar a certain time n 0 we have d n 0 > 0 because of [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF]; suppose that the stock price was increasing up to this time n 0 . Then, buy M stocks and your wealth at time n 0 + 1 will be positive since M S n 0 +1 > M S n 0 .

ii) We cannot expect to have no-arbitrage when H ∈ 1 2 as in the fractional Brownian motion case because now the limit process at H = 1 2 is not necessarily a martingale.

Laboratories in Mathematics of Genome and Stochastic Simulation". The first author was supported partially by the research project Fondecyt Reg. N. 1050843, Chile and Proyecto Anillo ACT-13: "Laboratorio ANESTOC".


Jennifer C Mabry
email: jennifer.mabry@earth.ox.ac.uk

Tf Lan

Christine Boucher

Peter G Burnard

Matthias S Brennwald

Ray Langenfelds

Bernard Marty

Tefang Lan

No evidence for change of the atmospheric helium isotope composition since 1978 from re-analysis of the Cape Grim Air Archive

come

No evidence for change of the atmospheric helium isotope composition since 1978 from re-analysis of the Cape Grim Air Archive

Introduction

The helium isotopic composition of air may be able to provide a new tracer for studying and quantifying environmental changes resulting from global warming and anthropogenic activity. Due to their light masses, the two isotopes of helium escape to space through thermal and non-thermal processes [START_REF] Ozima | Noble Gas Geochemistry[END_REF].

Consequently, the helium content of air is low (5.24 ppm vol., [START_REF] Gluckauf | A Micro-Analysis of the Helium and Neon Contents of Air[END_REF] and the atmospheric He isotope ratio is a transient value determined by variations of the strengths of sources (outgassing of the solid earth, extraterrestrial contributions) and sinks (loss to space) of 3 He and of 4 He over time [START_REF] Ozima | Noble Gas Geochemistry[END_REF]. The He isotope residence time in the atmosphere is of the order of 10 4 -10 6 yr [START_REF] Torgersen | Terrestrial helium degassing fluxes and the atmospheric helium budget: Implications with respect to the degassing processes of continental crust[END_REF][START_REF] Kockarts | Helium in the terrestrial atmosphere[END_REF], much longer than the atmospheric mixing time of 10 yr. Thus, the 3 He/ 4 He ratio of modern air (1.34-1.40 x 10 -6 , Mamyrin et al., 1969;Clarke et al., 1975;Davidson et al., 1988;Hoffmann and Nier, 1993;[START_REF] Sano | Precise measurement of helium isotopes in terrestrial gases[END_REF] should be constant on a decadal timescale. However, it has been suggested that this may no longer be the case [START_REF] Sano | A ten-year decrease in the atmospheric helium isotope ratio possibly caused by human activity[END_REF][START_REF] Pierson-Wickmann | Helium trapped in historical slags: a search for temporal variation of the He isotopic composition of air[END_REF][START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF][START_REF] Lupton | Changes in the atmospheric helium isotope ratio over the past 40 years[END_REF].

The exploitation of fossil fuels (coal, liquid and gaseous hydrocarbons) may have released large amounts of 4 He during the past couple of centuries and particularly the last few decades as natural gas production has more than doubled between 1971 and 2012 (IEA, 2014). The recent rapid release of gases from these reservoirs, which have concentrated crustal helium for tens to hundreds of Ma, could have decreased the 3 He/ 4 He in the atmosphere [START_REF] Sano | A ten-year decrease in the atmospheric helium isotope ratio possibly caused by human activity[END_REF][START_REF] Pierson-Wickmann | Helium trapped in historical slags: a search for temporal variation of the He isotopic composition of air[END_REF][START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF].

Other anthropogenic extraction industries such as mining or quarrying may also liberate unknown quantities of radiogenic He into the atmosphere [START_REF] Sano | Atmospheric helium isotope ratio: Possible temporal and spatial variations[END_REF].

The only potential anthropogenic source of excess 3 He that could increase the atmospheric 3 He/ 4 He ratio is He produced during induced U or Pu fission either in nuclear reactors or in nuclear weapons [START_REF] Lupton | The atmospheric helium isotope ratio: Is it changing?[END_REF]. However, anthropogenic nuclear He sources can be relatively well quantified and do not contribute significant He to the atmosphere [START_REF] Lupton | The atmospheric helium isotope ratio: Is it changing?[END_REF].

Studies on the atmospheric helium variations, summarized in Brennwald et al. (2013) (e.g., compilation in their Table 1), resulted in contradictory results, indicating either decreases in the 3 He/ 4 He ratio of the order of 0.1-0.3 ‰/yr during the last few decades [START_REF] Sano | Atmospheric helium isotope ratio: Possible temporal and spatial variations[END_REF][START_REF] Matsuda | Helium in old porcelain: The historical variation of the He isotopic composition in air[END_REF], or no detectable change since at least 1973 [START_REF] Lupton | Changes in the atmospheric helium isotope ratio over the past 40 years[END_REF]) or possibly 1956(Hoffmann and Nier, 1993).

The study of [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF] is particularly relevant, because it presents results of temporal atmospheric helium variations from samples periodically archived in stainless steel containers since 1978 (Cape Grim Air Archive (CGAA), Tasmania; [START_REF] Langenfelds | The Cape Grim Air Archive: the first seventeen years, 1978-1995[END_REF]. As these samples were collected from a single location, [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF] argued that potential problems due to geographical heterogeneities should not apply. Based on a statistical analysis of repeated runs, Brennwald et al. reported a mean decrease of the 3 He/ 4 He ratio of 0.23 to 0.30‰/yr during this period of time. In contrast, [START_REF] Lupton | Changes in the atmospheric helium isotope ratio over the past 40 years[END_REF] did not observe any significant difference between 3 He/ 4 He ratios measured in air samples taken at a single location (La Jolla, California, USA) in 1973 and 2013. Together with other data from different locations and periods of time, they derived a mean rate of decrease of the 3 He/ 4 He ratio of (0.014 ± 0.045)‰/yr (2σ), which is consistent with zero rate of change over a 40-year period.

We have developed a dedicated facility at CRPG Nancy (France) that permits the analysis of the 3 He/ 4 He ratio of air with a precision for individual measurements of about 2‰ (2σ, Mabry et al., 2014) which compares to about 14‰ (2σ, [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF]. In order to further investigate and better constrain the results reported by [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF], we have analyzed aliquots of the same air samples from the CGAA, as well as 5 further samples from years not analyzed by [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF].

Method

The CGAA is a series of air samples collected typically every three months since 1978 at Cape Grim on the north-west tip of Tasmania.

For each sample, 1-2 m 3 STP of air is collected and stored in 35-L stainless steel tanks. The sampling and storage conditions of the air samples were considered with great care to avoid pollution/contamination from cities and industry during sampling or modification of the sampled air during storage. This is necessary to provide a true baseline air sample for the southern hemisphere that could be used for many years. Details of the CGAA sampling procedure are documented in [START_REF] Langenfelds | The Cape Grim Air Archive: the first seventeen years, 1978-1995[END_REF] and [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF].

Subsamples for 3 He/ 4 He analysis were chosen from a selection of ten archive tanks spanning the time period 1978 -2011: 1978, 1979, 1984, 1988, 1993, 1997, 2000, 2004, 2008, 2011 (Table 1). Multiple subsamples were collected from each tank in 1.2-m-long, 3/8''-OD copper tubes, which were sealed by cold-welding with a crimping tool [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF].

For measurement, a roughly 10 to 15 cm 3 portion of the copper tube is sealed off with a steel clamp and attached to the extraction line using Swagelok Ultra-torr tube fittings and wrench-tightened. The sample manifold is then evacuated, the pump isolated, and the copper tube is opened to release the air sample into the extraction line for purification and then measurement. A full description of the sample processing and analysis is given in [START_REF] Mabry | High-precision helium isotope measurements in air[END_REF]. Everything aside from helium and neon is removed from the sample gas using a series of getters and charcoal fingers. Then, the neon is separated from helium using a cryo trap so that only the helium remains to be admitted into the mass spectrometer. After purification of the sample, a series of eleven standard-bracketed measurements are made. The standard used for the bracketing measurements is derived from a purified air standard collected from the Brittany coast. Note that this standard He is enriched by 3.3% relative to the He isotope composition of air, as a result of isotope fractionation during handling and purification of a large quantity of air. We attempted the purification procedure twice and obtained the same fractionation. The cause of this fractionation, possibly related to thermal effects between hot traps and traps at liquid N 2 temperature, are not fully elucidated. For our purpose however, the absolute value of the running standard has no bearing on the results (provided that its 3 He/ 4 He composition is close to that of air), because all samples are normalized to this common running standard and therefore can be inter-compared. In particular, we continuously calibrate atmospheric helium from local air (Brabois Park close to Nancy) against this standard (see below). Gas pressures of the purified sample aliquots are matched to within 2% of the standard aliquot pressures through the use of an adjustable volume in order to eliminate pressure effects in the mass spectrometer.

Additionally, local air sampled in a park near CRPG (Brabois Air) is analyzed periodically to compare all samples under the same analysis and sampling conditions, and so, to monitor the continued efficacy of the purification procedure and measurement stability. The measured 3 He/ 4 He values of Brabois Air samples were very stable (standard error 0.33 ‰, 2σ, N=83) over a year of measurements [START_REF] Mabry | High-precision helium isotope measurements in air[END_REF] over the same time period the CGAA sample measurements were made.

Results and Discussion

Helium in the Cape Grim Air Archive

In Table 1, we report measurements from a selection of coppertube subsamples from the CGAA. Particular effort was made to make multiple measurements from the earliest (1978,1979) and latest (2008,2011) subsamples since these would be the most likely to show a detectable difference and would best constrain the trend over the time period.

Three of the data points are statistically significant outliers. The 3 He/ 4 He from 1988 and 2004 are 10-20‰ lower than all the other data points, while the 3 He/ 4 He ratio of the 1984 subsample is about 4‰ lower than in the other samples (excluding 1988 and 2004). A generalized extreme Studentized deviate (GESD) test (Appendix A, [START_REF] Rosner | Percentage Points for a Generalized ESD Many-Outlier Procedure[END_REF] confirms the 1988 and 2004 data points as outliers with 99% confidence whether the test is run using the ratios or the residuals of a best-fit linear time trend. This test assumes the dataset is approximately normally distributed which we confirmed by a normal probability plot (not shown).

The 1984 tank at CSIRO is suspected of having a small leak as was noted in [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF]. The nitrogen isotopes in this particular storage tank were detectably fractionated and additional testing showed a change in the relative concentrations of H 2 , CH 4 , CO 2 , and N 2 O over time [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF]. These indicate a small leak in this storage tank which would lead to a decrease in the 3 He/ 4 He over time as slightly more 3 He effused out of the tank relative to 4 He, consistent with our data showing a lower 3 He/ 4 He(ratio(relative( to( the( other( samples. We can correct for this effect using observed changes in the concentrations of H 2 , CH 4 , CO 2 , and N 2 O and Graham's Law to calculate the expected fractionation of the helium isotopes (as in [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF]. The correction increases the 1984 3 He/ 4 He by about 3.5‰,( bringing( it to well within the scatter of the rest of the data excluding 1988 and 2004 (Table 1). Neither the 1988 nor 2004 tanks at CSIRO show any evidence of leakage. Thus, it seems likely that these samples were fractionated at some point after subsampling, e.g., during storage or aliquoting of the subsamples. In the following discussion, we will consider both the uncorrected and effusion-corrected 1984 data but will exclude the 1988 and 2004 data. a(linear(decrease(than(by(a(constant( 3 He/ 4 He,(their(data(does(not(rule( out(the(possibility(of(a(constant( 3 He/ 4 He(per(se. and thus taken together are consistent with no(statistically (significant(change(in(the(helium(isotopes( of( the( Cape( Grim( Air( Archive( over( the( measured( time( period,( but( the( new( data( better( constrain( the( upper( limit( of( possible( change. Furthermore,( this( result( is( in( good( agreement( with( recently( published( data(by(Lupton(and(Evans((2013),(who(report(no(significant(change(in( the(helium(isotopic(ratio((Fig(1b)(in(air(over(the(last(few(decades(and(a( 236 maximum(possible(decrease(of( 3 He/ 4 He(of(0.059‰/yr (2σ).( ( ( Brennwald(et(al.((2013)(also(measured(He/Ne(ratios(of(the(same( samples( and( did( not( observe( any( significant( trend( if( the( uncorrected( 1984(data(are(excluded.(If(the(change(in(helium(isotopic(ratio(were(due( to(excess( 4 He(released(into(the(atmosphere(as(theorized,(this(should( manifest(also(as(an(increasing(He/Ne(ratio(over(the(same(time(period.( The( fact( that( they( saw( no( change( in( this( ratio( was( at( odds( with( their( helium( isotope( data,( but( is( consistent( with( the( new( measurements( in( this(work.(( Helium'in'natural'gas' ( The( constancy( of( the( 3 He/ 4 He( ratio( over( the( last( few( decades( is( inconsistent( with( available( estimates( for( He( isotope( fluxes( during( this( period, which [START_REF] Sano | Atmospheric helium isotope ratio: Possible temporal and spatial variations[END_REF]. These predictions are dependent on the global average He content of natural gas, which has been estimated to be 0.1-0.25% [START_REF] Oliver | Helium concentration in the Earth's lower atmosphere[END_REF]. However, barring a credible source of excess 3 He, the data sets from this work and [START_REF] Lupton | Changes in the atmospheric helium isotope ratio over the past 40 years[END_REF] point to this being a significant overestimation of the mean helium content in natural gas produced globally.

These predicted He concentrations come primarily from the U.S.

Bureau of Land Management's (previously Bureau of Mines) survey of natural gas compositions (see e.g BLM, 2008). A straight average of the helium compositions contained in this database yield an average helium content of around 0.25 -0.3%. However, [START_REF] Lupton | Changes in the atmospheric helium isotope ratio over the past 40 years[END_REF] noted that this data shows an inverse correlation between helium content and reported production rate (open flow) of the well, indicating that there may be an inverse relationship between helium content and field size. Weighting the helium data by open flow brings the average down to around 0.15%. Additionally, this database may over-represent higher helium wells since it includes primarily data from U.S. wells (~16,700 U.S. records vs 133 non-U.S.) which may have a higher helium content then the rest of the world (non-U.S. average helium content ~ 0.06%). And this database may even over-sample higher helium wells within the U.S. as well since helium is an important commodity. These factors suggest that the true average helium composition is likely less than 0.15%.

If we look directly at helium production, [START_REF] Mohr | Helium Production and Possible Projection[END_REF] estimate that ~2.0 × 10 11 moles were produced world wide between 1921 and 2012. Assuming most of this helium made its way into the atmosphere within a few years, it would lead to a decrease in the atmospheric 3 He/ 4 He of about 0.0025‰/yr (or about 0.225‰ over the 91-year span). Narrowing the focus to just the period 1978-2011, we have about 1.5 × 10 11 moles produced world wide (U.S. Geological Survey, 2014), leading to a decrease in the atmospheric 3 He/ 4 He of about 0.005‰/yr (or about 0.15‰ over the 33-year span). If we further assume that helium is being extracted from most of the high-helium natural gas wells, then the average helium content for the remaining wells would be much lower. After accounting for helium production and using an average natural gas production over the last three decades of ~1 × 10 14 mol/yr (IEA, 2014), then we find the average helium content of natural gas consistent with our data is at most 0.034% (2σ maximum). This is roughly 3 -7× lower than the commonly quoted values of 0.1-0.25%.

'

Conclusion

We ( have( reTanalyzed( the( He( isotope( composition( of( a( series( of( air( aliquots( sampled( at( Cape( Grim,( Tasmania,( and( stored( in( stainless( steel( taken( by( CSIRO.( We do not observe any significant change in the( 3 He/ 4 He( ratio over this time period and find the maximum rate of decrease to be 0.033(-(0.042‰/yr (2σ); two outliers (1988 and 2004) have been excluded from the compilation. We do not reproduce the result( of ( Brennwald( et( al.( (2013),( who,( based( on( the( analysis( of( air( aliquots( from( the( same( series( reported( a( decrease( of( (0.23 -0.30 ± 0.16)‰/yr of( the( 3 He/ 4 He( ratio between 1978 and 2011. Our results are consistent with previous reports that the He isotope composition of air has not changed by more than 0.059‰/yr (2σ) between 1973 and 2013 [START_REF] Lupton | Changes in the atmospheric helium isotope ratio over the past 40 years[END_REF], and with the results of Hoffmann and Nier (1993) who concluded there was no significant change in the 3 He/ 4 He (ratio(based(on(measurements(done(on(air(collected(between( 1956(and(1988. Our data suggests that the commonly reported mean concentration of helium in natural gas is significantly over-estimated (at least 3×) and is likely no more than 0.034%.

Given the inconsistency of reported atmospheric 3 He/ 4 He trends in the literature, the discrepancy in measurements of the same CGAA samples reported here and by [START_REF] Brennwald | Concentrations and isotope ratios of helium and other noble gases in the Earth's atmosphere during 1978-2011[END_REF], and the unexplained fractionation of the 1988 and 2004 samples, it is proposed that a new subset of CGAA samples should be prepared by CSIRO and analyzed at CRPG. These samples will be filled in stainless steel flasks with valves, rather than the copper tubes used previously, and will extend the time series to at least 2015. ( (

Figure 1 :

 1 Figure 1: (a) The individual 2013 (solid circles) and those measured in this work (open squares). Data are normalized to the mean value of each respective data set to allow for direct comparison. (b) Shows the errorweighted means of the individual replicate measurements on subsamples. The effusion corrected 1984 mean is shown with an open diamond. The error-weighted best-fit lines are shown for Brennwald et al. 2013 (dashed line), this work with 1984 effusion corrected (solid line) and uncorrected (dotted line). Also shown for comparison are data from Lupton and Evans (2013, solid triangles). All errors are 2σ.

Table 1 :

 1 Helium isotope ratios from different subsamples of the Cape Grim Air Archive relative to our standard (R/Rstd) and then normalized to the overall mean of the data. Where more than one measurement was made, the error-weighted mean (x) and the error of the mean is reported. Numbers in italics have been identified as statistical outliers.

	Cape Grim		R/Rstd	2σ	Cape Grim		R/Rstd	2σ
	Archive Tank				Archive Tank
	7 Jul 1978 (UAN 780002)	x̄	1.0035 0.9992 1.0026 0.9979 1.0009	0.0020 0.0020 0.0022 0.0024 0.0011	2 Mar 1993 (UAN 930279) 7 Jan 1997 (UAN 970008)		1.0009 0.9989	0.0021 0.0021
	6 Feb 1979 (UAN 790001)		1.0015 1.0003 1.0001 0.9992 0.9997	0.0020 0.0022 0.0021 0.0020 0.0023	29 Sep 2000 (UAN 993562) 1 Dec 2004 (UAN 997089)		0.9988 0.9885	0.0020 0.0019
			0.9979	0.0023			1.0001	0.0023
			0.9997	0.0024	12 Aug 2008		1.0006	0.0022
		x̄	0.9999	0.0008	(UAN 999756)		0.9994	0.0022
	23 May 1984 (UAN 840004)	x̄	0.9963 0.9963 0.9963	0.0022 0.0022 0.0016	4 May 2011	x̄	1.0000 0.9990 0.9990	0.0013 0.0022 0.0020
	Effusion corrected x̄	0.9997	0.0016	(UAN		1.0007	0.0023
	21 Jun 1988 (UAN 880003)	x̄	0.9783 0.9771 0.9780	0.0020 0.0032 0.0017	20110462)	x̄	1.0017 1.0000	0.0022 0.0011
	An error-weighted least-squares fit (e.g. Baird, 1995) of the
	3 He/ 4 He with the uncorrected 1984 ratio (excluding 1988 and 2004)
	results in a trend line with a slope that is essentially zero: (0.000 ±
	0.033)‰/yr (2σ, Fig 1b). If the potential effusion effect on the 3 He/ 4 He
	value obtained from the 1984 tank is considered, the slope

  predict a decrease of the 3 He/ 4 He ratio of about 0.1 ‰/yr

		(b)
		1.005
	He/ 4 He normalized to mean	1.000			Effusion correction
	3
		0.995
		1970	1975	1980	1985	1990	1995	2000	2005	2010	2015
						Year

He values from replicate analyses reported inBrennwald et al.

Acknowledgments

This project has been funded by the French Agence Nationale de la Recherche (Project VIHA2) and by the European Research Council under the European Community's Seventh Framework Programme (FP7/2010-2016 grant agreement no. 267255). The Deep Carbon Observatory also provided funding support. We acknowledge the CSIRO and Cape Grim Baseline Air Pollution Station staff who assisted with collection, analysis and maintenance of the CGAA samples, and Dr.

Paul Steele for helpful discussions relating to this work. Thanks also to helpful comments from two anonymous reviewers, and to Editor Derek Vance.

(

Appendix'A:'Generalized'ESD'test'

For a data set which is approximately normally distributed, with mean (x̅ ) and standard deviation (σ) the generalized extreme Studentized deviate (GESD) test [START_REF] Rosner | Percentage Points for a Generalized ESD Many-Outlier Procedure[END_REF] can detect multiple outliers. The test null hypothesis is that there are no outliers in the data set, and the alternate hypothesis is that there are up to r outliers. The test statistic (R i )

)

is calculated for the most extreme outlier first (maximum |x i -x̅ |), and then that data point (x i ) is removed and the test statistic is recalculated until all potential outliers have been removed.

After calculating the test statistic for each potential outlier (R 1 , R 2 , …, R r ), the critical values can be calculated for significance level α:

* "$ = $ + -, -.,01"12 + -, -1 + -.,01"12 5 + -, + 1 $, = 1, 2, … , 8

where 9 = 1 -: 2 + -, + 1 ( ( and t p,ν represents the pth percentile of a t distribution with ν degrees of freedom.

The number of outliers is determined by finding the largest i such that R i > λ i .

'