L Delannoy

M Auzanneau

B Andrieu

O Vidal

P.-Y Longaretti

E Prados

D J Murphy

R W Bentley

M Carbajales-Dale

M Raugei

M Höök

V Court

C W King

F Fizaine

P Jacques

M K Heun

A Jackson

C Guay-Boutet

E Aramendia

J Wang

Hugo Le Boulzec

Cas Hall

Emerging consensus on net energy paves the way for improved integrated assessment modeling

Extracting, processing, and delivering energy requires energy itself, which reduces the net energy available to society and yields considerable socioeconomic implications. Yet, most mitigation pathways and transition models overlook net energy feedbacks, specifically related to the decline in the quality of fossil fuel deposits, as well as energy requirements of the energy transition. Here, we summarize our position across 8 key points that converge to form a prevailing understanding regarding EROI (Energy Return on Investment), identify areas of investigation for the Net Energy Analysis community, discuss the consequences of net energy in the context of the energy transition, and underline the issues of disregarding it. Particularly, we argue that reductions in net energy can hinder the transition if demand-side measures are not implemented and adopted to limit energy consumption. We also point out the risks posed for the energy transition in the Global South, which, while being the least responsible for climate change, may be amongst the most impacted by both the climate crisis and net energy contraction. Last, we present practical avenues to consider net energy in mitigation pathways and Integrated Assessment Models (IAMs), emphasizing the necessity of fostering collaborative efforts among our different research communities.

Broader context

The transition from fossil fuels to low-carbon energy is made difficult by several factors. One of which is the energy investments required by the transition, often examined through the lens of the EROI (Energy Return on Investment) metric. Although the concept of EROI is simple, its application has proven to be challenging due to theoretical and practical difficulties. To address this situation, we summarize our position with 8 key points, which approximate an emerging consensus around EROI, and identify key areas under investigation for the Net Energy Analysis research community. Our summary uncovers how net energy is critical for the assessment of equitable and feasible transition scenarios, and yet how it remains marginally addressed in the current use of Integrated Assessment Models. We therefore suggest avenues for improvements to make sure that energy-economy feedbacks are internally consistent in mitigation pathways

Introduction

On April 4 2022, IPCC Working Group III finalized its contribution to the Sixth Assessment Report. Reviewing progress and commitments for climate change mitigation, the report calls for more sustainable consumption habits and a shift away from fossil fuels towards low-carbon energy systems (IPCC, 2022). This transition nevertheless requires significant energy investments for the alternative low-carbon energy system, which can be examined through the lens of the Energy Return on (Energy) Invested or ERO(E)I metric [START_REF] Hall | Energy Return on Investment[END_REF][START_REF] Haberl | Contributions of sociometabolic research to sustainability science[END_REF]. Recent developments in the Net Energy Analysis (NEA) research community have highlighted EROI implications for socio-economic scenarios [START_REF] King | Implications of net energy-return-on-investment for a low-carbon energy transition[END_REF][START_REF] Capellán-Pérez | MEDEAS: a new modeling framework integrating global biophysical and socioeconomic constraints[END_REF]Delannoy et al., 2021a;[START_REF] Jacques | Assessing the economic consequences of an energy transition through a biophysical stock-flow consistent model[END_REF][START_REF] Slameršak | Energy requirements and carbon emissions for a low-carbon energy transition[END_REF], in particular regarding the practical challenges of the low-carbon transition. Yet, in part due to a lack of formal methodology prior to the 2010s [START_REF] White | The changing meaning of energy return on investment and the implications for the prospects of post-fossil civilization[END_REF][START_REF] Rana | Trends in Scientific Literature on Energy Return Ratio of Renewable Energy Sources for Supporting Policymakers[END_REF] and to a delay in the emergence of robust results, such studies have failed to influence transition scenarios. To remedy this situation, we provide an overview of the net energy approach, summarize the claimed emerging consensus around EROI, address how it relates to the low-carbon transition, and suggest ways to better integrate net energy in Integrated Assessment Models (IAMs).

The Net Energy Analysis approach

Net energy, i.e. the energy supplied to society in the form of energy carriers after subtracting the energy invested for the production and distribution of those energy carriers, is a fundamental prerequisite to allow the production and exchange of goods and services. For a given amount of net energy, a key metric of the energy system is the EROI -defined as the ratio between the total energy returned (E returned ) and the total energy invested to accomplish the conversion (E invested ) over the entire life cycle of the system under study, i.e. EROI = E returned /E invested .

As with all analyses that can be performed at the macro (economy-wide) and micro (technologyspecific) scales, EROI can have slightly different interpretations [START_REF] Brandt | A general mathematical framework for calculating systemsscale efficiency of energy extraction and conversion: Energy return on investment (EROI) and other energy return ratios[END_REF]King et al., 2015a,b). For example, at the scale of the global economy, the EROI has a minimum of 0 based on the first law of thermodynamics. When analyzing a single technology or energy subsystem that produces a final energy carrier, the EROI ratio can be less than one to one (1:1) (e.g., in Figure 1, E invested,2 is greater than E returned,2 ). Such systems can still be locally or temporarily useful when they have compelling properties, for example delivering a specific energy carrier that is in particular demand, e.g., the industrial food system, but they cannot be a main supplier of energy for society. Although the equations involved are simple, their application entails theoretical and practical difficulties that call for a rigorous definition of the system's boundaries [START_REF] Murphy | Energy return on investment, peak oil, and the end of economic growth[END_REF][START_REF] Raugei | Net energy analysis must not compare apples and oranges[END_REF].

E returned,1 E returned,2 E invested,1 E invested,2 E extracted,1
Energy system 1

Energy system 2 E extracted,2

Figure 1: Returned energy as a function of the extracted and the invested energy for two systems. EROI (global) = (E returned,1 + E returned,2 )/(E invested,1 + E invested,2 ). EROI (energy system 2) = (E returned,2 )/(E invested,2 ). Energy losses are omitted for clarity.

The "standard" (or primary stage) EROI accounts for the energy used in the extraction process only. It is useful for studying the energy demand of a primary energy extraction sector or technology.

The point-of-use (or final stage) EROI includes the energy used in not only extracting, but also processing and delivering an energy carrier. Therefore, for a given energy carrier, the pointof-use EROI is substantially lower than the standard EROI since additional energy inputs are considered. Focusing on the point of use is gaining in importance, as: (i) the energy requirements of processing, refining and other downstream processes for fossil fuels may be larger than that for their extraction, and (ii) most renewable energy systems directly deliver final energy carriers, i.e., typically electricity, making the analysis at the final energy stage essential to compare renewable and fossil fuel energy systems like for like 1 .

The dynamic EROI of the full energy system corresponds to the energy delivered by a country's (or the entire world's) energy system divided by its energy consumption at a given time, and is in that respect a Power Return on Investment (PROI) as the calculation is performed for a delimited time interval (one year usually) (King et al., 2015a;[START_REF] Carbajales-Dale | When is EROI not EROI?[END_REF]).

1 Discussion is still on-going in the research community about some subtle methodological issues on exactly how EROI at point of use should be formulated, but these do not affect the main argument being made here, i.e., that in order to be meaningful, all such comparisons should in fact be made at point of use.

Emerging consensus on net energy

As researchers in the field of NEA, we summarize our position with the following 8 key points, which approximate an emerging consensus around EROI:

1. The standard EROI of oil is usually lower than that of gas, which is lower than that of most coal [START_REF] Court | Long-term estimates of the energy-return-on-investment (EROI) of coal, oil, and gas global productions[END_REF].

2. Conventional fossil fuels (crude oil, natural gas liquids, etc.) may have lower standard EROIs than tight gas and oil produced from fracking [START_REF] Brandt | Net energy analysis of Bakken crude oil production using a well-level engineering-based model[END_REF][START_REF] Moeller | Net energy analysis of gas production from the marcellus shale[END_REF] but higher than other unconventional fuels (tar sands, mined shale oil, coal bed methane, etc.) (Brandt et al., 2015a;Delannoy et al., 2021a,b).

3. The standard EROI of new fossil energy resources is expected to improve initially as technology develops, before decreasing due to a decline in the quality of the extracted resource [START_REF] Dale | Global Energy Modelling: a Biophysical Approach (GEMBA)[END_REF][START_REF] Masnadi | Energetic productivity dynamics of global super-giant oilfields[END_REF]. For instance, the standard EROI of oil sands-derived crude has been increasing since the first bitumen-producing mines became operational (Guay-Boutet, 2023). On the contrary, many major conventional oil fields have already seen marked decreases in their standard EROI due to the requirements for enhanced recovery [START_REF] Tripathi | Estimating decades-long trends in petroleum field energy return on investment (EROI) with an engineering-based model[END_REF] and global resource depletion, as evidenced by the decline in the quantity of the remaining "2P" (proven and probable) reserves (The Shift Project, 2020;[START_REF] Laherrère | How much oil remains for the world to produce? comparing assessment methods, and separating fact from fiction[END_REF].

4. The aggregate EROI of fossil fuels at the point of use declines over time, albeit at a slower pace than at the point of extraction, since the largest investment (at the denominator of the EROI ratio) is not the energy required for extraction (that increases over time as resource quality decreases) but the subsequent energy required for processing and delivery (which is generally not much affected by the quality of the resource over the long-term) [START_REF] Brockway | Estimation of global finalstage energy-return-on-investment for fossil fuels with comparison to renewable energy sources[END_REF].

5. Today, the EROI of fossil-fueled electricity at point of end-use is often found to be lower than those of PV, wind and hydro electricity, even when the latter include the energy inputs for short-term (e.g., 8h) storage2 technologies [START_REF] Raugei | What are the energy and environmental impacts of adding battery storage to photovoltaics? a generalized life cycle assessment[END_REF][START_REF] Murphy | Energy return on investment of major energy carriers: Review and harmonization[END_REF]. Average EROI values however hide strong regional variability, particularly for solar and wind technologies [START_REF] Dupont | Global available wind energy with physical and energy return on investment constraints[END_REF][START_REF] Dupont | Global available solar energy under physical and energy return on investment constraints[END_REF].

6. The EROI of nuclear and hydropower have historically been high, however, the former is constrained by slow deployment times, the latter is limited in terms of availability of suitable locations, and both face many environmental considerations.

7. The point of use EROI for thermal fuels is usually low, specifically for liquid fuels (gasoline, biodiesel, bioethanol, etc.) compared to solid (coal, woodchips, etc.) or gaseous fuels [START_REF] Murphy | Energy return on investment of major energy carriers: Review and harmonization[END_REF].

8. A rapid large-scale deployment of renewable electricity and associated infrastructure will likely temporarily reduce the dynamic EROI (i.e., PROI) of the energy system as it requires a significant up-front energy investment embodied in infrastructure [START_REF] Dale | Energy balance of the global photovoltaic (PV) industry -is the PV industry a net electricity producer?[END_REF][START_REF] Sgouridis | The sower's way: quantifying the narrowing netenergy pathways to a global energy transition[END_REF][START_REF] Capellán-Pérez | Dynamic energy return on energy investment (EROI) and material requirements in scenarios of global transition to renewable energies[END_REF][START_REF] Jacques | Assessing the economic consequences of an energy transition through a biophysical stock-flow consistent model[END_REF][START_REF] Slameršak | Energy requirements and carbon emissions for a low-carbon energy transition[END_REF][START_REF] Fabre | Evolution of EROIs of electricity until 2050: Estimation and implications on prices[END_REF].

In parallel to this emerging consensus, several areas are under investigation, such as the future EROI trends of wind and solar. On one hand, their EROI might be negatively affected by the increase in energy requirements per unit of valuable mineral extracted due to geological depletion [START_REF] Fizaine | Renewable electricity producing technologies and metal depletion: A sensitivity analysis using the EROI[END_REF], whereby the quality of mineral deposits extracted (e.g., in terms of ore grade) decreases as a function of cumulative production. On the other hand, technological improvements may favorably affect the EROI of wind and solar PV [START_REF] Steffen | Historical and projected improvements in net energy performance of power generation technologies[END_REF]. The same is true for increasing the recycling capacity of renewable energy technologies, but the delay is significant because of the time required to build up a stock of materials suitable for recycling.

Another area under scrutiny is the extension of the analysis to the useful stage of energy use, i.e. at the stage when energy is actually exchanged for energy services [START_REF] Aramendia | Moving from final to useful stage in energy-economy analysis: A critical assessment[END_REF] (see Figure 2) as some energy carriers may be used for similar end-uses with very different final-to-useful efficiencies [START_REF] Aramendia | Exploring the useful energy impacts of the global energy transition: a net energy perspective[END_REF]. For example, electricity might fuel a car at a lower EROI point-of -use than gasoline, but an electric vehicle motor has a considerably higher final-touseful efficiency in converting its fuel input into mechanical drive when compared to a traditional internal combustion engine, such that an electric vehicle can have higher EROI at the useful stage.

Of particular interest is the use of net energy analysis at the useful stage for a comprehensive understanding of the rebound effect at different geographic and time scales. More precisely, this approach can help explain why global data shows energy use continuing to increase as individual technologies become more efficient, suggesting it is difficult to disprove that, to date, increased efficiency has enabled increased energy use. Models that attempt to quantify rebound show large rebound effects (typical economy-wide estimates are over 50%) [START_REF] Brockway | Energy efficiency and economy-wide rebound effects: A review of the evidence and its implications[END_REF].

Implications for the low-carbon transition

The net energy approach provides an enhanced understanding of the role of energy in economic processes, and as such, the EROI concept is increasingly used to model the energy-economy nexus. This growing modeling effort highlights two main net energy aspects which have implications for the low-carbon transition. On the one hand, the decline in the standard EROI of oil and gas may entail a rise in emissions per unit of net energy supplied to society [START_REF] Manfroni | The declining performance of the oil sector: Implications for global climate change mitigation[END_REF], and long-term energy price increases [START_REF] King | Relating financial and energy return on investment[END_REF][START_REF] Heun | Energy return on (energy) invested (EROI), oil prices, and energy transitions[END_REF], leading to periods of unfavorable growth or recession, especially for slow transition scenarios. On the other hand, theperhaps only temporary-reduction in net energy available for society in rapid transition scenarios may result in a high investment share and employment rate in low-emissions technologies, which could altogether generate inflation [START_REF] Režný | Energy transition scenarios and their economic impacts in the extended neoclassical model of economic growth[END_REF][START_REF] Jackson | Modelling energy transition risk: The impact of declining energy return on investment (EROI)[END_REF][START_REF] Jacques | Assessing the economic consequences of an energy transition through a biophysical stock-flow consistent model[END_REF], and thus raise questions of socio-political acceptance.

The pace of transition is bounded at the upper limit by the energy needed to sustain society without disruption (additional supply bottlenecks aside), and at the lower limit by the minimum speed required to meet climate targets (see Figure 3). Both limits are expected to move closer to each other as the transition is delayed, reducing the window of opportunity for a global transition compatible with ambitious climate targets. On one hand, the upper limit is likely to become more restrictive over time due to the geological depletion of fossil fuels, the fact that a more rapid low-carbon investment consumes a higher proportion of energy, and that more high-carbon investment needs replacement or becomes stranded. On the other hand, the lower limit will become more pressing because, trivially, the longer the transition delays, the less likely it is to comply with ambitious climate targets. The implementation of demand-side policies [START_REF] Creutzig | Towards demand-side solutions for mitigating climate change[END_REF] to reduce discretionary energy use, as highlighted by IPCC WG III (IPCC, 2022), is becoming increasingly relevant in this regard. Moving away from unnecessary uses and switching to more efficient conversion chains (e.g., from gasoline-powered to electric cars or bicycles) helps reduce discretionary energy use as long as rebound effects are mitigated. Further, recent research suggests that a decent life for all can be sustained at much lower levels of final energy use than at present within wealthy nations [START_REF] Cullen | Reducing Energy Demand: What Are the Practical Limits[END_REF][START_REF] Millward-Hopkins | Providing decent living with minimum energy: A global scenario[END_REF][START_REF] Kikstra | Decent living gaps and energy needs around the world[END_REF]. The energy transition has implications for equity. In particular, the upcoming reductions in net energy will necessarily amplify energy transition costs due to fossil fuel inflation and rapid lowcarbon investment. Such reductions will in turn exacerbate competition for the energy and material resources necessary for the transition, a competition in which low-income countries are already at a clear disadvantage [START_REF] Hall | Peak oil, EROI, investments and the economy in an uncertain future[END_REF]. Every Northern country that delays action thus risks compromising its ability to complete a transition and maintain or achieve high levels of material well-being, both for itself and other countries. This political situation raises inequity issues as countries from the Global North are likely to make their transition first. In this context, countries of the Global South are susceptible to lack access to energy, to the risk of getting slowed down -or even trapped -in their progress towards modern low-carbon energy, while being among the least responsible for and most affected by climate change [START_REF] Oswald | Large inequality in international and intranational energy footprints between income groups and across consumption categories[END_REF][START_REF] Konisky | The justice and equity implications of the clean energy transition[END_REF]Hickel et al., 2022b,a;[START_REF] Hickel | Existing climate mitigation scenarios perpetuate colonial inequalities[END_REF]. Accelerating the energy transition for the Global South is therefore a major stumbling block to a "just" transition, and requires massive financial support and technology transfers [START_REF] Cantarero | Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries[END_REF][START_REF] Newell | Toward transformative climate justice: An emerging research agenda[END_REF][START_REF] Poblete-Cazenave | Global scenarios of household access to modern energy services under climate mitigation policy[END_REF][START_REF] Fanning | Compensation for atmospheric appropriation[END_REF].

Time since the start of the transition

Proper consideration of net energy is required in mitigation pathways

While significant progress has been made in research on mitigation pathways, net energy has been addressed only marginally. The latest IPCC report (IPCC, 2022), for instance, mentions EROI issues in one paragraph (ch. 6, p. 44) and leaves out the evolution of the related literature, in part because the 8 key points developed earlier have only recently emerged. This situation results in insufficient discussion on the consequences of a decrease in the EROI of the energy system. The overlooking of net energy is also apparent in Integrated Assessment Models (IAMs), the main tools used to produce global, regionally disaggregated mitigation pathways (van Beek et al., 2020a).

First, most IAMs merely characterize exhaustible fossil fuel resources through cost-supply curves, whose limitations in terms of modeling and parameterization lead to significant loss of robustness for mitigation pathways. On one hand, these curves operate under the assumption that the supply of fossil fuel resources depends purely on economic criteria, which means that production fluctuates with price, but in reality prices increase also because production does not increase fast enough. They furthermore assume that the most economically viable reserves will be exploited first, regardless of the complex interplay of other socio-geopolitical factors that shape the reality of producing companies [START_REF] Heede | Potential emissions of co 2 and methane from proved reserves of fossil fuels: An alternative analysis[END_REF], countries [START_REF] Johnsson | The threat to climate change mitigation posed by the abundance of fossil fuels[END_REF], and regions [START_REF] Verbruggen | Peak oil supply or oil not for sale?[END_REF][START_REF] Norouzi | The fall of oil age: A scenario planning approach over the last peak oil of human history by 2040[END_REF]. On the other hand, typical upward sloping cost-supply curves are subject to criticism for potentially outdated, simplistic and overoptimistic assumptions in the recoverability of fossil resources [START_REF] Brecha | Emission scenarios in the face of fossil-fuel peaking[END_REF][START_REF] Kharecha | Implications of "peak oil" for atmospheric co 2 and climate[END_REF][START_REF] Nel | Implications of fossil fuel constraints on economic growth and global warming[END_REF][START_REF] Höök | Validity of the fossil fuel production outlooks in the IPCC emission scenarios[END_REF][START_REF] Verbruggen | Views on peak oil and its relation to climate change policy[END_REF][START_REF] Chiari | Constraints of fossil fuels depletion on global warming projections[END_REF][START_REF] Höök | Future coal production outlooks in the IPCC emission scenarios: Are they plausible?[END_REF][START_REF] Dale | Global energy modelling -a biophysical approach (GEMBA) part 2: Methodology[END_REF][START_REF] Mercure | An assessement of global energy resource economic potentials[END_REF][START_REF] Berg | Analysis of ultimate fossil fuel reserves and associated co 2 emissions in IPCC scenarios[END_REF][START_REF] Höök | Depletion of fossil fuels and anthropogenic climate change-a review[END_REF][START_REF] Murray | Peak oil and energy independence: Myth and reality[END_REF][START_REF] Capellán-Pérez | Fossil fuel depletion and socio-economic scenarios: An integrated approach[END_REF][START_REF] Chapman | The end of Peak Oil? Why this topic is still relevant despite recent denials[END_REF][START_REF] Mohr | Projection of world fossil fuels by country[END_REF][START_REF] Murray | Limitations of oil production to the IPCC scenarios: The new realities of US and global oil production[END_REF][START_REF] Bauer | Assessing global fossil fuel availability in a scenario framework[END_REF][START_REF] Capellán-Pérez | Likelihood of climate change pathways under uncertainty on fossil fuel resource availability[END_REF][START_REF] Ritchie | Why do climate change scenarios return to coal?[END_REF][START_REF] Wang | The implications of fossil fuel supply constraints on climate change projections: A supply-side analysis[END_REF][START_REF] Capellán-Pérez | MEDEAS: a new modeling framework integrating global biophysical and socioeconomic constraints[END_REF]. For instance, the MESSAGE (IIASA, 2020) and IMACLIM-R (Waisman et al., 2012a) (partly) models continue to depend on the data provided by Rogner et al.Rogner (1997) for global fossil fuel reserves and resources, while the EPPA model [START_REF] Chen | A multisectoral dynamic model for energy, economic, and climate scenario analysis[END_REF] includes simple recursive endogenous resource supply functions. The use of cost-supply curves also impedes the analysis of the economic consequences of a plateau or decline in oil production [START_REF] Bentley | The resource-limited plateau in global conventional oil production: Analysis and consequences[END_REF]-for example left out in the EMF27 [START_REF] Mccollum | Fossil resource and energy security dynamics in conventional and carbon-constrained worlds[END_REF] and RoSE [START_REF] Bauer | Assessing global fossil fuel availability in a scenario framework[END_REF][START_REF] Cherp | Global energy security under different climate policies, GDP growth rates and fossil resource availabilities[END_REF][START_REF] De Cian | The influence of economic growth, population, and fossil fuel scarcity on energy investments[END_REF][START_REF] Kriegler | Will economic growth and fossil fuel scarcity help or hinder climate stabilization?[END_REF] intercomparison exercises. The main problem of using technically simplistic and methodologically questionable cost-supply curves is not only overestimating the plausibility of highemission scenarios, but also making fossil fuels more attractive than they would be if depletion feedback effects were properly considered. This point of view is supported by the evaluation of AR5 scenarios against consistent growth rates of emissions from the fossil fuels industry [START_REF] Burgess | IPCC baseline scenarios have over-projected co 2 emissions and economic growth[END_REF][START_REF] Pielke | Plausible 2005-2050 emissions scenarios project between 2 °C and 3 °C of warming by 2100[END_REF], and the analysis of the GCAM-MAGICC integrated assessment model's sensitivity to revised cost-supply curves [START_REF] Capellán-Pérez | Likelihood of climate change pathways under uncertainty on fossil fuel resource availability[END_REF]. It is also backed by the comparison of WoLiM [START_REF] Capellán-Pérez | Fossil fuel depletion and socio-economic scenarios: An integrated approach[END_REF], 2015) or MEDEAS [START_REF] Capellán-Pérez | MEDEAS: a new modeling framework integrating global biophysical and socioeconomic constraints[END_REF] energy-constrained model results with scenarios from the literature, the incorporation of thorough oil production profiles in IMACLIM-R [START_REF] Rozenberg | Climate policies as a hedge against the uncertainty on future oil supply[END_REF][START_REF] Waisman | Peak oil profiles through the lens of a general equilibrium assessment[END_REF], and the examination of various fossil resource availabilities in the RoSE exercise.

Another critical modeling assumption is the fact that the energy used by the industry for a given scenario is not calculated in relation to the demand for the raw materials necessary for the completion of that scenario. To be able to calculate the raw material requirements, IAMs would have to represent the stocks of all infrastructures, combined with data on lifetimes and material intensities. The potential inconsistency between the industrial energy calculated from elasticities in IAMs and the industrial energy that would be calculated using a stock and raw materials approach adds further uncertainty to the net energy requirements [START_REF] Edelenbosch | Comparing projections of industrial energy demand and greenhouse gas emissions in long-term energy models[END_REF][START_REF] Bataille | Industry in a net-zero emissions world: New mitigation pathways, new supply chains, modelling needs and policy implications[END_REF].

Moreover, IAMs dismiss comprehensive energy-economic feedbacks. They indeed assume that decreasing (net) energy supply, or increasing energy costs, do not influence economic growth whatsoever, as in the IMAGE [START_REF] Stehfest | Integrated assessment of global environmental change with IMAGE 3.0: Model description and policy applications[END_REF], GCAM [START_REF] Calvin | GCAM v5.1: representing the linkages between energy, water, land, climate, and economic systems[END_REF] or POLES (European Commission Joint Research Centre, 2018) models, or have minimal impact when the output is recursively calculated, for instance using nested constant elasticity of substitution (CES) production functions found in models like EPPA, GTEM-C [START_REF] Cai | A hybrid energy-economy model for global integrated assessment of climate change, carbon mitigation and energy transformation[END_REF], REMIND, and WITCH [START_REF] Bosetti | A world induced technical change hybrid model[END_REF]. These functions have indeed faced criticism for their inability to accurately align with historically observed patterns in the dynamics of energy transition [START_REF] Heun | From theory to econometrics to energy policy: Cautionary tales for policymaking using aggregate production functions[END_REF][START_REF] Kaya | Constant elasticity of substitution functions for energy modeling in general equilibrium integrated assessment models: a critical review and recommendations[END_REF]. We find this lack of energy-economic feedbacks particularly troublesome as the decrease in the EROI of the energy system will influence the impact of make demand-side measures in mitigation scenarios. The lack is even more problematic since some authors have found that IAMs favor Bioenergy with Carbon Capture and Storage (BECCS) over the use of renewable energy, notably by underestimating the cost reduction potential of renewables and especially PV [START_REF] Creutzig | The underestimated potential of solar energy to mitigate climate change[END_REF][START_REF] Victoria | Solar photovoltaics is ready to power a sustainable future[END_REF][START_REF] Grant | Cost reductions in renewables can substantially erode the value of carbon capture and storage in mitigation pathways[END_REF][START_REF] Wigley | Value-added diagnostics for the assessment and validation of integrated assessment models[END_REF][START_REF] Xiao | Plummeting costs of renewables -are energy scenarios lagging?[END_REF][START_REF] Way | Empirically grounded technology forecasts and the energy transition[END_REF], while in fact bioenergy and CCS technologies result in a significant decline in net energy [START_REF] Fajardy | The energy return on investment of BECCS: is BECCS a threat to energy security?[END_REF][START_REF] Sekera | Assessing carbon capture: Public policy, science, and societal need[END_REF]. The importance of the net energy-economy feedback becomes even more apparent when considering the substantial energy requirements associated with the deployment of Direct Air Carbon Capture and Storage (DACCS), which are estimated to consume up to 300 EJ/yr by 2100 in some scenarios [START_REF] Realmonte | An inter-model assessment of the role of direct air capture in deep mitigation pathways[END_REF].

A considerable exception to current IAMs is the MEDEAS model [START_REF] Capellán-Pérez | MEDEAS: a new modeling framework integrating global biophysical and socioeconomic constraints[END_REF][START_REF] Samsó | Scenario analysis and sensitivity exploration of the MEDEAS europe energy-economy-environment model[END_REF][START_REF] Solé | Modelling the renewable transition: Scenarios and pathways for a decarbonized future using pymedeas, a new open-source energy systems model[END_REF])-now developed as the WILIAM model in the scope of the LOCOMOTION project-which appears to be the sole multi-scale3 IAM that explores, from a heterodox perspective [START_REF] Nieto | An ecological macroeconomics model: The energy transition in the EU[END_REF], the implications that the energy required for the transition may have on the energy system and the economy [START_REF] Hafner | Emergence of new economics energy transition models: A review[END_REF]. Unlike other IAMs, MEDEAS includes an energy-economy feedback that allows energy availability to limit GDP growth in the event that it falls short of demand [START_REF] Nieto | An ecological macroeconomics model: The energy transition in the EU[END_REF]. When compared with AR5 business-as-usual scenarios, the results obtained with MEDEAS show a larger primary energy intensity of GDP, as well as lower CO 2 intensity of primary energy, GDP per capita, and temperature change over pre-industrial levels [START_REF] Capellán-Pérez | MEDEAS: a new modeling framework integrating global biophysical and socioeconomic constraints[END_REF]. Such a modeling approach not only enables the characterization of the interaction between energy and the economy, such as the rebound effect (de Blas et al., 2020), in a more historically consistent way (de [START_REF] Blas | Modelling of sectoral energy demand through energy intensities in MEDEAS integrated assessment model[END_REF], but also allows the user to assess the probability of GHG scenarios taken from other IAMs [START_REF] Huard | Estimating the likelihood of GHG concentration scenarios from probabilistic integrated assessment model simulations[END_REF], as well as degrowth scenarios (de Blas et al., 2020;[START_REF] Nieto | An ecological macroeconomics model: The energy transition in the EU[END_REF][START_REF] Pulido-Sánchez | Material and energy requirements of transport electrification[END_REF].

The reasons why net energy is not comprehensively accounted for in IAMs are multiple and, in our view, fall primarily into three categories.

First, most IAMs lack proper representation of the energy and material flows of the goods and services provided [START_REF] Pauliuk | Industrial ecology in integrated assessment models[END_REF][START_REF] Pedersen | IPCC emission scenarios: How did critiques affect their quality and relevance 1990-2022?[END_REF][START_REF] Desing | Mobilizing materials to enable a fast energy transition: a conceptual framework[END_REF][START_REF] Aramendia | Exploring the useful energy impacts of the global energy transition: a net energy perspective[END_REF], making them structurally unable to consider the industrial energy embodied in the infrastructures, and thus the energy-economy linkages brought to the fore by net energy analysis. Overlooking these flows may lead to an overestimation of the potential for reducing energy intensity (thus assuming possibly unrealistic decoupling rates between GHG and energy/material use), a greater focus on supply-side solutions for mitigating climate change [START_REF] Scott | Demand vs supply-side approaches to mitigation: What final energy demand assumptions are made to meet 1.5 and 2 °c targets?[END_REF], and underestimating the impact of rebound effects on energy demand [START_REF] Brockway | Energy efficiency and economy-wide rebound effects: A review of the evidence and its implications[END_REF][START_REF] Andrieu | Energy intensity of final consumption: the richer, the poorer the efficiency[END_REF][START_REF] De Blas | The limits of transport decarbonization under the current growth paradigm[END_REF][START_REF] Semieniuk | Plausible energy demand patterns in a growing global economy with climate policy[END_REF][START_REF] Gambhir | Assessing the feasibility of global long-term mitigation scenarios[END_REF].

Second, most IAMs -either energy system models coupled with macroeconomic growth models or multi-sectorial Computable General Equilibrium (CGE) models -still utilize optimal growth theory from neoclassical economics. However, in these models, increases in energy costs cannot significantly affect GDP growth, either because GDP or technological change are assumed to be exogenous, or because the cost share of energy (as a percentage of GDP) is assumed to have negligible feedback on GDP. Thus, the current crop of IAMs not only downplays the contribution of energy and exergy in economic processes [START_REF] Kümmel | How energy conversion drives economic growth far from the equilibrium of neoclassical economics[END_REF][START_REF] Santos | Useful exergy is key in obtaining plausible aggregate production functions and recognizing the role of energy in economic growth: Portugal 1960-2009[END_REF][START_REF] Spangenberg | Like blending chalk and cheese -the impact of standard economics in IPCC scenarios[END_REF], but also sets aside its interaction with money and the financial sector, as these are both largely unmodeled in IAMs [START_REF] Pollitt | The role of money and the financial sector in energy-economy models used for assessing climate and energy policy[END_REF][START_REF] Espagne | Money, finance and climate: The elusive quest for a truly integrated assessment model[END_REF][START_REF] Sanders | Representation of financial markets in macro-economic transition models-a review and suggestions for extensions[END_REF][START_REF] Valcke | Macrodynamics and climate: a reformulation[END_REF]. This omission further precludes any attempt to understand how high levels of debt, which can increase financial instability risks, can be associated with net energy constraints or high energy costs (e.g. the global financial crisis of 2007-2008) [START_REF] Svartzman | From ecological macroeconomics to a theory of endogenous money for a finite planet[END_REF].

Third, the current climate change scenarios framework (illustrated by the Shared Socioeconomic Pathways, SSP [START_REF] Riahi | The Shared Socioeconomic Pathways and their energy, land use, and greenhouse gas emissions implications: An overview[END_REF]) nurtures a simplistic and technocratic vision of the economy, that assumes little in the way of interdependence among population, economic growth, and other socio-economic parameters such as net-energy [START_REF] Kuhnhenn | Economic growth in mitigation scenarios: A blind spot in climate science global scenarios from a growth-critical perspective[END_REF][START_REF] Buhaug | On growth projections in the shared socioeconomic pathways[END_REF][START_REF] Asefi-Najafabady | The failure of integrated assessment models as a response to 'climate emergency' and ecological breakdown: the emperor has no clothes[END_REF][START_REF] Purvis | Modelling global futures: a comparison of 'limits to growth' and the use of integrated assessment models within the climate literature[END_REF][START_REF] Court | A representation of the world population dynamics for integrated assessment models[END_REF]Pielke and Ritchie, 2021a,b). This lack of explicit interdependence hinders existing climate mitigation scenarios from adequately assessing societal transformations [START_REF] Trutnevyte | Societal transformations in models for energy and climate policy: The ambitious next step[END_REF], (in)justice [START_REF] Rao | Income inequality projections for the shared socioeconomic pathways (ssps)[END_REF][START_REF] Rivadeneira | in)justice in modelled climate futures: A review of integrated assessment modelling critiques through a justice lens[END_REF][START_REF] Pachauri | Fairness considerations in global mitigation investments[END_REF] including large-scale shifts in energy use between Global North and South [START_REF] Hickel | Existing climate mitigation scenarios perpetuate colonial inequalities[END_REF], and systemic risks [START_REF] Rising | The missing risks of climate change[END_REF]. The roots of the aforementioned limitations can be found in a lack of reflexivity, imaginative flexibility, plurality, transparency and transdisciplinarity within the IAM community, as acknowledged by some of its own members [START_REF] Doukas | From Integrated to Integrative: Delivering on the Paris Agreement[END_REF][START_REF] Anderson | Debating the bedrock of climate-change mitigation scenarios[END_REF][START_REF] Foster | Concrete utopianism in integrated assessment models: Discovering the philosophy of the shared socioeconomic pathways[END_REF][START_REF] Hirt | A review of linking models and socio-technical transitions theories for energy and climate solutions[END_REF][START_REF] O'neill | Achievements and needs for the climate change scenario framework[END_REF][START_REF] Raskin | Excluded futures: the continuity bias in scenario assessments[END_REF][START_REF] Robertson | Transparency, trust, and integrated assessment models: An ethical consideration for the intergovernmental panel on climate change[END_REF][START_REF] Skea | Intergovernmental panel on climate change: Transparency and integrated assessment modeling[END_REF][START_REF] Sgouridis | Visions before models: The ethos of energy modeling in an era of transition[END_REF][START_REF] Pedersen | IPCC emission scenarios: How did critiques affect their quality and relevance 1990-2022?[END_REF][START_REF] Koasidis | Why integrated assessment models alone are insufficient to navigate us through the polycrisis[END_REF], but also in the gradual erosion of IAMs' neutrality [START_REF] Ellenbeck | How modelers construct energy costs: Discursive elements in energy system and integrated assessment models[END_REF][START_REF] Beek | Navigating the political: An analysis of political calibration of integrated assessment modelling in light of the 1.5 ○ c goal[END_REF] due to political influence, and the community's interest in playing an increasingly normative role in climate governance and policy-making [START_REF] Beck | The politics of anticipation: the ipcc and the negative emissions technologies experience[END_REF][START_REF] Beek | Anticipating futures through models: the rise of integrated assessment modelling in the climate science-policy interface since 1970[END_REF][START_REF] Beck | Imagining the corridor of climate mitigation -what is at stake in ipcc's politics of anticipation?[END_REF].

Avenues of improvement to consider net energy in IAMs

Several initiatives are underway to better account for industrial energy, and represent the interactions between energy and the economy in macroeconomic models and/or IAMs [START_REF] Keppo | Exploring the possibility space: taking stock of the diverse capabilities and gaps in integrated assessment models[END_REF]. Some IAMs, for instance, have adopted the use of more reliable data pertaining to fossil fuel energy resources, as exemplified by the incorporation of a comprehensive bottom-up dataset from Rystad Energy in E3ME-FTT-GENIE [START_REF] Mercure | Reframing incentives for climate policy action[END_REF][START_REF] Semieniuk | Stranded fossil-fuel assets translate to major losses for investors in advanced economies[END_REF] or the construction of detailed field-level analysis supply curves in TIAM-UCL [START_REF] Welsby | Unextractable fossil fuels in a 1.5 °C world[END_REF]. Dynamic constraints on extraction rates have also been introduced, as in REMIND [START_REF] Baumstark | REMIND2.1: transformation and innovation dynamics of the energy-economic system within climate and sustainability limits[END_REF] or TIAM-UCL [START_REF] Pye | The TIAM-UCL Model (Version 4.1.1) Documentation[END_REF]) models, as well as specific rules trying to mimic the behavior of swing producers, as in IMACLIM-R or IMAGE, albeit in a very simplified way and mostly for the oil market [START_REF] Faehn | Capturing key energy and emission trends in CGE models: Assessment of status and remaining challenges[END_REF][START_REF] Foure | Macroeconomic drivers of baseline scenarios in dynamic CGE models: review and guidelines proposal[END_REF]. In an attempt to bridge the gap with Industrial Ecology (IE), several IAMs (notably REMIND and MESSAGE) have explored the implications of incorporating life cycle assessment coefficients from input-output (I-O) tables such as THEMIS [START_REF] Pehl | Understanding future emissions from low-carbon power systems by integration of lifecycle assessment and integrated energy modelling[END_REF][START_REF] Arvesen | Deriving life cycle assessment coefficients for application in integrated assessment modelling[END_REF][START_REF] Luderer | Environmental co-benefits and adverse side-effects of alternative power sector decarbonization strategies[END_REF] or EXIOBASE [START_REF] Budzinski | Coupling energy system models with multi-regional input-output models based on the make and use framework -insights from MESSAGEix and EXIOBASE[END_REF], highlighting the potential for sustainability research areas such as the energy-industry nexus and post-growth scenarios [START_REF] Lefèvre | Integrated assessment models and input-output analysis: bridging fields for advancing sustainability scenarios research[END_REF]. Efforts are also underway for examining the contribution of improved Energy System Models (ESM) [START_REF] Huang | Appending material flows to the national energy modeling system (NEMS) for projecting the physical economy of the united states[END_REF][START_REF] Kullmann | Combining the worlds of energy systems and material flow analysis: a review[END_REF]. Still, most IAMs operate within the neoclassical equilibrium framework, and thus do not properly capture the feedback from the energy system on the economy. For instance, [START_REF] Pehl | Understanding future emissions from low-carbon power systems by integration of lifecycle assessment and integrated energy modelling[END_REF] [START_REF] Pehl | Understanding future emissions from low-carbon power systems by integration of lifecycle assessment and integrated energy modelling[END_REF] integrate a life-cycle assessment perspective in the REMIND model, and find that "fully considering life-cycle greenhouse gas emissions has only modest effects on the scale and structure of power production in cost-optimal mitigation scenarios". However, the authors rely on a model that uses a CES production function, with limited feedback from the energy system (including its energy requirements) on the economy, and focus exclusively on the power sector.

In an attempt to remedy this situation, we highlight six avenues for improving IAMs:

1. the integration of Industrial Ecology methods (e.g. Material Flow Analysis) or modules such as DyMEMDS [START_REF] Vidal | Modelling the Demand and Access of Mineral Resources in a Changing World[END_REF][START_REF] Boulzec | Dynamic modeling of global fossil fuel infrastructure and materials needs: Overcoming a lack of available data[END_REF][START_REF] Boulzec | Material requirements and impacts of the building sector in the shared socioeconomic pathways[END_REF], ODYM-RECC [START_REF] Pauliuk | ODYM-an open software framework for studying dynamic material systems: Principles, implementation, and data structures[END_REF][START_REF] Pauliuk | Linking service provision to material cycles: A new framework for studying the resource efficiency-climate change (RECC) nexus[END_REF], or QTDIAN [START_REF] Süsser | The QTDIAN modelling toolbox-Quantification of social drivers and constraints of the diffusion of energy technologies[END_REF], in order to better capture stocks and flows of energy and materials associated with the industrial subsector;

2. the adoption of a multi-sectoral energy framework, for instance relying on primary-finaluseful (PFU) energy databases [START_REF] Heun | The energy and exergy of light with application to societal exergy analysis[END_REF][START_REF] Steenwyk | The contributions of muscle and machine work to land and labor productivity in world agriculture since 1800[END_REF][START_REF] Pinto | The rise and stall of world electricity efficiency:1900-2017, results and insights for the renewables transition[END_REF][START_REF] Marshall | TA Multi-Regional Primary-Final-Useful (MR-PFU) energy and exergy database v1[END_REF] and consistent energy services narratives;

3. the use of an ecological macroeconomic framework (as in HARMONEY [START_REF] King | An integrated biophysical and economic modeling framework for long-term sustainability analysis: the HARMONEY model[END_REF][START_REF] King | Interdependence of growth, structure, size and resource consumption during an economic growth cycle[END_REF] or TranSim [START_REF] Jackson | Modelling energy transition risk: The impact of declining energy return on investment (EROI)[END_REF]) or ESM which deal with environmental and biophysical indicators [START_REF] Sherwood | Putting the biophysical (back) in economics: A taxonomic review of modeling the earth-bound economy[END_REF] in a more comprehensive way, such as ENBIOS [START_REF] Martin | Overlooked factors in predicting the transition to clean electricity[END_REF], EnergyScope [START_REF] Limpens | EnergyScope TD: A novel opensource model for regional energy systems[END_REF] or the one developed by Crownshaw [START_REF] Crownshaw | A modelling framework for evaluating the dynamic metabolic feasibility of energy transition scenarios[END_REF]; 4. a common reporting template to include the energy consumption of the energy sector as well as useful energy in mitigation scenarios [START_REF] Neumeyer | Dynamic EROI assessment of the IPCC 21 st century electricity production scenario[END_REF][START_REF] Palmer | A biophysical perspective of IPCC integrated energy modelling[END_REF][START_REF] Slameršak | Energy requirements and carbon emissions for a low-carbon energy transition[END_REF];

5. an explicit modelling of energy-economy feedbacks (including rebound effects) using net energy, at least with the aim of understanding how a net energy feedback would affect IAMs results to assess to what extent energy and economic feedbacks are internally (in)consistent across mitigation pathways [START_REF] Palmer | A biophysical perspective of IPCC integrated energy modelling[END_REF];

6. the exploration of new mitigation pathways achieving high wellbeing levels with low resource use [START_REF] Wilson | The 'high-with-low' scenario narrative: Key themes, cross-cutting linkages, and implications for modelling[END_REF], limiting the deployment of energy intensive carbon dioxide removal(van [START_REF] Vuuren | Alternative pathways to the 1.5 °C target reduce the need for negative emission technologies[END_REF]Pye et al., 2020a;[START_REF] Diesendorf | Scenarios for mitigating co2 emissions from energy supply in the absence of co2 removal[END_REF][START_REF] Hollnaicher | On economic modeling of carbon dioxide removal: values, bias, and norms for good policy-advising modeling[END_REF][START_REF] Sers | Ecological macroeconomic assessment of meeting a carbon budget without negative emissions[END_REF], and equitable low-growth [START_REF] Burgess | Multidecadal dynamics project slow 21st-century economic growth and income convergence[END_REF] and post-growth scenarios [START_REF] Floyd | Energy descent as a post-carbon transition scenario: How 'knowledge humility' reshapes energy futures for post-normal times[END_REF][START_REF] Otero | Biodiversity policy beyond economic growth[END_REF][START_REF] Keyßer | 1.5 °C degrowth scenarios suggest the need for new mitigation pathways[END_REF][START_REF] Warszawski | All options, not silver bullets, needed to limit global warming to 1.5 °C: a scenario appraisal[END_REF][START_REF] Hickel | Urgent need for post-growth climate mitigation scenarios[END_REF][START_REF] Kikstra | Towards degrowth in integrated assessment models[END_REF][START_REF] Li | Integrated assessment modelling of degrowth scenarios for australia[END_REF][START_REF] Moyer | Modeling transformational policy pathways on low growth and negative growth scenarios to assess impacts on socioeconomic development and carbon emissions[END_REF].

However, if these measures are to be properly implemented, they must be carried out simultaneously and without neoclassical economics theories [START_REF] Brand-Correa | Economics for people and planet-moving beyond the neoclassical paradigm[END_REF][START_REF] Stern | The economics of immense risk, urgent action and radical change: towards new approaches to the economics of climate change[END_REF], which we see as incompatible. This point particularly addresses the IAM community's appeal that "further studies should at least aim at better reflecting the plurality of the visions of the economy" [START_REF] Keppo | Exploring the possibility space: taking stock of the diverse capabilities and gaps in integrated assessment models[END_REF] and take advantage of the robust development of heterodox economics [START_REF] Proctor | Expanding the possible: exploring the role for heterodox economics in integrated climate-economy modeling[END_REF], especially ecological macroeconomics [START_REF] Cattan | A macroeconomic critique of integrated assessment environmental models: the case of Brazil[END_REF][START_REF] Althouse | Ecological Macroeconomics for a Shared Planet: towards a Global Political Ecology of Money[END_REF].

As countries seek to develop new nationally determined contributions (NDCs) and the IPCC currently considers reforming itself to produce more relevant knowledge for climate action [START_REF] Asayama | Three institutional pathways to envision the future of the ipcc[END_REF][START_REF] Hermansen | Climate action post-paris: how can the ipcc stay relevant? npj[END_REF][START_REF] Noy | Economists are not engaged enough with the ipcc[END_REF], current momentum is towards the development of a new generation of IAMs and scenarios [START_REF] Pedersen | Variability in historical emissions trends suggests a need for a wide range of global scenarios and regional analyses[END_REF][START_REF] Gambhir | Climate change mitigation scenario databases should incorporate more non-IAM pathways[END_REF]Pedersen et al., 2022a;[START_REF] Ranjan | A new scenario framework for equitable and climate-compatible futures[END_REF][START_REF] Savin | Tired of climate targets? shift focus of ipcc scenarios from emission and growth targets to policies[END_REF][START_REF] Meinshausen | A perspective on the next generation of earth system model scenarios: towards representative emission pathways[END_REF][START_REF] Peters | Ar6 scenarios database: an assessment of current practices and future recommendations[END_REF][START_REF] Szetey | Improving the relevance of the shared socioeconomic pathways for sustainability science[END_REF]. In this regard, we believe that fostering collaborative efforts among our different research communities is timely, and could help improve integrated assessment modeling, with these dynamics being all the more supported by the convergence of views on demand-side measures and alternative economic pathways.

Conclusion

Consideration of net energy is crucial to assess and design comprehensive and coherent climate mitigation scenarios. Yet, in part due to the late emergence of robust results in the EROI literature, such consideration has not yet spread beyond the Net Energy Analysis community. Here we try to address this issue by outlining the emerging EROI consensus, exploring key areas under investigation, and identifying further work.

Our summary underlines that, in a fossil fuel dominated world, the initial energy investment to power the transition to a low-carbon future will inevitably come from fossil fuels. This does not mean, though, that renewables cannot eventually support themselves. However, net energy constraints may still limit the energy available to invest in energy infrastructure and the energy available for discretionary uses, absent more sustainable production and consumption habits. This situation may be particularly destabilizing for industrializing countries, which might stay at the doorstep of the energy transition, unable to increase their reliance on modern low-carbon energy, while being among the least responsible and among the most impacted by climate change. These dynamics should not be ignored in transition scenarios, and we therefore call on fellow researchers to integrate net energy into Integrated Assessment Models using theories outside of the neoclassical economics paradigm. In this respect, we believe that fostering collaborative efforts among our different research communities could prove decisive.

Figure 2 :

 2 Figure 2: Standard, or primary stage EROI (EROI standard ), point-of-use or final stage EROI (EROI point-of -use ) and useful EROI (EROI usef ul ).

Figure 3 :

 3 Figure 3: Sketch of principle of an evolving window of opportunity for the global inclusive transition to low-carbon energy as a function of normalized time at the start of the transition. The solid lines delineate the current window of opportunity. The dashed lines represent a future window in which action has not been taken quickly enough such that climate and net energy increasingly constrain the window of opportunity.

The inclusion of storage devices in the system boundary (rather than at the level of an individual power generation technology), however, is more relevant at the country, regional, or grid level, because each technology, if deployed in isolation, would require some storage capacity to successfully keep up with the pattern of electricity demand.

In this regard, it is worth noting that it has been estimated that, due to residual fossil emissions, 640-950 GtCO 2 carbon dioxide removal (CDR), i.e., BECCS, DACCS and afforestation, will be required for a likely chance of limiting end-of-century warming to 1.5 ○ C, when strengthened pre-2030 mitigation action is combined with very stringent long-term policies[START_REF] Luderer | Residual fossil CO 2 emissions in 1.5-2 °C pathways[END_REF].

The SFCIO-IAM[START_REF] Sers | Ecological macroeconomic assessment of meeting a carbon budget without negative emissions[END_REF] and WORLD7[START_REF] Sverdrup | Development of a biophysical economics module for the global integrated assessment model WORLD7[END_REF] are for instance only global models.

Acknowledgments

This work benefited from the careful reading of Paul Brockway, Jarmo Kikstra and Loïc Giaccone. We also thank Iñigo Capellán-Pérez for comments on a previous version of the manuscript. Louis Delannoy, Pierre-Yves Longaretti and Emmanuel Prados acknowledge the support of the French National Institute for Research in Digital Science and Technology (INRIA). Victor Court acknowledges the support of the Chair Energy and Prosperity, under the aegis of La Fondation du Risque.

Author Contributions

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Zhao Dan

Jean-Louis Thomas

Régis Marchiano

Computation of the radiation force exerted by an acoustic tweezers using pressure field measurements

Acoustic tweezers allow to manipulate small objects like elastic spheres with a force generated by the radiation pressure which arises from the nonlinear interaction between the incident and scattered waves by the object. The accurate control of the object by acoustic tweezers requires the study of the components of the threedimensional force.If the physical properties of the elastic sphere are known, then the 3D components of the force can be calculated thanks to a decomposition of the incident acoustic field in the spherical functions basis. In this work, we propose to evaluatethe expansion coefficients. Three methods are used and compared. The first one consists in measuring the acoustic field on a spherical surface centered on the theoretical position of the object and to calculate the spherical functions decomposition by Lebedev quadratures. The second method is based on the measurement of the acoustic field at random points in a spherical volume and on the resolution of the inverse problem by a sparse method called the orthogonal matching pursuit. In the third method, the incident beam is measured on a transverse plane, decomposed into a sum of plane waves and then the expansion coefficients are calculated. The results of the three methods will be presented and compared.

Acoustic tweezers like optical and magnetic tweezers are used for 3D manipulation of small objects such as cells and molecules. These micro manipulators are in a strong demand in the field of biophysics. Among these tweezers, optical tweezers [START_REF] Ashkin | Acceleration and trapping of particles by radiation pressure[END_REF][START_REF] Ashkin | Observation of a single-beam gradient force optical trap for dielectric particles[END_REF][START_REF] Ashkin | Optical trapping and manipulation of single cells using infrared laser beams[END_REF] have been developed first and demonstrated their precision and trap capabilities. However, the high intensity at the focus of optical tweezers can lead to photo-damage and heating of the target object, especially in vivo samples. Magnetic tweezers are not limited by photo-damage and are widely used in the biology field especially for manipulation and analyses of DNA and RNA [START_REF] Lionnet | Wringing out dna[END_REF]32 . The main limitation of magnetic tweezers is the limited range of constant-force that can be exerted due to their low trap stiffness. Moreover, applying large force requires high-current electromagnets which would cause heating or produce non constant-force. The radiation force is proportional to the field intensity divided by the speed of propagation.

As the velocity of light is 5 orders of magnitude larger than sound speed, acoustic tweezers are able to afford large forces with much smaller intensity than optical tweezers and are a solution for the heating issue. This advantage enables a wide range of applications of acoustic tweezers in various domains such as materials science, study of biophysical properties of cells and molecules, micro-rheology, biophysical characterization of DNA, etc. Different kinds of acoustic traps exist. Most of them are based on standing waves either in the bulk [START_REF] Xie | Acoustic method for levitation of small living animals[END_REF] or propagating at the surface of a solid substrate [START_REF] Collins | Acoustic tweezers via sub-time-of-flight regime surface acoustic waves[END_REF][START_REF] Ding | On-chip manipulation of single microparticles, cells, and organisms using surface acoustic waves[END_REF] . In these schemes, all pressure nodes (or anti-nodes depending on the object density and compressibility) act as potential traps. On the contrary, optical tweezers are selective traps with a single position of equilibrium. Radial selectivity was achieved by Wu 34 using two counterpropagating focused ultrasonic beams. Single beam acoustic tweezers are characterized by the ability to pick up, trap and manipulate a single small elastic particles in three dimensions [START_REF] Baresch | Acoustic tweezers: trapping and manipulation of an object by radiation pressure of a progressive wave[END_REF][START_REF] Baresch | Spherical vortex beams of high radial degree for enhanced single-beam tweezers[END_REF][START_REF] Baresch | Observation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers[END_REF][START_REF] Baudoin | Folding a focalized acoustical vortex on a flat holographic transducer: Miniaturized selective acoustical tweezers[END_REF] as its optical counterpart. For all the possible applications, the calibration of the force provided by acoustic tweezers is of importance, especially for micro-rheology studies. The optical and magnetic tweezers' forces calibration is achieved mainly by two methods: the first one consists in studying the Brownian motion of trapped objects with sizes comparable to the wavelength, then the force is determined from Hooke's law [START_REF] Carter | Precision surface-coupled optical-trapping assay with one-basepair resolution[END_REF][START_REF] Mack | Practical axial optical trapping[END_REF] ; the second one is to use the viscous drag force generated by a controlled fluid flow [START_REF] Malagnino | Measurements of trapping efficiency and stiffness in optical tweezers[END_REF] .

Nonetheless, the first method is not applicable for acoustic tweezers since the wavelength and the object can be much larger and the Brownian motion disappears at these scales. As for the second method, the fluid drag forces are also used to calibrate the acoustic trapping force 22 , but difficulties arise for single beam acoustic tweezers due to the Magnus effect [START_REF] Goldstein | Classical Mechanics[END_REF] caused by the rotation of bead in an acoustic vortex beam [START_REF] Baresch | Orbital angular momentum transfer to stably trapped elastic particles in acoustical vortex beams[END_REF] . Previously, the three dimensional force exerted on a spherical particle was modeled using the incident beam expansion on spherical functions. This model depends on the expansion coefficients, A m n , dubbed beam shape coefficients (BSC), and the scattering coefficients of the particle, R n 6 .

In this article, we are interested in characterizing the radiation pressure exerted by acoustic tweezers on a spherical particle using this model. Thus, this is not a direct measurement of the force. The acoustic field is measured with a calibrated hydrophone and then the force is deduced. Section II is a short reminder of the model focused on the equations needed to calculate the radiation pressure exerted on an elastic sphere. To obtain the BSC of the incident pressure field, three methods are numerically investigated in section III: Lebedev quadrature, inverse problem regularization by sparsity and angular spectrum method (ASM). In section IV, a focused Gaussian beam and a focused acoustic vortex are synthesized and measured. The BSC are recovered by applying the three methods. Then, the acoustic field is reconstructed and compared with measurements and finally the radiation pressure is determined by the three methods.

II. THEORETICAL BACKGROUND: RADIATION PRESSURE

Let us consider a 3D Cartesian system of coordinates (x, y, z). The three components of radiation pressure exerted on an arbitrarily located elastic sphere in a perfect fluid by an arbitrarily incident beam are [START_REF] Baresch | Three-dimensional acoustic radiation force on an arbitrarily located elastic sphere[END_REF] :

F x = - V k 2 0 ∞ n=0 |m|<n (Q -m n A m * n A m-1 n+1 C n + Q m n A m n A m+1 * n+1 C * n ), (1)
F y = + V k 2 0 ∞ n=0 |m|<n (Q -m n A m * n A m-1 n+1 C n + Q m n A m n A m+1 * n+1 C * n ), (2)
F z = +2 V k 2 0 ∞ n=0 |m|<n (G m n A m * n A m n+1 C n ). (3)
With :

V = p 2 0 /(4ρ 0 c 2 0 ), C n = R * n + R n+1 + 2R * n R n+1 , Q m n = (n + m + 1)(n + m + 2)/ (2n + 1)(2n + 3), G m n = (n + m + 1)(n -m + 1)/ (2n + 1)(2n + 3).
p a (x, y, z, t) is the linear component of the acoustic pressure, its time average is zero. ρ 0 and c 0 are respectively the fluid density and speed of sound at rest. Coefficients A m n are the coefficients of the expansion into spherical functions:

p a (r, θ, ϕ, t) = p 0 ∞ n=0 |m|<n A m n j n (kr)Y m n (θ, ϕ)exp(-iωt) (4)
where the spherical harmonics are defined by: From the theoretical expressions of the radiation pressure, two sets of coefficients are required to determine the forces:

Y m n (θ, ϕ) = (2n + 1) 4π (n -m)! (n + m)! P m n (cos θ)e imϕ = N m n P m n (cos θ)e imϕ
• Scattering coefficients: R n • Incident BSC: A m n
The scattering coefficients R n for an arbitrary incident beam are known and depend only on the physical characteristics of the object and propagation medium [START_REF] Baresch | Three-dimensional acoustic radiation force on an arbitrarily located elastic sphere[END_REF] . Therefore, our problem of determining the radiation forces can be solved by searching the incident BSC A m n . These coefficients are known analytically for several beams (plane wave, Bessel beam, vortex beam). Nevertheless, in real applications the pressure field is sampled in time and in space on a finite set of points and make the determination of BSC a challenge. In the following part, three different methods for calculating these coefficients are investigated.

III. CALCULATION OF BEAM SHAPE COEFFICIENT

A m n FROM THE PRES- SURE FIELD
For each method, the determination of the BSC is tested on the same incident field: a focused vortex beam of topological charge m = 1. This beam is very important because, it is the keystone to create acoustic tweezers [START_REF] Baresch | Acoustic tweezers: trapping and manipulation of an object by radiation pressure of a progressive wave[END_REF][START_REF] Baresch | Spherical vortex beams of high radial degree for enhanced single-beam tweezers[END_REF][START_REF] Baresch | Observation of a single-beam gradient force acoustical trap for elastic particles: acoustical tweezers[END_REF][START_REF] Marzo | Holographic acoustic elements for manipulation of levitated objects[END_REF] . As this paper proposes to address the problem of the characterisation of the radiation force and especially the one exerted by acoustic tweezers, it is reasonable to validate the methodology on a field as close as possible to the final target. The BSC for a focused vortex beam are 5 :

A m n = δ m,m 4π(kr 0 ) 2 h (1) n (kr 0 ) π π-α 0 P m n (cosθ )N m n dθ . (6)
with δ m,m the Kronecker delta, h

n the spherical Hankel function of first kind, m the topological charge of the vortex and α 0 , a 0 , r 0 are respectively the aperture half angle, the radius of the transducer and the focal distance as illustrated on top of Fig. 1. In this paper, we use the expression of normalized spherical harmonics with a factor N m n (see Eq. 5), hence A m n are different from previous works 6 , [START_REF] Baresch | Spherical vortex beams of high radial degree for enhanced single-beam tweezers[END_REF] .

- Using Eq. 4, the pressure field can be computed anywhere. Fig. 1 shows the pressure field for a focused vortex beam of charge m = 1 (top row), and the BSC (bottom). We selected the case of water, c 0 = 1500m • s -1 , as propagating medium and a frequency of 1.2M Hz. The wavelength is λ = 1.25mm. The pressure field has a zero amplitude along the axis of propagation. This is a common feature associated with vortex beam [START_REF] Hefner | An acoustical helicoidal wave transducer with applications for the alignment of ultrasonic and underwater systems[END_REF][START_REF] Marchiano | Synthesis and analysis of linear and nonlinear acoustical vortices[END_REF] . The focusing is sharp because the aperture of the transducer is 110mm(88λ), 75mm(60λ) away from the focus (corresponding to an aperture half angle equal to α 0 = 43 o ). As expected the BSC are restricted to the column m = 1 with a non intuitive variation in function of the radial degree. We assessed numerically the error on the radiation force, Eq. 1-3 by decreasing the truncation order. Thereafter, all series are truncated at n ≤ N = 25 and the error on the force is 0.001.

A. Lebedev Quadrature

The first method is based on the orthogonality of the spherical harmonics:

< Y m n , Y m n >= δ n,n δ m,m
where the scalar product on the sphere is defined by < f (θ, ϕ), g(θ, ϕ) >= f (θ, ϕ)g(θ, ϕ) sin θdθdϕ. By applying this property, BSC A m n can be expressed by integrals over a spherical surface:

A m n = 1 p 0 j n (kr) < p, Y m n > = 1 p 0 j n (kr) θ ϕ p(θ, ϕ)Y m * n (θ, ϕ) sin θdθdϕ. (7)
The integrals in Eq. 7 are computed by numerical integration. Using a numerical quadrature to perform this integration in the context of acoustic radiation force or torque have already been investigated [START_REF] Silva | Off-axial acoustic radiation force of repulsor and tractor bessel beams on a sphere[END_REF][START_REF] Silva | Radiation torque produced by an arbitrary acoustic wave[END_REF] . In these previous works, the sphere was sampled with a very fine grid incompatible with actual measurements of the pressure fields and there was no noise. Different quadrature rules have been tested: Legendre-Gauss quadrature, Chebyshev quadrature 10 and Lebedev quadrature. Among these quadratures, the Lebedev quadrature 31

gives the best precision for a given number of points and only that quadrature is used here [START_REF] Lebedev | A quadrature formula for the sphere of the 131st algebraic order of accuracy[END_REF] .

The number, position and the weight of Lebedev grids defined on an unit sphere ((x i , y i , z i ) and weights w i ) have been derived by Sobolev [START_REF] Sobolev | Cubature formulas on the sphere invariant under finite groups of rotations[END_REF] . Therefore, measuring the pressure field at the Lebedev points (see Fig. 2 to visualize their positions) and using this quadrature give the BSC A m n .

A m n = 1 p 0 j n (kr)

I-1 i=0 p(θ i , ϕ i )Y m * n (θ i , ϕ i )w i . (8)
Lebedev quadrature has optimal efficiency, i.e. the number of points required, I = (N + 1) 2 /3, is the smallest, where N is the highest order of the polynomials integrated on the sphere. Moreover, the distance between the Lebedev points is roughly constant. This feature is very interesting since it provides an optimal sampling of the sphere in regards with the finite size of an hydrophone and thus it optimizes the signal-to-noise-ratio (SNR)

The integrand is the product of spherical harmonics, Eq. 4, 7. If the series is truncated at n ≤ N , the integrand is a polynomial of order smaller than 2N and hence N = 2N . For our case, N = 25 and this yields I = 867. It must be noted that the number of Lebedev points is not arbitrary. Here, we use I = 974 Lebedev points on a sphere with radius 7mm(5.6λ).

This choice amendellows to perfectly retrieve the high order modes (here up until n = 25) as shown in Fig. 2, the BSC obtained by the Lebedev quadrature for a focused vortex beam. Of course, this previous estimation does not take into account the noise in the measurements.

It is known that the determination of the BSC are prone to errors in the presence of noise 29 .

To assess the robustness of the method in presence of noise, we proceed in three steps. First, the BSC of Eq. 6, named thereafter A m n th , are computed and the corresponding pressure field calculated with Eq. 4 on the Lebedev grid and in the focal plane to determine the maximum pressure. Second, a noise with a uniform distribution in an interval of amplitude 5% of this maximum pressure is added to the pressure field calculated on the Lebedev grid. Third, the BSC of this noisy pressure field, noted A m n , are estimated with Eq. 8 and shown on Fig. 2.

In Fig. 2, the lines where the BSC are very different from the original ones correspond to 10 the values closest to zero for the Bessel function (Fig. 3). Indeed, since the scalar product with spherical harmonics is a linear operation, the result is the scalar product with the ideal pressure fields plus the scalar product with the noise. Hence the error is proportional to 1/j n (kr). On Fig. 3, we can observe a first oscillating part up to n = 35 followed by a fast decrease converging to 0. We selected a sphere radius large enough, 7mm(5.6λ), so that the truncature order N = 25 is located in the oscillating part.

To assess the numerical performance of the method, we compute the relative error:

err = 1 (N + 1) 2 N n=0 m=n m=-n |A m n -A m n th | max(|A m n th |) (9)
(N + 1) 2 is the total number of BSC A m n of order n ≤ N . Here the relative error is 0.061. An upgrade in order to mitigate the detrimental effect of noise is to use a double layer Lebedev sphere [START_REF] Rafaely | The spherical-shell microphone array[END_REF] . The idea is to use two spheres with different radii and to apply the Lebedev quadrature, for a given ordrer n, to the sphere for which the Bessel function has the greatest absolute value. Using the asymptotic behavior of Bessel function for large x compared to n, j n (x) ≈ cos(x -π/2)/x, in Fig. 3, we selected the radius of the second sphere such that j n (x ) ≈ sin(x -π/2)/x to optimize the estimation, this leads to r 1 = Instead of solving Eq. 7 with its discretized version Eq. 8, another strategy consists in solving Eq. 4 whose discretized counterpart can be reformulated under a matrix/vector form:

P = M A + . ( 10
)
with vector P whose components are the Fourier transform of the pressure field at frequency f 0 at points of discretization (x i , y i , z i ) : p(x i , y i , z i , f 0 ) of length I, vector A whose compo-nents are the BSC A l = A m n with l = n(n + 1) + m of length L = (N + 1) 2 , the matrix M whose elements are (j n (kr i )Y m n (θ i , ϕ i )) with (r i , θ i , ϕ i ) the points (x i , y i , z i ) written in spherical coordinates of size (I ×L)and vector the additive noise on points (x i , y i , z i ). To compare this method with the double layer Lebedev sphere quadrature, we set the same number of points: 1948 dispersed in a spherical volume of identical radius r = 7.11mm(5.7λ). As previously, the truncature order is set to 25. At this stage the points distribution is free and this can be used to avoid an ill-conditioned matrix. Then, the best choice is a set of random points distributed in a spherical volume as illustrated on Fig. 5.

Because of the noise, the matrix M is always full rank, the direct inversion is then always possible but unstable in regard of a small change in the noise. This ill-posed problem required regularization to get a meaningful solution. As can be seen on Fig. 1 for a focused vortex beam, a large number of BSC are null. So, vector A is sparse. This a priori can be used to regularize the inversion:

à = argmin ||A|| 0 such as P = M A (11)
With this formulation, vector à is searched with a particular constraint: it must contain a minimum of non-zero terms.

To solve Eq. 11, we choose to use Orthogonal Matching Pursuit algorithm (OMP) [START_REF] Cai | Orthogonal matching pursuit for sparse signal recovery with noise[END_REF] .

This algorithm is iterative. For each iteration, the component of M with the highest inner product with the remaining part of vector P is selected. Then its contribution is subtracted and the iterations continue on the residue. This procedure stops when the iteration reaches the number of non-zero elements of the BSC (25 in our case) or when the residual reaches a limit. With the OMP method, we should be able to recover the BSC on the column m = 1 with 25 iterations. However, tests for a vortex beam have shown the inefficiency of this stopping criterion. Thus, the stopping criterion of the OMP procedure will be the residual limit (lower than 0.001). Fig. 5 shows the BSC obtained with the OMP algorithm for the same noisy pressure field as before.There is a very good agreement with the original set of BSC even if some BSC laying outside the column m = 1 are not exactly set to zero. Here, the relative error is 0.0141 close to the 0.015 obtained with the Lebedev method. A drawback is the number of iteration required, the computation can be very long. A method to improve this is the Block version of OMP. It's the same procedure but with the matrix M in a block version [START_REF] Eldar | Block-sparse signals: Uncertainty relations and efficient recovery[END_REF] . Fig. 5 shows the BSC obtained by applying the Block-OMP method, the matrix M is divided into blocks of (N × 10), and the iteration number is 35 only. An unexpected result is that the relative error is then twice better : 0.006

C. Angular spectrum method (ASM)

A third approach is to use a transformation from angular spectrum to spherical harmonics. [START_REF] Sapozhnikov | Radiation force of an arbitrary acoustic beam on an elastic sphere in a fluid[END_REF] .

The Fourier transform of the pressure in plane z can be seen as a superposition of plane waves:

p(x, y, z, ω) = 1 4π 2 k 2 x +k 2 y ≤k 2 S(k x , k y )e ikxx+ikyy+i √ k 2 -k 2 x -k 2 y z dk x dk y , (12)
where the angular spectrum S(k x , k y ) is the 2D spatial Fourier transform of the pressure in plane z = 0:

S(k x , k y ) = +∞ -∞ +∞ -∞
p(x, y, z = 0, ω)e -i(kxx+kyy) dxdy,

According to [START_REF] Sapozhnikov | Radiation force of an arbitrary acoustic beam on an elastic sphere in a fluid[END_REF] , the pressure field can be rewritten:

p(x, y, z, ω) = 1 π ∞ n=0 i n j n (kr) n m=-n Y nm (θ, ϕ) k 2 x +k 2 y ≤k 2 S(k x , k y )Y * nm (θ k , ϕ k )dk x dk y , (14)
with:

k x = k sin(θ k ) cos(ϕ k ) k y = k sin(θ k ) sin(ϕ k ) and k z = k cos(θ k ). The comparison of
Eq. 4 and Eq. 14 shows that the A m n can be written as:

A m n = i n π k 2 x +k 2 y ≤k 2 S(k x , k y )Y * nm (θ k , ϕ k )dk x dk y . (15)
The noisy pressure field is simulated here on a square grid of dimension 7mm × 7mm(5.6λ × 5.6λ) regularly sampled with a total of 2500 points. Note that the mesh grids of ASM should be very fine otherwise errors of integration in Eq. 15 arise.The square grid is located at the focal distance (x, y, z = 0). This field is Fourier transformed (Eq.

13) with a discrete Fourier transform (DFT). The sampling after a DFT can be refined by zero-padding for instance. We performed a polynomial interpolation instead. Indeed, knowing the polynomial coefficients, numerical integration of Eq. 15 can be achieved with a variable step method to increase accuracy. Therefore, all methods except the Lebedev quadrature on a single sphere allow to retrieve the BSC for a focused vortex beam with a good confidence even in presence of noise. The results are synthesized in Tab. I.

D. Estimation of the radiation pressure by the three methods

To assess the efficiency of each method, we calculate the radiation forces exerted on a polystyrene sphere of radius r = 0.1λ using equations 1, 2, 3. These forces are then decom- posed into three components: radial, azimuthal and axial forces (F ρ , F φ , F z ) in a cylindrical basis (ρ, φ, z). On the left of Fig. 7, the forces are calculated by using all the BSC obtained by the three methods, while on the right of Fig. 7, the forces are computed with A 1 n only.

All methods yield accurate estimations of the radial force. On the contrary, the azimuthal force, F φ , has a much weaker amplitude and all methods give poor estimates. Nevertheless, OMP method roughly recovers the original shape of the force. These differences originate from the estimated BSC with finite value outside column m = 1. It is shown in Fig. 7 that after filtering out these BSC, all methods recover perfectly the theoretical force. In the case of the axial force, F z , both OMP and the ASM turn out to provide good estimations while again, the Lebedev method is less efficient and leads to fast oscillations around the expected curve. However, these errors can not be reduced by filtering BSC outside m = 1. We may assume that the error is hidden in column m = 1. The oscillations on the axial force calculated by Lebedev method are periodical, similar to those caused by a standing wave. Moreover, we know that the radiation pressure due to a standing wave is much stronger than the one due to a progressive wave [START_REF] King | On the acoustic radiation pressure on spheres[END_REF] . Any error on the estimated BSC in this regard should lead to large discrepancies on the force estimation.

1 0 1 2 z/ 20 0 20 F z (nN)
In order to investigate this assumption, a weak amplitude wave propagating in the opposite direction is superposed to the incident wave. The BSC of the counter propagating wave can be computed as follow. The symmetry z → -z transforms cos(θ) into cos(π -θ) = -cos(θ).

Then, considering that the associated Legendre functions P m n (cos(θ)) satisfy the relation :

P m n (-x) = (-1) (n+m) P m n (x). ( 16
)
and Eq. 4, 5, the BSC of the wave propagating in the opposite direction can be computed by multiplying the BSC by (-1) (n+m) . Taking into account the mean relative error on the estimated BSC 0.006 for block OMP, we fixed the amplitude of this weak counterpropagating wave at 0.005 so that the new BSC are (1 + 0.005(-1) (n+m) )A m n th . On Fig. 8 is plotted the axial force for the progressive wave only and with the counterprogating wave superposed.

Comparing with Fig. 7, oscillations with the same periodicity but weaker amplitudes are obtained.

There remains to explain why the Lebedev method is more sensitive to the noise than the other two methods. In our simulation, the random noise amplitude is evenly distributed between -5% and 5% of the maximum pressure of the incident beam in all three cases.

However for Lebedev quadrature, the pressure field is sampled at the surfaces of two spheres with radius r = 7.11mm(5.7λ) and r = 7.45mm(6λ) where the wave is either yet converging, z < 0, or diverging, z > 0. Since the wave is sharply focused, on these spheres its amplitude and hence the SNR is 10dB lower.On the contrary, the set of points used either inside a spherical volume (OMP) or on a focal plane (ASM) contains locations where the pressure amplitude is maximum. By calculating the SNR at a measurement point where the signal is the maximum for each method, we obtain the results of 22.5dB, 32.2dB and 32.5dB for the Lebedev quadrature, OMP and ASM respectively. To confirm the role played by the SNR, the OMP method is now applied in conditions similar to the ones used for Lebedev quadrature. The pressure field is sampled on a set of points randomly distributed on a spherical surface of r = 7mm(5.6λ). The axial force obtained by the two methods are now similar with oscillating errors of about the same period and amplitude. Besides, if we increase the radius to 10mm(8λ), the fluctuations become stronger as expected since the SNR is even more degraded.

To compare the numerical estimation of the forces by different methods (with all the BSC A m n ), the relative errors between the force calculated with the A m n , F , in the presence of noise and the force computed with the A m n th , F th , are then calculated :

err f orce = 1 K K 0 |F -F th | max|F th | (17)
K is the total number of positions where the forces are estimated. The results are presented in Tab. II. We can conclude that both the OMP and angular spectrum are effective methods for estimating the radiation force from pressure field measurements with very low relative error as presented in Tab. II. The task is nevertheless difficult since small errors potentially result in spurious standing waves and the radiation pressure exerted by standing waves is much stronger than for progressive waves.

IV. EXPERIMENTAL MEASUREMENTS

In this section, the three methods are applied on experimental data. A focused Gaussian beam, with charge m = 0, and a focused vortex beam with charge m = 1 have been synthesized using a large antenna made of 120 piezoelectric transducersdistributed on a hexagonal pattern on a concave surface with a radius of curvature of 45cm. The array aperture is 11cm(88λ). An acoustic lens is used to reduce this focal to 7.5cm(56λ). With the lens, the half-angle, α 0 , is comparable to the simulated cases. The pressure field is synthesized by selecting the electric signals fed to each transducers by a multichannel electronics made of 120 arbitrary signals generators. These signals are calculated using the inverse filter technique [START_REF] Marchiano | Synthesis and analysis of linear and nonlinear acoustical vortices[END_REF][START_REF] Marchiano | Doing arithmetic with nonlinear acoustic vortices[END_REF][START_REF] Thomas | Pseudo angular momentum and topological charge conservation for nonlinear acoustical vortices[END_REF] . The set-up and procedure are identical to the ones used in previous results of the previous section, the noise is not the single source of discrepancy between A m n and A m n th and as consequence between F and F th , Eq. 17. The inverse filtering while very efficient does not achieve a perfect synthesis of the looked for pressure fields, Fig 10 . For instance, the experimental measurements are not perfectly axisymmetric and this will have an impact on the azimuthal force. For the axial plane, (xz), OMP and ASM methods also provide quite a good reconstruction on main and secondary lobes. The "X-shape" and high pressure at the focus, features expected for sharply focused beam, are perfectly reproduced, while the Lebedev quadrature estimation has some spurious amplitude oscillations. These oscillations have a comparable period with the one observed on the axial force, F z , Fig. 7, in the numerical simulation when noise was present. We provided an explanation for this phenomenon in the previous section. As a result, both the OMP and ASM are able to estimate the incident BSC A m n and hence the acoustic pressure field in the volume of interest around the focus.

Finally, we use these obtained BSC of the focused Gaussian beam and the vortex beam m = 1 to calculate the radiation forces exerted on polystyrene spheres of radius r = 0.1λ with Eq. 1, 2, 3. The results are shown on Fig. 11 and Fig. According to Fig. 11, for a focused Gaussian beam and for each method the radial force is in good agreement with the theoretical one. Theoretically no azimuthal force is applied, but a weak rotational force exists in the force estimation by three methods. This dissimilarity of azimuthal force can be due to the difference between the theoretical and experimental field synthesised by inverse filtering, as well as the presence of the noise in the measurements. As for the axial force, the force estimated by ASM is very close to the theoretical one with a very low relative error of 0.07. However, the one calculated by OMP appears to be oscillating though in the reconstruction on plane (xz) no oscillations are visible. Note that the trap slope is positive for both radial and axial forces. Therefore at the origin the force is null but the equilibrium is unstable. To achieve acoustical tweezers for a stiffer and denser particle compared to water, cancellation of the pressure field at the focus is required as the case studied below.

For the focused vortex beam of charge m = 1, the forces are quite similar with the ones obtained by adding noise in the numerical assessment of the three methods (see previous section). First, the computation of the radial force agrees with the theoretical force whatever the method. Secondly, the azimuthal forces computed by the experimental BSC are different from the theoretical one. These differences are caused by the value of A m n coefficients outside column m = 1. If we keep only column m = 1 and recalculate the azimuthal force, then, all the forces for different methods superpose with the theoretical force. This filtering makes the pressure modulus axisymmetric and eliminates any anisotropy in the transverse plane whatever its origin inverse filtering imperfection or biased estimation of BSC. The third observation is that both OMP and ASM provide a good estimate of the axial force which has the same negative slope as the theoretical one, i.e same trap strength and stiffness. The OMP axial force has some fluctuations, but performs much better than the Lebedev method, the force obtained with ASM is smooth and close to the theory but with a shift about 0.2λ of the equilibrium position where F z = 0. This shift can be caused by a slight shift 0.2λ (0.25mm) of the focal point in the experimental measurements. As reported in the Tab. IV, even with this shift, the relative error is small (0.14), by cancelling the shift, the relative errors will decrease to 0.091 and 0.085 for the OMP and ASM respectively which are very similar to the numerical estimations in the Tab. II of section III.

V. CONCLUSION

In this paper, the measurement of the radiation pressure on an elastic sphere exerted by acoustic tweezers is presented. The radiation force is not measured directly but is obtained from measurements of the pressure field associated with a model (see Eqs.1-3). To obtain the forces, it is necessary to estimate the BSC from experimental sampling of the pressure field.

Three methods were developed in section III: the Lebedev double layer sphere quadrature, the OMP method, and the ASM. First, we assessed the methods by simulating an acoustic vortex of charge m = 1 with 5% noise. The results show that all these methods can recover the BSC. In spite of that, the forces computed by the different methods agree well with the theory, except the Lebedev quadrature for which the axial force is fluctuating.

The experimental verification of these methods is done in section IV. Compared with the With the help of these methods, we are able to anticipate the radiation forces by mea-393 suring the acoustic field. As presented in previous sections, a very important component of 394 three dimensional radiation force is the axial force which is much smaller than the transverse 395 ones. In our experiments, we were capable to measure the axial force with a low relative 396 error of 0.14. For further measurements, this accuracy can be improved by increasing the axial force, for example, in the case of an acoustic vortex we can increase the aperture angle α 0 (Fig. 1) to get larger axial forces. At the same time, reducing the noise by any methods: shielding, averaging, filtering would help. The SNR leading to spurrious standing wave is the main limitation for axial or azimuthal force measurements.

( 5 )

 5 with (r, θ, ϕ) the spherical coordinates linked to the Cartesian coordinates by x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ. j n designates the spherical Bessel function, P m n (cos(θ)) are the Legendre polynomials and k is the wave number. The azimuthal number m and the radial degree n satisfy |m| ≤ n.

FIG. 1 .

 1 FIG. 1. (Top) Modulus of the complex pressure field at frequency corresponding to a wavelength

FIG. 2 .FIG. 3 .

 23 FIG. 2. (Top) points on a Lebedev sphere, and (Middle) reconstructed BSC for an incident focused

7. 11 FIG. 4 .

 114 FIG. 4. (Top) measurement points of a double layer Lebedev sphere, (Bottom) reconstructed

FIG. 5 .

 5 FIG. 5. (Top) randomly distributed measurement points in a sphere, and (Middle) reconstructed

Fig. 6 FIG. 6 .

 66 Fig.6shows the BSC obtained with the ASM method The agreement with the original

FIG. 7 .

 7 FIG. 7. (Left) radiation force exerted on a polystyrene sphere of radius r = 0.1λ with all the BSC,

FIG. 8 .

 8 FIG. 8. Axial force exerted on a polystyrene sphere of radius r = 0.1λ with theoretical BSC A m n

FIG. 9 .

 9 FIG.9. Axial force exerted on a polystyrene sphere of radius r = 0.1λ with BSC computed by

work 7 .

 7 FIG. 10. Reconstruction of the incident beam in Fourier domain, (pressure modulus is shown),

F

FIG. 12 .

 12 FIG.12. Experimental radial F ρ , azimuthal F φ , axial F z forces exerted on an polystyrene sphere

TABLE II .

 II Relative error of the forces for three methods

	Relative error Lebedev OMP ASM
	F ρ	0.036 0.012 0.018
	F φ	0.87	0.23 0.22
	F z	0.29 0.056 0.096

TABLE III .

 III 12. The axial force obtained with Lebedev method is not presented on the Figures since it's very fluctuating like its reconstruction on (xz) plane. As in the previous section, the relative error of the experimental forces for Gaussian beam and vortex are calculated and reported in tables Tab.III, IV. Relative error of the experimental forces for three methods (Gaussian beam)

	Relative error Lebedev OMP ASM
	F ρ	0.036 0.036 0.012
	F z	0.23	0.11 0.07

TABLE IV .

 IV Relative error of the experimental forces for three methods (Acoustic vortex m = 1)

ACKNOWLEDGMENTS

This work was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02, and more specifically within the framework of the Cluster of Excellence MATISSE led by Sorbonne Universités.

VI. REFERENCE

measurements in (xy) plane, the reconstructions of the field by the three methods are similar and almost identical. Nevertheless, for (xz) plane reconstructions, the field rebuilt by the Lebedev quadrature contains a lot of oscillations. Apart from that, both the OMP method and the angular spectrum method are in good agreement with the experimental data. From the analysis of the forces, the OMP and angular spectrum (ASM) methods allow to predict the radial and axial forces with a good precision and the azimuthal force with a lower