Andre Chandesris

Jean-Rene Malavoi

Nicolas Mengin

Jean-Gabriel Wasson

HYDROMORPHOLOGY AUDITING: A GENERALIZED FRAMEWORK AT A NATION SCALE TO VIEW STREAMS AND RIVERS IN THEIR LANDSCAPE CONTEXT

Keywords: Stream, Rivers, landscape, human pressures, cumulative impacts, geomorphology auditing

In Europe, there are less than 10 % of total stream and river length that can be considered as relatively free of human pressures. Historical and contemporary human uses of landscape and watershed has produced situations where several cumulative impacts is very common. In that context, it is essential to analyse properly the risks of hydromorphological impairment and its causes, before defining appropriate restoration measures. We propose a multi-scale hierarchical framework named Syrah for "The relational system of water course hydromorphology auditing". It is based on geomorphology functioning principles. This "top down" approach proposed relies on an assessment of large scale "damage risk": damage to processes (flow and sediment transport in particular) and structures (resulting morphology) are at the heart of the assessment. Fourteen types of hydromorphological damage have been identified. These are the most common types and are most likely to be the cause of impact on the ecological state of water courses. To process them, the audit relies on an evaluation of the layers of geographical data, existing databases, and on cross references between this information and the data required for management, programming, decision-making and assessment of restoration actions. "Land Use and Activities" (urbanization, agriculture, transport, energy) and resulting "Artificial Features and Uses", which are identifiable are quantified. Syrah multi-scale approach has been validated on selected regions, representing a diversity of hydromorphological situations and around 10 % of the France territory.

THE CURRENT CONTEXT

In order to implement the appropriate measures to achieve the objectives set by the European Water Framework Directive (WFD, 2000/60/EC), an analysis tool for the hydro morphological functions of water courses is required. A "Good Ecological State", which is the common objective assigned to all water bodies, is based on an assessment of the biological compartments (fish, macro-invertebrates, macrophytes and diatoms) and of chemical parameters. The physical characteristics of the water bodies, only provided to qualify the "Very Good Ecological State", are taken into account indirectly, by their affect on the quality of aquatic biocoenosis, which are in themselves capable of influencing the biological state.

AUDIT PRINCIPLES

The primary determinants on a regional scale (relief, climate, geology) define the hydro morphological control variables (hydrological and sedimentary regimes, width and gradient of valley bottoms). The key factors of ecological status are dependent on these variables, as well as on the structure of the riparian vegetation and the correct state of the water course's lateral and vertical connectivity: physical habitat, aquatic "climate", food webs. To understand and diagnose hydro morphologically driven ecological dysfunctions, we must necessarily take account of this hierarchical and multi-scale organization of the operation of hydro systems.

The "top down" approach proposed in the SYRAH-CE audit system relies on an assessment of large scale "damage risk" which makes it possible to focus analysis work on the lower level if high probabilities of damage are identified. The Audit is analogous in essence to the recent work in Australia [START_REF] Norris | Very-broad-scale assessment of human impacts on river condition[END_REF]. For technical reasons (short turnaround for the performance of the audit) and economic reasons (relatively limited budget), the assessment of the hydro morphological function as a function of the constraints exerted by the primary determinants along water courses has been given precedence over a more conventional approach involving a description of the "state" confined solely to the station level. Damage to processes (flows and sediment fluxes in particular) and structures (resulting morphology) are at the heart of the assessment:

-they are in fact closely linked to the intensity of anthropogenic pressures in a given geomorphologic context (at the scale of a section of the water course),

-they are clearly at the root of direct and indirect disturbances of aquatic habitats and also of their regeneration processes.

Fourteen types of hydro morphological damage have been identified. These are the most common types and are most likely to be the cause of impact on the ecological state of water courses.

To process them, the audit relies on an evaluation of the layers of geographical data, existing databases, and on cross references between this information and the data required for management, programming, decision-making and assessment of restoration actions.

PRINCIPLE

We direct our first scale of analysis of hydro morphological dysfunctions at the level of a higher compartment called "Land use and activities" (urbanization, agriculture, transport, energy). These activities and land use interact, according to their nature, with the operation of water courses at several different spatial, lateral and longitudinal scales (watershed, floodplain, river channel). They are set out in concrete terms in "Artificial features and uses", which are identifiable and often quantifiable objects, with direct and indirect effects on the operation of water courses. These effects translate into "damage to processes" (modification of flow and sediment fluxes, stream erosion processes, hydrodynamic components) and "structural alterations" (plane, longitudinal and transversal geometry, morphological units, substrates) of the physical environment (fig. 1). These different types of damage are in fact modifications (detrimental) of natural forms of water courses and consequently of their habitats.

TYPES OF DAMAGE: DISTURBANCE OF PHYSICAL FUNCTIONING AND STRUCTURES

The main aim of the audit is to detect hydro morphological damage of a "nonnatural origin" which can be clearly linked to deterioration of the "Ecological state", particularly through a deterioration of aquatic and riparian habitats. Structural damage (morphological in the main) is generally translated by a modification of the "fluvial forms" (main channel and secondary branches, succession of morphological units, river channel geometry, substrate). This necessitates descriptions or measurements taken by direct observation in the field. As regards damage to processes (flow and sediment fluxes), we must factor in a notion of time, which requires the use of time series. It is difficult to assess the damage directly, especially in terms of structural damage (it is necessary to carry out measurements in the field on the entire river system), and perhaps even impossible (complicated and voluminous systems to be set up and kept up to date). It has therefore been necessary to propose an indirect assessment method.

ARTIFICIAL FEATURES AND USES

The final aim of the audit is to foster the implementation of measures designed to correct dysfunctions, if possible from the outset. It seemed to us to be appropriate to propose audit methods starting "upstream" in the causal chain, and thus at the level of "Artificial features and Uses" (fig. 2). A list of developments and uses likely to cause hydro morphological damage has been drawn up, taking account of the various spatial scales involved: watershed (agriculture, urban area), floodplain (agriculture, urban area, transport), river channel (transport, energy, perhaps tourism). It is possible to analyze all these features and uses identified as being on a large scale using nationally available geographical databases.

Figure 3. Examples of maps taken from the analysis of large-scale developments and uses

The resulting maps (fig. 3) can be used for management and programming purposes but their precision is limited, particularly regarding "local" morphological aspects. This scale of analysis is therefore inadequate to enable a precise diagnosis of the dysfunctions and to design restoration measures, but does nevertheless provide us with an overview of a large area.

The analysis at the scale of geomorphologic sub-sections provides a description of these "features and uses" to a level of precision compatible with the search for causes of deterioration of the observable ecological state. This level of precision in the analysis is made possible by the existence of precise geographical databases such as BDTOPO IGN® (fig. 4).

RESULTS

We test the methodology on 6 watersheds, chosen as representative of a large diversity of geographical situations. The total area of the test zones is around 10 % of the total France metropolitan area. We compare data and associated probability to encounter physical impairment from the filter process on maps and thematic national databases with observed situations on field. A qualitative assessment has been built for each of the component of the methodology (see Appendix). There is a good to very good agreement between the two sets of information. The 2 levels of screening are complementary. Some components like dams, dikes, levees and weirs are closed dependent from the quality of databases. Some observed components are more widespread than expected: siltation and embeddedness are also encountered in zones where the land use was not intensive agriculture and due for example to the presence of numerous ponds in the watershed (1 case on 2); this is also the case of the stream straightening, which is also frequent, due to historical anthropogenic use, and only visible at small scale (1 case to 5). The general screening process is now validated.

We obtain gross indicator values for the identified "features and uses" of each analytical unit (geomorphologic sub-section). These results can be stored in georeferenced mapped databases (fig. 5). A further step will be necessary to reinterpret these results according to the geomorphologic characteristics of the section in which they were collected. For example, an identical density of thresholds does not have the same consequences in a fast-flowing mountain torrent as in a mature river with a gentle slope. Another example: infrastructures in the floodplain immediately next to the river channel will only have really negative consequences on geodynamically active rivers.

POTENTIAL USES OF THE AUDIT

Apart from mapping the risks of hydromorphological deteriorations undergone by rivers, the SYRAH-CE audit provides aid for management decision and functional restoration. The raw results of the audit allow an easy identification of the parts of the river network that are undergoing limited pressure. This information, combined with the knowledge of the chemical quality of the water, is necessary to delineate the sectors likely to be classified in "High status" according to the WFD, which are thus a priority for conservation. The method used allows the rivers analysed to be considered in a more general context, and focuses the analysis on the hydromorphological functioning, considered on a scale that exceeds the site of investigation. Mapping the indicators that represent the pressures causing geomorphological dysfunction allows to identify the most prevailing ones; the problems can be located, and even categorized in a ranking order. An analysis of this set of information could provide support for the establishment of management plans to be considered on several scales, with an easier identification of desirable restoration actions, and help for the decision-makers.

Figure 1 .

 1 Figure 1. SYRAH-CE conceptual framework

Figure 2 .

 2 Figure 2. Pressure and physical damage risk variables

Figure 4 .

 4 Figure 4. Example of an analysis at the scale of a section using BDTOPO IGN®: communication infrastructures in floodplain

Figure 5 .

 5 Figure 5. Examples of results of the analysis of features and uses (Malavoi, 2007).

Acknowledgments

We acknowledge the numerous people involved in this national audit, in particular the members of the National expert group in the Ministry in charge of Ecology, the 6 Water Agencies, Onema and Regional Services of Environment.

APPENDIX

Field observation versus "aerial" view and national databases Engineering works and uses

Watershed and Riparian Scale

Intensive agriculture

At risk situations: 100 % true No large scale risk identified: 50 % with siltation by fine sediment (different origins: ponds, reservoirs, …)

Dams

Difficult to verify by spot field observations -Great dependency on data quality of the national dam database

SEDIMENT FLUXES

Sediment mining

True to the expected pattern at this scale of investigation

Impervious area

True to the expected pattern at this scale of investigation

Reservoir water storage

Difficult to verify by spot field observations -Great dependency on data quality of the national dam database Water abstraction

Effective variable -often at a wider range than expected

FLOW

Minimum flow (by pass section) Dikes and levees Riparian area removal

Efficient for the floodplain vegetation, lees for the detail of the corridor Stabilization Efficient for the large scale constraints

MORPHOLOGY Straightening

At risk situations: 100 % true No large scale risk identified: 20 % with observed straightening

Channnelization

Good identification in areas at risk. Other impacted situations still exist.

Weirs

Minimum flow (by pass section) Methodology and relevant variable to calibrate

Dikes and levees Heterogeneous data, qualitative information.

Riparian area removal

True to the expected pattern at this scale of investigation (riparian and channel vegetation in the floodplain)

Stabilization

True to the expected pattern at this scale of investigation

Straightening

True to the expected pattern at this scale of investigation. Sensitivity function of the typology of streams Channelization

MORPHOLOGY

Weirs

Great dependency on data quality of the national dam database


"Technology and management to ensure sustainable agriculture, agro-systems, forestry and safety" Agrotechnologies towards Ecotechnologiesthe three pillars for developing Eco-design

Keywords:

To boost agrotechnologies towards ecotechnologies ("environmental technologies" according to ETAP programme of EU, or "more ecologically productive technologies" in the context of agriculture), we need to strengthen a "triple bottom" system:

To take into account, in "Life Cycle Analysis" methodologies, the natural variability in time and space of these applications in land use.

To develop an overall approach for realistic machinery qualification, in order to feed the environmental burdens accurately through relevant data bases collected on agrotechnologies in real action.

To work on Eco-innovation processes, by deepening specific innovation tools and methods, for implementation of innovative solutions chosen according to LCA results. This paper presents the concept, develops the methods and illustrates them by examples of results on organic spreading technologies.

Introduction

The concept of Environmental Technologies1 really emerged in 2004 thanks to ETAP (Environmental Technologies Action Plan), a programme launched by the European Commission. In France, this concept has been transformed in "Ecotechnologies" (term more ambiguous) and a national programme has been led since 2005, called "Ecotechnologies and Sustainable Development", in which the project ECODEFI (Ecodesign and development of eco-assessment and eco-innovation methodologies) is operational with now, some results that can be presented.

A "system" approach for boosting agrotechnologies in ecotechnologies

Agricultural technologies have a big potential for being classified in "environmental technologies" because agricultural activity allows reducing environmental burdens of human activities by using organic wastes and sludge in biomass production. By this way, it's a traditional "pollution treatment" technology, but it can be also a "cleaner technology" as are able to design new technologies which reduce pollutant flows at all stages of life cycle, especially during the use phase, dedicated to agricultural production. We are going to highlight some main research tracks enable to boost this "eco-design" concept 2 .in this sector. Without a multicriteria and quantitative assessment of environmental burdens on a clear application field, it's impossible to claim an objective environmental benefit. So, we promote a "top-down" approach which use, at first, "Life Cycle Analysis3 (LCA) methodologies in order to classify and measure the main impacts occurred by the classical technology that we want to improve. Thus, we are able to identify the good tracks for innovation researches. Moreover, to be relevant, this quantitative environmental assessment need to be fed by realistic and specific datas, provided by some practical measurements on machineries in work situation. It is the technology qualification. Besides, innovation process can be strengthened and more efficient by using specific creativity methodologies, supported by a strong knowledge of operational mechanisms. However, the practice of this procedure on real situations has identified some scientific and technical bottlenecks in three research areas:

Accounting for variability in the Life Cycle Assessment methodology.

Data collection is the most time consuming and costly part of LCA studies. This is especially true for agriculture as field emissions are highly variable, depending on climate, soil type and farming practice [START_REF] Audsley | Harmonisation of environmental life cycle assessment for agriculture[END_REF]. Therefore, there is a challenge to rate a technology in a generic and accurate way when its performance depends on many inter-related factors. 2 The product eco-design aims to optimise environmental performances at the same time than keeping quality of use (UE official newspaper of 15/02/05 -C38E/45) -Eco-design is a preventive approach which takes into account environment as soon as the design work phase.

3 Life Cycle Assessment (LCA) is since 1997 an internationally standardised method (ISO 14040) for the evaluation of the potential negative impacts of goods and services (products) along their life cycle on the environment, human health, and resource availability. LCA helps to avoid the "shifting of burdens" among life cycle phases, among the various impacts, and among countries -extract of " European Platform on Life Cycle Assessment" -LCA@JRC.it -http://lca.jrc.ec.europa.eu/ Figure 1: A "system" approach for boosting Ecodesign process.

The case of manure application techniques illustrates the complexity of such an assessment. Fig. 2 shows NH 3 losses following slurry application with different techniques expressed as a percentage of NH 3 losses with the band spreading technique [START_REF] Langevin | Implication of field and climate variability in the Life Cycle Assessment (LCA) of slurry application techniques: a scoping study[END_REF]. It clearly shows the variability of the technique performance, even if rough trends can be outlined. One possibility to overcome the challenge of rating such agrotechnologies would be to draw up the environmental profile of the studied technology, i.e. its performance in a number of contrasted situations. In the case of manure application, agro-ecological modelling at the field scale could be used to quantify the polluting emissions in contrasted situations of soil, climate and practice, and these data could be further integrated into a complete LCA for all selected situations, constituting an environmental profile for the technology under study.

Developing machinery qualification by an overall approach

Metering is a basic principle to enhance environmental performances of a machine. The development of sensors has largely contributed to the advances in precision agriculture enabling to correct the inputs according to agro-ecological needs. But the good achievement of the modulations would require a strict command of the machine to accurately execute its task.

To verify the functioning of the machines, it has been necessary to set up sophisticated test benches measuring technological performances under accurately controlled conditions. The European Network for Testing Agricultural Machines (ENTAM) connects most of the European test centres and actively participates to the sharing of experience and the sharing of test results. By this way ENTAM encourages the development of up to date installations. Specific tools have been set up in order to bridge the gap between indoor benches metering conditions and field working conditions. The "Spreading Simulator" [START_REF] Piron | Environmental evaluation of organic spreading machines -a spreading simulator[END_REF] has been designed to assess the performances of a spreader in real-world geographic context. It is supplied by the results of the test benches obtained with standard procedures. It delivers a realistic application rate mapping which could not be acquired by any other available mean. Fig. 3 illustrates the simulation results and their analysis provided by the software. A further step is needed to link the measured technological indicators with potential environmental impacts. Field emissions contribute to the total potential impacts; as they depend on practice and soil and climate conditions. It is then necessary to specify a set of reference conditions to assess these impacts. Considering that the future working conditions of the machine are unknown at the time of its manufacturing, it is necessary to take into account the occurrence of sensitive environments and ignore ideal situations where, for example, the soil will absorb damages due to hazardous practices The choice of the reference conditions will be guided by the results of the LCA sensitivity study, by the questioning of agronomical and environmental experts and also by the integration of good practice codes requirements. Once the reference conditions are stated, models may be used to quantify the emissions and the subsequent potential impacts accordingly.

Working on Eco-innovation processes

The term "environmental innovation (eco-innovation)" has been introduced by the interdisciplinary project 'Innovation Impacts of Environmental Policy Instruments' [START_REF] Klemmer | of publications from a Joint Project on Innovation Impacts of Environmental Policy Instruments[END_REF]. To eco-innovate, several leads are possible such as dematerialization, moving from products to services, durability and versatility. In our case study of spreading technologies, we can take example of the increase of machine versatility, in order to spread properly not only mineral but also organic fertilizers specially the cohesive ones. The major noted problem with this type of product is the flow rate decrease during the time of discharge. To identify the origin of this problem, we have recourse to the Root Cause Analysis by utilizing the cause-effect model (Figure 4). Because one event in the chain causes the next, solving any of the preceding events will solve the problem. Once the ''core'' problem is identified, we generate concepts by using the Theory of Inventive Problem Solving (TRIZ). TRIZ is a knowledge -based methodology, for accelerated development of design concepts. Also, it is a creative problem-solving methodology especially tailored to scientific and engineering problems. To seek solutions to each category of problem, it is essential to scan several (or all) TRIZ tools and methods. The most important tools and methods of the TRIZ methodology are the Contradiction Matrix, the 40 Inventive Principles, the 39 TRIZ Parameters, the ARIZ Algorithm, the 76 Standard Solutions, the Su-Field analysis, the laws and trends of technology evolution, the System of Operators, the Effects and Anticipatory Failure Determination [START_REF] Serban | A TRIZ approach to design for environment[END_REF][START_REF] Savransky | Engineering of Creativity[END_REF]. As an illustrative explanation, the scientific effects tool is introduced. According to TRIZ, the scientific effect is one of the principles for resolving physical contradictions. There are Figure 4. Cause-effect model -"flow rate decrease" can be occurred by "product adhesion to hopper walls", "jamming", … about 10,000 effects described in the scientific literature [START_REF] Kraev | Kraev's Korner: Scientific Effects -Lesson 6[END_REF]. Each effect may be a key to solving a large group of problems [START_REF] Savransky | Engineering of Creativity[END_REF]. For "product adhesion to hopper walls" problem, several effects have been found so that reduce sliding frictional force. Among these effects, we may mention: normal force increases friction between objects / vibration / surface roughness affects sliding friction / solid lubricant reduces friction coefficient / Adsorption and desorption affect adhesive force between solids. Eventually, the designer chooses one or more effects that have the highest likelihood of efficiency in a defined situation. The choice of effect is contingent upon the industrial expertise and practice of the company. In fact, to succeed in root cause analysis and to establish the "cause-effect" tree, a good understanding of occurred mechanisms is required. Especially, a deepened mastering in fluid mechanics is necessary for technological problems which concern matters (liquid, pasty or heterogeneous solid) with movement in interaction with devices and machines.

Conclusion:

Figure 2 :

 2 Figure 2: Relative NH 3 losses factors for slurry application techniques

"Environmental technologies include both integrated technologies that prevent pollutants being generated during the production process, and end-of-pipe technologies that reduce the release into the environment of any pollutants. They can also include new materials, energy, and resourceefficient production processes" -extract from M. Galatola's presentation in European commission conference on "Towards European methodology for environmental technology assessment" -october

2006.

So, in Eco-design, we need to have a "system" approach, associating environmental assessment (LCA), fed by realistic and global machinery qualification processes, in order to guide relevant innovations. However, whatever the novel solution, even more so if it's a disruptive innovation, this one must be adaptable and compatible with end-user practices. So, this dimension must be taken into account in early stage. At the same way than for Eco-design concept, a participatory design concept can be adopted as soon as the beginning of the work, for targeting effective solutions.


Silva D Da
email: philippe.balandier@cemagref.fr

Forestry in achieving millennium goals -Novi Sad

There is a current trend towards managing forests with multiple objectives, in particular to preserve or increase biodiversity and sustainability. There is renewed interest in understorey vegetation as a way both to increase the number of species and, indirectly, to favour fauna, including game, or improve soil quality. However, this stratum of herbaceous or shrubby species can also compete with young tree seedling and jeopardise tree regeneration. Hence a compromise has to be found among the different management objectives for the forest and in particular the understorey.

Light is one of the main environmental factors controlling ecological and biological processes in forests. For example, light quantity and light quality control the success of seed germination and the establishment and growth of tree seedlings in the understorey. Light also promotes the development of the floor vegetation, the composition of which varies with site conditions in addition to light. Modifying light availability in the understorey, for example with thinning operations, thus interferes with tree regeneration and flora cover and composition. The understorey plants can intercept a significant fraction of the light and so the resulting amount of light available for tree seedlings varies greatly depending on species and composition of the floor vegetation. Light is successively intercepted and transmitted by the different strata composing the forest stand, i.e. overstorey trees, midstorey trees, shrubs and herbaceous species. The different strata interact, the development of each one being controlled by the others, often through modification of light availability. Foresters have to control this chain of light sharing to steer forest stands towards different objectives. We report results concerning the light sharing chain and discuss their implications for management.

Introduction

For a long time forests of the temperate zone have been managed solely for the production of wood. In his book on sylviculture, [START_REF] Perrin | Sylviculture[END_REF] noted that "earlier foresters considered herbs as enemies and managed stands for a clean soil", a clean soil meaning without any vegetation, "evidence that the trees were using all the available radiation". The current trend is more towards managing forests with multiple objectives, in particular to preserve or increase biodiversity and sustainability. The understorey vegetation has received renewed attention as a way both to increase the number of species in the forest, and indirectly, to favour fauna, including game, or improve soil quality. However, this stratum of herbaceous or shrubby species can also compete with young tree seedling and so jeopardise tree regeneration. In addition, a dense layer of vegetation, particularly thorny species, is not always appreciated by urban forest walkers. Hence a compromise has to be found among the different managing objectives for the forest and in particular the understorey. Foresters need tools to implement such forest management.

Light is one of the main environmental factors controlling ecological and biological processes in forests. Clearly, factors such as soil water availability, nutrients and temperatures are also fundamental, but these are all linked to some extent to light availability [START_REF] Barbier | Influence of tree species on understory vegetation diversity and mechanisms involved -a critical review for temperate and boreal forests[END_REF]. Light and water from rainfall are both related to tree crown density and gap percentage in the canopy, so that an increased light penetration implies an increase in rainfall reaching the forest floor. Less light or water interception by the canopy in turn means a smaller leaf area index (LAI) and so lower tree transpiration [START_REF] Bréda | L'indice foliaire des couverts forestiers: mesure, variabilité et role fonctionnel[END_REF]. This condition can improve soil water content, but the effect is offset by the water consumption of the understorey vegetation, which can increase with greater light availability. Light also drives the energy balance of the stand. More light means a significant increase in air and soil temperatures during the day and a small decrease in night temperatures by radiation loss. The net increase in day temperatures and a higher soil water content improve the biological activity of the soil; the humus is therefore of better quality and nutrients are recycled more efficiently. Hence light can be considered as a synthetic indicator of resource availability and microclimate in forests [START_REF] Barbier | Influence of tree species on understory vegetation diversity and mechanisms involved -a critical review for temperate and boreal forests[END_REF].

Light quantity and quality are among the processes that control the success of seed germination and the establishment and growth of tree seedlings in the understorey. The shadetolerance of young tree species varies according to their successional status, the more shadetolerant species being potentially better able to regenerate in the dark understorey than the more intolerant ones. Light is consequently a fundamental key to forest regeneration. Light also promotes the development of the ground vegetation, composed of graminoids, forbs, shrubs, and young trees, the composition of which varies with site conditions and overstorey tree species, in addition to light. The understorey vegetation can compete with tree seedlings for resources to varying degrees depending on the species present, and may seriously impede tree regeneration in some cases (Balandier et al., 2006). These understorey plants can intercept a significant fraction of the light in the understorey and so the remaining light available for tree seedlings can vary greatly depending on species and composition of floor vegetation.

The light path into the forest is as follows: the incident light above the canopy is first intercepted by the overstorey trees, and then by the midstorey trees when these are present. The light that reaches the understorey controls the development and composition of the floor vegetation, which in turn intercepts some light. Finally, the tree seedlings use what light remains. The forester has to control this chain of light sharing to direct forest stands towards different objectives. We present here some results on this light sharing chain and their implications for management.

Materials and Methods

Light transmitted by the overstorey and understorey vegetation was measured using different methods. Hemispherical photographs were used to indirectly estimate light transmission through the overstorey of adult trees. The equipment used was a camera fitted with a 180° aperture lens with which it was possible to take a photograph of the whole sky hemisphere, with the zenith at the centre of the photograph and the horizon at the edge. The photograph was taken without direct radiation from the sun, i.e. under full cloud cover, or at sunrise or sunset, which considerably restricted the time periods when photographs could be taken. The camera was set perpendicular to the sky zenith about 1.5 m above the ground, if possible above the understorey vegetation. The principle for computation of transmitted light from the photograph was as follows; the colour photograph was converted into a black (vegetation) and white (sky) picture using a local thresholding procedure [START_REF] Balandier | Six-year evolution of light use efficiency, carbon gain and growth of beech saplings (Fagus sylvatica L.) planted under Scots pine (Pinus sylvestris L.) shelterwood[END_REF]. The ratio of white to black pixels (sky to vegetation) gave the gap fraction from which the transmitted light could then be computed [START_REF] Adam | PiafLA -software to calculate transmitted light by canopies[END_REF]. The transmitted light could be divided into diffuse light from the whole sky vault using the whole photograph, and direct light from the sun using the fraction of the photograph corresponding to the path of the sun, which could be calculated for any day of the year. The method thus enabled us to compute the transmission of light for the whole year, certain months or a particular day. The drawback of the method is the difficulty obtaining a quality photograph with no over-exposure in which all parts of trees are clearly represented. This is possible only if no direct sun is present on the photograph and the sky is perfectly homogeneous, and the colour pixels are correctly classified into sky and vegetation. This last is the most severely limiting part of the method.

Transmitted light can also be measured directly using light sensors. In many experiments we used either linear (length 1 m) or point light sensors in the total solar radiation spectrum (300-3000 nm) or the PAR spectrum (400-700 nm). Sensors were set 1.5 m above the ground and if possible above the understorey vegetation for at least 24 h, to account for the complete sun path over the entire day. In the same 24 h period a sensor of the same type was set nearby in an open place to measure incident radiation. The transmittance of the forest stand was then calculated as the ratio of light in the understorey to incident light during the 24 h period. This process enabled us to compare, at least to some extent, measurements made on different days of the year (i.e., with different sun flux) and with different weather conditions (i.e., cloudy or sunny). The drawback of the method is that a single day of measurement cannot account for light interception by the overstorey during several months or the whole year, particularly if the canopy is very heterogeneous, i.e., with large gaps where direct sun radiation can reach the ground on some days or in some months in the year.

Light transmitted by the understorey vegetation was measured using a linear PAR sensor of length 80 cm and width 1 cm (Accupar, Decagon). Its long-stemmed shape was easy to slide into the vegetation, generally a few cm above the ground, without disturbing the structure of the plant cover. It was composed of 80 photocells to measure local variations in light interception. In the same way as overstorey transmittance, light recorded under the understorey vegetation is related to a measure of incident radiation during the same period. The drawback of the method is the same as for the overstorey, and a hemispherical photograph is not easy to take from within the vegetation.

Light measured either under the overstorey or under the understorey vegetation was related to vegetation characteristics such as tree height, diameter, basal area, crown characteristics, or herbaceous height or cover.

Results

Light interception by the overstorey

In regular, even-aged stands, composed of a single species, light interception by the overstorey is relatively easy to predict. Light measurements and stand characteristics showed that transmittance (T) of the overstorey was related exponentially to stand basal area (G), by a Beer-Lambert law (Balandier et al., 2006;[START_REF] Sonohat | Predicting solar radiation transmittance in the understorey of even-aged coniferous stands in temperate forests[END_REF], T = exp(-kG), where k is the extinction coefficient: this is a function of species, stand age, and time lag and intensity of the last thinning [START_REF] Sonohat | Predicting solar radiation transmittance in the understorey of even-aged coniferous stands in temperate forests[END_REF]. Figure 1 illustrates the effect of species identity; for the same stand basal area and age, light intercepted by different species during the leafy period can vary by a factor of 1 to 4. In these regular, even-aged stands, the quality of light, i.e. its spectral composition, can be deduced from measurements of a considered wavelength band as there are relationships between total solar radiation (300-3000 nm), photosynthetically active radiation (PAR, 400-700 nm) and red (R, 660 nm) or far-red (FR, 730 nm) radiation (Balandier et al., 2006).

Of particular interest is the R / FR ratio, which drives many plant morphological processes.

In uneven, irregular stands, composed of one or more species, light measurement, prediction and simulation is not easy because of the large gaps in the overstorey canopy, which leads to a very wide variability of transmitted light, in both quantity and quality. In that case, a single variable at the stand scale, such as basal area, is no longer sufficient, and more local variables must be considered such as individual tree size and position, or at least information on tree spatial distribution [START_REF] Goreaud | Simuler des peuplements de structures variées pour faciliter l'utilisation des modèles "arbre" spatialisés[END_REF]. The difficulty of the approach is staying at a level of description that allows forest managers to actually use the model, i.e. it must not be too complex, at least for the information required as input. With this in mind, we built a model with a multiscale representation of the forest (Da [START_REF] Silva | Multiscale framework for modeling and analyzing light interception by trees[END_REF], i.e. from the tree branch scale, crown scale, tree scale to stand scale, that can be used to study the worsening of light prediction with the scale of representation [START_REF] Silva | Modeling of light transmission under heterogeneous forest canopy: model description and validation. 5 th international workshop on functional structural plant model[END_REF]. This model is being evaluated.

Development and composition of the understorey vegetation

Light availability in the understorey promotes the development of the floor vegetation. Globally, at the community level, there is a logarithmic increase in the vegetation cover with light availability, from no vegetation for very low levels of transmittance (≅ 0.02) to a maximum for an asymptote of 0.4 of transmittance, which means that increasing light above this value does not correspondingly augment vegetation cover or biomass [START_REF] Balandier | La lumière, outil sylvicole pour favoriser la diversité végétale ou la gestion cynégétique des peuplements de mélèze (Larix sp.)[END_REF]. At the species level, patterns of plant cover or biomass with light availability are not simple and depend on the light requirements of species. For example, a recent study showed a near-linear increase in Cytisus scoparius, a light-requiring species, with light availability, whereas Rubus idaeus, a more shade-tolerant species, showed a bell-shaped curve in response to light, with a maximum cover for a transmittance of 0.45 [START_REF] Gaudio | Light-dependent development of two competitive species (Rubus idaeus, Cytisus scoparius) colonizing gaps in temperate forest[END_REF].

We are studying the response patterns of other common species that colonise gaps in temperate forests, such as Rubus fruticosus, Molinia caerulea, Calluna vulgaris and Pteridium aquilinum.

Light availability modifies not only the cover of the floor vegetation but also its species composition. However, the response pattern of species composition with respect to light is not simple. Some studies have reported a bell-shaped curve with a maximum number of species for intermediate light levels (e.g., [START_REF] Balandier | La lumière, outil sylvicole pour favoriser la diversité végétale ou la gestion cynégétique des peuplements de mélèze (Larix sp.)[END_REF], others a continuous increase in species richness with light. The first response is observed more frequently at a local scale, whereas the second is often recorded for larger geographical scales [START_REF] Rajaniemi | Explaining productivity-diversity relationships in plants[END_REF]. The bell-shaped curve from shade to full light is explained by the progressive enrichment of a flora made up of shade-tolerant species by more and more light-requiring species as light increases up to a maximum species richness; thereafter one or more monopolistic, strongly light-requiring species colonise the whole space at the expense of all the other species. In terms of tree regeneration management, those monopolistic species (e.g. Rubus fruticosus, Molinia caerulea and Pteridium aquilinum) are highly competitive for resources and can jeopardise tree seedling establishment and growth.

Light interception by the understorey vegetation

Depending on their development, the understorey plants can intercept a non-negligible amount of light (fig. 2). For example, the transmitted light recorded under Cytisus scoparius, Rubus fruticosus, and Epilobium angustifolium can be as low as 0.01, 0.03 and 0.05, respectively [START_REF] Sivade | Evaluation de la compétitivité des principales espèces de plantes affectant la régénération des forêts de moyenne montagne[END_REF]. These are very low values, which can seriously compromise tree regeneration when tree seedlings are small and surrounded by such species. Another study on Rubus fruticosus for different stands in France and England suggested that transmittance could range from 0.5 to less than 0.02 for plant LAI values ranging from 0.5 to 4, respectively (unpublished data).

Discussion: implications for tree regeneration

Tree seedling establishment and growth are affected to varying degrees by light availability according to species shade-tolerance; for example Fagus sylvatica, a late successional shadetolerant species, is able to survive and grow under very limited amounts of transmitted light, as low as 0.03. It grows more vigorously with increasing light up to a value of about 0.4, beyond which no further improvement in growth is usually seen (e.g. [START_REF] Balandier | Six-year evolution of light use efficiency, carbon gain and growth of beech saplings (Fagus sylvatica L.) planted under Scots pine (Pinus sylvestris L.) shelterwood[END_REF]. By contrast, an early successional species that is not very shade-tolerant, such as Pinus sylvestris, needs more light to establish and grows more vigorously with more light linearly up to a transmittance of 1. Hence forest managers have to run the stand differently, especially as regards thinning operations, to regenerate and promote particular species. However, as shown above, depending on the light availability in the understorey, vegetation of varying density can develop and intercept a large amount of the available light (in addition to water and nutrients, Balandier et al., 2006). This understorey vegetation, depending on its density and species composition, can strongly compete with tree seedlings, severely impeding their growth, or even causing their death [START_REF] Balandier | Régénération naturelle du pin sylvestre en forêt mélangée chêne -pin de l'Orléanais[END_REF][START_REF] Coll | Competition for water and light between beech seedlings and surrounding vegetation in abandoned meadow colonised by woody species[END_REF][START_REF] Provendier | Compared effects of competition by grasses (Graminoids) and broom (Cytisus scoparius) on growth and functional traits of beech saplings (Fagus sylvatica)[END_REF]. Therefore the forest manager has to find a compromise between a light level that promotes optimal tree seedling growth, but also generates a dense competing layer of understorey vegetation that can jeopardise tree regeneration, and a very low light level that prevents all understorey plant growth, including tree seedlings. The point of equilibrium that allows significant tree seedling growth while preventing too-dense understorey vegetation is often also the point where species richness is at its maximum; for a natural sustainable management, this point should be sought by forest managers adjusting light sharing among the different strata of the forest.

Fig. 1 .

 1 Fig. 1. Simulations of light transmitted during the leafy period by different species according to their stand basal area for stands in France and Belgium. Transmittance is expressed as the quantity of light measured in the stand understorey relative to the incident light above the canopy.

Fig. 2 .

 2 Fig. 2. Light (PAR) transmitted to soil level by three common plants colonising gaps in temperate forests in May (top) and July (bottom) 2005 at different sites in uplands of central France (for each species the centre box covers 50% of the data, the vertical line in the box being the median).


S Nascimbène

N Navon

K J Jiang

F Chevy

C Salomon

Exploring the thermodynamics of a universal Fermi gas

Keywords: numbers: 03.75.Ss, 05.30.Fk, 32.30.Bv, 67.60.Fp

From sand piles to electrons in metals, one of the greatest challenges in modern physics is to understand the behavior of an ensemble of strongly interacting particles. A class of quantum manybody systems such as neutron matter and cold Fermi gases share the same universal thermodynamic properties when interactions reach the maximum effective value allowed by quantum mechanics, the so-called unitary limit [START_REF] Ho | Universal thermodynamics of degenerate quantum gases in the unitarity limit[END_REF][START_REF] Inguscio | Ultracold Fermi Gases[END_REF]. It is then possible to simulate some astrophysical phenomena inside the highly controlled environment of an atomic physics laboratory. Previous work on the thermodynamics of a two-component Fermi gas led to thermodynamic quantities averaged over the trap [START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF][START_REF] Luo | Measurement of the entropy and critical temperature of a strongly interacting Fermi gas[END_REF][START_REF] Luo | Thermodynamic Measurements in a Strongly Interacting Fermi Gas[END_REF], making it difficult to compare with many-body theories developed for uniform gases. Here we develop a general method that provides for the first time the equation of state of a uniform gas, as well as a detailed comparison with existing theories [START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF][START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF][START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF][START_REF] Combescot | Particle distribution tail and related energy formula[END_REF][START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF][START_REF] Liu | Virial expansion for a strongly correlated Fermi gas[END_REF][START_REF] Rupak | Universality in a 2-Component Fermi System at Finite Temperature[END_REF][START_REF] Combescot | Normal state of highly polarized Fermi gases: simple many-body approaches[END_REF][START_REF] Combescot | Normal state of highly polarized Fermi gases: full many-body treatment[END_REF][START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic monte carlo method for divergent signalternating series[END_REF]. The precision of our equation of state leads to new physical insights on the unitary gas. For the unpolarized gas, we show that the low-temperature thermodynamics of the strongly interacting normal phase is well described by Fermi liquid theory and we localize the superfluid transition. For a spin-polarized system [START_REF] Shin | Observation of phase separation in a strongly interacting imbalanced Fermi gas[END_REF][START_REF] Partridge | Pairing and phase separation in a polarized Fermi gas[END_REF][START_REF] Nascimbene | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF], our equation of state at zero temperature has a 2% accuracy and it extends the work of [START_REF] Shin | Phase diagram of a two-component Fermi gas with resonant interactions[END_REF][START_REF] Shin | Determination of the equation of state of a polarized fermi gas at unitarity[END_REF] on the phase diagram to a new regime of precision. We show in particular that, despite strong interactions, the normal phase behaves as a mixture of two ideal gases: a Fermi gas of bare majority atoms and a non-interacting gas of dressed quasi-particles, the fermionic polarons [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF][START_REF] Nascimbene | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF][START_REF] Shin | Determination of the equation of state of a polarized fermi gas at unitarity[END_REF][START_REF] Chevy | Universal phase diagram of a strongly interacting fermi gas with unbalanced spin populations[END_REF][START_REF] Schirotzek | Observation of fermi polarons in a tunable fermi liquid of ultracold atoms[END_REF].

In this letter we study the thermodynamics of a mixture of the two lowest spin states (i = 1, 2) of 6 Li prepared at a magnetic field B = 834 G (see Methods), where the dimensionless number 1/k F a characterizing the s-wave interaction is equal to zero, the unitary limit. k F is the Fermi momentum and a the scattering length. Understanding the universal thermodynamics at unitarity is a challenge for many-body theories because of the strong interactions between particles. Despite this complexity at the microscopic scale, all the macroscopic properties of an homogeneous system are encapsulated within a single equation of state P (µ 1 , µ 2 , T ) that relates the

1 0 1 Η Μ 2 Μ 1 0 0.5 Ζ Μ 1 k B T h FIG. 1: Schematic representation of the universal function h(η, ζ).
It fully describes the thermodynamics of the unitary gas as a function of chemical potential imbalance η = µ2/µ1 and of the inverse of the fugacity ζ = exp(-µ1/kBT ). In this paper we measure the function h over the black lines (1, ζ) and (η, 0) which correspond to the balanced unitary gas at finite temperature and to the spin-imbalanced gas at zero temperature, respectively.

pressure P of the gas to the chemical potentials µ i of the species i and to the temperature T . In the unitary limit, this relationship can be expressed as [START_REF] Ho | Universal thermodynamics of degenerate quantum gases in the unitarity limit[END_REF]:

P (µ 1 , µ 2 , T ) = P 1 (µ 1 , T )h η = µ 2 µ 1 , ζ = exp -µ 1 k B T , (1) where
P 1 (µ 1 , T ) = -k B T λ -3 dB (T )f 5/2 -ζ -1
is the pressure of a single component non-interacting Fermi gas and

f 5/2 (z) = ∞ n=1 z n /n 5/2 . h(η, ζ
) is a universal function which contains all the thermodynamic information of the unitary gas (Fig. 1). In cold atomic systems, the inhomogeneity due to the trapping potential makes the measurement of h(η, ζ) challenging. However, this inhomogeneity of the trap can be turned into an advantage as shown in [START_REF] Shin | Determination of the equation of state of a polarized fermi gas at unitarity[END_REF][START_REF] Ho | Obtaining phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases[END_REF].

We directly probe the local pressure of the trapped gas using in situ images, following the recent proposal [START_REF] Ho | Obtaining phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases[END_REF]. In the local density approximation, the gas is locally homogeneous with local chemical potentials:

µ i (r) = µ 0 i -V (r) (2)
(µ 0 i is the chemical potential at the bottom of the trap for species i). Then a simple formula relates the pressure P to the doubly-integrated density profiles [START_REF] Ho | Obtaining phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases[END_REF]: where n i (z) = n i (x, y, z)dxdy, n i being the atomic density. ω r (resp. ω z ) is the transverse (resp. axial) angular frequency of a cylindrically symmetric trap (see Fig. 2) and µ iz = µ i (0, 0, z) is the local chemical potential along the z axis. From a single image, we thus measure the equation of state (1) along the parametric line (η, ζ) = (µ 2z /µ 1z , exp(-µ 1z /k B T )), see below.

P (µ 1z , µ 2z , T ) = mω 2 r 2π (n 1 (z) + n 2 (z)) , (3)
The interest of this method is straightforward. First, one directly measures the equation of state (EOS) of the uniform gas. Second, each pixel row z i gives a point h(η(z i ), ζ(z i )) whose signal to noise ratio is essentially given by the one of n 1 (z) + n 2 (z); typically one experimental run leads to ∼ 100 points with a signal to noise between 3 and 10. With about 40 images one gets ≃ 4000 points h(η, ζ), which after averaging provides a low-noise EOS of standard deviation σ = 2%. In the following we illustrate the efficiency of our method on two important sectors of the parameter space (η, ζ) in Fig. 1: the balanced gas at finite temperature (1, ζ) and the zerotemperature imbalanced gas (η, 0).

We first measure the equation of state of the unpolarized unitary gas at finite temperature, P (µ 1 , µ 2 , T ) = P (µ, T ). The measurement of h(1, ζ) through the local pressure (3) can be done provided one knows the temperature T of the cloud and its central chemical potential µ 0 .

In the balanced case, model-independent thermometry is notoriously difficult because of the strong interactions. Inspired by [START_REF] Spiegelhalder | Collisional Stability of 40 K Immersed in a Strongly Interacting Fermi Gas of 6 Li[END_REF], we overcome this issue by measuring the temperature of a 7 Li cloud in thermal equilibrium with the 6 Li mixture (see Methods).

µ 0 is fitted on the hottest clouds so that the EOS agrees in the classical regime ζ ≫ 1 with the second-order virial expansion h(1, ζ) ≃ 2(1+ζ -1 / √ 2) [START_REF] Ho | High temperature expansion applied to fermions near Feshbach resonance[END_REF]. For colder clouds we proceed recursively. The EOS of an image recorded at temperature T has some overlap with the previously determined EOS from all images with T ′ > T . In this overlap region µ 0 is fitted to minimize the distance between the two EOS's. This provides a new portion of the EOS at lower temperature. Using 40 images of clouds prepared at different temperatures, we thus reconstruct a low-noise EOS from the classical part down to the degenerate regime, as shown in Fig. 3a.

We now comment the main features of the equation of state. At high temperature, the EOS can be expanded in powers of ζ -1 as a virial expansion [START_REF] Liu | Virial expansion for a strongly correlated Fermi gas[END_REF]:

h(1, ζ) 2 = ∞ k=1 (-1) k+1 k -5/2 + b k ζ -k ∞ k=1 (-1) k+1 k -5/2 ζ -k
, where b k is the k th virial coefficient. Since we have b 2 = 1/ √ 2 in the measurement scheme described above, our data provides for the first time the experimental values of b 3 and b 4 . b 3 = -0.35( 2) is in excellent agreement with the recent calculation b 3 = -0.291-3 -5/2 = -0.355 from [START_REF] Liu | Virial expansion for a strongly correlated Fermi gas[END_REF] but not with b 3 = 1.05 from [START_REF] Rupak | Universality in a 2-Component Fermi System at Finite Temperature[END_REF]. b 4 = 0.096 [START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic monte carlo method for divergent signalternating series[END_REF] involves the 4-fermion problem at unitarity and could interestingly be computed along the lines of [START_REF] Liu | Virial expansion for a strongly correlated Fermi gas[END_REF].

Let us now focus on the low-temperature regime of the normal phase ζ ≪ 1. As shown in Fig. 3b, we observe a T 2 dependence of the pressure with temperature. This behavior is reminiscent of a Fermi liquid and indicates that pseudogap effects expected for strongly-interacting Fermi superfluids [START_REF] Chen | BCS BEC crossover: From high temperature superconductors to ultracold superfluids[END_REF] do not show up at the thermodynamic level within our experimental precision. In analogy with 3 He or heavy-fermion metals, we fit our data with the EOS:

P (µ, T ) = 2P 1 (µ, 0) ξ -3/2 n + 5π 2 8 ξ -1/2 n m * m k B T µ 2 , (4)
P 1 (µ, 0) = 1/15π 2 (2m/ 2 ) 3/2 µ 5/2
being the pressure of a single-component Fermi gas at zero temperature. m * is the quasi-particle mass and ξ -1 n is the compressibility of the normal gas extrapolated to zero temperature, and normalized to that of an ideal gas of same density. We deduce two new parameters m * /m = 1.13(3) and ξ n = 0.51 [START_REF] Inguscio | Ultracold Fermi Gases[END_REF]. Despite the strong interactions m * is close to m, unlike the weakly interacting 3 He liquid for which 2.7 < m * /m < 5.8, depending on pressure. Our ξ n value is in agreement with the variational Fixed-Node Monte-Carlo calculations ξ n = 0.54 in [START_REF] Carlson | Superfluid Fermi gases with large scattering length[END_REF], ξ n = 0.56 in [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF] and with the Quantum Monte-Carlo calculation ξ n = 0.52 in [START_REF] Bulgac | Quantum Monte Carlo simulations of the BCS-BEC crossover at finite temperature[END_REF]. This yields the Landau parameters F s 0 = ξ n m * /m -1 = -0.42 and F s 1 = 3(m * /m -1) = 0.39. In the lowest temperature points (Fig. 3c) we observe a sudden deviation of the data from the fit (4) at (k B T /µ) c = 0.32(3) (see supplementary materials). We interpret this behavior as the transition from the normal phase to the superfluid phase. This critical ratio has been extensively calculated in the recent years. Our value is in close agreement with the diagrammatic Monte-Carlo calculation (k B T /µ) c = 0.32(2) of [START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF] and with the Quantum Monte-Carlo calculation (k B T /µ) c = 0.35(3) of [START_REF] Bulgac | Quantum Monte Carlo simulations of the BCS-BEC crossover at finite temperature[END_REF] but differs from the self-consistent approach in [START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF] giving (k B T /µ) c = 0.41, from the renormalization group prediction 0.24 in [START_REF] Gubbels | Renormalization group theory for the imbalanced Fermi gas[END_REF], and from several other less precise theories. From eq. ( 4) we deduce the total density n = n 1 + n 2 = ∂P (µ i = µ, T )/∂µ and the Fermi energy

E F = k B T F = 2 /2m(3π 2 n) 2/3
at the transition point. We obtain (µ/E F ) c = 0.49( 2) and (T /T F ) c =0.157 [START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic monte carlo method for divergent signalternating series[END_REF], in very good agreement with [START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF]. Our measurement is the first direct determination of (µ/E F ) c and (T /T F ) c in the homogeneous gas. It agrees with the extrapolated value of the MIT measurement [START_REF] Shin | Phase diagram of a two-component Fermi gas with resonant interactions[END_REF].

Below T c , advanced theories [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF][START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF] predict that P (µ, T )/2P 1 (µ, 0) is nearly constant (Fig. 3b). Therefore at T = T c , P/2P 1 ≃ ξ -3/2 s ≃ 3.7, and is consistent with our data. Here ξ s = 0.42 [START_REF] Ho | Universal thermodynamics of degenerate quantum gases in the unitarity limit[END_REF] is the fundamental parameter characterizing the EOS of the balanced superfluid at zero temperature, a quantity extensively measured and computed in the recent years [START_REF] Inguscio | Ultracold Fermi Gases[END_REF].

Our data is compared at all temperatures with the calculations from [START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF][START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF][START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF][START_REF] Combescot | Particle distribution tail and related energy formula[END_REF] (Fig. 3a). The agreement with [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF] is very good for a large range of temperatures. Concerning [START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF], the deviation with our data is about one error bar of the Monte-Carlo method below ζ = 0.2 and the deviation increases with temperature (Fig. 3a). Furthermore, we show in the supplementary material that h(1, ζ)/2 must be greater than 1, an inequality violated by the two hottest Monte-Carlo points of [START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF].

From our homogeneous EOS we can deduce the equation of state of the harmonically trapped unitary gas by integrating h(1, ζ) over the trap (see supplementary material). In particular, we find a critical temperature for the trapped gas (T /T F ) c = 0.19 [START_REF] Inguscio | Ultracold Fermi Gases[END_REF], where

T F = (3ω 2 r ω z N ) 1/3
. This value agrees very well with the recent measurement of [START_REF] Riedl | Superfluid Quenching of the Moment of Inertia in a Strongly Interacting Fermi Gas[END_REF], and with less precise measurements [START_REF] Luo | Thermodynamic Measurements in a Strongly Interacting Fermi Gas[END_REF][START_REF] Greiner | Emergence of a molecular Bose-Einstein condensate from a Fermi gas[END_REF][START_REF] Inada | Critical temperature and condensate fraction of a fermion pair condensate[END_REF].

Let us now explore a second line in the universal diagram h(η, ζ) (Fig. 1) by considering the case of the T = 0 spin-imbalanced mixture µ 2 = µ 1 , i.e. η = 1. Previous work [START_REF] Shin | Observation of phase separation in a strongly interacting imbalanced Fermi gas[END_REF][START_REF] Partridge | Pairing and phase separation in a polarized Fermi gas[END_REF][START_REF] Nascimbene | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF] has shown that phase separation occurs in a trap. Below a critical population imbalance a fullypaired superfluid occupies the center of the trap. It is surrounded by a normal mixed phase and an outer rim consisting of an ideal gas of the majority component. In two out of the three previous experiments including ours [START_REF] Shin | Observation of phase separation in a strongly interacting imbalanced Fermi gas[END_REF][START_REF] Nascimbene | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF], the local density approximation has been carefully checked. We are therefore entitled to use [START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF] to analyze our data.

As in the previous case, the relationship between the pressure and the EOS requires the knowledge of the chemical potentials µ 0 1 and µ 0 2 at the center of the trap. µ 0 1 is determined using the outer shell of the majority spin component (i = 1). The pressure profile P (µ 1z , µ 2z , 0) corresponds to the Fermi-Dirac distribution and is fitted with the Thomas-Fermi formula

P 1 = α(1 -z 2 /R 2 1 ) 5/2 , providing µ 0 1 = 1 2 mω 2 z R 2 1 .
Using P 1 for the calculation of h = P/P 1 cancels many systematic effects on the absolute value of the pressure. Moreover, fitting the outer shell using a finite-temperature Thomas-Fermi profile [START_REF] Shin | Phase diagram of a two-component Fermi gas with resonant interactions[END_REF], we measure a temperature k B T = 0.03(3)µ 0 1 . µ 0 2 is fitted by comparison in the superfluid region with the superfluid equation of state at zero temperature [START_REF] Chevy | Universal phase diagram of a strongly interacting fermi gas with unbalanced spin populations[END_REF]:

h(η, 0) = (1 + η) 5/2 /(2ξ s ) 3/2 .
(5)

Our measured equation of state h(η, 0) is displayed in Fig. 4. By construction our data agrees for η 0.1 with The red curves are the successive virial expansions up to 4 th order. The blue triangles are from [START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF], the green stars from [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF], the purple diamonds from [START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF], and the blue solid line from [START_REF] Combescot | Particle distribution tail and related energy formula[END_REF]. The grey region indicates the superfluid phase. (b) Equation of state P (µ, T )/2P1(µ, 0) as a function of (kBT /µ) ). (c) Expanded view of (b) near Tc. The sudden deviation of the data from the fit occurs at (kBT /µ)c = 0.32(3) that we interpret as the superfluid transition. The black dashed line indicates the mean value of the data points below Tc.

Ideal Fermi Gas Ideal Fermi Gas Virial2 Virial2 Virial3 Virial3 Virial4 Virial4
eq.( 5). In Fig. 4 the slope of h(η, 0) displays an obvious discontinuity for η = η c = 0.065 [START_REF] Shin | Determination of the equation of state of a polarized fermi gas at unitarity[END_REF]. This is a signature of a first-order quantum phase transition to the partially polarized normal phase. The error bar is dominated by the uncertainty on ξ s . This value is slightly higher than the prediction η c = 0.02 given by the fixed-node Monte-Carlo [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF] and than the value η c = 0.03(2) measured in [START_REF] Shin | Phase diagram of a two-component Fermi gas with resonant interactions[END_REF].

From the relations n i = ∂P/∂µ i we deduce from h(η, 0) the density ratio n 2 /n 1 (see inset in Fig. 4). This ratio is discontinuous at the phase transition, from a maximum value in the normal phase (n 2 /n 1 ) c = 0.5(1) to n 2 = n 1 in the superfluid phase. Our value is close to the zero-temperature calculation 0.44 [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF] and agrees with the coldest MIT samples [START_REF] Shin | Phase diagram of a two-component Fermi gas with resonant interactions[END_REF][START_REF] Shin | Determination of the equation of state of a polarized fermi gas at unitarity[END_REF]. It confirms that the temperature is much smaller than the tricritical point temperature T = 0.07T F [START_REF] Shin | Phase diagram of a two-component Fermi gas with resonant interactions[END_REF] where the discontinuity vanishes, justifying our T = 0 assumption made above. For η < η c our data displays a good agreement with a simple polaron model, based on the pioneering work in [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF]. A polaron is a quasi-particle describing a single minority atom immersed in the majority Fermi sea [START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic monte carlo method for divergent signalternating series[END_REF][START_REF] Nascimbene | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF][START_REF] Chevy | Universal phase diagram of a strongly interacting fermi gas with unbalanced spin populations[END_REF][START_REF] Schirotzek | Observation of fermi polarons in a tunable fermi liquid of ultracold atoms[END_REF]. It is characterized by a renormalized chemical potential µ 2 -A µ 1 and an effective mass m * p [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF]. Following this picture, we write the pressure as the sum of the Fermi pressures of ideal gases of majority atoms and of polarons:

Η c Η c Η Η A A 1.0 0.
P = 1 15π 2 2m 2 3/2 µ 5/2 1 + m * p m 3/2 (µ 2 -Aµ 1 ) 5/2 , (6)
which can be written as

h(η, 0) = 1 + m * p m 3/2 (η -A) 5/2 . ( 7
)
A and m * p have recently been calculated exactly [START_REF] Combescot | Normal state of highly polarized Fermi gases: full many-body treatment[END_REF][START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic monte carlo method for divergent signalternating series[END_REF]: A = -0.615, m * p /m = 1.20 [START_REF] Inguscio | Ultracold Fermi Gases[END_REF] and with these values inserted in [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF] the agreement with our data is perfect. Note that our data lies slightly above the variational fixednode Monte Carlo calculation [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF]. We therefore con- clude that interactions between polarons are not visible at this level of precision. Alternatively, we can fit our data with m * p /m as a free parameter in [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF]. We obtain m * p /m = 1.20 [START_REF] Inguscio | Ultracold Fermi Gases[END_REF]. The uncertainty combines the standard error of the fit and the uncertainty on ξ s . This value agrees with our previous measurement m * p /m = 1.17 [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF] [START_REF] Nascimbene | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF] (with a 5fold improvement in precision), with the theoretical value m * p /m = 1.20( 2) in [START_REF] Combescot | Normal state of highly polarized Fermi gases: full many-body treatment[END_REF][START_REF] Prokof'ev | Fermi-polaron problem: Diagrammatic monte carlo method for divergent signalternating series[END_REF] and with the variational calculation [START_REF] Combescot | Normal state of highly polarized Fermi gases: simple many-body approaches[END_REF]. It differs from the values 1.09(2) in [START_REF] Pilati | Phase separation in a polarized Fermi gas at zero temperature[END_REF], 1.04(3) in [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF], and from the experimental value 1.06 in [START_REF] Shin | Determination of the equation of state of a polarized fermi gas at unitarity[END_REF].

We arrive at a simple physical picture of the T = 0 spin-polarized gas: the fully paired superfluid is described by an ideal gas EOS renormalized by a single coefficient ξ s ; the normal phase is nothing but two ideal gases, one of bare majority particles and one of polaronic quasi-particles.

In conclusion, we have introduced a powerful method for the measurement of the equation of state of the unitary and homogeneous Fermi gas, that enables direct comparison with theoretical models and provides a set of new parameters shown in Tab.I. The method can readily be extended to any multi-component cold atom gas in three dimensions that fulfills the local density approximation (see supplementary discussion). We have shown that the normal phase of the unitary Fermi gas is a strongly correlated system whose thermodynamic properties are well described by Fermi liquid theory, unlike high-T c cuprates.

Note added in proof : Since this paper was accepted for publication, we have become aware of the measurement of a similar equation of state for the balanced unitary Fermi gas at finite temperature by different methods [START_REF] Horikoshi | Measurement of Universal Thermodynamic Functions for a Unitary Fermi Gas[END_REF]. METHODS SUMMARY Our experimental setup is presented in [START_REF] Nascimbene | Collective Oscillations of an Imbalanced Fermi Gas: Axial Compression Modes and Polaron Effective Mass[END_REF]. We load into an optical dipole trap and evaporate a mixture of 6 Li in the |1/2, ±1/2 states and of 7 Li in the |1, 1 state at 834 G. The cloud typically contains N 6 = 5 to 10 × 10 4 6 Li atoms in each spin state and N 7 = 3 to 20 × 10 3 7 Li atoms at a temperature from T = 150 nK to 1.3 µK. The 6 Li trap frequencies are ω z /2π = 37 Hz, ω r /2π varying from 830 Hz to 2.20 kHz, and the trap depth is 25 µK for our hottest samples, with T ≃ 2T F . 6 Li atoms are imaged in situ using absorption imaging, while 7 Li atoms are imaged after time of flight, providing the temperature in the same experimental run (Fig. 4). Since the scattering length describing the interaction between 7 Li and 6 Li atoms, a 67 = 2 nm, is much smaller than k -1 F , the 7 thermometer has no influence on the 6 Li density profiles. The 7 Li-6 Li collision rate, Γ 67 = 10 s -1 , is large enough to ensure thermal equilibrium between the two species.

METHODS

Construction of the equation of state by successive patches.

A typical image at high temperature provides about 100 pixels corresponding to ζ values varying from 2 at the trap center to 6 at the edges, with a signal-to-noise from 3 to 10. 7 such images are fitted in the wings using the second-order virial expansion and averaged to obtain a low-noise EOS up to ζ = 2. Then images of clouds where the evaporation has been pushed to a slightly lower temperature are recorded. They show about 75% overlap in ζ with the previous EOS. After minimization of the distance between a new image and the previously determined EOS in the overlap region, we obtain the value of µ 0 for a single image with 3% statistical uncertainty. This process is repeated for 6 successive trap depths. When averaging one image with typically 10 previous images, we obtain a new EOS with an error on ζ of about 0.03/ √ 10 ≃ 1%. The EOS experiences a random walk error on the 40 images of 0.01 × √ 40 ≃ 5% for the coldest data. An independent check of the maximum error is provided by the good agreement with the superfluid equation of state for temperatures lower than T c [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF][START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF].

Evaluation of the systematic uncertainties. For the measurement of h(1, ζ), the combined uncertainties on the radial frequency of the trap, trap anharmonicity, magnification of our imaging system, and atom counting affect the pressure measurement given in (3) at ≃ 20% level. However, two measurements, one at relatively high temperature and one at very low temperature, enable us to show that the overall error does not exceed 6%. In the temperature range ζ > 0.5, the agreement between the experimental value b 3 = -0.35 [START_REF] Inguscio | Ultracold Fermi Gases[END_REF] and the theoretical value b 3 = -0.355 of the third virial coefficient indicates that the global systematic error is smaller than 6%. Second, at very low temperature, theory [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF][START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF] predicts that the variation of P/2P 1 as a function of k B T /µ in the superfluid phase remains smaller than 5%. Our value of P/2P 1 = 3.75 below the critical point is within 5% of the T = 0 prediction ξ -3/2 s = 3.7(2). This confirms that systematic errors for our coldest samples are also smaller than 6%.

For the determination of the critical transition to superfluidity we fit the low-temperature data P (µ, T )/2P 1 (µ, 0) with a variable horizontal line for T < T c and with the Fermi-liquid equation (4) for T > T c . The result of the fit is the dashed black line in Fig. 3c, which intersects equation ( 4) at (k B T /µ) c = 0.315 [START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF]. This statistical error is negligible compared to the error induced by the 6% systematic uncertainty discussed above, justifying our very simplified fit procedure. Indeed a 6% error on the pressure induces a 10% error on µ for images recorded in the vicinity of the critical temperature, leading to (k B T /µ) c = 0.32 [START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF].

For the measurement of h(η, 0), the fit of the fully polarized wings of the cloud serves as a pressure calibration for the rest of the cloud, cancelling many systematic effects.

In order to estimate temperature effects in the polarized gas, let us first remark that in the superfluid phase corrections scale as T 4 for the bosonic excitations and are exponentially suppressed by the gap for the fermionic ones [START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF]. So in our temperature range k B T = 0.03µ 0 1 their contributions will be very small. On the other hand, in the partially polarized normal phase, we expect a typical Fermi liquid T 2 scaling. In order to obtain an estimate of the error on the EOS, we develop the following simple model. In equation ( 9) which describes a mixture of zero-temperature ideal gases, we replace the Fermi pressures by the finite-temperature pressures of ideal gases (see equation ( 1)):

P (µ 1 , µ 2 , T ) = P 1 (µ 1 , T ) + m * p m 3/2 P 1 (µ 2 -Aµ 1 , T ),
and run the analysis described in the main text. At T = 0.05µ 0 1 , the correction on h is less than 1%, half of our current error bar. Limit of 7 Li Thermometry. As the scattering length between the 7 Li atoms, a 77 = -3 nm is negative, the 7 Li cloud becomes unstable when a BEC forms. This occurs at T ∼ 150 nK with typically 3500 atoms. Precise thermometry with lower atom numbers becomes difficult. For the measurement of the zero-temperature equation of state of the imbalanced gas, we do not use 7 Li thermometry but rather the fit of the wings of the majority spin component. E/E 0 as a function of (T /TF ) 0 (black dots) and the EOS measured on 40 K in [START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF] (open red squares). The grey regions correspond to the superfluid phase.

Equation of State of the Trapped Unitary Gas

In this work, we have measured the equation of state of the homogeneous unitary gas. We can deduce from our data the EOS of the trapped balanced unitary gas, which has been measured in [START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF][START_REF] Luo | Thermodynamic Measurements in a Strongly Interacting Fermi Gas[END_REF].

the local density approximation, the total atom number N = n dr 3 is expressed as a function of the temperature T and the chemical potential µ 0 at the center, involving the function h(1, ζ):

N = -2 √ π k B T ω 3 ∞ ζ0 d log 1/2 (ζ/ζ 0 ) dζ f 5/2 (-ζ -1 )h(ζ)dζ, (8)
where ζ 0 = exp(-µ 0 /k B T ) and ω = (ω 2 r ω z ) 1/3 . We use for the calculation a discretized version of (8) taken solely on our experimental values of h, i.e. without using any interpolating or fitting function. Similar expressions are used to calculate the Fermi temperature E F = k B T F = ω(3N ) 1/3 , the total entropy S and energy E of the cloud. The equation of state E/N E F as a function of S/N k B , displayed in Supplementary Fig. 5a, is in very good agreement with [START_REF] Luo | Thermodynamic Measurements in a Strongly Interacting Fermi Gas[END_REF].

The normal-superfluid phase transition for the trapped gas occurs when at the trap center ζ 0 = ζ c = exp(-(k B T /µ) -1 c ), with (k B T /µ) c = 0.32(3), as measured on the homogeneous EOS h (1, ζ). At this point we get (T /T F ) c = 0.19(2), (S/N k B ) c = 1.5 [START_REF] Ho | Universal thermodynamics of degenerate quantum gases in the unitarity limit[END_REF] and (E/N E F ) c = 0.67 [START_REF] Luo | Thermodynamic Measurements in a Strongly Interacting Fermi Gas[END_REF].

In order to make the comparison with [START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF], we also express the equation of state E/E 0 as a function of (T /T F ) 0 , where the superscript 0 refers to the quantities evaluated on a non-interacting Fermi gas having the same entropy (Supplementary Fig. 5b). The good agreement with the measurement in [START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF], performed on 40 K clouds, illustrates the universality of the unitary gas.

Physical interpretation of the pressure in the normal mixed phase

We have shown that the pressure in the normal mixed phase can be described as the sum of the Fermi pressures of ideal gases of majority atoms and of polarons: (9) Here, we evaluate the corresponding canonical EOS relating the energy E to the densities n 1 , n 2 , in order to compare with the Fixed Node Monte Carlo prediction [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF]. Since at unitarity we have E = 3P V /2 [START_REF] Ho | Universal thermodynamics of degenerate quantum gases in the unitarity limit[END_REF], we just express the chemical potentials in terms of densities by using the thermodynamical identities n i = ∂ µi P , which yield respectively:

P = 1 15π 2
n 2 = 1 6π 2 2m * 2 (µ 2 -Aµ 1 )
3/2 (10)

n 1 = 1 6π 2 2mµ 1 2 3/2 -An 2 . (11)
The last term in equation [START_REF] Liu | Virial expansion for a strongly correlated Fermi gas[END_REF] clearly indicates the increased majority density due to the presence of the minority component. Expressing the pressure as a function of n i in (9) yields the energy:

E = E FP (1 + Ax) 5/3 + m m * x 5/3 ,
where E F P is the energy of the fully polarized gas and x = n 2 /n 1 . Expanding E to order x 2 finally leads to an expression similar to that obtained in [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF]:

E(x) = E FP 1 + 5 3
Ax + m m * x 5/3 + Bx 2 + .... , with B = 5A 2 /9 = 0.2. Our value of B is close to the calculated value B ≃ 0.14 from [START_REF] Lobo | Normal state of a polarized Fermi gas at unitarity[END_REF].

Trap Anharmonicity

First, in the axial direction z, the confinement is produced magnetically and the corresponding anharmonicity is negligible. In the radial direction, we develop the gaussian potential to fourth order around ρ = 0:

V r (ρ) = V 0 1 -exp -ρ 2 σ 2 ≃ 1 2 mω 2 r ρ 2 + ǫρ 4 ,

FIG. 2 :

 2 FIG. 2: Schematic representation of our atomic sample. The 6 Li atomic cloud is imaged in the direction y; the column density is then integrated along the direction x to give n(z). The 7 Li atoms are imaged after a time of flight along the z direction.

FIG. 3 :

 3 FIG. 3: (color online) Equation of state of a spin-balanced unitary Fermi gas. (a) Finite-temperature equation of state h(1, ζ) (black dots). The error bars represented at ζ = 0.14 and ζ = 2.3 indicate the 6% accuracy in ζ and h of our EOS.The red curves are the successive virial expansions up to 4 th order. The blue triangles are from[START_REF] Burovski | Critical temperature and thermodynamics of attractive fermions at unitarity[END_REF], the green stars from[START_REF] Bulgac | Spin 1/2 fermions in the unitary regime: A superfluid of a new type[END_REF], the purple diamonds from[START_REF] Haussmann | Thermodynamics of the BCS-BEC crossover[END_REF], and the blue solid line from[START_REF] Combescot | Particle distribution tail and related energy formula[END_REF]. The grey region indicates the superfluid phase. (b) Equation of state P (µ, T )/2P1(µ, 0) as a function of (kBT /µ) 2 , fitted by the Fermi liquid equation of state[START_REF] Luo | Measurement of the entropy and critical temperature of a strongly interacting Fermi gas[END_REF]. The red dashed line is the non-interacting Fermi gas (NIFG). The horizontal dot-dashed (resp. dotted) line indicates the zero-temperature pressure of the superfluid phase ∝ ξ

  µ)c (µ/EF )c (T /TF )c -0.35(2) 0.096(15) 0.32(3) 0.49(2) 0.157(15) ξn m * /m ηc (n2/n1)c m * p /m 0.51(2) 1.13(3) 0.065(20) 0.5(1)1.20[START_REF] Inguscio | Ultracold Fermi Gases[END_REF]

FIG. 5 :

 5 FIG.5:(color online) Equations of state of the trapped unitary gas. (a) Comparison between our EOS E/N EF as a function of S/N kB (black dots) and the EOS measured in[START_REF] Luo | Thermodynamic Measurements in a Strongly Interacting Fermi Gas[END_REF] (open red squares). The red solid line is the second-order virial equation of state. (b) Comparison between our EOS E/E 0 as a function of (T /TF ) 0 (black dots) and the EOS measured on 40 K in[START_REF] Stewart | Potential Energy of a 40 K Fermi Gas in the BCS-BEC Crossover[END_REF] (open red squares). The grey regions correspond to the superfluid phase.

2 (µ 2 -

 22 Aµ 1 ) 5/2 .

TABLE I :

 I Table of quantities measured in this work.

Acknowledgements We are grateful to R. Combescot, X. Leyronas, Y. Castin, A. Recati, S. Stringari, S. Giorgini, M. Zwierlein and T. Giamarchi for fruitful discussions and to C. Cohen-Tannoudji, J. Dalibard, F. Gerbier and G. Shlyapnikov for critical reading of the manuscript. We acknowledge support from ESF (Euroquam), SCALA, ANR FABIOLA, Région Ile de France (IFRAF), ERC and Institut Universitaire de France.

Author Contributions S. Nascimbène and N. Navon contributed equally to this work. S.N., N.N. and K.J. took the experimental data and all authors contributed to the data analysis and writing of the manuscript.

Author Information Correspondence and requests for materials should be addressed to S. N. (email:sylvain.nascimbene@ens.fr).

where mω 2 r = 2V 0 /σ 2 and ǫ = -V 0 /2σ 4 . In the balanced case, we have

Introducing n = ∂P/∂µ and defining u = mω 2 r ρ 2 /2 + ǫρ 4 we obtain, to lowest order,

The error on the measurement of h is then

We evaluate the integral in [START_REF] Rupak | Universality in a 2-Component Fermi System at Finite Temperature[END_REF] using the experimental values of h(1, ζ). In our shallowest trap, the worst case anharmonicity effect is 5%.

An exact inequality on the equation of state of an attractive Fermi gas

Writing the hamiltonian as H = H 0 + U , where H 0 is the single-particle part of the hamiltonian and U is the inter-particle interaction, one has the general inequality Ω ≤ Ω 0 + V 0 , where Ω 0 is the grand potential associated with H 0 and • 0 is the thermal average related to H 0 [START_REF] Feynman | Statistical Mechanics: A set of lectures[END_REF]. Taking for U a short range square potential of depth U 0 < 0 recovering the true scattering length, one has trivially V 0 < 0, hence Ω ≤ Ω 0 . Using the thermodynamic identity Ω = -P V , and recalling that Ω 0 = -2P 1 V and h = P/P 1 , we finally get the inequality

Extension to a Multi-Component System

We extend the equation ( 2) to a mixture of species i, of mass m i , trapped in a harmonic trap of transverse frequencies ω ri , following the calculations in [START_REF] Ho | Obtaining phase diagram and thermodynamic quantities of bulk systems from the densities of trapped gases[END_REF]. Using Gibbs-Duhem relation at a constant temperature T , dP = i n i dµ i , then

where we have used local density approximation (µ i (r) = µ 0 i -V (r)) to convert the integral over space to an integral on the chemical potentials. The integral is straightforward and yields to