G Varoquaux

R A Nyman

R Geiger

P Cheinet

A Landragin

P Bouyer

How to estimate the differential acceleration in a two-species atom interferometer to test the equivalence principle

Keywords: numbers: 37, 25, +k, 03, 75, Dg

We propose a scheme for testing the weak equivalence principle (Universality of Free Fall) using an atom-interferometric measurement of the local differential acceleration between two atomic species with a large mass ratio as test masses. A apparatus in free fall can be used to track atomic free-fall trajectories over large distances. We show how the differential acceleration can be extracted from the interferometric signal using Bayesian statistical estimation, even in the case of a large mass and laser wavelength difference. We show that this statistical estimation method does not suffer from acceleration noise of the platform and does not require repeatable experimental conditions. We specialize our discussion to a dual potassium/rubidium interferometer and extend our protocol with other atomic mixtures. Finally, we discuss the performances of the UFF test developed for the free-fall (0-g) airplane in the ICE project (http://www.ice-space.fr) .

Introduction

The Einstein equivalence principle is fundamental to the standard model of particle physics and all metric theories of gravity [START_REF] Will | The confrontation between general relativity and experiment[END_REF][START_REF] Lämmerzahl | The search for quantum gravity effects I[END_REF]. It can be broken into three elementary principles: the local Lorentz invariance; the local position invariance (also known as universality of red shift); and the universality of free-fall (UFF), stating that all freely-falling point particles follow identical trajectories independent of their internal composition. The Lorentz invariance and the position invariance are local properties and can be tested to unmatched precision using atomic clocks [START_REF] Wolf | Cold Atom Clock Test of Lorentz Invariance in the Matter Sector[END_REF][START_REF] Ashby | Testing local position invariance with four cesiumfountain primary frequency standards and four NIST hydrogen masers[END_REF][START_REF] Blatt | New limits on coupling of fundamental constants to gravity using [sup 87]sr optical lattice clocks[END_REF] and ultra-stable cavities [START_REF] Muller | Tests of relativity by complementary rotating michelson-morley experiments[END_REF]. On the contrary, the UFF can only be tested by tracking trajectories, ideally of freely-falling test masses. Various extensions to the current theoretical-physics framework predict violations of the UFF [START_REF] Damour | Testing the equivalence principle: why and how?[END_REF]. It is thus important to look experimentally for such violations and push the limits of experimental tests of the UFF. Moreover, laboratory experiments are an important complement to astrophysical observations in testing fundamental physics, because of the possibility of controlling the environment and repeating the experiments in varying conditions.

In this article, we show that using an atom interferometer with two different atomic species in free fall can lead to an accurate test of the UFF, even for different laser wavelength and mass (i.e. different scaling factors for the interferometers). We present a protocol that allows us to accurately extract the acceleration difference and show that this measurement is almost insensitive to strong vibrational noise or platform movement, which usually limit the atom interferometer accuracy, or platform movement. We show how Bayesian statistical methods introduced in [START_REF] Stockton | Bayesian estimation of differential interferometer phase[END_REF] for two identical atom interferometers can be extended to extract the acceleration difference between the two atomic species, taking advantage of phase-correlated measurements between both interferometers. For the sake of clarity, we focus on the simultaneous use of rubidium 87 and potassium 39 atoms in a light-pulse gravimeter [START_REF] Kasevich | Atomic interferometry using stimulated Raman transitions[END_REF]. Comparing the acceleration of these two different atomic species constitutes a meaningful test of the UFF, as they combine a large mass ratio (almost a factor of two), very different nuclear compositions (37 protons and 50 neutrons for 87 Rb and 20/19 for 39 K) and almost equal laser wavelength and thus interferometer scale factors. We show that, with reasonable experimental parameters, a test could reach the accuracy of η ∼ 5•10 -11 in a one-day parabolic flight campaign in a zero-g airplane, and, if extended to long-duration experiments, could compete with the best available apparatus. Our discussion can be generalized to other atomic species such as the mixture proposed in [START_REF] Dimopoulos | Testing general relativity with atom interferometry[END_REF] or lighter atomic species.

Testing the Universality of Free-Fall

A figure of merit often used to characterize a test of the UFF is the Eötvös ratio η, giving the fractional difference in acceleration between two test masses in free-fall: η = ∆a/a. Alternate quantum gravitation theories predict deviations from the UFF for η 10 -13 [START_REF] Damour | Violations of the equivalence principle in a dilatonrunaway scenario[END_REF][START_REF] Sandvik | A simple cosmology with a varying fine structure constant[END_REF][START_REF] Wetterich | Probing quintessence with time variation of couplings[END_REF]. Current experimental limits on violations of the UFF (η < 10 -13 ) are set by lunar laser-ranging measurements [START_REF] Williams | Progress in lunar laser ranging tests of relativistic gravity[END_REF] and torsion-balance laboratory experiments [START_REF] Schlamminger | Test of the equivalence principle using a rotating torsion balance[END_REF]. Tests of the UFF by monitoring the acceleration difference between two objects freely falling simultaneously have shown η < 10 -10 [START_REF] Kuroda | Test of a composition-dependent force by a free-fall interferometer[END_REF]. While all these experiments test the validity of the UFF on macroscopic objects, in the quest for a quantum gravity theory, it is interesting to look for deviations on elementary, or microscopic, particles, where quantum mechanics is needed to describe their evolution [START_REF] Audretsch | Matter Wave Interferometry and Why Quantum Objects are Fundamental for Establishing a Gravitational Theory[END_REF][START_REF] Adunas | Probing Quantum Violations of the Equivalence Principle[END_REF].

The accuracy and sensitivity of local-acceleration measurements using atominterferometry nowadays rival state-of-the-art conventional accelerometers using macroscopic test masses [START_REF] Kasevich | Atomic interferometry using stimulated Raman transitions[END_REF][START_REF] Peters | Measurement of gravitational acceleration by dropping atoms[END_REF][START_REF] Peters | High-precision gravity measurements using atom interferometry[END_REF]. With such sensors, the quantity measured directly relates to the acceleration of weakly-interacting particles via experimentally wellcontrolled quantities, such as laser wavelengths [START_REF] Bordé | Atomic interferometry with internal state labeling[END_REF]. In addition, the evolution of these particles in the gravitational field can be modeled within a covariant quantum field theory [START_REF] Borde | Relativistic Phase Shifts for Dirac Particles Interacting with Weak Gravitational Fields in Matter-Wave Interferometers[END_REF]. Recent results using atom-interferometric gravimetry to compare the acceleration between two isotopes [START_REF] Peters | Measurement of gravitational acceleration by dropping atoms[END_REF][START_REF] Fray | Atomic Interferometer with Amplitude Gratings of Light and Its Applications to Atom Based Tests of the Equivalence Principle[END_REF] have demonstrated the possibility of atominterferometric tests of the UFF. Ongoing efforts to extend the size of inertial-sensing atom interferometers by increasing the interrogation time [START_REF] Nyman | a transportable atomic inertial sensor for test in microgravity[END_REF][START_REF] Dimopoulos | Testing general relativity with atom interferometry[END_REF][START_REF] Stern | Light-pulse atom interferometry in microgravity[END_REF] open the door to high-accuracy atom accelerometers which will be very sensitive to smaller accelerations, thus pushing the limits of these tests. These long interrogation times, i.e. large free-fall heights, can be achieved when using a large experimental chamber to launch the atoms such as a ten-meter-high fountain, as suggested in [START_REF] Dimopoulos | Testing general relativity with atom interferometry[END_REF].

Compact apparatuses can also be used in reduced-gravity environments, such as drop towers [START_REF] Vogel | Bose Einstein condensates in microgravity[END_REF][START_REF] Könemann | A freely falling magneto-optical trap drop tower experiment[END_REF], orbital platforms [START_REF] Cacciapuoti | Atomic Clock Ensemble in Space: Scientific Objectives and Mission Status[END_REF], or atmospheric parabolic flight [START_REF] Stern | Light-pulse atom interferometry in microgravity[END_REF]. However, increasing the interrogation time also increases the sensitivity of the interferometer to acceleration noise [START_REF] Cheinet | Measurement of the sensitivity function in time-domain atomic interferometer[END_REF] which can scale from ∼ 10 -5 m.s 2 in drop towers to ∼ 10 -2 m.s 2 in the 0-g Airbus [START_REF] Varoquaux | An Ultra-Cold Atom Source for Long-Baseline Interferometric Inertial Sensors in Reduced Gravity[END_REF] and on the International Space Station [START_REF] Penley | The international space station as a microgravity research platform[END_REF]. Atoms, isolated in a vacuum chamber, are truly in free-fall in the Earth's local gravity field, as long as they do not hit the chamber walls, or experience field gradients (optical or static magnetic). However, their acceleration is recorded relative to an ill-defined experimental frame. This can compromise the increase in sensitivity. Measuring differential phase between similar interferometers using the same light has been shown to reject common-mode inertial noise up to large scaling-factors [START_REF] Snadden | Measurement of the earth's gravity gradient with an atom interferometer-based gravity gradiometer[END_REF][START_REF] Bertoldi | Atom interferometry gravity-gradiometer for the determination of the newtonian gravitational constant g[END_REF][START_REF] Herrmann | Noise-immune conjugate large-area atomm interferometers[END_REF], however, in the case of an UFF measurement, the two interferometers compared do not share the same sensitivity to inertial effects and the common-mode rejection is not straightforward.

Differential atom interferometer in free-fall

The acceleration-measurement process on each single species can be pictured as marking successive positions of freely-falling atoms with a pair of Raman lasers, pulsed in time. The resulting atomic phase shift φ is the difference between the relative phase of the Raman lasers at the atom's successive classical positions [START_REF] Peters | Measurement of gravitational acceleration by dropping atoms[END_REF][START_REF] Bordé | Theoretical tools for atom optics and interferometry[END_REF]. When the Raman lasers are used in a retroreflected configuration, this phase simply relates to the distance between the atomic cloud and the reference retroreflecting mirror. In such a three-pulse interferometer, using atoms without initial velocity, the inertial phase shift varies with the acceleration a as:

φ = k a T 2 , ( 1
)
where a is the acceleration of the atoms relative to the mirror, k is the effective wavevector of the Raman lasers, and T is the time between successive pulses.

To test the UFF, we need to extract the difference in acceleration between the two species. However, in atom interferometers with internal-state labeling [START_REF] Bordé | Atomic interferometry with internal state labeling[END_REF], the experimental signal is the population ratio n between the two output arms of a single interferometer n ∼ cos(φ+Φ), with Φ related to the phase noise of the Raman lasers [START_REF] Nyman | a transportable atomic inertial sensor for test in microgravity[END_REF] and contributions due to vibrations and unwanted inertial effects on the retroreflecting mirror [START_REF] Gouet | Limits to the sensitivity of a low noise compact atomic gravimeter[END_REF]. To extract an absolute value of the interferometric phase, accumulating data to scan a fringe is required. This accumulation of several experimental points can thus be hindered by the acceleration noise of the platform, as the acceleration measured may vary from one measurement to another. To have access to long integration, we thus want to extract as much information as possible about the phase difference with a minimum set of independent measurements.

We now focus on the Potassium(K)-Rubidium(Rb) atom interferometer described in [START_REF] Nyman | a transportable atomic inertial sensor for test in microgravity[END_REF]. The UFF experimental signal that we are interested in is the Eötvos ratio, η = 2(a Ka Rb )/(a K + a Rb ) with a K and a Rb the accelerations of potassium and rubidium. We thus want to extract the acceleration difference δa = a Ka Rb . This can be related directly to the difference of the inertial phase of each interferometer: using Equation ( 1

), δφ = φ K -φ Rb = k K a K T 2 K -k Rb a Rb T 2 Rb
where k K = 4π/767 nm -1 and k Rb = 4π/780 nm -1 are the effective wave-vectors of the Raman transitions, and T K and T Rb are the interrogation times for the K and Rb interferometers respectively. In order to directly read out δa, we need to adjust the respective interferometer interrogation times so that they have the same scale factor:

k Rb T 2 Rb = k K T 2 K = S, i.e. T Rb /T K ∼ 1.
008. In this case, we simply have δφ = Sδa.

(

As in [START_REF] Stockton | Bayesian estimation of differential interferometer phase[END_REF] we make a statistical description of the measurement process in a twospecies interferometer. Two quantities are measured:

n K = A + B cos(φ K + Φ K ) n Rb = C + D cos(φ Rb + Φ Rb ) (3)
where the capital letters represent fluctuating quantities: A and C are the offsets of the population measurement, B and D are the fringe visibilities, and Φ K and Φ Rb are related to the phase and acceleration noises on the two interferometers. In the following, we neglect the laser-induced phase noise, due e.g. to finite laser linewidth [START_REF] Le Gouët | Influence of lasers propagation delay on the sensitivity of atom interferometers[END_REF] or microwave source jitter [START_REF] Nyman | a transportable atomic inertial sensor for test in microgravity[END_REF], since it can be reduced with appropriate phase-locking techniques [START_REF] Gouet | Limits to the sensitivity of a low noise compact atomic gravimeter[END_REF]. The interferometric phase noise, due to vibrations or other uncontrolled inertial effects, appears as shifts of the local Raman phase for each interferometer. Following [START_REF] Cheinet | Conception et réalisation d'un gravimètre à atomes froids[END_REF] and assuming a white acceleration noise of power spectral density (PSD) S 0 α , the standard deviation of Φ K,Rb can be written

σ Φ i = k i T 3/2 i 2S 0 α 3 , i = K, Rb. (4)
This phase noise can be expressed as a random spatial displacement X(t) of deviation σ X = σ Φ /k of the retroreflecting mirror. The effect of other common-mode, spatial, Raman-phase fluctuations such as optical aberrations [START_REF] Fils | Influence of optical aberrations in an atomic gyroscope[END_REF] can also be included in this fluctuating variable X.

When calculating the differential response of the two-species atom interferometer, the relative phase noise between the two interferometers can be written ∆Φ = Φ K -Φ Rb = k K X where the standard deviations of X and X are simply related by a vernierscale relation

σ X = σ X δk 2k (5)
with σ X calculated for T = T K and δk/2k = (k Kk Rb )/2k K ≈ 8.5•10 -3 . The derivation of this scale relation and its physical explanation are detailed in Appendix B. We can now rewrite Eq. ( 3) as a simplified measurement model, highlighting differential effects:

n K = A + B cos( ΦRb + δφ + k K X) n Rb = C + D cos ΦRb (6)
The phase ΦRb = Sa Rb + Φ Rb of the rubidium interferometer is taken as the reference. The phase offset in the potassium interferometer includes the UFF signature, δφ and the effect of vibrations of the retroreflecting mirror, k K X.

Extracting the differential phase using Bayesian analysis

To extract the differential phase δφ from statistically independent data acquired during different measurement sequences, we use recursive Bayesian estimation [START_REF] Wasserman | All of statistics[END_REF][START_REF] Zawisky | Phase estimation in interferometry[END_REF].

Although several solutions for signal processing in atom-interferometric inertial sensors have been studied [START_REF] Stockton | Bayesian estimation of differential interferometer phase[END_REF][START_REF] Foster | Method of phase extraction between coupled atom interferometers using ellipse-specific fitting[END_REF], Bayesian estimators make best use of a noise model for estimating information from experimental data [START_REF] Stockton | Bayesian estimation of differential interferometer phase[END_REF] and they have been successfully used in quantum optics [START_REF] Zawisky | Phase estimation in interferometry[END_REF][START_REF] Guerlin | Progressive field-state collapse and quantum nondemolition photon counting[END_REF] or in optical interferometers [START_REF] Dupuis | Bayesian estimation of pulsar parameters from gravitational wave data[END_REF]. Moreover, Kalman filtering, a restricted version of Bayesian estimation, plays a critical role for proper use of fiber-optic gyroscope data [START_REF] Gaiffe | From R&D brassboards to navigation grade fog-based instruments : the experience of photonetics/ixsea[END_REF]. In the Bayesian framework, the parameters to be estimated are considered as random variables, whose probability distribution is deduced from the measurements by inverting the measurement model. Using Bayes' theorem, we calculate the probability distribution function p(δφ | (n K ,n Rb ) i ) for the parameter δφ given the results of a coupled measurement on both interferometers (n K , n Rb ) i , for each measurement i. The probability distribution p(δφ | (n K ,n Rb ) 1 , (n K ,n Rb ) 2 , . . .) for δφ, given all measurements, is the product of all these conditional probabilities: The non linearities of the measurement model introduced by the trigonometric functions in Eq. ( 6), with different periods k Rb and k K , make the analytical calculation of the conditional probability required for the Bayesian estimation tedious and the resulting expression is computationally costly to evaluate. We calculate this probability distribution (i.e. the "posterior") by using Monte Carlo sampling of the state space with the noise model (Eq. ( 6)). We use the probability law to estimate the reversed conditional probability (i.e. the "prior") p(n K , n Rb |δφ). A kernel density estimator [START_REF] Wasserman | All of statistics[END_REF] can be used to reduce the number of sampling points required, though this may not increase overall numerical efficiency. The posterior, used for Eq. ( 7), is obtained from the prior using Bayes' theorem [START_REF]Bayes An Essay towards solving a Problem in the Doctrine of Chances[END_REF].

p(δφ | (n K ,n Rb ) 1 , (n K ,n Rb ) 2 , . . .) = p(δφ | (n K ,n Rb ) 1 ) p(δφ | (n K ,n Rb ) 2 ) . . . (7)
We have run Monte Carlo simulations of the estimation process for different values of displacement noise amplitude σ X with the measurement model described in Eq. ( 6), as pictured in figure 1. We use Gaussian noise of standard deviation σ = 0.05 for the various parameters. The common-mode phase ΦRb is taken to be uniformly random in the interval [0, 2π]. As seen in figure 2, the estimator converges to a precise value of the differential phase with fewer than 10 uncorrelated measurements for fluctuations of X up to a few micrometers, which corresponds to a drift of the one-species interferometric signal by several complete fringes. This surprising result comes from the vernier scale relation in Eq. ( 5) that reduces the differential measurement sensitivity to Raman-phase fluctuations.

One should note that if δφ ∼ 0, the measurements performed do not contain enough information for good estimation (we have checked that the Fisher information matrix [START_REF] Wasserman | All of statistics[END_REF] is zero for δφ = 0). This can be easily understood since the data n K and n Rb are then distributed along a single line and we lose the sensitivity to phase/vibration noise [START_REF] Foster | Method of phase extraction between coupled atom interferometers using ellipse-specific fitting[END_REF]. It is thus necessary to introduce a controlled phase jump of π/2 in one of the two interferometers, e.g. on the phase of the Raman lasers. This choice of interferometric phase shift corresponds to working on the side of a fringe in a standard interferometer: the two parameters are maximally independent, distributed on a large-ellipticity curve rather than a flat line (see figure 1), giving optimal measurement sensitivity.

We have performed other simulations for different choices of noise distribution for X, which can be related to acceleration-noise power spectral density using the interferometer sensitivity function as described in Appendix B. We find that the convergence rate is not dependent on the nature of the bell-shaped noise distribution, nor on its behavior in its wings, but only on the RMS amplitude of the fluctuations. In addition, we have performed simulations for different dual-interferometer configurations in which the distance between the effective Raman transition wave vectors is reduced, thus reducing the Vernier-scale effect. Specifically we have investigated using the potassium D1 transition (at 770 nm) with rubidium, or the two isotopes of rubidium, with transitions separated only by 3 GHz ∼ 0.03 nm (see tab. 1). We find that for small variations of X, (σ X < 0.5 µm typ.), the phase-estimation is not limited by the Vernier scale effect as the convergence speed is similar for all three intereferometer configurations.

Higher-order inertial effects

We now consider the contribution of additional inertial effects on the differential measurement strategy. Equation ( 1) gives only dependence of the phase shift to an acceleration, and does not include rotation, gravity gradients, and higher-order effects. The effect of rotation can, in principle, be rejected by a feed-forward on the phase of the lasers [START_REF] Yver-Leduc | Reaching the quantum noise limit in a high-sensitivity cold-atom inertial sensor[END_REF] and a accurate control of the atom cloud velocity after release. We are then left with the major second-order contribution to the phase shifts, the effect of the gravity gradients. Since the atomic trajectories in the two interferometers explore slightly different altitudes, gravity gradients will add a contribution to the inertial phase shift [START_REF] Wolf | Gravimetry using atom interferometers: Some systematic effects[END_REF][START_REF] Ch | Atomic clocks and inertial sensors[END_REF]:

φ γ ∼ k γ T 2 7 12 T 2 a -(v - v r 2 )T , (8)
where v is the relative velocity between the center of mass of the atom cloud and the reference mirror. v r = k m is the recoil velocity and γ the gravity gradient. If the scale factors S of both interferometers are kept constant, the numerical value of the total differential phase shift (Eq. 2) then becomes δφ tot = δφ + δφ γ with δφ γ the residual gradient-induced phase shift. In general, the contribution to δφ γ of first term in Eq. ( 8) is negligible and v can be chosen to compensate for the recoil velocity k/m.

Conclusion

We finally turn to the specific case of the experiment in the 0-g airplane described in [START_REF] Stern | Light-pulse atom interferometry in microgravity[END_REF]. We can use the results from our simulation to give an order of magnitude of the precision achievable by a campaign of measurements in Zero-G flights. For a residual displacement from free fall X such as σ X ∼ 1 µm (achieved with proper decoupling from the low frequency vibrations), after 30 data points, the standard deviation on δφ is ∼ 3•10 -2 rad. Thus, for an interrogation time of T = 2 s, the differential-acceleration resolution is ∼ 2.5•10 -10 m•s -2 , and the η parameter characterizing a test of the UFF can be measured to a precision of η ∼ 5•10 -11 . For the experimental conditions encountered in a 0-g plane, an initial velocity v ≤ 1 cm•s -1 and T = 2 s, the effect of gravity gradients is negligible.

In conclusion, we have shown that Bayesian estimation can be efficient to perform a differential measurement between two inertial sensors using atoms of different mass and interrogation wavelength. Even for large vibrational noise, and large interrogation times, the Bayesian estimator converges rapidly. The measurement of the differential phase shift , i.e. the acceleration difference can thus be measured to a high precision. This opens new perspectives for the development of high precision test of fundamental physics such as tests of the equivalence principle. For example, we predict a precision of η ∼ 5•10 -11 when using only 30 experimental data points with a free-fall time of 4 s in the Zero-G Airbus, such as for the ICE experiment [START_REF] Nyman | a transportable atomic inertial sensor for test in microgravity[END_REF][START_REF] Stern | Light-pulse atom interferometry in microgravity[END_REF]. In the future, freefall and integration times may be increased by deploying atom-interferometric inertial sensors on dedicated orbital platforms for next-generation tests of the UFF, at the price of an increased sensitivity to vibrational noise. The use of fast-convergence estimators will help rejecting this acceleration noise and thus relax the requirement on drag-free vibration isolation performance. A rough estimate indicates that for 20 seconds of interrogation time and an integration over 1 year [START_REF] Ertmer | Matter Wave Explorer of Gravity (MWXG)[END_REF], a target accuracy of η ∼ 8•10 -15 , close to that of the project µSCOPE [START_REF] Touboul | Microscope, testing the equivalence principle in space[END_REF], is reachable with no specific drag-free platform. In the case of a differential acceleration measurement, the two interferometers have different sensitivities h K (t) and h Rb (t) (because T K = T Rb ). Additionally, the impact of the acceleration noise is slightly different on the K-interferometer (∝ k K X (t)) and on the Rb-interferometer (∝ k Rb X (t)). The relative interferometric phase noise ∆Φ = Φ K -Φ Rb can be easily calculated by replacing kf (t) by k K f K (t)k Rb f Rb (t) in Eq. (B.3) . We find to leading order in δk/k ≈ 0.017:

σ 2 ∆Φ = 2k 2 T 3 S 0 α 3 δk 2k 2 + O δk k 3 . (B.5)
Combining Eq. (B.5) and Eq. (B.4), we can estimate the vibration noise rejection ratio between the one-species and the two-species interferometer:

σ X σ X = k K σ ∆Φ k K σ Φ ≈ δk 2k ≈ 0.00852. (B.6)
It is also interesting to visualize this noise rejection of the two-species interferometer in the frequency domain since one often has access to the acceleration noise spectrum of the experimental platform. In this formalism, the interferometer sensitivity functions H K (ω) and H Rb (ω) result from the Fourier transforms of h K (t) and h Rb (t). The variance of the relative interferometric phase noise can be written: This feature is all the more interesting that the amplitude of the acceleration noise of the platform (such as the Zero-G Airbus) is usually the highest at low frequencies.

σ 2 ∆Φ = +∞ 0 |H(ω)| 2 S α (ω) ω 4 dω 2π , (
Appendix C. Noise rejection ratio in the aircraft A300-0G experimental platform

In this appendix we evaluate the noise rejection of the two-specices accelerometer in the experimental platform where the I.C.E. experiment is performed, namely the A300-0G aircraft carrying out parabolic flights. We take into account the measured acceleration noise spectrum S α (ω) in the plane during the quiet part of the parabola , in the direction of the Raman beams propagation. With the notations introduced in Appendix B, the noise rejection ratio is now given by

σ X σ X = σ ∆Φ σ Φ = +∞ 0 dω |H(ω)| 2 S α (ω)/ω 4 +∞ 0 dω |H K (ω)| 2 S α (ω)/ω 4 1/2 , (C.1)
with H(ω) the transfer function of the differential accelerometer (H K (ω) for the Kaccelerometer). In figure C1, we have plotted the evolution of the rejection ratio with the total interrogation time of the interferometer. We obtained these data by numerical integrations in equation (C.1), performed on one hand for the real acceleration noise in the plane, and on the other hand for a white noise (S α (ω) = S 0 α ). In the case of a white acceleration noise, we find that the noise rejection ratio is independant of the interrogation time and is equal to the vernier factor δk/2k ≈ 0.0085, as it was derived in Appendix B. This rejection ratio can be seen as the weighted average rejection on a limited bandwidth around 1/T which contributes the most to the interferometric phase noise. When T is increased, this bandwidth shifts to lower frequencies together with the rejection efficiency. On a white noise, this leads to a constant rejection ratio. On a structured noise such as the real noise spectrum measured in the plane (figure C1), this is no longer true. For T K = 100 ms, the relevant bandwidth is around 10 Hz where the spectrum is relatively flat and we find a rejection similar to the white spectrum case. For larger values of T K , the noise spectrum gives more weight to the low frequencies for which rejection is more efficient. The rejection ratio is then improved up to a factor 3 as one can see in figure C1. In other words, both accelerometers (one species and two-species) operate where the noise is stronger for longer T , but the two-species accelerometer phase noise will increase less than the one-species one.

nFigure 1 .

 1 Figure 1. Simulation of the model of Eq. (6). The random variable ΦRb is generated from a uniform distribution taken in the interval 0, 2π. We generate X from a normal distribution centered around 0 with σ X = 1.2 µm. A, B, C and D are generated from a normal distribution with σ = 0.05. The mean values are 0 for A and C and 1 for B and D. For each simulation are displayed (going clockwise, starting bottom left) : the measurement probability in the (n Rb , n K ) plane for the value of the hidden parameter δφ choosen in the simulation, the different measurements actually drawn, the successive probability-distribution estimations for δφ, and the distance between estimated values of δφ and the actual value.

Figure 2 .

 2 Figure 2. Standard deviation of the Bayesian estimation of the differential phase δφ between K and Rb interferometers. The standard deviations of the estimation of the differential phase decrease at different rates with repeated measurements for different mirror-displacement amplitudes (dotted lines). The convergence of the points towards the dotted lines shows how correlated noise is handled by the algorithm.

4 Figure B1 .

 4B1 Figure B1. Comparison of the transfer functions of the two-species (K-Rb) and one-species (Rb, dashed line) accelerometers in the low frequency range for T = 2 s.

B. 7 )

 7 with S α (ω) the acceleration noise PSD and H(ω) = H K (ω) -H Rb (ω) the transfer function of the differential interferometer. In figureB1, we have plotted |H(ω)| 2 /ω 4 and the transfer function |H Rb (ω)| 2 /ω 4 of a one-species (Rb) interferometer, for perfect π/2 pulses, an effective Rabi frequency Ω = 2π × 50kHz of the Raman transitions and T = 2 s. This figure shows that the relative phase noise ∆Φ is considerably reduced with respect to the phase noise of a single interferometer in the low frequency domain.

Figure C1 .

 C1 Figure C1. Left : measured acceleration noise spectrum of the A300-0G aircraft in the direction of the Raman beams. Right : noise rejection ratio between the onespecies and the differential accelerometers for various interrogation times T K (various scale factors). The squares correspond to a white acceleration noise whereas the dots account for the real noise spectrum in the plane.

Table 1 .

 1 Standard deviation on the phase estimate, in radians, after 30 measurements, for different mirror-displacement amplitudes σ X , and for different pairs of Raman transitions. The species and transition lines considered here are: the potassium D2 line at 767 nm, the potassium D1 line at 770 nm, and the rubidium 85 and 87 D2 lines at 780 nm, with a distance of 3 GHz ∼ 0.03 nm between the two isotopes.

		K-D2/Rb	K-D1/Rb	85 Rb/ 87 Rb
	σ X ∼ 0.1 µm	.022	.022	.021
	σ X ∼ 0.5 µm	.037	.033	.021
	σ X ∼ 1.5 µm	.101	.075	.021
	σ X ∼ 3.0 µm	.229	.166	.022

Acknowledgments

The authors would like to thank Jonas Kahn and Peter Wolf for enlightening discussions. The I.C.E. collaboration is funded by the Centre National d'Etudes Spatiales (CNES), IFRAF and RTRA "Triangle de la Physique". Further support comes from the European Union STREP consortium FINAQS and the European Space Agency MAP program A0-2004-064/082.

Appendix A. Simple derivation of the single-atom interferometer spatial-displacement noise

It can be shown [START_REF] Peters | High-precision gravity measurements using atom interferometry[END_REF][START_REF] Storey | The Feynman path integral approach to atomic interferometry: a tutorial[END_REF] that the major contribution to the interferometric phase shift is due to the interaction with the Raman beams. Whenever the state of the atom changes during such an interaction, it acquires an additional phase φ i (ξ i , t i ) = k ξ iω t i . The sign of the phase depends on the initial state of the atom. The position of the atom ξ i with respect to the retroreflecting mirror, taken at the time t i of the pulses i = {1, 2, 3}, can be written ξ i = ξ(t i ) = x(t i ) + X (t i ) where x(t) is the absolute position of the atom and X (t i ) is a random mirror position related to the vibration noise. Taking ξ 2 = t 2 = 0, t 3 = -t 1 = T and tracing all the state changes leads to a phase difference

Without gravitational field the trajectories are straightlines and the inherent symmetry of the situation leads to φ = k [X (-T ) + X (T )]. Here, the interferometric phase shift is only related to the vibrational noise and can be written φ = Φ = k X(T ) where X(T ) is a random variable representing the amplitude of the vibrational noise phase shift Φ for an interrogation time T , as calculated in Appendix B.

The introduction of a gravitational field breaks the symmetry. The atom now falls three times as far during transit in the second half of the interferometer as in the first half and we find an additional contribution to the phase shift φ = k a T 2 proportional to the gravitational acceleration, so that the interferometric signal becomes :

Appendix B. Interferometric phase noise of the two-species accelerometer

In this section, we derive the vernier scale relation in Eq. ( 5) between the interferometric phase noise standard deviations of the two-species (K-Rb) and one-species (Rb) accelerometers. Following [START_REF] Cheinet | Measurement of the sensitivity function in time-domain atomic interferometer[END_REF][START_REF] Cheinet | Conception et réalisation d'un gravimètre à atomes froids[END_REF], the interferometric phase noise can be written

where h(t) is the sensitivity function of the interferometer (see reference [START_REF] Cheinet | Measurement of the sensitivity function in time-domain atomic interferometer[END_REF] for its definition) and X (t) represents the retroreflecting mirror position at time t so that d 2 X (t)/dt 2 = α(t) with α(t) the acceleration noise of the experimental platform.

Integrating by parts equation (B.1) leads to

Neglecting the duration of the Raman light pulses (τ ∼ 20µs typically) with respect to the interrogation time (T ∼ 2s), the function h(t) is an odd and piecewise constant function and the function f (t) is an even and piecewise linear function; moreover, h(t) and f (t) are equal to zero out of the window [-T, T ]. Thus, the first term in the above equation vanishes and the variance of the interferometric phase noise can be written:

Assuming a white acceleration noise, i.e. α(t 1 )α(t 2 ) = S 0 α δ(t 1t 2 ) results in a major simplification of equation (B.3), and a straightforward single-integration of f (t) 2 finally leads to:


C F De Angelis
email: caroldeangelis3@hotmail.com

M P Soares

David J Mckenzie
email: david.mckenzie@cnrs.friarada

E W Taylor
email: tedwtaylor@hotmail.com

I L Cardoso

I C Souza

Daniel A Wunderlin
email: danielwunderlin@gmail.com

M V Monferrán
email: mmonferran@unc.edu.ar

M N Fernandes

Cleo A C Leite
email: cleo.leite@ufscar.br

De Angelis

; Soares

; Mckenzie

; Taylor

; Cardoso

; Souza

; Leite

Settleable atmospheric particulate

Keywords: iron processing, industry, metal/metalloid, nanoparticle, swimming challenge, physiological responses

Introduction

Small amounts of metals/metalloids occur naturally in the aquatic environment through biogeochemical cycles. However, industrial activity causes a continuous and persistent production of atmospheric particulate matter (APM), affecting land and water bodies in diverse ecosystems [START_REF] Callén | Comparison of receptor models for source apportionment of the PM10 in Zaragoza (Spain)[END_REF][START_REF] Park | Source contributions to fine particulate matter in an urban atmosphere[END_REF][START_REF] Salgado | Informações gerais e ecotoxicológicas de material particulado[END_REF][START_REF] Tsai | Characterization of chemical species in atmospheric aerosols in a metropolitan basin[END_REF]. APM is usually composed of a complex mixture of particles that vary in shape and size. Its final composition can be variable and contains emerging metal compounds with unpredictable new combinations of metals and metal nanoparticles [START_REF] Nordberg | Handbook on the Toxicology of Metals[END_REF][START_REF] Salgado | Informações gerais e ecotoxicológicas de material particulado[END_REF][START_REF] Soriano | Deposition of heavy metals from particulate settleable matter in soils of an industrialised area[END_REF].

Although, recent publication has revealed potential risk to fish biota (e.g., [START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF][START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF], the presence of metallic particles from APM deposition (sedimentable APM -SePM) in water bodies has not been properly monitored.

Several environmental regulatory agencies still do not consider air-water crosscontamination in their environmental monitoring programs (e.g., CONAMA, 2018, Environmental Protection Agency -EPA, and European Union and Canada). For example, estuarine waters close to the steel industrial complex in Vitoria Bay in the state of Espírito Santo, Brazil, receive constant deposition, of varying intensity, of metallic particles from SePM such as Al, Cr, Mn, Fe, Ni, Cu, Zn, As, Se, Ag, Cd, Hg, and Pb (μg Souza et al., 2021bSouza et al., , 2021a)).

 L -1 ,
Previous studies in the environs of Vitoria city have found that SePM can affect the aquatic environment and cause bioaccumulation [START_REF] Souza | Matching metal pollution with bioavailability, bioaccumulation and biomarkers response in fish (Centropomus parallelus) resident in neotropical estuaries[END_REF] and biomagnification processes through the food web [START_REF] Santos | Source apportionment of settleable particles in an impacted urban and industrialized region in Brazil[END_REF][START_REF] Souza | Interrogating pollution sources in a mangrove food web using multiple stable isotopes[END_REF].

Environmentally relevant levels of SePM contamination have adverse effects on fishes, such as deleterious impacts on hematological status and innate immunity, and stress responses (O. niloticus, [START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF]. In particular, SePM had effects that would compromise blood oxygen-carrying capacity, causing a reduction in hematocrit, erythrocyte numbers, and mean corpuscular hemoglobin concentration [START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF]. Moreover, exposure to SePM also caused gill damage and ventilatory alterations in the same species [START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF]. These findings indicate that SePM may cause limitations to physiological performance when aerobic metabolic demand increases, with consequences for activities such as swimming, feeding, reproducing, and diverse other components of population functioning. This hypothesis can be investigated with measurements of swimming capacity and associated aerobic metabolism [START_REF] Mckenzie | Complex physiological traits as biomarkers of the sub-lethal toxicological effects of pollutant exposure in fishes[END_REF][START_REF] Plaut | Critical swimming speed: its ecological relevance[END_REF][START_REF] Priede | Metabolic scope in fishes[END_REF].

Swimming capacity is a key physiological variables; that have direct ecological relevance because they define fish capacity to forage, evade predators, protect territory, and migrate, among many other activities that can influence survival and fitness [START_REF] Mckenzie | Complex physiological traits as biomarkers of the sub-lethal toxicological effects of pollutant exposure in fishes[END_REF][START_REF] Plaut | Critical swimming speed: its ecological relevance[END_REF][START_REF] Priede | Metabolic scope in fishes[END_REF]. At the same time, the associated aerobic metabolism indicates the resting metabolic rate, the minimum and maximum aerobic capacity, and the respective aerobic scope. Fish activities are energetically located inside its aerobic scope. A scope alteration just changes the overall animal capacity to perform. Therefore, that is directly correlated to its ecology; since aerobic scope is considered a parameter of animal performance in relation to environmental conditions, especially related to instantaneous energy-demanding activities [START_REF] Domenici | The Effect of Hypoxia on Fish Swimming Performance and Behaviour[END_REF][START_REF] Lucas | Effects of metals and persistent organic pollutants on the fitness and health of juveniles of the endangered european sturgeon Acipenser sturio Exposed to W1ater and sediments of the garonne and dordogne rivers[END_REF][START_REF] Milinkovitch | Effect of dispersed crude oil exposure upon the aerobic metabolic scope in juvenile golden grey mullet (Liza aurata)[END_REF].

Swimming performance and elements of aerobic metabolism have long been known to be sensitive to environmental factors and stressors and have been used extensively as biomarkers of the toxicological effects of pollutants [START_REF] Beamish | Swimming capacity[END_REF][START_REF] Brett | Swimming Performance of Sockeye Salmon ( Oncorhynchus nerka ) in relation to Fatigue Time and Temperature[END_REF][START_REF] Sprague | Measurement of pollutant toxicity to fish-III[END_REF].

We assessed the result of short-term (96h) exposure to an environmentally relevant level of SePM contamination on the swimming performance and aerobic metabolism of Nile tilapia. This tropical freshwater species provides a good model for ecotoxicology because its physiology is well-studied [START_REF] Solis | A comparative study of phagocytic activity and lymphoproliferative response in five varieties of tilapia Oreochromis spp[END_REF]. It has been introduced into many tropical and sub-tropical countries across the globe, where it provides important fisheries [START_REF] De Silva | Tilapias as alien aquatics in Asia and the Pacific: a review[END_REF][START_REF] Schofield | Survival, growth and reproduction of non-indigenous Nile tilapia, Oreochromis niloticus (Linnaeus 1758). I. Physiological capabilities in various temperatures and salinities[END_REF], and it inhabits the SePM-contaminated sites in the Vitória Bay area [START_REF] Gallão | The Impact of Sublethal Concentrations of Treated Leachate on Nile tilápia (Oreochromis niloticus)[END_REF]. Our approach allowed us to identify and assess sublethal effects of SePM on functional biomarkers that are relevant to animal performance. This allows projection of consequences for individuals and populations of an important food fish, with implications that are relevant both to environmental quality and human health.

Materials and Methods

Settleable Atmospheric Particulate Matter -SePM

SePM collection and analysis methods have been described previously (Soares et al., 2022). It is composed of a complex mixture of particles, especially emerging metal compounds with unpredictable new combinations of metals and metal nanoparticles; its composition varies in shape and size. [START_REF] Nordberg | Handbook on the Toxicology of Metals[END_REF][START_REF] Salgado | Informações gerais e ecotoxicológicas de material particulado[END_REF][START_REF] Soriano | Deposition of heavy metals from particulate settleable matter in soils of an industrialised area[END_REF]. The SePM used in this experiment was collected at Ilha do Boi (20º18'32"S, 40º16'33"O), near the industrial complex of Tubarão in Vitória City (5-8 g/m 2 of SePM each month over an area of 5,000 m 2 , Souza et al., 2021a). This location has an intensive influence of SePM emitted by this industrial complex, which is integrated by an iron pelleting industry and an iron and steel exportation harbor [START_REF] Santos | Source apportionment of settleable particles in an impacted urban and industrialized region in Brazil[END_REF]. The SePM presence in the water environment may be related to input from air and water run-off from hard-standing areas [START_REF] Mclusky | The estuarine ecosystem: ecology, threats and management[END_REF]. Recently, isotopic analyses demonstrated that the metals present in that SePM (emitted from the nearby industrial complex) were in fact the main source of aquatic contamination in those water bodies. [START_REF] Souza | Interrogating pollution sources in a mangrove food web using multiple stable isotopes[END_REF]. Industrial activity may vary seasonally, influencing SePM deposition [START_REF] Santos | Source apportionment of settleable particles in an impacted urban and industrialized region in Brazil[END_REF][START_REF] Souza | Nanoparticle transport and sequestration: Intracellular titanium dioxide nanoparticles in a neotropical fish[END_REF]. Therefore, we sampled the contaminant from a pool of raw SePM collected over a year in order to assure experimental repeatability [START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF][START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF].

Similar to previous experiments, we used 1 gL -1 of raw SePM mass to produce environmentally realistic contamination in the exposed group [START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF][START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF], this level of contamination is similar to that observed in estuarine waters close to Vitória Bay observed in environmental records (aquatic environment) and reported by the Institute of the Environment and Water Resources for the estuarine area near the city of Vitória, ES -Brazil (IEMA, 2021;[START_REF] Santos | Source apportionment of settleable particles in an impacted urban and industrialized region in Brazil[END_REF][START_REF] Souza | Matching metal pollution with bioavailability, bioaccumulation and biomarkers response in fish (Centropomus parallelus) resident in neotropical estuaries[END_REF]. [START_REF] Arrivabene | Effect of pollution by particulate iron on the morphoanatomy, histochemistry, and bioaccumulation of three mangrove plant species in Brazil[END_REF] reported SePM in the estuarine waters of Santa Cruz and Vitória Bay, which is related to air-water cross-contamination from metallurgical activity. [START_REF] Souza | Matching metal pollution with bioavailability, bioaccumulation and biomarkers response in fish (Centropomus parallelus) resident in neotropical estuaries[END_REF] established chemical characterization of metallic contaminants in these estuarine waters and observed significant amounts of metals and metalloids from SePM in surface waters, interstitial water, and sediments (Al, Cr, Mn, Fe, Ni, Cu, Zn, As, Se, Ag, Cd, Hg, and Pb -μgL -1 ). The raw SePM was diluted in the experimental water (see below). A sample of the SePM used for contamination was stored for any further analyses and evaluations. The Control group was not exposed to SePM.

We collected water samples from the experimental aquaria for quantitative and qualitative metal analyses. We used mass spectrometry with inductively coupled plasma (Q-ICPMS, Agilent 7500 Series CX technology) equipped with an Awe SX-100 autosampler (CETAC-technologies, Omaha, NE, USA) at the Food Science and Technology Institute of Córdoba (YCITAC -Universidad Nacional de Cordoba, Argentina).

Fish supply and acclimation

Juveniles of O. niloticus were provided by the Polettini fish farm (Mogi-Mirim -SP, Brazil) and transported to the Department of Physiological Sciences at UFSCar (São Carlos -SP), where they were acclimated in 1,000 L holding tanks during 30 d. Each tank had a continuous supply of dechlorinated, well-aerated water (25 ± 1°C; DO 6.0-7.3 mg.L -1 ; hardness 48-53 mg.L -1 ; pH 6.7-7.3, (as CaCO3); alkalinity 40-43 mg.L -1 as CaCO3; chloride 42 mg.L -1 ; ammonium 0.1 mg.L -1 ; and conductivity 110 µS.cm -1 ). Fish remained under natural photoperiod (~12 h:12 h) and were fed ad libitum with specific commercial pellets: Ø 5-8 mm, 6 % fat, 32 % protein and 12 % moisture (Supra ® Acqua line) during acclimation period. Food supply was withdrawn 24h before contamination protocol and experiments. The UFSCar Committee of Ethics in Animal Experimentation (CEUA-UFSCar) analyzed and approved all procedures.

Experimental Design

After acclimation, 20 fish (body mass, MB = 180 ± 98g, length = 17.9 ± 0.17 cm, width = 1.9 ± 0.02 cm, and height = 6.7 ± 0.09 cm) were divided into two groups of similar mass, which underwent the exposure protocol. For 96h, each fish was put in an individual aquarium (200 L) with clean water (Control, n = 10) or water with SePM (SePM, n = 10). Each aquarium was shielded with a black plastic sheet to avoid visual disturbance and respected the recommended OECD fish:water volume ratio (1g:1L, (OECD 203, 2019). After 48h, 50% of aquarium water was replaced with the relevant water. After 96 h, each fish underwent an incremental critical swimming speed (Ucrit) protocol with concomitant respirometry [START_REF] Brett | The Respiratory Metabolism and Swimming Performance of Young Sockeye Salmon[END_REF], in a Steffensen-type acrylic swimtunnel respirometer as described by [START_REF] Mckenzie | Physiological determinants of individual variation in sensitivity to an organophosphate pesticide in Nile tilapia Oreochromis niloticus[END_REF], see details below.

In the present protocol, fish swam in clean, biofiltered, and aerated water, having acclimated to the tunnel for 24h. Therefore, this would allow the SePM group to partially recover from exposure effects, if that were possible. To further investigate the potential for fish to recover from the effects of SePM, a second swimming trial was performed at 96 h after the first, with all animals held in uncontaminated water with the same individual aquarium exposure protocol (Rec group). This Rec test was only applied to the SePM group. It has previously been demonstrated that the swimming performance of Nile tilapia is constant over two sequential 96h periods in clean water [START_REF] Mckenzie | Physiological determinants of individual variation in sensitivity to an organophosphate pesticide in Nile tilapia Oreochromis niloticus[END_REF].

Swimming Performance

After exposure protocol, each fish was put into a bucket which was flushed with clean biofiltered water for 5 min. After that, the fish was transferred to the respirometry chamber of the swim tunnel (volume 13.4 L) and left to recover for 24h in aerated clean water at 25 ± 1°C. During recovery, they were kept swimming at 1 body length  s -1 (BLs - 1 ). In that gentle speed, fish could maintain body position using some pectoral sculling and gentle tail beats (see [START_REF] Mckenzie | Effects of growth hormone transgenesis on metabolic rate, exercise performance and hypoxia tolerance in tilapia hybrids[END_REF]. The anterior portion of the swim chamber was covered with opaque black plastic sheeting to avoid visual disturbance of the fish, which spontaneously occupied the covered area.

The swimming trial was based on the protocol of [START_REF] Mckenzie | Physiological determinants of individual variation in sensitivity to an organophosphate pesticide in Nile tilapia Oreochromis niloticus[END_REF]. Each fish underwent increasing levels of swimming speed. Each level increased 1 BLs -1 in swimming speed and had 30 min duration. After reaching 4 BLs -1 , increments were reduced to 0.5 BLs -1 until exhaustion (collapsing against the back of the swim chamber, [START_REF] Mckenzie | Physiological determinants of individual variation in sensitivity to an organophosphate pesticide in Nile tilapia Oreochromis niloticus[END_REF].

Swimming speeds were corrected for the blocking effect of each fish [START_REF] Bell | Water tunnel design for fisheries research[END_REF]. The second swimming trial (Rec group) was held with the same characteristics as the first. The Ucrit (BLs -1 ) was calculated by adding the velocity of the most recently completed swimming step to the product of the velocity incremental increase and the time proportion of swimming until fatigue at the last current velocity, as described by [START_REF] Brett | The Respiratory Metabolism and Swimming Performance of Young Sockeye Salmon[END_REF]:

(1) Ucrit = Us + (Uss  Tss)

where Us is the last completed velocity of the swimming step, Uss is the incremental increase in velocity, and Tss is the time proportion of swimming until fatigue at the last current velocity.

Swimming respirometry

During swimming, oxygen uptake was measured by intermittent closed respirometry in two cycles of 15 min at each speed (measuring time -9 minutes, waiting time -1 minute, and flushing time -5 minutes). An optical oxygen probe and the associated software (Fibox, GmbH and Oxyview, Pre-sens) allowed for continuous recordings of the oxygen tension in the respirometry chamber. The oxygen uptake rate was calculated at each swimming speed based on the gradient of decline in O2 during the closure period, at a sampling rate of once each 15 s. The rate of oxygen uptake (M ̇O2 -mMolO2  kg -1  h -1 ) was then calculated as

(2) M ̇O2 = [βO2 VRE  W0 -1  ( 𝛿𝑝O2 𝛿𝑡 )]
where βO2 denotes the coefficient for O2 solubility in water (mg of O2  L -1 ); VRE, the chamber volume (L); Wo -1 , fish body mass (Kg); and (

𝛿𝑝O2 𝛿𝑡

) indicates the O2 redution over time (% of O2  h -1 ). After each swimming protocol, blank tests were performed to measure rates of background microbial oxygen uptake, then the respirometer was thoroughly washed with clean water.

For each swim test, a least-squares exponential regression was applied to the relationship between swimming speed and the ṀO2 measurements. Extrapolation back to the y-intercept, a notional swimming speed of zero, was employed to estimate standard metabolic rate (SMR), the minimum rate of ATP use required to sustain life, in the absence of voluntary muscular movements and digestion/absorption of nutrients (Chabot et al., 2016). Maximum metabolic rate (MMR) was measured as the highest M ̇O2 achieved during the swimming trail, which is assumed to represent the maximum rate of oxygen uptake that an animal can achieve to create ATP aerobically [START_REF] Norin | Measurement and relevance of maximum metabolic rate in fishes[END_REF]. Aerobic metabolic scope (AS) was measured as:

(3) AS = MMR -SMR which estimates the maximum capacity to supply oxygen to sustain metabolic activities beyond SMR, activities such as digestion, locomotion, growth, reproduction, and so on [START_REF] Fry | The effect of environmental factors on the physiology of fish[END_REF]. We also assessed the net metabolic cost of swimming (COS -mMolO2  kg -1  h -1 at a sustained speed of 2 BLs -1 as:

(4) COS = M ̇O2 (at 2 BLs -1 ) -SMR an estimate of the metabolic cost of activity at low velocity [START_REF] Allen | Physiological insights for aquaculture diversification: Swimming capacity and efficiency, and metabolic scope for activity in cojinoba Seriolella violacea[END_REF].

The swimming efficiency (Eswim -µmolO2  Kg -1  BL -1 ) was evaluated as described by McKenzie et al. ( 2017):

(5) Eswim = COS2 -1 (speed of 2 BLs -1 ) Finally, we analyzed the maximum energetic efficiency of swimming performance (Emax -µmolO2  Kg -1  BL -1 ), a variable of significant ecological relevance [START_REF] Ohlberger | Swimming efficiency and the influence of morphology on swimming costs in fishes[END_REF], was calculated based on Kunz et al. ( 2018

Results

Settleable Atmospheric Particulate Matter

SePM contamination introduced several metal elements into the water. The main bioavailable components were iron and aluminum, followed by titanium and cerium (Table 1). Fe and Al represented composed approximately 97% of the total metals available in the contaminated water. However, despite their high concentration, both metals were below the reference LC50 for fishes (Fe -1.46-1.71 mg  L -1 and Al -0.095-235 mg  L -1 -Kennedy, 2011). Additionally, these metals were below the upper limit (Fe, 0.3-5.0; Al, 0.1-0.2 mg. L -1 ) indicated for protected freshwater bodies (for drinking supply, aquatic communities, aquaculture, and fishing, CONAMA, 2005).

SePM affected swimming performance and aerobic metabolic rates

The Control group had a mean Ucrit of 5.68 ± 0.21 BL  s -1 at 25 o C. This swimming performance was associated with maintenance of significant respiratory support for activity (Figure 1). SMR was 4.49 ± 0.32 mMolO2  kg -1  h -1 in the Control group. Activity increased metabolic rate so that MMR was 3.2 times higher at 14.58 ± 0.82 mMolO2  kg -1  h -1 , such that Control AS was 10.09 ± 0.61 mMolO2  kg -1  h -1 (Figure 2). The Control group had COS at 3.21 ± 0.51 mMolO2  kg -1  h -1 , Eswim of 1.58 ± 0.25 µmolO2  Kg -1  BL -1 , while Emax was 2.75 ± 0.12 µmolO2  Kg -1  BL -1 (Figure 3).

The SePM group showed impaired performance with a Ucrit of 4.15 ± 0.26 BL  s -1 , 27% lower than the Control fish (p < 0.001 -Figure 1). This reduction in performance was associated with effects of SePM on respiratory metabolism. SePM contamination reduced SMR by 26 % (3.32 ± 0.17 mMolO2  kg -1  h -1 , p = 0.001 -Figure 2A) and MMR by 27 % (10.51 ± 0.78 mMolO2  kg -1  h -1 , p = 0.002 -Figure 2B) when compared to the Control group. Additionally, SePM contamination decreased AS by 29 % compared to the control group (7.19 ± 0.77 mMolO2  kg -1  h -1 , p = 0.003 -Figure 2C).

SePM did not change the energetics of sustained swimming at low velocity, with values of COS of 3.72 ± 0.31 mMolO2  kg -1  h -1 (p = 0.333 -Figure 3A) and Eswim of 2.09 ± 0.23 µmolO2  Kg -1  BL -1 (p = 0.102 -Figure 3B) that were similar to the Control group. However, SePM contamination did alter Emax, which was 11% lower in the SePM group compared to the Control, at 2.46 ± 0.07 µmolO2  Kg -1  BL -1 (p = 0.011 -Figure 3C).

Second swimming trial

The second swimming trial provided no evidence of recovery in swimming performance or metabolic rates (Table 2). Swimming performance was 18% further reduced in the Rec group, with a Ucrit of 3.41 ± 0.17 (Table 2). All other variables remained stable when compared to the first swimming trial. Although all 10 fish in the SePM group survived the 96h contamination and the first swim test, 3 of these fish did not survive the additional 96h recovery period before the second swim test. All fish that underwent the second swim test survived to the end of the experimental protocol.

Discussion

Our experimental approach used a functional test to address the effects of sublethal levels of contamination on a widely-studied fish species. An environmentally relevant concentration of SePM limited the swimming performance of O. niloticus, an effect that was associated with substantial alterations in respiratory metabolism. This is clear evidence of effects on functional integrity, an impairment that indicates significant limitations on respiratory metabolism that may reduce fish fitness due to direct and indirect impacts on mobility and energy supply.

SePM and general metal toxicology

Our exposure protocol used an environmentally relevant level of SePM contamination and was able to replicate the presence and concentration of bioavailable metals as reported in Santa Cruz and Vitoria Bay (IEMA, 2021;Souza et al., 2021a;Souza et al., 2021;[START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF][START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF]. The analysis of SePM confirmed that the contamination is highly complex in metal composition. The observed range of metals corroborates the results of previous studies [START_REF] Galvão | The mineralogical composition of coarse and fine particulate material, their fate, and sources in an industrialized region of southeastern Brazil[END_REF][START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF]Souza et al., 2021a;Souza et al., 2021). Metals have a broad range of impacts on fish physiology, including oxidative stress, genotoxicity, hematological alterations, histopathological alterations in many organs, and problems in diverse physiological systems [START_REF] Lushchak | Environmentally induced oxidative stress in aquatic animals[END_REF]Pyle et al., 2002;Taylor et al., 1996;Giles, 1984). Moreover, previous analyses also described the presence of different-sized particles, which may vary in shape and composition. Furthermore, part of these are in nanometric scale [START_REF] Salgado | Informações gerais e ecotoxicológicas de material particulado[END_REF][START_REF] Santos | Source apportionment of settleable particles in an impacted urban and industrialized region in Brazil[END_REF]Souza et al., 2021b).

Therefore, the bioavailable metals in experimental water comprise a complex and unpredictable contaminant due to the range of different elements, possible combinations of metallic/metalloid components, and the presence of metallic nanoparticles [START_REF] Salgado | Informações gerais e ecotoxicológicas de material particulado[END_REF][START_REF] Souza | Interrogating pollution sources in a mangrove food web using multiple stable isotopes[END_REF]2021a;2021b;[START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF]. This makes it difficult to identify the main components responsible for functional alterations observed in tilapia.

Presently, using raw SePM in experimental protocols is the only way to replicate such environmental contamination and study its impacts [START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF][START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF]. Although the precise mechanisms underlying the SePM effect have yet to be revealed, the observed limits to performance are of obvious interest since the level of individual metals was below their lethal concentration [START_REF] Kennedy | The Toxicology of Metals in Fishes, Encyclopedia of Fish Physiology: From Genome to Environment[END_REF] and under the maximum limits set by the Brazilian regulatory agency [START_REF] Conama | Conselho Nacional do Meio Ambiente -CONAMA -Resolução n 357[END_REF]. It seems that we are dealing with an emerging problem that might be highly ecologically relevant.

SePM exposure limits swimming performance

Short-term swimming performance tests have generated valuable data to evaluate fish health after physiological challenges, such as infections [START_REF] Tierney | The relationships between fish health, metabolic rate, swimming performance and recovery in return-run sockeye salmon, Oncorhynchus nerka (Walbaum)[END_REF] and sublethal toxicant exposures [START_REF] Beaumont | Plasma ammonia concentration in brown trout in soft acidic water and its relationship to decreased swimming performance[END_REF][START_REF] Mckenzie | Complex physiological traits as biomarkers of the sub-lethal toxicological effects of pollutant exposure in fishes[END_REF].

Swimming performance has been used to identify physiological damage caused by diverse dissolved metals in salmonids, such as copper [START_REF] Waiwood | Effects of copper, pH and hardness on the critical swimming performance of rainbow trout (Salmo gairdneri Richardson)[END_REF], zinc [START_REF] Alsop | The effects of temperature and swimming speed on instantaneous fuel use and nitrogenous waste excretion of the Nile tilapia[END_REF], and nickel [START_REF] Pane | The physiological consequences of exposure to chronic, sublethal waterborne nickel in rainbow trout ( Oncorhynchus mykiss ): exercise vs resting physiology[END_REF]. This was the first investigation that used fish swimming performance to evaluate the effect of SePM.

Previous experiments using a similar swimming protocol at the same temperature in Nile tilapia reported Ucrit values similar to our Control group [START_REF] Mckenzie | Physiological determinants of individual variation in sensitivity to an organophosphate pesticide in Nile tilapia Oreochromis niloticus[END_REF], confirming the consistent quality of the fish population used and the utility of the swimming trials. Consequently, the reduced values of Ucrit and Emax in SePMcontaminated fish identify clear impairment of swimming performance and metabolic effectiveness.

It is parsimonious to attribute the Ucrit reduction to the associated changes in aerobic metabolism. Functional cardiorespiratory integrity is necessary to support a high level of aerobic swimming [START_REF] Mckenzie | Swimming and other activities | Energetics of Fish Swimming[END_REF]. Therefore, a break in the integrity of mechanisms for oxygen uptake and delivery may account for the reduction in swimming capacity. The observed decrease in Emax supports such an argument.

Our experimental approach allowed us to reveal that SePM limited the ability to swim, which has obvious ecological relevance. Swimming relates to the capacity to perform essential behavioral tasks for interactions with the environment, such as maintaining a position against currents, foraging, migrating, reproducing, and many others. The evidence that swimming capacity was limited by impaired metabolic support is also relevant because it indicates that other aerobically demanding activities would be similarly impacted.

SePM exposure affected aerobic metabolism

Metabolic measures provide a significant description of energy mobilization in individual animals [START_REF] Rosewarne | Measuring maximum and standard metabolic rates using intermittent-flow respirometry: A student laboratory investigation of aerobic metabolic scope and environmental hypoxia in aquatic breathers[END_REF], which is considered a direct reflection of the functional integrity of an individual and its potential to operate in nature [START_REF] Rice | Bioenergetics modelling approaches to evaluation[END_REF].

The assessment of aerobic profile has been reported as ecologically relevant as a potential bioindicator of metabolic performance, growth rate, and survival [START_REF] Killen | The intraspecific scaling of metabolic rate with body mass in fishes depends on lifestyle and temperature[END_REF] as they bring a unifying currency of animal performance [START_REF] Wikelski | The physiology of life histories[END_REF]. The swimming performance protocol allowed for calculating robust metabolic indexes regarding the fish metabolic profile and revealed aspects of the energy used for basal maintenance (SMR) and the relative cost of the activity under low (COS, Eswim) and high demand reaching up its limit and the related increase scope (MMR, AS, and Emax).

Control tilapia had a metabolic profile very similar to previous reports for the species. For instance, using a similar swimming technique at the same temperature, [START_REF] Mckenzie | Physiological determinants of individual variation in sensitivity to an organophosphate pesticide in Nile tilapia Oreochromis niloticus[END_REF] reported similar mean values for SMR, MMR, and AS (respectively, 4.1 ± 0.5, 16.8 ± 1.3, 12.6 ± 0.1 mMolO2  Kg -1  h -1 ) in the same strain of Nile tilapia.

The 96h exposure to SePM clearly limited both the total energy used and the efficiency of the use of this energy. The exposed fish had reduced standard metabolic rates. Although they were capable of matching the aerobic demands of low levels of sustained swimming (COS and Eswim), SePM-contaminated fish could not achieve the same metabolic rate to support high swimming speeds. Therefore, the overall aerobic metabolic phenotype was affected, with a reduction in basal metabolic processes and an impaired ability to meet the increased metabolic rate required to support intense activity such as swimming at high speeds.

Xenobiotics are reported to cause apparently conflicting metabolic effects, such as: metabolic limitations and metabolic masking [START_REF] Hammer | Fatigue and exercise tests with fish[END_REF]. The first condition would impair both metabolite supply and removal. Therefore, it would limit the SMR and would have an additional contribution to the reduction in the energetic efficiency of swimming and the critical speed. Additionally, it would reduce the maximal energy production capacity and aerobic scope. Even at sublethal level, metals have been reported to induce alterations with an ultimate metabolic consequence and long-term effects such as reduction in development, growth, and reproduction [START_REF] Lucas | Effects of metals and persistent organic pollutants on the fitness and health of juveniles of the endangered european sturgeon Acipenser sturio Exposed to W1ater and sediments of the garonne and dordogne rivers[END_REF]. For instance, a low level of Cu and Cd was observed to reduce the metabolic rate in the freshwater fish species Pimephales promelas [START_REF] Pistole | Influence of metal concentrations, percent salinity, and length of exposure on the metabolic rate of fathead minnows (Pimephales promelas)[END_REF]. Zn also reduced the metabolic activity in Rhamdia quelen [START_REF] Leitemperger | The bioaccumulation of waterborne zinc in tissues of silver catfish (Rhamdia quelen) and its effect on biochemical parameters[END_REF].

Previous experiments reported that SePM contamination has a deleterious effect on both capacities for oxygen uptake and delivery in the tilapia [START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF][START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF]. That was also associated with a compensatory ventilatory increase and ended up reducing the tolerance to hypoxia [START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF]. Such effects are coherent with the current findings of reduced swimming performance and alterations in aerobic metabolism. The same authors reported that tilapia could sustain its routine metabolic rate (RMR) after a similar SePM contamination [START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF]. We suggest that the combination of these factors (maintenance of RMR while reducing SMR with high ventilation cost) indicates problems with energy allocation. A larger part of the total energy would be used for ventilation, while other processes potentially have limited energy support. That interpretation corroborates the observed energetic limitation of swimming. A higher ventilatory effort is expected to increase the energy used for active ventilation at each swimming level [START_REF] Freadman | Swimming Energetics of Striped Bass (Morone Saxatilis) and Bluefish (Pomatomus Saltatrix): Hydrodynamic Correlates of Locomotion and Gill Ventilation[END_REF]. Therefore, such an energy reallocation was revealed as inefficient energy use (i.e., low Emax), lower MMR and AS when the fish were forced to increase their metabolism for swimming.

Additionally, AS has been proposed as an index of animal performance in relation to environmental conditions, since it is related to instantaneous energy-demanding activities and may demonstrate the physiological consequences of several environmental constraints such as temperature increase, hypoxia, and pollutants [START_REF] Domenici | The Effect of Hypoxia on Fish Swimming Performance and Behaviour[END_REF][START_REF] Lucas | Effects of metals and persistent organic pollutants on the fitness and health of juveniles of the endangered european sturgeon Acipenser sturio Exposed to W1ater and sediments of the garonne and dordogne rivers[END_REF][START_REF] Milinkovitch | Effect of dispersed crude oil exposure upon the aerobic metabolic scope in juvenile golden grey mullet (Liza aurata)[END_REF]. Therefore, the observed reduction in AS on exposure to SePM may restrain a range of relevant energetically demanding activities such as growth, reproduction, and offspring size [START_REF] Claireaux | Linking environmental variability and fish performance: integration through the concept of scope for activity[END_REF]. Furthermore, the capacity to support increased energy demand levels is also supposed to impact recovery after exhausting activity. Part of the SePM group (30%) died during the additional recovery period after the first swimming trial. This may be related to SePM negative effects previously reported; SePM caused significant damage to the gill structure, aerobic capacity, oxygen-carrying capacity of the blood, and energy allocation in Nile tilapia [START_REF] De Angelis | Settleable atmospheric particulate matter affects cardiorespiratory responses to hypoxia in Nile tilapia (Oreochromis niloticus)[END_REF][START_REF] Soares | Settleable atmospheric particulate matter induces stress and affects the oxygencarrying capacity and innate immunity in Nile tilapia (Oreochromis niloticus)[END_REF], impairing O2 uptake from the water, the metabolism and swimming performance as observed in the present experiment.

MMR can be associated with intrinsic and extrinsic factors [START_REF] Norin | Measurement and relevance of maximum metabolic rate in fishes[END_REF]. Thus, the maximum swimming activity may cause a transitory debilitation in every vertebrate since it depletes cellular energy reserves and pH alteration due to lactic acid produced, and others. It also causes ionic unbalance in fish associated with the increased ventilatory effort. Post-activity recovery might cause long-term demands for an aerobic effort to sustain the biochemical effort to recover osmolarity, pH, and energetic balance.

Apparently, SePM-contaminated fish was unable to recover as fast as control fish.

Therefore, these casualties reinforce the apparent severity of this type of pollution and suggest that if such limitations are not compensated in the long term, SePM will impact a fish's overall use of energy. Consequently, it may have effects on development, growth, reproduction, post-prandial metabolic increment, and any other metabolically demanding activities [START_REF] Mckenzie | Complex physiological traits as biomarkers of the sub-lethal toxicological effects of pollutant exposure in fishes[END_REF][START_REF] Killen | The intraspecific scaling of metabolic rate with body mass in fishes depends on lifestyle and temperature[END_REF][START_REF] Plaut | Critical swimming speed: its ecological relevance[END_REF][START_REF] Priede | Metabolic scope in fishes[END_REF][START_REF] Rice | Bioenergetics modelling approaches to evaluation[END_REF] with potential consequent reductions in the overall fitness of the fish population and ecological impact.

Final Conclusions

An environmentally relevant concentration level of SePM significantly limited the swimming performance of O. niloticus and caused significant reductions in metabolic indexes, SMR, MMR, Emax, and AS. Although exposure to environmentally relevant levels of SePM did not lead to immediate death, it did induce significant limitations on physiological performance. Such a combination between swimming and metabolic limitations may affect activities that support fish population dynamics. SePM has the potential to affect multiple levels of fish performance, and therefore, it is still a rather unexplored toxic component that may lead to a growing and silent threat to aquatic biota.

It does require further study to understand toxicity thresholds, the mechanisms of toxicity to several organisms, and forms of risk assessment. Future work should also address the effect of long-term contamination by SePM over a range of natural physicochemical variations such as salinity and temperature.

  ): (6) Emax = MMR  Ucrit -1 Statistical analyses All data were analyzed for normality and homogeneity (Shapiro-Wilk and Levene). Differences between experimental groups (Control and SePM) were analyzed by Student's T-test. Recovery was tested by comparing the mean values of SePM (first swimming test) and Rec (second swimming test) with paired Student's T-test. All data were analyzed with R v.4.1.1 (R Core Team 2021). We assumed the significance level of 95% (P<0.05). The relevant data are presented as means  SE.

Acknowledgments

The authors thank Piscicultura Polettini for donating the fish for the present study, Gustavo Marega Oda (DCF) for his valuable assistance in animal care and experiment performing, the NGO "JUNTOS SOS -ES" for collecting the SePM, and the Brazilian Agricultural Research Corporation (EMBRAPA) for the analysis provided.

Financial support