Exoplanets detection limits using spectral cross-correlation with spectro-imaging. An analytical model applied to the case of ELT-HARMONI

The combination of high-contrast imaging and medium to high spectral resolution spectroscopy offers new possibilities for the detection and characterization of exoplanets. The molecular mapping technique uses the difference between the planetary and stellar spectra. While traditional post-processing techniques are quickly limited by speckle noise at short angular separation, it efficiently suppresses speckles. Its performance depends on multiple parameters such as the star magnitude, the adaptive optics residual halo, the companion spectrum, the telluric absorption, as well as the telescope and instrument properties. Exploring this parameter space through end-to-end simulations to predict potential science cases and to optimize future instrument designs is very time-consuming, making it difficult to draw conclusions. We propose to define an efficient methodology for such an analysis. Explicit expressions of the estimates of signal and noise are derived, and they are validated through comparisons with end-to-end simulations. They provide an understanding of the instrumental dependencies, and help to discuss optimal instrumental choices with regard to the targets of interest. They are applied in the case of ELT/HARMONI, as a tool to predict the contrast performance in various observational cases. We confirm the potential of molecular mapping for high-contrast detections, especially for cool planets at short separations. We provide guidelines based on quantified estimates for design trade-offs of future instruments. We discuss the planet detection performances of HARMONI observing modes. While they nicely cover the appropriate requirements for high detection capability of warm exoplanets, a transmission extended down to J band would be beneficial. A contrast of a few 1E-7 at 50mas should be within reach on bright targets in photon noise regime with molecular mapping.


Introduction 1
The spectral information that is carried by the light reflected off 2 or emitted by an exoplanet is of primary importance for studies 3 of its nature.It makes it possible to probe the physical properties 4 and the composition of the atmosphere; this is a proxy for in-5 ferring the formation and evolution processes (Oberg & Bergin 6 2016; Mollière et al. 2022) and a key piece of information in 7 assessing their habitability (Turbet 2018), as well as the poten-8 tial existence of life (Wang et al. 2018).The brightness of the 9 host star and the huge star-to-planet flux ratio induce noise and 10 systematics on the planet's signature, which are two major chal-11 lenges to these observations.12 Such a detection and characterization of the exoplanetary 13 light can be attempted with or without angularly resolving the 14 planet itself, and, in each case, considering the total, broadband Starting from the spatially unresolved case, in the favorable configuration of transiting planets, the planetary signal can be extracted thanks to temporal differential spectro-photometry (Tinetti et al. 2018), namely, by observing the planet before, during, and after the transit.The performance of this technique is ultimately limited by systematics related to measurement stability and by the limited-time window associated with transits (once every orbit).
The spectral diversity between the star and the planet signal, and/or the use of specific spectral signatures can be used to get rid of these instrumental systematics to the point that even non-transiting planets can be detected and characterized without spatially resolving them.This has enabled the detection and characterization of β-Pic b observed with VLT/CRIRES (Snellen et al. 2014), for instance.The planetary signal detection remains strongly limited by the photon noise associated with the huge stellar flux.reduced if the star's diffracted signal at the location of the planet is attenuated using coronagraphic techniques.As for unresolved characterizations, differential signal extraction can be performed without using any prior knowledge of the exoplanet spectrum.
The key limitation is associated with systematics on images (in particular, the speckle noise), in spite of various studies and substantial efforts that have been made to handle it (e.g., improved AO correction methods, novel focal plane sensing techniques, improved signal extraction algorithms using data diversity).One way to get rid of the confusion between the speckles in the stellar halo and the planetary signal is the interferometric coherence difference, as used, in particular, in the ExoGRAVITY program (Lacour et al. 2020); this approach is telescope-time expensive, however, since it requires the four UTs of the VLT.In addition, it has a low global transmission.As said before, another way to deal with these systematics is to use the spectral information to look for specific spectral features, over the field of view.This technique referred to as molecular mapping.This requires a high enough spectral resolution (which is discussed below) to distinguish the absorption features of interest from the speckle-induced spectrum modulation.The integral field spectrographs (IFS) used with high-contrast imaging instruments such as VLT/SPHERE (Beuzit et al. 2019), GPI (Macintosh et al. 2014), and Subaru/SCExAO (Lozi et al. 2018) focus on spatial information and have a limited spectral resolution (R=50).However, molecular mapping has also recently been demonstrated with higher resolution IFS and spectrographs that were not originally dedicated to high-contrast, for instance: with VLT/SINFONI (R=5000) (Hoeijmakers et al. 2018), Keck/OSIRIS (R=4000) (Ruffio et al. 2019), and VLT/MUSE (R=3450) (Haffert et al. 2019).
There is a plentiful parameter space to explore when following this approach, with various balances and trade-offs between the spectral information content, noise regimes, and observation context (ground vs. space), as well as star and exoplanet types.
We should emphasize the fact that, in practice, when designing an instrument, the main design choices usually favor some capabilities at the expense of others.A classical example is, for instance, the necessary trade-off for an IFS with a given detector size, between its field-of-view (FoV), spectral bandwidth, and spectral resolution.There is certainly not a single, universal sweet spot and the choice depends on various parameters that include the telescope specifications, image quality, stellar type and brightness, total observing time, exoplanet contrast, projected separation, spectral properties, and so on, with the chosen priority given to a given spectral signature.Some IFS using this high spectral resolution approach are already operational, but are not necessarily equipped with the optimal observing modes for the detection of exoplanets (VLT/ERIS (Kravchenko et al. 2022), VLT/MUSE, Keck/OSIRIS, JWST/MIRI (Patapis et al. 2022;Mâlin et al. 2023)).Others are currently being designed with these trade-off considerations in mind, such as SPHERE+ MedRES (Boccaletti et al. 2022), ELT/HARMONI, ELT/METIS (Carlomagno et al. 2020), or are planned in the more distant future (ELT/PCS) (Kasper et al. 2021).Such discussions on the instrument trade-offs and the different high-resolution instrument modes have already been addressed 103 in some specific cases, such as for space-based, very-high-104 contrast instruments (Wang et al. 2017) or when coupling 105 VLT/SPHERE with CRIRES+ (Otten et al. 2021).This will 106 need to be extended, and to cover a variety of future instruments.107 108 We argue that a new performance analysis approach is 109 needed to explore this parameter space and we propose the use 110 of a semi-analytical tool that we have developed.Until now, 111 a huge effort has been made to develop end-to-end simulation 112 tools to estimate the performance of various high-contrast im-113 agers; these simulations are necessary to model and to estimate 114 the impact of complex effects (e.g., speckle noise, instrumental 115 instabilities, or correlations generated by the detector) on the fi-116 nal achievable contrast, with various post-processing techniques.117 A strong drawback of this computationally heavy approach is 118 the difficulty to explore wide parameter spaces and to fully trace 119 and understand the role of each of the various parameters that 120 contribute to the performance budget, however.The approach 121 presented here proposes an answer to this difficulty.
We first recall in Section 2 the principles of molecular map-124 ping and how different types of noise limit its efficiency.We 125 verified our analytical model with comparisons with end-to-end 126 simulations in Section 3. Section 4 details the bandwidth ver-127 sus spectral resolution trade-off, and how tellurics, and rotational 128 broadening may influence it.We then consider in Section 5 the 129 specific case of the ELT/HARMONI instrument to first compare 130 the performance derived through an end-to-end simulation tool 131 and the one derived with our semi-analytical tool.We then use 132 the latter to discuss the interest of the different observing modes 133 of this instrument with respect to different types of planets.Fi-134 nally, a conclusion is drawn in Section 6.We consider the observational information of spectro-imagers, 138 after the first steps of data reduction and calibration in the form 139 of a 3D collected flux, S , expressed as the measured integrated 140 flux as a function of the wavelength, λ, and spaxel position (x,y).141 It can be decomposed into several components when the ob-142 served scene is a stellar halo as in Eq.1: (2) with γ atm the atmosphere transmission, S star as the star spec-144 trum, S planet as the planet PSF as a function of wavelength, and 145 n as the noise.The sky background spectrum does not appear 146 here, as we assume that it has been perfectly subtracted from 147 the data.The total noise n is further broken down in Eq. 2 into 148 several standard noise sources: the stellar photon noise, n halo , 149 the background noise, n bkgd , which is the photon noise com-150 ing from the sky emission, the telescope, and the instrument, 151 and the readout noise of the detector, n RON .Also, M speckle de-152 scribes the stellar PSF over the FoV and depends on the wave-153 length and it is normalized for each spectral channel such that 154 M(λ, x, y)dxdy = 1.Spectra quantities (S star , S planet , and con-155 sequently also S ) are expressed in detected photo-electrons, in-156 tegrated over the total integration time for each spectral chan-157 nel.Therefore, those quantities consider the intrinsic star and 158 (5) Mspeckle (λ, x, y) = M speckle (λ, x, y) + S planet (λ, x, y) Ŝ star (λ) LF + n(λ, x, y) γatm (λ).Ŝ star (λ) LF  .
We note that Eq. 5 should be an approximation in real life, since the model of the spectrum of the star and telluric absorptions cannot be perfect.Then, the modulated stellar spectrum must be removed from the spaxels.We note that the planet's continuum will be removed, as the estimated modulation Mspeckle (λ, x, y) will take it into account.Here, S res (λ, x, y), the residual spectrally high-pass filtered signal after stellar component subtraction is expressed as: where In practice, n (λ, x, y) is approximately equal to n(λ, x, y) and we will use this approximation in the rest of the paper.The detailed calculation and discussion can be found in Appendix B.
With these operations, the high-frequency content of the planet (and possibly from the star, if it has a rich absorption lines content) is isolated from the speckles at the location of the planet.We note that the planet signal is still affected by telluric lines due to the atmospheric absorption.Now that the planet is spectrally disentangled from the speckle contamination, molecular mapping (Snellen et al. 2015;Hoeijmakers et al. 2018) consists in cross-correlating, in the spectral dimension, the resulting signal S res (λ, x, y) in each spaxel (x,y) from the IFS datacube with templates (see Fig. 1) computed in regard to various companion properties (temperature, C/O ratio, metallicity, etc.).When cross-correlating a template with a spaxel corresponding to a companion, a correlation peak can be observed: its position corresponds to the radial velocity and its width can be interpreted as being induced by the spin of the companion (Snellen et al. (2014)).
The spectral template, t, used for cross-correlation is userdefined, according to the type of planet atmosphere that is considered.It is expected that various templates will be tested on observation data and they will come either from empirical libraries or from theoretical models.A specific molecular spectrum can also be used to probe the presence of a molecule in the observed atmospheres (Hoeijmakers et al. 2018;Ruffio et al. 2019;Wang et al. 2021;Cugno et al. 2021).Therefore, at a given radial velocity, the cross-correlation is just the scalar product of the residual signal, S res , and the Doppler-shifted and normalized template, t RV , such that λ max λ i =λ min t 2 RV (λ i ) = 1.This operation is described in Eq.7: where v RV is the radial velocity, and A, B = A B cos(θ) denotes the scalar product of A and B, and θ is the angle between A and B.
The expression of the correlation signal of interest is derived using Eq.6 with the approximate expression of the propagated noise n(λ, x, y) and the definition of the scalar product: where θ planet,RV and θ star,RV are the angle between the high- where N λ is the number of spectral channels sampling the spectra, and the Fourier conjugate variable is associated with the spectral resolution, R, because here it is homogeneous to 1/λ.
Using the representation on the Fourier domain, Fig. 2 presents the PSD of the high-pass filtered spectrum for a companion with two different spectral resolutions, while keeping the same bandwidth (from 1 µm to 3 µm) and the same amount of photons.The area under the PSD curves is then equal to α 2 .In our study, this visualization allows us to better identify the spectral content location, and to directly compare it with the capabilities of an instrument with any given spectral resolution.Increasing the resolution expands the Fourier spectral range, and thus increases the signal of interest α.Indeed, thin absorption lines in companion spectra induce a very high-resolution content (even if the majority of the information seems at first glance to be found between a resolution of 100 and 10000).Having such a high resolution thus facilitates companion detection.In the same way, these representations in Fourier space allow us to visualize (in Fig. C.1) alpha dependencies as a function of planet temperature and spectral range.
Therefore, this factor α quantifies the spectral richness of the companion spectrum and is homogeneous to photons.Analogously to the use of ADI techniques, where the signal of interest is the total number of photons coming from the planet, the signal of interest for molecular mapping lies in this factor.We note that unlike the case of ADI, where the signal remains complete, because we are filtering out the low frequencies here, we sacrifice about 60% to 90% of the planetary signal.We can define the fraction of the useful high frequency planetary spectrum over the total spectrum as described in Eq.10.This quantity, as a function of the maximum resolution is expressed as follows (and represented in Fig. 3): In other words, this ratio quantifies the cost of molecular mapping in terms of signal to be detected; for instance, with respect to ADI, it is used to get rid of the speckle noise limitation.
As seen above, α can vary significantly depending on the planet temperatures (we considered a 500-1700K range in this study), as the high frequency spectral content is not the same.It can be directly estimated with spectra models, with the resolution and magnitude of the planet as inputs, and it does not require any end-to-end simulations.It depends on the companion spectrum, resolution, bandwidth, and exposure time.We note that the cosine factor is not estimated here and we assume it to be equal to 1 hereafter.However, this could be a limitation of molecular mapping in real life since we have very few data to check the similarity between templates and real spectra.We should always keep in mind that differences likely exist between the template and the signal, and that the detection estimates that are derived when using identical information for the signal and the template are therefore computed in the most optimistic scenario.
However, not all of this information may be usable for detection if the host star has lines in common with the planet.Indeed, subtracting the stellar spectrum will subtract those common lines as well.The term beta quantifies this effect, as it expresses the projection of the star's lines onto the template.Figure 4 shows the proportion of this self-subtraction for M-and A-host stars, and as a function of the planet type.M-stars and T planets have many CO lines, which explains the 30% subtraction factor in K band with a 1700K planet.In contrast, A-stars have much less content in common with the companion spectra and the subtrac-tive term is almost zero.We note that beta can be negative if the 338 cosine between the template and the stellar spectrum is negative, 339 meaning that the spectra have a negative correlation.This ex-340 plains the negative values in the figure.This negative correlation 341 is very weak, however, as it has no strong physical origin.
342 Also, various effects may induce some structure in the CCF 343 as a function of radial velocity.The first effect is the intrinsic 344 structure of the template autocorrelation.Radial velocity depen-345 dencies on the β term is a secondary potential sources, even 346 though it is minor in most of the cases.These facts are the pri-347 mary motivation to assess the detection levels in the spatial di-348 mension rather than in the radial velocity dimension.However, 349 these effects can still affect the position of the peak as well as its 350 shape.Therefore, one should take caution when deriving refined 351 estimates of the companion orbital velocity or spin velocity.

353
The signal of interest, as quantified above, now has to be com-354 pared to the level of the noise term (σ 2 CCF ), which is the variance 355  ).We notice that the colder the planet, the higher content of molecular absorption lines and the larger fraction of signal preserved at high resolution.
of the projected noise onto the template n(λ, x, y), t RV in Eq. 8.
It corresponds to the expected amount of noise in the correlation map (see Fig 1).It is paramount to estimating this variance in order to later derive the analytical expression of the signal-to-noise ratio (S/N) and qualifying a detection with a sufficiently high confidence level.Assuming that the PSF is centrosymmetric, the noise, n(λ, x, y), will only depend on λ and on the separation, ρ, as the star photon noise is the only source of variable noise in the FoV (Eq.2).Therefore, σ 2 CCF will depend on the separation, ρ, and it can be decomposed with each noise contributor of Eq. 2.
We also make the assumption that except for the speckle noise, the noise is uncorrelated at the pixel scale.
noted as σ 2 halo,CCF (ρ) for the projected photon noise from the stellar halo at the separation, ρ, while σ 2 bkgd,CCF is for the projected photon noise from the sky and instrumental background, respec-371 tively.
where (x 0 , y 0 ) is the location of the planet and ρ 0 is the star-378 planet separation.Injecting the definition of the projected noise 379 given in Eq.11 into Eq.12, the S/N becomes: With the signal of the planet being spatially distributed on 381 several pixels, we can then integrate on a spatial box that is a 382 few pixels wide (Eq.14) to optimize the S/N as done with a 383 matching template technique (Ruffio et al. (2019)): where w is the width of the box.

385
Assuming here that we cross-correlate planet spectra with a 386 perfect template, the cosine would equal to 1 and we can use Eq.13 to derive the expression of the highest contrast that leads 388 to a 5σ detection, as a function of separation, and for a given 389 exposure time: where ρ is the separation, N exp is the number of exposures,

391
A FWHM is the area of the FWHM in pixels, and α 0 is the com-  we assumed that these modulations were on a finite frequency support, and that low-pass filtering could effectively separate the modulations from the high-frequency content of the companion.However, if the frequency support of the modulations is not limited and/or a filter with a poor cut-off resolution is applied, residual modulations would result in another stellar residue (which would vary in the FoV this time) and add a new source of noise in Eq.11.It is therefore important to check whether the assumption is valid by analyzing the PSD of the speckles modulation.This will also allow us to discuss the necessary cut-off resolution.
Figure 5 shows the PSD of a speckle modulation taken at several angular separations, the white noise from photon and read-out noise, as well as from the planet.For illustration purposes, we used the speckles derived from the system analysis of HARMONI, which are based on an end-to-end light propagation diffractive model of the instrument that reproduces their chromatic evolution.This reflects what we would expect to have with a good AO system, and partially calibrated non-common path aberrations (Carlotti et al. (2022b), Jocou et al. (2022)).We do not discuss here the relative levels between the curves, which strongly depend on the observation case, but we are here interested in the PSD shapes.The speckle modulation PSD has a large peak at low spectral resolution and often dominates the others.This is why the speckle modulation has to be removed with a high-pass filter with a chosen cut-off frequency.In the same figure, the PSD of the speckle modulation still shows a high-frequency tail.This is because modulation of speckles is observed on a finite bandwidth window.The Fourier transform of this window is responsible for this PSD tail (i.e., the PSD of the speckle is convoluted by a sinc function corresponding to the Fourier transform of the rectangular function) dominating the intrinsic level of the speckle modulation.
To better visualize the frequency content, we propose to study the residual fraction of the speckle modulation and of a planet (T=1700K) as a function of the cut-off frequency of the high-pass filter (Fig. 6).To do this, we can use Eq.10, this time fixing the resolution but varying the cutoff frequency of the highpass filter.We notice that the fraction of residual speckle modulation increases with separation.This is well understood from the fact that the radial expansion rate of the speckles is proportional to the separation and thus makes the modulations faster at larger separations.We consider that if this parameter was critical, different cut-off frequencies could therefore be chosen depending on the separation.We note that on every curve, regardless of the separation, the main part of the speckle modulation is removed with a frequency cut corresponding to a spectral resolution of 100.This explains that on IFS like SPHERE and GPI (Macintosh et al. ( 2014)), a high-pass spectral filter would be inefficient to disentangle the speckles modulation from the planet features.However, having a spectral resolution greater than a few hundreds makes it feasible.It can also be seen that the planet spectra do indeed have a frequency content over a different frequency domain than the speckle modulations, as the planet PSD ratio decreases slowly as a function of the cut-off resolution.
Thus, frequency cut can be tuned up to optimize the S/N.An overly high frequency cut would remove almost totally the speckles, but also the features of the planet, and thus degrade the S/N.On the contrary, an overly low frequency cut would better preserve the planet, but also the speckle noise, degrading the S/N as well.
Studies of the capability of removing the impact of speckle noise as a function of the cut-off frequency can be done empirically through the test of fake companion detection in a simulated data cube.Under the same astronomical conditions as in Fig. 5, in Fig. 7 we indicate an optimal cut-off frequency slightly above 50.As expected, above a cut-off resolution of 100, speckles are almost entirely removed but the remaining planetary signal to be detected is slowly reduced.On the other hand, when the resolution cut-off is under 30, the influence of speckles is present, which degrades the S/N.For the following, we consider the case of a R c = 100 cut-off as a reasonable assumption.A second reason to choose a high enough cut-off resolution is the noise statistics.Indeed, every noise component is Gaussian except for the speckle noise, so if the speckles are efficiently removed, the noise would be essentially Gaussian, and we could apply the 5σ threshold to the S/N to validate a detection with a 99.99994% confidence level.The quantile-quantile plot in Fig. 8 is used to compare quantiles from the normal distribution with quantiles empirically estimated from the data.The closer the data are from the Gaussian distribution, the closer the blue dots are from the red line.It shows here that the spatial correlation matches well with a Gaussian distribution after the speckle suppression, and that the 5σ threshold can be applied.
The bottom panel in Fig. 5 represents the PSD of the same components as in the top panel, but after the high-pass filtering.We see that the speckle noise is just under the photon and  read-out noise.With this cut-off frequency, one millionth of the 492 speckle PSD remains in the filtered spaxels.The term σ 2 speckles is 493 therefore negligible compared to the other noises.We therefore 494 continue to neglect this noise in the following sections.We go on 495 to check our S/N model with end-to-end simulations in various 496 cases to conclude on the robustness of the validation.We propose to verify here our semi-analytical performance anal-500 ysis in the specific case of the future ELT/HARMONI spectro-501 imager.We have a specific interest in this instrument as it is a ma-502 jor upcoming facility for the international community that will 503 combine both a high-contrast imaging capability and medium-504

514
This visible and near-IR IFS is a general-use workhorse instru-515 ment that is designed to address a wide range of science goals 516 (Thatte et al. 2020(Thatte et al. , 2021(Thatte et al. , 2022)).discuss the additional capabilities of molecular mapping hereafter.

HARMONI high contrast mode properties
The list of the possible spectral set-ups in the case of NIR high contrast observations are summarized in Table 1.
Concerning the image properties, we used the result of a realistic model of the apodized PSF of the telescope, as seen through the HCM.It considers the adaptive optics turbulence residuals (Neichel et al. (2016)) in the case of good or median seeing conditions for which high-contrast observations are considered, resulting in good image quality (Fig. 9), with a Strehl ratio in K-band going from 85% for median seeing to 95% for the best conditions.The model also reproduces the non-common path aberrations by considering the specifications of the optics and the way they are seen (or not) by the wavefront sensors, to reproduce the temporal and spectral evolution of the residual wavefront errors.The detailed end-to-end image formation simulation provides a long exposure time-averaged image (which is sufficient for the estimation of the main noise contributors in our approach); it also produces time-series that can feed an extensive end-to-end simulation (Houllé et al. (2021)).In this latter case, fake planets are injected to produce end-to-end contrast curves as a function of separation.We specify that when comparing the obtained planet-to-star contrast in various spectral configurations, the contrast is defined here as the total flux ratio between the faintest detectable planet and the star over a fixed bandwidth from 1.4 µm to 2.5 µm.We argue that it is the best way to discuss the most favorable observation set-up for a given astronomical target.
In high-contrast observations, one of two coronagraph apodization patterns can be selected depending on the target brightness and separation range of interest (Table 2).Photon noise and read-out noise were added to background noise from the sky and the instrument estimated with the HAR-MONI simulator pipeline (HSIM) (Zieleniewski et al. (2019)).We assume a detector read-out noise of 10 e-/pixel/read, typical of Teledyne's H4RG (Beletic et al. (2008)) for short detector integration times (DIT).Read-out noise can be attenuated if we 3.2.Validation of FastCurves with respect to end-to-end simulations As a first and preliminary step, we want to validate our semianalytical approach against end-to-end simulations for a specific case.In particular, this is aimed at checking the absolute values of the estimated useful signal of interest α and of the noise variance based on the halo brightness and detector noise, whereas the end-to-end simulation implements many realizations of noisy spectro-images, used as inputs to the molecular mapping signal extraction algorithm.We used several test cases to try out the robustness of the method.We ran end-to-end simulations for different scenarios: 1) in the case of different types of planets and a broad spectral range; 2) in the case of an M-star with numerous absorption lines contaminating the planet spectrum; and 3) in the extreme case of a poor AO correction to test whether speckles are still not limiting.
Figure 10 compares the contrast curves from our own endto-end simulations and those from FastCurves.Those test cases are an important validation of our general approach and, more specifically, of the quantitative absolute estimates of the useful signal and noise.Finally, it also confirms the assumption that we made, namely, that the residual speckle noise does become negligible after high pass-filtering it in the spectral dimension.
We also compared our estimates to the results published by Houllé et al. (2021).However, Houllé et al. used ATMO templates for fake planets injection and used BT-Settl for the cross correlation step, as a way to introduce some differences between the planet spectrum and the template spectrum.This leads to the combination of two effects affecting the estimation of detection performance: 1) a lower spectral content for ATMO resulting in a lower signal of interest α and 2) a reduced similarity between templates, leading to a cosine smaller than 1.We took this into account to ensure the coherence of our work.We have computed the term α from the ATMO models, and we have estimated the mismatch with the cosine term which was computed as detailed in Eq.16: We found the same orders of magnitude as the above-cited authors reported in their work, as illustrated in Fig. 11.We note that relying only BT-Settl spectra, we would have estimated a deeper contrast by one magnitude .

Trade-off for spectro-imaging
We have seen that α, which represents the amount of useful signal of the companion after continuum removal (Eq.]9), greatly depends on the resolution, the spectral range, and the type of planets.We want to explore this parameter space in regard to various trade-offs and to evaluate the fluctuation of the S/N in the photon-limited case.Considering only the photon noise allows the study to take into account only relative values and to not depend on both a particular star magnitude and integration time.We go on to discuss the case of read-out noise in the HAR-MONI application case in Section 5.

646
We note that here the noise factor, as mentioned above, only 647 includes the propagated star photon noise contribution.We have 648 chosen to take an A-star (T=7200K) to have a negligible β term.

649
The S/N maps have been normalized to only appreciate the vari-  Quantitatively, in these examples, the gain in detectable signal 654 can be as high as a factor 10 from low to high resolution.This 655 can also be evidenced in Fig. 13, which shows the same spec-656 trum at two different spectral resolutions and the corresponding 657 high number of lines detected at higher resolution.In addition, we point out that this effect is quite inhomogeneous from a bandwidth to another, as there can be large disparities in the S/N gain for the same resolution.

Trade-off between bandwidth and resolution
The limited pixel resources of a detector imposes a trade-off between resolution, bandwidth, and field of view (discussed later in Section 4.3).To respect the Nyquist criterion, a higher resolution mode requires more pixels to sample the spectra correctly and leads to a narrower spectral range.On the one hand, having a wider bandwidth provides a larger total number of photons and more companion absorption lines, which both increase α.On the other hand, higher resolution allows for better separation of absorption peaks, which become more numerous, as well as deeper and more distinct, which also increases α.
This trade-off, under the constraint of a fixed number of pixels, will obviously lead to different S/N variations than those presented in Fig. 12, where the bandwidth was fixed and identical  1) with a star magnitude of 4.7, L, and T spectral type companions, as well as a rotational broadening of 20 km/s.These curves are simulated with median seeing conditions with a Strehl ratio of about 80% in K band.The total exposure time is set to 2 hours.We note that the green dashed curve on the left panel is the contrast curve obtained with the ADI ANDROMEDA algorithm (Cantalloube et al. (2015)), and the orange curve is the raw contrast with a standard deviation of 1σ.
for any resolution.For example, in Fig. 13, it can be seen that the value of α is higher at low resolution than at high resolution, but this factor is to be compared with the photon noise.Thus, a higher resolution coupled with a narrower bandwidth can result in either a better or a worse overall S/N, depending on the local wealth of spectral lines at high resolution, as illustrated in Fig.

13.
Thus, Fig. 14 thus illustrates the variations of the S/N when assuming that the product of the resolution and of the bandwidth remains constant.Black dots represent the location of HAR-MONI IFU setups for future discussion (see Sect. 5.1).We note that the higher the resolution, the more sensitive we are to the central wavelength.This is because the spectral range is limited; therefore, we must be fully focused on the most interesting lines, and this is what induces the specific triangular shapes.Thus, choosing too high a spectral resolution could be risky.The red area corresponding to the optimal instrumental setup seems to point out a region where the resolution is around 8000 to 20000 and which moves with temperature from the H-band to the Kband.
We note that in the present trade-offs discussion, the detection boost is the only decision criterion.Obviously, the choice to go to higher resolution can also be motivated by the desire to Fig. 12. S/N variation (defined in Eq.17) as a function of the resolution and the central wavelength of the spectral range, with a constant 0.5 µm large bandwidth.The α quantity has been estimated for a BT-Settl template corresponding to a temperature of 500K (top) and 1700K (bottom).The S/N has been normalized to the maximum, while the absolute values depend on many observing parameters, starting with the stellar magnitude and planetary contrast.
have a better characterization of the physico-chemical properties 699 of the planet in restricted bandwidths of interest.Another parameter involved in the trade-off discussion is the 702 FoV.The larger the FoV, the fewer pixels will be available to 703 either increase the resolution or increase the bandwidth.In the 704 context of a blind search for a planet over a given FoV, one way 705 to provide a quantitative metric for this trade-off is to compare 706 the required total observation time needed to cover the whole 707 FoV with a given S/N goal: either a larger FoV at once (with little 708 spectral information, resolution×bandwidth, inducing a longer 709 integration time per FoV) or mosaïcking over several smaller 710 FoV, with richer spectral information.We show in Fig. 15, the 711 equivalent gain in observation time to reach the same detection 712 level to cover a whole FoV by mosaïcking, either by increasing 713 the resolution, or by increasing the bandwidth.(bottom).Each red rectangle delimits the observable spectral bands by keeping the same number of pixels for both resolutions.In each case, the corresponding alpha quantities and the S/N, both normalized to the top panel, are indicated.We note that some areas have more absorption lines, and thus benefit more than others from an increased resolution (e.g., there is a forest of lines corresponding to the presence of CO around 2.3 µm).Both α and the photon noise decrease as the resolution increases and the bandwidth decreases; comparing the two variations may lead to a lower (left rectangle) or higher (right rectangle) resulting S/N.Fig. 14.Evolution of the S/N (relative to its maximum) with the spectral resolution, for a fixed number of pixels (3300 pixels per spectrum here, corresponding to the case of HARMONI) and a correspondingly variable bandwidth.The largest spectral range is set to cover the H and K bands H (from 1 µm to 2.8 µm) for three different templates of planets signature.These maps include only the photon noise, not considering any wavelength-dependant instrumental transmission, which makes this result quite generic, informative of the planetary spectra properties and not of the instrument specificities.Three different planet temperatures are considered: 500 K (left), 1200 K (middle), and 1700 K (right).
In these examples, above the modest bandwidth (> 0.1 µm), oplanetary lines, resulting in a smaller value of α. Figure 14 already takes into account the transmission of tellurics in the estimation of α thanks to the ESO simulator Skycalc (Noll et al. (2012)).Figure 16 shows the impact of such tellurics on the S/N optimal zones.Despite a slight shift of these optimal zones, the impact of telluric absorption is moderate on young planets emitting their own light.This brings, on average, a degradation of the S/N by a factor of 0.9.This is because the correlation between the telluric absorption lines and those of the young companions is weak.
However, the effect might be stronger when observing planets with an atmospheric composition more similar to the Earth.In such a case, the correlation is much stronger and our telluric absorption becomes troublesome by specifically removing the signal of interest from the exoplanet.Such a problem is partially mitigated if the exoplanet overall radial velocity shifts the spectral features and if the spectral resolution is high enough to deblend the signal of interest from our atmosphere absorption.Relying on molecules opacities from the HITRAN database (Rothman et al. 2009), Fig. 17   (with respect to Earth).As the resolution increases, the lines of interest appear in between the telluric absorption lines: a higher resolution allows a faster S/N gain (as a function of radial velocity) up to a high plateau value.The gain can even decrease as for some molecules, such as CO 2 , the absorption lines are almost evenly spaced in the spectral dimension.This is what causes the periodicity of the gain for a resolution of 3000 in Fig 17 .On the contrary, in the case of a young giant planet spectrum, there is no strong correlation with the exoplanet spectrum of interest and we found no significant S/N gain with the radial velocity of the planet.

Exoplanet reflected light vs stellar spectrum
If we consider the (worst) case of a featureless exoplanetary albedo, the exoplanet's reflected light is essentially a fraction of the stellar spectrum shifted by the exoplanet orbital velocity (with respect to its host star).Despite the gray albedo, this Doppler shift will also enable molecular mapping to reveal the exoplanet within the stellar halo.Using a high-resolution spec- The telluric absorption spectrum has been computed for an airmass of 1 using the ESO Skycalc module.The telluric absorptions do not impact the S/N distribution very much, with the exception of making it slightly more sensitive to the central wavelength, thus delimiting the spectral domains of interest more clearly.For instance, we see that near 2 µm the tellurics separate more the two zones of interest in H band and in K band by making the S/N vary more quickly.
trum of an M3 type star as an example -GJ388, as from the 774 SPIRou database (Donati et al. 2018) -we quantify the inter-775 est in going to a higher resolution for this scientific case in Fig. 776 18.This difficult case of high contrast reflected light exoplanets 777 will definitely push for high spectral resolution at the innermost 778 separations.The exoplanet spin rotation induces a convolution of its intrinsic 781 spectrum, broadening the specific lines.This attenuates the high 782 frequency end of the spectrum PSD and reduces the advantages 783 of observing at higher resolution.This effect is observed in Fig. 784 19, which compares colormaps representing the relative S/N of 785 the same companion at a temperature of 1200K, but with two 786 different rotational velocities.As expected, the most interesting 787

FastCurves contrast predictions in various observing cases
Here, we explore a few cases of planet or companion types.The point here is not to be exhaustive but to cover a diversity of spectra cases, with very different effective temperatures with the BT-Settl model ranging from 500 to 1700K.We note that, for these cases, HARMONI modes mainly cover the most interesting part of the parameter space (Fig. 14) and are complementary with respect to different planet temperatures and noise regimes.However, it suffers from not observing in J-band for the coldest companions.In each case, we present the obtained contrast curves for a two-hour exposure and a sixth-magnitude star, for various spectral setups (see Table 1) with the apodizer SP1 (Table 2).The results are displayed in Fig. 20.These quickly generated contrast curves can be used as an exposure time calculator (ETC).They also help to determine the best observation mode that one should select according to the prior knowledge of the companions.
We find that the ultimate contrast achievable with the actual design seems to be a few 10 −7 in two hours of effective integration (without including the system operation overheads).As expected, a given contrast is more rapidly obtained on cool companions: warmer planets have wider and fewer lines, and thus a smaller signal to be detected in molecular mapping.Also, T-type companions are more easily detected in the H band, unlike the Ltype companions which are more easily detected in the K band.
In addition to the ability of the ELT to observe young planets as close as a few dozens of mas, it might also enable the observation of planets in reflected light (Vaughan et al. ( 2022)).

Dominant noise regimes
Since speckle noise is negligible in this approach, detection is either limited by photon noise or by read-out noise.Figure 21 shows the dominant noise regimes as a function of the star magnitude, separation, and spectral resolution in the case of HAR-MONI.In this plot, we notice that for the high resolution mode (R=17385), the frontier between the photon noise limited case and the read-out noise limited case is located around a magnitude of eight in the short-separation, high-contrast region.This frontier is achieved thanks to the up-the-ramp detector read mode, which significantly reduces the impact of read-out noise.Unless the DIT can be significantly increased above 1 min, while remaining compatible with smearing and within the backgroundlimited regime, these plots provide a good indication that the maximum spectral resolution can fully benefit from the collected photons, while avoiding the read-out noise degradation of contrast.Quantitatively, these values will be slightly modified according to the image quality and the corresponding level of the stellar halo.

Revisit of the coronagraphic set-up
Finally, we revisit the definition and the selection of the optimal coronagraphic setup.The high-contrast module includes two apodizers, designed for the more demanding assumption of ADI post-processing to recover the full companion spectrum.In such a case, it is interesting to minimize the intensity of residual diffracted light in the dark hole, not only to reduce the corresponding photon noise level, but also to limit any pinned speck- les associated to the remaining diffraction pattern.This pushes 849 the apodizers design towards higher contrast at the cost of trans-850 mission.This trade-off can be revisited in the case of molecu-851 lar mapping where speckle noise is not anymore the main lim-852 itation2 .In such a case, a new metric can be applied to design 853 apodizers, comparing the transmission to photon noise, as de-854 tailed in Eq.18.
where S/N apo and S/N 0 are the S/N obtained with and without 856 the apodizer, respectively.Also, t apo and t 0 are the throughput 857 with and without apodizer.Finally, σ apo and σ 0 are the bright-858 ness levels of the adaptive optics halo for the two setups.

859
This new metric only takes photon noise into account.860 In practice, FastCurves provides the comparison of various 861 apodizer cases currently considered for HARMONI (see Fig. 862 22).At short separations (100 mas), the non-coronagraphic case 863 (no apodizer) actually outperforms the apodized setups with a 864 0.5 mag contrast gain.The gain becomes even larger at higher 865 angular separation, with a gain of one magnitude.

Conclusions
15 planetary flux or by attempting to focus on some specific spectral 16 features.Each case involves different dominant limitations that 17 lead to different potential ultimate performances, as we briefly 18 review hereafter.

Fig. 1 .
Fig. 1. 2D correlation map processed on simulated PSF with a fake planet injected, shown in the top-left panel.The colorbar represents the correlation strength between spaxels and the template.The top right panel shows the histogram of the correlation strength map, displaying the distribution of the correlation values in the annulus.Here, µ CCF and σ CCF denote the mean and the standard deviation of these values, respectively.Also, α(x0, y0) is the correlation at the location of the planet.Bottom panel shows the cross-correlation function on (blue) and off (orange) the planet's location. 352

Fig. 2 .
Fig. 2. PSD of high-pass filtered (R c = 100) planet spectrum (T=500K, log(g)=4, [M/H]=0) for two different resolutions on the total spectral range offered by BT-Settl spectra (0.3 − 12 µm).The hatched areas correspond to the areas under the curves equal to the quantity α 2 .Note: the PSD has been smoothed only for better visualization purposes.The y-axis is relative to the total flux of the companions.Only the ratios between the curves explaining the α variation from a given resolution to another one should be taken into consideration.

Fig. 3 .
Fig.3.Fraction of the planetary spectrum remaining after high-pass filtering with a resolution cut R c = 100 as a function of the spectral resolution of the instrument (see text).Templates are chosen for different temperatures but with a constant gravitational surfaces (log(g)=4) and a constant metallicity ([M/H]=0).We notice that the colder the planet, the higher content of molecular absorption lines and the larger fraction of signal preserved at high resolution.

Fig. 4 .
Fig.4.Percentage of information subtracted from the planetary signal due to the high-frequency content of the host star as a function of spectral resolution for several star and planet types.The stellar spectra that are used here come from the BT-NextGen database.The M-type star was chosen with a temperature of 3200K, and the A-type star with a temperature of 7200K.Both have a surface gravity of log(g) = 4, and a metallicity of [M/H]=0.This signal subtraction is obtained with non-Doppler-shifted spectra to correspond to the worst-case scenario, where absorption lines common to both object are overlaid.
N at the location of the planet can be computed by com-374 paring the signal of interest described earlier with the variance 375 of the CCF values at the same separation as the planet (as shown 376 in the upper panels of Fig 1).It can be derived as: 387 392panion spectrum PSD integral over the FWHM area for a planet-393 to-star flux ratio equal to 1.The same applies to β 0 , which is the 394 projection of the template spectrum over the star spectrum for a 395 planet-to-star flux ratio equal to 1.The typical FWHM size for 396 HARMONI's apodized PSF is a bit more than 3 pixels wide.This 397 core area covers 20% to 30% of the total flux from the planet de-398 pending on the AO correction.
2. Discussion of the speckle noise 400The result obtained in Eq.6 is only valid if the estimation of the 401 modulation functions (Eq.5) is accurate and if the estimations 402 of the star spectrum, the sky background spectrum, and the tel-403 lurics absorption are as well.For the reasons mentioned above, 404

Fig. 5 .
Fig. 5. Effect of high-pass filtering on speckles.Top panel: PSDs of the speckle modulation (taken at a several separations), photon and read-out noise, and planet spectrum.Bottom panel: Same PSDs after the highpass filtering with a resolution cut-off of R c = 100.

Fig. 6 .
Fig. 6.Residual fraction of the speckle modulation in respect to several angular separation (solid lines) and a planet spectrum (dashed line) as a function of the resolution cut-off.The planet spectrum is a BT-Settl template corresponding to T=1700K, log(g)=4, [M/H]=0 observed in the K-band degraded to a spectral resolution of 7000.

Fig. 7 .
Fig. 7. Plot shows the empirical S/N obtained with simulations for different cut-off frequencies obtained with the injection of a fake planet with the same HARMONI-like observation conditions as in Fig. 5.

Fig. 8 .
Fig. 8. Study of noise statistics before and after high-pass filtering.Top panel: Quantile-quantile plot without speckle filtering.Bottom panel: Quantile-quantile plot with speckle high-pass filtering with a resolution cut-off of 100.

517
The high-contrast module (HCM) takes benefit from the 518 combination of a good AO image quality on bright stars in 519 the NIR and spectral resolutions from 3500 to 17000.Using a 520 dedicated fine correction of quasi-static instrumental aberrations 521 (N'Diaye et al. (2013); Hours et al. (2022)) and a coronagraph, 522 it has been designed to characterize planets as close as 100 mas 523 from their host star (goal 50 mas) and presenting a 10 −6 flux ratio 524 with it as described in Carlotti et al. (2022b).It will allow inter-525 esting detections of the full (including continuum) spectrum of 526 faint companions thanks to differential imaging and we further 527

Fig. 9 .
Fig. 9. Monochromatic high contrast long exposure PSFs with the two HARMONI current apodizers.It has been simulated with the best seeing condition (Strehl ratio of 90% in K-band).

Fig. 10 .
Fig. 10.Comparison of contrast curves obtained through our method with respect to end-to-end simulations.The comparison has been made for 1 hour exposures with respect to several cases: (top left panel) H-band HARMONI observation mode (Table 1) with an A-star (7200K, log(g)=4, [M/H]=0) magnitude of 6 with T-type planet (500K, log(g)=4, [M/H]=0) and L-type planet (1700K, log(g)=4, [M/H]=0) with median seeing condition (0.65" for λ = 0.5 µm, Strehl=94%).Top-right panel: Same templates and conditions on K-band HARMONI observation mode.Bottom-right panel: Same templates on H-band with JQ3 seeing condition (0.74" for λ = 0.5 µm, Strehl=73%).Bottom-left panel: Same planetary templates but with an M-star (3200K, log(g)=4, [M/H]=0) on K-band with median seeing condition.The solid lines are the prediction of FastCurves with these parameters, the dashed lines are the result of molecular mapping processing with time-series simulated PSF and fake planets injection.

650
ations of the S/N.The dependency over the resolution (along the 651 y-axis) in Fig 12 is directly related to Fig. 3, showing that the 652 level of useful information increases with the spectral resolution. 653

Fig. 11 .
Fig. 11.Comparison between our approach (right column) and results fromHoullé et al. (left column).The comparison was made for the H, K, and K2 high bands observation modes (see Table1) with a star magnitude of 4.7, L, and T spectral type companions, as well as a rotational broadening of 20 km/s.These curves are simulated with median seeing conditions with a Strehl ratio of about 80% in K band.The total exposure time is set to 2 hours.We note that the green dashed curve on the left panel is the contrast curve obtained with the ADI ANDROMEDA algorithm(Cantalloube et al. (2015)), and the orange curve is the raw contrast with a standard deviation of 1σ.
Spatial versus spectral information trade-off 701

Fig. 13 .
Fig.13.BT-Settl spectrum of a T=1700K companion at two different spectral resolutions: R=7000 (top) and R=17000 (bottom).Each red rectangle delimits the observable spectral bands by keeping the same number of pixels for both resolutions.In each case, the corresponding alpha quantities and the S/N, both normalized to the top panel, are indicated.We note that some areas have more absorption lines, and thus benefit more than others from an increased resolution (e.g., there is a forest of lines corresponding to the presence of CO around 2.3 µm).Both α and the photon noise decrease as the resolution increases and the bandwidth decreases; comparing the two variations may lead to a lower (left rectangle) or higher (right rectangle) resulting S/N.
illustrates this effect by estimating the gain in detection for spectral absorption lines caused by H 2 O and CO 2 molecules as a function of the exoplanet radial velocity

Fig. 15 .
Fig. 15.Gain of time as a function of the bandwidth (top panel) or the spectral resolution (bottom panel) at the expense of mosaïcking the FoV for detecting a 500K planet (log(g)=4, [M/H]=0).The spectral range has been chosen to start at λ min = 1.2 µm for the FoV/Bandwidth trade-off and a fixed spectral range between 1.2 µm and 1.8 µm for the FoV/Resolution trade-off.

Fig. 16 .
Fig. 16.Comparison of S/N variations for a planet with temperature of 1700K with (bottom panel), and without (top panel) telluric absorption.The telluric absorption spectrum has been computed for an airmass of 1 using the ESO Skycalc module.The telluric absorptions do not impact the S/N distribution very much, with the exception of making it slightly more sensitive to the central wavelength, thus delimiting the spectral domains of interest more clearly.For instance, we see that near 2 µm the tellurics separate more the two zones of interest in H band and in K band by making the S/N vary more quickly.

Fig. 17 .
Fig. 17.S/N gain as a function of radial velocity shift for H 2 O and CO 2 molecules.The three curves represent this gain for different spectral resolutions: 3000, 10000, and 40000.

Fig. 18 .
Fig. 18.S/N gain as a function of a radial velocity shift for featureless exoplanetary albedo.The three curves represent this gain for different spectral resolutions: 3000, 10000, and 40000).

Fig. 19 .
Fig.19.Evolution of the S/N with the resolution for the same exoplanet spectrum template (1200 K) but with two different rotational broadening results, corresponding to rotational speed of 3km/s and 30km/s.The optimal area goes toward lower resolution as the rotational speed of the planet increases.

Fig. 20 .
Fig. 20.Predicted contrast curves for HARMONI generated with the FastCurves package for three different planet temperatures.The exposure time is set to be 2h on a 6 magnitude A-star with a Strehl ratio of about 60% in the H band.

Fig. 21 .
Fig. 21.Limit star magnitude above which photon noise becomes smaller than read-out noise as a function of separation and spectral resolution.

Fig. 22 .
Fig. 22. Contrast curves with and without apodizers.The comparison is made while considering the best seeing conditions (Strehl ratio of 95% in the K band).

Fig. B. 1 .
Fig. B.1.Relative weight of the low frequency and high frequency part of the noise and stellar signal.Top panel: PSDs of the high-pass filtered noise (in blue) and of the complementary low-pass filtered noise (in red).Bottom panel: Stellar spectrum PSD with the high-frequency part in blue and the low-frequency part in red.The star spectrum corresponds to a M-star and the tellurics to present a case where we have a strong high-frequency content.

Fig. C. 1 .
Fig. C.1.Visualization of the amount of information useful for molecular mapping detection.Top panel: PSD of two high-pass filtered planet spectra (T=500K and T=1200K, log(g)=4, [M/H]=0) at the same resolutions and with the same amount of photons.Bottom panel: PSD of high-pass filtered planet spectra (T=1700K, log(g)=4, [M/H]=0) on two different spectral ranges with a 300 nm bandwidth (center on λ 0 = 1.35 µm in red and center on λ 0 = 1.75 µm in purple).