S Pires

S Plaszczynski

A Lavabre

Towards a fast, model-independent Cosmic Microwave Background bispectrum estimator

Keywords: Cosmology, CMB, Method, Data Analysis

The measurements of the statistical properties of the Cosmic Microwave Background (CMB) fluctuations enable us to probe the physics of the very early Universe especially at the epoch of inflation. A particular interest lays on the detection of the non-Gaussianity of the CMB as it can constrain the current proposed models of inflation and structure formation, or possibly point out new models. The current approach to measure the degree of non-Gaussianity of the CMB is to estimate a single parameter f local NL which is highly model-dependent. The bispectrum is a natural and widely studied tool for measuring the non-Gaussianity in a model-independent way. This paper sets the grounds for a full CMB bispectrum estimator based on the decomposition of the sphere onto projected patches. The mean bispectrum estimated this way can be calculated quickly and is model-independent. This approach is very flexible, allowing exclusion of some patches in the processing or consideration of just a specific region of the sphere.

Introduction

Inflation is the currently favored theory of early Universe which predicts an early short period of rapid expansion and explains the origin of primordial perturbations. Many models of inflation predict weakly non-Gaussian primordial curvature perturbations and the primordial curvature perturbation Φ(x) is parametrized using the local model [9] as follows:

Φ(x) = Φ L (x) + f local NL (Φ 2 L (x) -Φ 2 L (x) ), (1)
where f local NL is the non-linear coupling constant in the local model. Due to its simplicity the local model is highly favored especially because all the higher-order moments are determined in terms of this f local NL parameter. Most of models of inflation only predict a value for f local NL to characterize the non-Gaussianity of the CMB. However, there exists other models of inflation that predict different types of deviations from Gaussianity, detailed calculation of which have been investigated in [1,5].

The observed CMB temperature fluctuations ∆T /T are related to the primordial curvature perturbation Φ(x) through the following non-linear relation [START_REF] Komatsu | The Ninth Marcel Grossmann Meeting[END_REF]:

∆T T ≈ g T (Φ(x)), (2)
where g T is the radiation transfer function. On small scales, this function is very complex and it is evaluated numerically by solving the Boltzmann transport equation. On very large scales, in the Sachs-Wolfe regime (l < 10), this relationship simplifies to ∆T T = Φ(x) 3 . A number of theories of inflation have been proposed that make different predictions about the CMB fluctuations. Measurements of the statistical properties of the CMB are a direct test of inflation which can help to rule out the many models of inflation that have been proposed.

The observed CMB anisotropies ∆T /T can be expanded onto spherical harmonics:

∆T /T = lm a lm Y lm .

If the CMB is Gaussian, it is fully described by its angular power spectrum:

C l = 1 2l + 1 +l m=-l |a lm | 2 . ( 4
)
But, as explained previously, many realistic models predict deviations from Gaussianity. Even if, the level of non-Gaussianity is predicted to be very small in single field slow-roll inflation model, there is a large class of more general models that predict a substantially high level of primordial non-Gaussianity. If these models are true, the power spectrum provides a limited insight to the physics of the very early Universe and higher-order estimators are needed to probe the CMB non-Gaussianity. The bispectrum is a natural model-independent approach to probe the small departure from Gaussianity that could originate during inflation. The CMB angular bispectrum may be calculated from product of three spherical harmonic coefficients of the CMB temperature field. For Gaussian fields, the expectation value is exactly zero. Given statistical isotropy of the universe, the angular bispectrum B l1l2l3 is given by (Komatsu et al, 2001):

B l1l2l3 = m1,m2,m3 l 1 l 2 l 3 m 1 m 2 m 3 a l1m1 a l2m2 a l3m3 , (5)
where the matrix denotes the Wigner-3j symbol, B l1l2l3 satisfy the triangle condition: |l 1 -l 2 | ≤ l 3 ≤ l 1 + l 2 and the parity invariance:

l 1 + l 2 + l 3 = even.
Then, a non-zero bispectrum is a signature of a more complicated inflationary period than a simple inflation slow-roll model predicts.

However, computation of a full bispectrum is very time-consuming and it is usually assumed that the signal is too weak for each of the multipoles to be measured individually. Instead a least squares fit to compare the observed bispectrum with a particular (separable) theoretical bispectrum is used. Thus, most non-Gaussianity studies focus on estimating the f local NL parameter because the bispectrum is fully specified by this parameter in the case of a local model (1). A fast estimator for f local NL has been developed by [START_REF] Komatsu | [END_REF] and improved by [1]:

f local NL ≃   lmax l1≤l2≤l3 (B prim l1l2l3 ) 2 C l1 C l2 C l3   -1 S prim , (6)
where C l is the theoretical power spectrum and B prim l1l2l3 is the theoretical bispectrum [9] for f local NL = 1. The statistics S prim defined in [8] has only a complexity of O(N 3/2 ) whereas the full bispectrum analysis is O(N 5/2 ). A detection of f local NL > 10 will rule out most of the existing inflation models.

However, an expression like equation ( 1) is not general and there are many other inflationary models that predict different types of deviations from Gaussianity. Other models have been investigated and detailed calculations of other form of non-Gaussianity have been carried out (see for example [1,5]). However, all these estimators are highly model-dependent and may not be able to constrain, many of the non-Gaussiannity signatures. In this paper, we will investigate a more general and model-independent approach to probe the CMB non-Gaussiannity by measuring the full bispectrum. If the non-Gaussianities are of the local type, the bispectrum will reach a maximum in case of squeezed configurations (i.e. one wave vector is much smaller than the other two).

Full bispectrum estimation on spherical data using the spherical harmonics definition (5) is very time consuming. This paper tries to tackle this problem by introducing a promising approach for accelerating the calculation of the fully general bispectrum. This method is based on the projection of the sphere onto small-field projected maps for which a Fourier decomposition is used to estimate the bispectrum. A mean bispectrum estimator can then be obtained by combining results from all the patches. This paper sets the grounds for this method however for full optimization of the estimator, we need more accurate non-Gaussian simulations which are not yet available and an analytical prediction of a full bispectrum for a given level of non-Gaussianity which is very complex to code up.

The outline of the paper is as follows. In §2, the main issues of the method are pointed out and the method is optimized to do power spectrum estimation. In §2.4 the same method is used to accelerate the full bispectrum estimation. Some adaptations of the method are required to optimize the bispectrum estimation.

Decomposition of the sphere onto rectangular Cartesian maps

In this section, we will describe a method to speed up the spectral analysis by decomposing the sphere onto patches. Such method is similar to the Welch's method [START_REF] Opennheim | Digital Signal Processing[END_REF]15] that is commonly used to reduce the variance in spectral analysis of 1D data. For a sphere, the size and the repartition of the patches have to be decided. Then, the pixels of the patches have to be projected onto rectangular Cartesian maps. A spectral estimator is then obtained by averaging the result of the spectral analysis on each rectangular Cartesian map. Such methods suffer from the limited size of the projected patches which introduces a bias. In a future paper, we will consider multi-taper techniques [2] that consist in averaging over different tapers using the full data. These methods reduce the bias since the data length is not shortened.

In this section, we will follow the basic approach that consists in decomposing the sphere onto patches. A number of issues has to be solved before having an optimal estimator.

Some issues of the method

The first issue to be tackled is to find an optimal tessellation of the sphere. To facilitate the post-processing, especially the FFT required to do spectral analysis, the patches ought to be rectangular and preferably have the same orientation. However, no regular tessellation of the sphere by rectangles exists. Instead, a pseudo regular tessellation with rectangles can be obtained by allowing the patches to overlap. The overlapping also improves the power spectrum estimation : according to the Welch's overlapped segment method [15,[START_REF] Opennheim | Digital Signal Processing[END_REF], estimating the power spectrum of a signal by splitting this one into overlapping windowed patches reduces the noise in the estimated power spectra by reducing the frequency resolution. There still remains the size and position of the patches to be determined. To reduce the distortions, the size of the patches should depend on the map projection and is a compromise between the distortions introduced by the projection and the window function effects.

The effect of the window function on the final power spectrum is another issue to take into account. The limited size of the patches forces us to use window functions. Thus, instead of analyzing the signal s(x, y), we will be analyzing the truncated signal: s h (x, y) = s(x, y)h(x, y). In the frequency domain, we obtain the following convolution product

S h (u, v) = S(u, v) * H(u, v)
where H(u, v) is the Fourier transform of the window function. By default the window function is a rectangular window, constant inside the patch and zero elsewhere (see the left panel of the Fig. 5). But, it appears that a window function with better frequency response has to be used. The ideal window function is one whose frequency response is a Dirac delta function. It corresponds to an infinite rectangular window function which is impossible in practice. Instead, the frequency response normally consists of a main lobe and side lobes (see the right panel of the Fig. 5). To be close to a Dirac function, the main lobe ought to be the highest and the narrowest to increase the frequency resolution and the side lobes have to be the lowest to limit the mode-to-mode interaction. The rectangular window function is usually not recommended because of its significant sides lobes.

An additional issue comes from the projection effects. The method that we have developed is based on the decomposition of the sphere onto a few rectangular Cartesian maps. It assumes to do pixel projections from the HEALPix map to rectangular Cartesian maps. No matter how sophisticated the projection process will be, distortions are inherent in flattening the sphere. Some classes of map projections maintain areas, and others preserve local shapes, distances, and/or directions... No projection, however, can preserve all these characteristics. Choosing a projection thus always requires compromising accuracy in some way, and that is one reason why so many different map projections have been developed.

Whatever the projection, if we want to keep this projection exact, we will have to deal with rectangular Cartesian maps having a non-regular grid. This will be a problem especially for the power spectrum and bispectrum estimation because we need to estimate the Fourier coefficients from a signal f (x, y) at arbitrary nodes (x, y).

Assuming the signal f is a periodic function, it can be decomposed into Fourier series as follows:

f (x, y) = +∞ n1=-∞ +∞ n2=-∞ C n1,n2 (f )e 2πi n 1 T x e 2πi n 2 T y (7)
The approximation that has been used in this paper consists in finding the coefficients C n1,n2 (f ) of the Fourier series (7). The problem can be solved efficiently by noting the system matrix is a Toeplitz matrix i.e. a diagonalconstant matrix [see 4, for more details about the use of Toeplitz matrices in irregular sampling problems]

Optimization of the method for power spectrum estimation

As discussed in the previous section, a number of issues has to be solved in order to use the previous method for spectral estimation. In this section, the method is optimized to perform power spectrum estimation. For this purpose, we have generated simulations of the full sky CMB temperature as realizations of a random Gaussian field with a prescribed power spectrum. The cosmology adopted in these simulations is consistent with the WMAP parameters. We use the HEALPix pixelisation as WMAP and Planck missions with a resolution parameter of n side = 1024.

The tiling of the sphere

Following the ideas discussed in §2, several decompositions have been tested.

An interesting tiling consists in decomposing the sphere onto rectangular patches distributed into lines of same latitude (see Fig. 1). This decomposition Figure 1: Sphere tiling using an equi-latitude decomposition introduces a substantial overlapping at the pole. For power spectrum estimation, this effect can be neglected by assuming the CMB field is isotropic. But this decomposition should not be used to detect non-Gaussianity because some types of non-Gaussianity can produce localized hot spots or other structures.

Another tiling consists in decomposing the sphere onto rectangular patches located at the HEALPix centers of a lower resolution (see Fig. 2). This decomposition insures a good repartition of the patches in the sky and should be preferred for bispectral estimation. Once, we have selected the optimal tiling of the sphere, the pixels of the patches are projected onto rectangular Cartesian maps using a projection that will be described in §2.2.2. To perform the decomposition, the size of the patches has to be fixed in such a way that keeps the induced distortions to minimum. To reduce the spectral leakage, the size of the patch has to be increased to narrow the main lobe of the frequency response. But increasing the size of the patch will increase the distortions effects due to projection effects.

In Fig. 3, the mean power spectrum has been estimated using three different sizes of field: 10 • x10 • , 20 • x20 • and 30 • x30 • . A Hann window has also been used to reduce the spectral leakage. Then, these power spectra (red) have been compared to the theoretical power spectrum used to simulate the full-sky CMB (black). As expected, an important spectral leakage is observed on the mean power spectrum estimated with patches of 10 • x10 • (see top panel of Fig. 3). With patches of 30 • x30 • (see bottom panel of Fig. 3), the spectral leakage is severely dampened but some distortions are visible at large scales, most certainly due to projection effects. The best result is obtained with a field of 20 • x20 • (see middle panel of Fig. 3) which is a good compromise to both minimize the spectral leakage and reduce the distortions introduced by the map projection.

Map projection

The map projection introduces distortions of various classes, as it has been discussed in §2.1. The choice of the map projection has to be done in such a way that it reduces the error in the estimated power spectrum. Two projections have been compared:

1. The gnomonic projection is constructed by projecting every point of the sphere onto patches from the center of the sphere. Assuming the patch is tangent to the point S (θ • , ψ • ), the coordinate transformation is the following:

x y = cos θ sin θ•-sin θ cos θ• cos(ψ-ψ•) cos θ cos θ•+sin θ sin θ• cos(ψ-ψ•) sin θ sin(ψ-ψ•) cos θ cos θ•+sin θ sin θ• cos(ψ-ψ•) , ( 8
)
where θ is the longitude and ψ is the latitude.

2. The stereographic projection is constructed by projecting every point of the sphere onto patches from the sphere north pole in a plane tangent to the south pole. Assuming the patch is tangent to the point S (θ • , ψ • ), the coordinate transformation is the following:

x y = 2(cos θ sin θ•-sin θ cos θ• cos(ψ-ψ•)) 1+cos θ cos θ•+sin θ sin θ• cos(ψ-ψ•) 2 sin θ sin(ψ-ψ•) 1+cos θ cos θ•+sin θ sin θ• cos(ψ-ψ•) . (9)
In Fig. 4, we show the mean power spectra estimated from patches of 20 • x20 • with a Hann window using two different projections: the gnomonic projection (left) and the stereographic projection (right). The two mean power spectra (red) are compared to the theoretical power spectrum (black). The better result is obtained with the gnomonic projection that will be used as the default projection for power spectrum estimation. However, we have to note that the map projection errors have already been reduced by fixing the size of the field to 20 • x20 • (see §2.2.1).

Windowing

After the projection, the full-sky CMB is decomposed onto rectangular Cartesian maps of 20 • x20 • , but, as already discussed in §2.1, the analysis of a finite signal affects the frequency analysis. The simplest way to model a patch of a finite size is through the usage of a rectangular window. But, this default window introduces an important spectral leakage. There is a lot of possible other window functions that can be used to reduce the spectral leakage in the power spectrum estimation. A few of the more common window functions have been compared in this study (see Fig. 5): The map projection effect in the power spectrum estimation: Power spectra estimated with two different projections: the gnomonic projection (left) and the stereographic projection (right). These power spectra (red) are compared to the theoretical power spectrum (black). The right column corresponds to a zoom of the left power spectrum.

Figure 5: Window functions commonly used in FFT power spectrum estimation (left) and its frequency response (right). A signal whose frequency is actually located at zero offset leaks into neighboring frequencies with the amplitude shown. The rectangular window which is equivalent to no windowing is the least recommended because of its large side lobes.

1. The Rectangular window that is the default window:

h(x, y) = 1. ( 10
)
2. The Hann window that has been used in §2.2.1:

h(x, y) = cos(πx) 2 cos(πy) 2 . (11)
3. The Bartlett window:

h(x, y) = 1 -| x 2 | -| y 2 |. (12)
In Fig. 6, the mean power spectrum has been estimated by decomposing the CMB full-sky onto patches of 20 • x20 • using the three windows described above. These mean power spectra (in red) have been compared to the theoretical power spectrum that has been used to simulate the CMB full-sky (in black). As expected, a spectral leakage is observed with the default rectangular window due to the finite size of the signal (see the top panel of Fig. 6). The spectral leakage has been reduced by applying the previously described non-rectangular window functions (see the middle and bottom panel of Fig. 6). However, the Hann window does the best job (see the bottom panel of Fig. 6). Indeed, the Hann window is known to produce moderate side lobes (see the bottom panel of Fig. 5) and to have high frequency resolution which is close to an ideal window function (see §2.1). Therefore, the Hann window will be used as the default window function for power spectrum estimation. As said previously, in this paper, we haven't tested the multitaper approach [2] that seems to be optimal to reduce the spectral leakage in the power spectrum estimated from small patches of the sky. This will be done in a future work.

Validation of the method with the power spectrum

In the previous section, we have done an optimization of the method for power spectrum estimation. There still room for improvements but is outside the scope of this paper. The best power spectrum estimation has been obtained by decomposing the sphere onto patches of 20 • x20 • , by projecting the pixels of the patch onto rectangular Cartesian maps using a gnomonic projection and by multiplying each projected map by a Hann window. The power spectrum estimated by this method is now compared with the power spectrum estimated from the spherical harmonics formula (equation 4).

Figure 7: Mean power spectra estimated from 100 full-sky CMB simulated maps using two different methods: using the full-sky method based on the spherical harmonic coefficients (top left panel in red) and using the method based on the decomposition of the sphere onto patches (top right panel in pink). These two mean power spectra are compared to the theoretical power spectrum used to produce the simulations (in black). The bias in the mean power spectrum estimated with the full-sky method is about 0.5% while it is about 3% with the per patch method. The empirical standard deviation is given for the two methods in the bottom panel.

In Fig. 7, we have estimated a mean power spectrum from 100 full-sky CMB simulated maps using the two different methods. On top left, we have the mean power spectrum estimated from spherical harmonics coefficients (in red) and in the top right, the optimized mean power spectrum obtained by the method based on the decomposition of the sphere onto patches (in pink). These two mean power spectra have been compared to the theoretical power spectrum used to do the simulations (in black). The two curves on the left panel lie on top of each other while the two curves in the right panel show a small shift. There still remains a small amount of leakage after windowing. This kind of bias can be corrected [7] or reduced using a multitaper approach [2]. The standard deviation is significantly smaller in the per patch method than in the full-sky method (see the bottom panel of Fig. 7), this is because the frequency resolution has been reduced.

It could be thought, the per patch method is not recommended for power spectrum estimation because the decomposition of the sphere onto patches takes some time and there already exists a fast method for a fully CMB power spectrum estimation. However, in practice, we never have access to a full-sky CMB map because of the contamination by residual foregrounds. For this reason, the per patch method can be preferred for some applications.

Application of the method

As mentioned previously, we never have access to a full-sky CMB map. The CMB map obtained by a method of component separation is always partially masked to discard contaminated pixels which introduces problems in power spectrum estimation. About 15-20% of the most contaminated data is removed mostly in the galactic plane. Several methods exist to overcome the missing data problem, the most common one to be the MASTER method [7] that uses apodization windows. The method that has been proposed in this paper is an alternative to methods such as MASTER that try to correct the power spectrum from the effect of missing data. The approach is totally different because the problem of missing data is solved by avoiding the patches with important contaminations on the power spectrum estimation.

Following the same idea, we can imagine to use this method as a diagnosis of the component separation quality. An estimator of the power spectrum per latitude can be obtained by averaging the result of the spectral analysis on each patch located at the same latitude (see Fig. 1). The contamination by residual foregrounds introduces distortions on the power spectrum that should increase close to the galactic plane. The level of contamination per latitude is an indicator of the efficiency of the component separation method. Different component separation methods can be evaluated and compared by this way using simulated data. Fig. 8 shows the effect of residual foregrounds in the power spectrum estimation. The left panel shows the power spectra estimated from a simulated CMB map purely Gaussian for different latitudes. The variance is only due to the number of patches per latitude that decreases moving to the poles (see Fig. 2). This explains that the statistical variance becomes important for power spectra estimated for latitudes close to the poles (larger than |70 • |).

The right panel of Fig. 8 shows the power spectra per latitude estimated from a simulated CMB map with a significant level of residual foregrounds in the galactic region. This CMB map has been obtained by applying an optimized component separation method to simulations of Planck observations. The quality of the component separation method can be evaluated by comparing this result to the previous result. As previously, we observe distortions in the power spectrum close to the pole due to the statistical variance (latitude larger than |70 • |). But some important distortions are also present close to the galactic plane (latitude equal to 0 • -red line and 9.5 • -pink line) clearly due to residual foregrounds. This method can be used to select the better component separation method and help to define the region to be masked. Obviously, this only could be accomplished on simulated data for which the true power spectrum is known.

Full bispectrum estimation

The primary goal of this paper is to provide a method for accelerating the calculation of the full bispectrum that cannot be estimated with spherical harmonics in a reasonable time. In the previous section, we have introduced a promising approach that have been tested on the power spectrum estimation. The previous methodology will now be applied to estimate a full bispectrum. Obviously, the method needs now to be optimized for the bispectral estimation.

Limitations of the study

There will be two limitations to the optimization of this method. The first limitation is the resolution of the non-Gaussian simulations of CMB currently available. The bispectrum of a Gaussian field being null, non-Gaussian CMB simulations have to be used to test the validity of the method. For that purpose, we have used the CMB non-Gaussian simulations of local type provided at the following address: http://planck.mpa-garching.mpg.de/cmb/fnl-simulations. A non-Gaussian CMB temperature map can easily be computed with any desired level of non-Gaussianity (f local NL ) by linear combination of the a lm provided. More details about the simulations can be found in [3]. A major problem in our study is that the HEALPix resolution parameter of these simulations is n side = 512 which is quite low compared to Planck data (n side = 2048) which has limited our study. But, there exists no other non-Gaussian simulations of CMB publicly available with a better resolution.

The second limitation is that no publicly available codes exist for a theoretical computation of the full bispectrum for a given level of non-Gaussianity, due to its complexity. The comparison with theory is required to be sure that the processing is not introducing important errors in the mean bispectrum. An analytic prediction of the bispectrum has been given in [START_REF] Komatsu | The Ninth Marcel Grossmann Meeting[END_REF] for non-Gaussianities of the local-type (F local NL ) but only for the equilateral configuration. In this paper, the comparison with the theoretical bispectrum has been barely done on the equilateral configuration. However, even focusing on a particular configuration like the equilateral configuration, our method still remains more powerful than f local NL estimation methods because all modes of the equilateral bispectrum are reconstructed compared to a single parameter.

Optimization of the method for bispectrum estimation

In this section, the optimization of the method for bispectral estimation will be done by comparing the equilateral bispectrum estimated by our method on the CMB non-Gaussian simulations described previously with the analytic prediction given in [START_REF] Komatsu | The Ninth Marcel Grossmann Meeting[END_REF]. However, the low resolution of the non-Gaussian simulations will considerably limit this study. Consequently, more work will have to be done as soon as non-Gaussian simulations with a better resolution will be available.

As for power spectrum estimation, a number of issues has to be solved in order to adapt the previous method to do bispectral estimation. The first issue is the tiling of the sphere. As said previously, for bispectral estimation, the decomposition of the sphere onto rectangular patches located at the HEALPix centers of a lower resolution (see Fig. 2) is preferred because this decomposition ensures a good repartition of the patches in the sky. But the size of the patches has to be reduced significantly compared to previous decomposition because the bispectrum is very sensitive to the non-Gaussianities introduced by the projection effects. As a result, the gnomonic projection will be preferred to do the projection from the pixels of the patches onto the rectangular Cartesian maps because this projection introduces very little distortions for small patches [6]. However, the size of the patches has to be fixed in such a way the distortions are reduced. In Fig. 9, the mean bispectrum has been estimated using three different size of field: 7

• x7 • , 10 • x10 • , 17 • x17 • .
The pixel projection introduces non-Gaussianities in the projected maps that appears in the bispectrum. For patches of 7 • x7 • , the amplitude and the location of the acoustic oscillations are quite well detected. But, the larger the field is, the more important the projection effects are. For patches of 17 • x17 • , the CMB non-Gaussianities of the local type almost disappear. There is still more power at the location of the acoustic oscillations but the distortion effects introduce an important noise in the mean bispectrum. This experiment should be redone with simulations with a better resolution.

Another issue is the window function. As for spectral estimation, it doesn't exist a perfect window function for bispectral estimation. It should be a trade-off between bispectral resolution and leakage effect. In Fig. 10, we have compared the Hann window to the rectangular window to do bispectral estimation and we obtain a better result with the default rectangular window function (left). Indeed, the amplitude of the acoustic oscillations are better recovered with the default rectangular window. In a future work, a window function dedicated to the CMB data has to be designed to improve the bispectral estimation. Some authors have already tried to address the problem of finding an optimal bispectral window [see 13, 16].

Preliminary Results

The best equilateral bispectral estimation has been obtained by decomposing the sphere onto patches of 7 • x 7 • and by projecting the pixels of the patch onto rectangular Cartesian maps using a gnomonic projection. The code FASTLens [14] publicly available has been used to compute the bispectrum in the Cartesian maps. This study has been limited to the equilateral configuration of the bispectrum. As soon as a code to compute analytical prediction for the full CMB bispectrum will be released, the same study could be extended to all the configurations of the bispectrum. Furthermore, the study will be improved as soon as non-Gaussian simulations of CMB with a better resolution will be available.

Anyway, the equilateral bispectrum computed (see the left panel of Fig. 10) shows the expected acoustic oscillations and the Silk damping predicted by theory for a CMB with non-Gaussianity of the local-type [START_REF] Komatsu | The Ninth Marcel Grossmann Meeting[END_REF]. Therefore, this preliminary result shows that we can compute an equilateral bispectrum which is in good agreement with the analytical predictions.

Conclusion

The goal of this paper is to present a fast way to compute a full-sky CMB bispectrum. It should be pointed out that it is currently too hard to directly measure this full CMB bispectrum. This paper introduces a promising approach for accelerating the estimation of the full bispectrum on the sphere. This method involves the decomposition of the HEALPix map onto small projected Cartesian maps. A mean full bispectrum can then be estimated by combining results from all the projected maps.

First, this approach has been used to estimate the power spectrum of a fullsky CMB map. A number of optimizations have been done to obtain the best power spectrum estimator. An interesting application of the method in order to test the quality of the CMB component separation on the galactic region has then been presented.

The approach has then be applied to the full bispectrum estimation to accelerate its computation. The approach presented in this paper enables a fast reconstruction of the whole bispectrum directly from the observational data. A number of optimizations have been performed to improve the quality of the bispectral estimation. However, these optimizations has been only tested on the equilateral configuration of the bispectrum because of the lack of analytical predictions for the full bispectrum. Anyway, this study could be easily extended to other configurations as soon as a code to compute analytical predictions for the full CMB bispectrum will be released. Another limitation of the study comes from the resolution of the non-Gaussian CMB simulations used for the analysis. This is a preliminary result that should be improved as earlier as is feasible.

However, the equilateral bispectrum that is computed from the non-Gaussian CMB simulations using this approach is in very good agreement with the analytical predictions. Indeed, the features expected by the theory are present despite the poor resolution of the estimated bispectrum. Thus, this approach appears very promising to constrain the CMB non-Gaussianity in a model-independent way.

Figure 2 :

 2 Figure 2: Sphere tiling using the HEALPix centers of a lower resolution.

Figure 3 :

 3 Figure 3: The effect of the size of the patches in the power spectrum estimation: a power spectrum (with error bars) has been estimated using three different size of patches (Note: a Hann window function is used to reduce the spectral leakage): 10 • x10 • (top left), 20 • x20 • (middle left) and 30 • x30 • (bottom left). Then, these three power spectra (red) have been compared to the theoretical power spectrum (black). The right column corresponds to a zoom in on the left power spectrum.

Figure 4 :

 4 Figure4: The map projection effect in the power spectrum estimation: Power spectra estimated with two different projections: the gnomonic projection (left) and the stereographic projection (right). These power spectra (red) are compared to the theoretical power spectrum (black). The right column corresponds to a zoom of the left power spectrum.

Figure 6 :

 6 Figure 6: The window function effect in the power spectrum estimation: a mean power spectrum (with error bars) is obtained by decomposing the sphere onto patches of 20 • x20 • using a rectangular window (top left), a Bartlett window (middle left) and a Hann Window (bottom left). These power spectra (red) are compared to the theoretical power spectrum (black). The right column corresponds to a zoom in on the left power spectrum.

Figure 8 :

 8 Figure 8: Power spectrum per latitude (top) and corresponding errors (bottom) (patches of 10 • x10 • ) estimated from a simulated Gaussian CMB map (n side = 2048) without (left) and with (right) contamination of foregrounds. The north and south contributions have been merged.

Figure 9 :

 9 Figure 9: The effect of the size of the patches in the bispectrum estimation: a mean bispectrum has been estimated using three different size of patches (7 • x7 • , 10 • x10 • , 17 • x17 • ) from 100 full-sky non-Gaussian CMB maps (F local NL = 100).

Figure 10 :

 10 Figure 10: The effect of the window function in the bispectrum estimation: a mean bispectrum has been estimated by decomposing the sphere onto patches of 7 • x7 • using a rectangular window (left) and a Hann window (right) from 100 full-sky non-Gaussian CMB maps with F local NL = 100 and F local NL = 500.

Acknowledgment

This work has been supported by the European Research Council grant SparseAstro (ERC-228261). Some of the results in this paper have been derived using the HEALPix package (Gorski, Hivon, and Wandelt 1999). We wish to thank F. Sureau, P. Paykari and Y. Moudden for useful discussions and comments.


C Tapia

E Bellet-Amalric

D Aldakov

F Boudoire

K Sivula

Laurent Cagnon

V Artero

E Bellet-Amalr

Achieving visible light-driven hydrogen evolution at positive bias with a hybrid copper-iron oxide| TiO 2 -cobaloxime photocathode †

Achieving visible light-driven hydrogen evolution at

positive bias with a hybrid copper-iron oxide|TiO2-cobaloxime photocathode

Introduction

Hydrogen production via the water splitting reaction is a promising solution to store massive amounts of renewable energy and allow the essential ecological transition. 1 However, direct sunlight-driven, i.e. photoelectrochemical, H 2 evolution must overcome significant challenges in terms of performance, stability and scalability in order to be considered as a credible alternative to current technologies, 2 which rely on photovoltaics coupled to water electrolyser. Water-splitting photoelectrochemical cells (water-splitting PECs) hold promise to exceed current technology since they are based on photoelectrode(s) that combine solar energy absorption, energy conversion and catalysis for hydrogen evolution or water oxidation. [START_REF] Van De Krol | Photoelectrochemical Hydrogen Production[END_REF][START_REF] Pan | [END_REF] The choice of the photoelectrode material(s) determines the efficiency of the PEC. Light absorption is generally achieved by a semiconductor material that should combine (i) suitable bandgap energy and band positions to allow catalysis to proceed, (ii) efficient charge carrier separation and transpor-tation and (iii) good stability in contact with aqueous electrolytes. Here, we focus on the design of an efficient photocathode for hydrogen evolution, for which fewer solutions exist. Indeed, the large majority of studies have focused on Cu 2 O [5][6][7][8] and p-doped Si 9,10 as photocathode candidates due to their narrow band-gaps, adequate band positions, and Earth abundancy. However, the flat-band potential of p-Si combined with the low photovoltage associated to its low band-gap directly affects the onset photocurrent potential, which is not positive enough for standalone photoelectrochemical watersplitting. In addition, both p-Si and Cu 2 O suffer from the lack of photostability and require ultra-efficient protective coating to avoid oxidation and photo reduction, respectively. Other solutions exploit critical elements such as indium and gallium. 11,12 Silicon and III-V semiconductors exhibit limited stability in aqueous solutions and must be thoroughly passivated after HF etching of native oxide layer. 9,10,[13][14][15] Metal oxides hold promises in that prospect. 5,16 Unfortunately, copper oxides suffer from photocorrosion if they are not efficiently protected. To overcome this limitation, the interest has shifted to ABO x ternary metal oxides, in which the insertion of a B cation (such Fe, V, Bi, Cr) can perturb the electronic band structure enough to stabilize copper oxides against photocorrosion. 17 In particular CuFeO 2 delafossite, exclusively based on Earth-abundant elements, has a 1.5-1.6 eV bandgap suitable for absorption of visible light and a conduction band positioned negative enough to allow for hydrogen evolution. [18][19][20][21] Only few studies were reported so far toward the use of this material in a photocathode for H 2 evolution, and combining this photoactive material with a surface catalyst to enhance its photochemical properties has not been explored so far. Recently, we and others have demonstrated that cobalt diimine-dioxime [22][23][24][25] or cobaloximes, [25][26][27][28][29] with proper anchoring groups can be used as molecular electrocatalysts for H 2 evolution in metal oxide-based photocathodes. Therefore, we decided to exploit the CoHEC cobaloxime catalyst (Fig. 1) to decorate photoactive copper-iron oxide materials towards the elaboration of a H 2 -evolving photocathode and found that good performance could be reached with the deposition of a thin layer of amorphous TiO 2 as a protective coating. 23,26 Fig. 1 displays the structure and electronics of such a hybrid architecture. The protection of the Cu-Fe oxide layer with a thin layer of amorphous TiO 2 allows to photoproduce H 2 with faradaic efficiencies of 54-88% and with onset potential as high as 0.4 V vs. RHE in neutral aqueous electrolyte under pure visible light irradiation. Such a hybrid approach 30 thus proves a competitive and greener alternative to dye-sensitized photoelectrodes 25,31,32 with the advantage of avoiding the dye preparation and sensitizing steps.

Experimental section

Materials

Fluorine-doped tin oxide (FTO) coated glass substrates were purchased from Solems (Palaiseau, France). Acetone, absolute ethanol, copper nitrate trihydrate, iron nitrate nonahydrate, ethylene glycol, methanol, poly(ethylene glycol)-block-poly( propylene glycol)-block-poly(ethylene glycol) (F-108) and potassium phosphate were purchased from Sigma-Aldrich. Chloro ([4,4′-bipyridine]-2,6-dicarboxylic acid)bis(dimethylglyoximato) cobalt(III) (CoHEC) was purchased from Dyenamo (Sweden). Titanium(IV) isopropoxide (TTIP) was purchased from Strem Chemicals Inc. All the reagents were used as received without further purification.

Photocathode preparation

Cu-Fe oxide (CuFe x O y ) deposition on FTO glass substrates. The procedure was adapted from previously reported studies. 18,33 Glass|FTO substrates (20 × 40 mm 2 ) were cleaned first with soap and water, then with ethanol and finally with acetone, each step during 15 minutes in an ultrasonic bath. Then, they were treated in an UV-ozone cleaner during 15 minutes. A mixture of 0.2 M Cu(NO 3 ) 2 •3H 2 O, 0.2 M Fe (NO 3 ) 3 •9H 2 O and 0.45 M ethylene glycol in ethanol was spin coated on glass|FTO substrates at 3300 rpm during 1 minute followed for 600 rpm during 30 seconds, following a protocol previously reported. 21 The substrates were placed in a flat titanium furnace and heated up to 200 °C in the air during 30 minutes to stabilize the film. The spin coating and the heating processes were repeated 5 times for each glass|FTO substrate. A last annealing step at 450 °C for 30 minutes and then at 600 °C for 2 hours in a tubular furnace under argon flow was performed to eliminate all organic residues, reduce Cu(II) to Cu(I) and crystalize the Cu-Fe oxide films (CuFe x O yFLAT ). 18,33 In some cases, the sixth layer was structured in order to increase the surface area of the CuFe x O y electrode. Here, F-108 polymer (120 mg mL -1 ) was added to the solution used for fabricate the sixth layer. 25 We will refer to these layers as CuFe x O ySTRUCT .

Deposition of amorphous TiO 2 by atomic layer deposition (ALD). TiO 2 was deposited on CuFe x O yFLAT electrodes by ALD (Savannah 100 ALD system, Cambridge Nano Tech Inc.) with precursor temperature at 95 °C, pump line temperature at 150 °C and sample chamber temperature at 255 °C in continuous mode under 5 sccm nitrogen flow. TTIP was used as Ti source and water as O source. 355 cycles were applied, each cycle being composed of 1 s pulse of TTIP, 5 s pause, 0.015 s pulse of water and 5 s pause. XRD measurement where performed on a Smartlab Rigaku diffractometer in glancing incidence: the incident beam was fixed at low angle (1 degree) to maximize the signal of the thin layer. The measurements correspond to a 2θ scan. Film-thickness measurements were performed with a laser scanning microscope VK-X120 (Keyence), using VK Viewer and VK Analyser softwares.

An ellipsometer (M 2000 model) from J. A. Woollam Co., In. was used to measure the thickness of the ALD-TiO 2 substrate, using the CompleteEASE software (V4.72). It was used in the "off-sample" configuration with an incident angle of attack of 75°with respect to the film plane normal, and the light wavelength was scanned between 200 nm and 1700 nm. The obtained data were fitted with the CompleteEASE software (V4.72).

X-ray photoelectron spectroscopy (XPS) analyses were carried out with a Versa Probe II spectrometer (ULVAC-PHI) equipped with a monochromated Al Kα source (hν = 1486.6 eV). The core level peaks were recorded with constant pass energy of 23.3 eV. The XPS spectra were fitted with CasaXPS 2.3 software using Shirley background. Binding energies are referenced with respect to the adventitious carbon (C 1s BE = 284.6 eV).

Inductively coupled plasma atomic emission spectroscopy (ICP-AES) was performed using a Shimadzu ICPE-9000 instrument with a mini plasma torch in axial reading mode. The sample preparation involves the digestion overnight of the samples CuFe x O y , CuFe x O y -CoHEC, CuFe x O y |TiO 2 -CoHEC and CuFe x O y -CoHEC before and after chronoamperometry treatment in 1 mL of HNO 3 (64%).

Photoelectrochemical and hydrogen production measurements

Electrochemical data were acquired with a Biologic VSP 300 (Linear scan voltammetry), an IVIUM compact potentiostat (chronoamperometry) and Autolab potentiostast with FRA module (electrochemical impedance spectroscopy -EIS). All measurements were carried out in a three electrode ( photo) electrochemical cell, filled with pH 6.7 0.2 M phosphate buffer as electrolyte. A titanium wire was used as counter electrode (CE) and a Ag/AgCl (3 M KCl) electrode as reference electrode (RE). Potentials are referred to RHE by using the equation (E = 0.200 V + 0.059 × pH). Calibration of the reference electrode was realized with [K 4 Fe(CN) 6 ] in 0.1 M potassium phosphate buffer at pH 7. 34 Argon or nitrogen was bubbled to the electrolyte for 20 minutes to remove the oxygen before any electrochemical measurement. The geometrical active surface area was 0.5 cm 2 . The electrical contact in the working electrode was from the FTO part. EIS measurements were conducted in 0.2 M phosphate pH 6.7 at 10 kHz and 0.4-0.7 V vs. RHE potential range.

For photoelectrochemical measurements, a 280 W Xe lamp was used to irradiate the electrodes with 1 sun filtered so that only the visible part of the spectrum reaches the photoelectrode (400-780 nm, 65 mW cm -2 ). UV irradiation was filtered with a Spectra-Physics 59472 (λ < 400 nm) cut off filter and IR irradiation was filtered by a Spectra-Physics 6123NS circulating water filter kept at 25 °C. Chronoamperometric measurements were performed in a previously described cell with 4 mL electrolyte volume and 2.5 cm 2 geometrical active surface area. 35 The working electrode was connected with conductive silver paste to a wire, and the cell was completely isolated from air. Hydrogen production was determined by a hydrogen probe from Unisense, polarized at 1 V vs. Ag/AgCl, and calibrated following the procedure described in our previous publication. 35 It was placed inside the photo-electrochemical cell before and after the chronoamperometry to measure the amount of hydrogen in solution. 35

Results and discussion

Different photocathodes for the hydrogen evolution reaction (HER) were prepared following the three-step strategy displayed in Fig. 2. First, the copper-iron oxide (CuFe x O y ) thin layers were prepared by sol-gel method from a mixture of copper and iron salts and spin-coated on glass|FTO substrates, following previous reports 18,33 as described in the Experimental section. Then, amorphous TiO 2 was deposited on CuFe x O y layers by ALD. Afterwards, the CoHEC cobaloxime catalyst was grafted on the top of CuFe x O y |TiO 2 samples by overnight soaking into methanolic solution. On another batch of samples, CuFe x O y layer was nanostructured (CuFe x O ySTRUCT ) using the F108 triblock co-polymer template and the catalyst is grafted at the surface of CuFe x O ySTRUCT without amorphous TiO 2 deposition.

Characterization of the Cu-Fe oxide (CuFe x O y ) layers

Brown colored CuFe x O yFLAT six-layered samples were obtained after several spin-coating deposition and annealing cycles as described in the Experimental section. Observation by SEM revealed a single thin film of 221 ± 30 nm thickness (Fig. 3A) without nanostructuration (Fig. 3Ac). The low active surface area of the electrode could result in poor catalyst loading in further steps. The addition of a triblock polymer known as F-108 in the sol-gel used for the sixth and last cycle resulted in higher nanostructuration (CuFe x O ySTRUCT , Fig. 2 and3Ad). This top layer with a thickness of 137 ± 9 nm (Fig. S1 †) displayed a grain size around 100-200 nm in diameter (Fig. 3Ad). The total thickness of CuFe x O ySTRUCT was estimated to 320 ± 40 nm.

The Cu : Fe atomic ratio measured by EDX on CuFe x O yFLAT thin film was 1 : 1.3 (Fig. S2 †), indicating a copper-deficient delafossite CuFeO 2 structure. XRD revealed that CuFe x O yFLAT electrodes contain a rhombohedral delafossite CuFeO 2 (JCPDS 04-015-2087) phase together with cubic spinel CuFe 2 O 4 (JCPDS 01-074-8585) and Fe 2 O 3 (JCPDS 04-003-2900) parasitic phase (Fig. 3B). These results were confirmed by Raman spectroscopy performed in CuFe x O yFLAT and CuFe x O ySTRUCT (Fig. S3 †). CuFe 2 O 4 is first formed from CuO and Fe 2 O 3 , which are formed from the same sol-gel precursor solution before reaching 180 °C and around 300 °C, respectively. 33,36 Afterwards, the spinel CuFe 2 O 4 phase can be converted with CuO during the last annealing step and form CuFeO 2 delafossite with no need of additional reductant agent. 37 The presence of CuFe 2 O 4 in our films suggests this last step is not fully achieved. The XRD spectrum of CuFe x O ySTRUCT samples revealed a higher ratio between CuFe 2 O 4 and CuFeO 2 phases. Nevertheless, CuFe 2 O 4 has been also reported as a good candidate for HER under visible light, with a similar bandgap and band position as CuFeO 2 delafossite. 38,39 Bandgap energy values (E g ) for direct and indirect allowed transitions were determined by UV-visible absorbance spectrum. Tauc plots were measured based on the UV-visible absorbance data of CuFe x O yFLAT corrected for FTO absorption and taking into account the thickness of the layers. As shown in Fig. 4A, CuFe x O yFLAT thin film exhibited an indirect bandgap The flat band potential (E FB ) was determined through Mott-Schottky plot by performing EIS measurements. The p-type nature of the CuFe x O y layers on FTO was confirmed by the negative slope of the Mott-Schottky plot. E FB was estimated at 0.72 V vs. RHE in 0.2 M phosphate buffer ( pH 6.7) and 10 kHz (Fig. S5 †). 41 This value is more negative than the already reported value of 1 V vs. RHE for delafossite CuFeO 2 18 which again suggested the presence of impurities. The flat-band potential also corresponds to the potential at which the semiconductor starts to show photocurrents (onset potential) [START_REF] Van De Krol | Photoelectrochemical Hydrogen Production[END_REF] and a similar value of E FB 0.70-0.75 V vs. RHE is determined from LSV measurements on CuFe x O ySTRUCT back illuminated with 1 sun visible light (65 mW cm -2 , 400-780 nm) in the same electrolyte (Fig. 4B). This electrolyte was selected based on previous studies on Cu-based oxides conducted in neutral pH. 42,43 The open circuit potential (OCP) of the CuFe x O y layers when illuminating the sample was 0.70 V vs. RHE, and a shift in the photopotential towards less positive potential was recorded in dark (0.63 V vs. RHE), shown in Fig. S6. † When illuminating, the bands tended to flatten and the OCP became more positive. [START_REF] Van De Krol | Photoelectrochemical Hydrogen Production[END_REF] This behaviour also confirms the p-type nature of the semiconductor and its flat band potential measured by EIS. We acknowledge that OCP should be monitored for increasing light intensities until saturation is reached. Indeed if the light intensity is not high enough to flatten the bands, the flat band potential value may be underestimated. [START_REF] Chen | Photoelectrochemical Water Splitting -Standards[END_REF] We note however that the OCP value measured under 1 sun illumination matches both onset photocurrent potential and E FB value determined using Mott-Schottky analysis. When potentials more negative than the onset potential are applied to the CuFe x O ySTRUCT electrodes, the energy bands at the surface of the semiconductor in contact with the electrolyte bend towards more negative potential allowing electrons to travel to the electrolyte. According to the reported data, 18 the diffusion length of the charge carriers of CuFeO 2 delafossite is ≈300 nm. Therefore, the thickness of the CuFe x O y layers was convenient to allow both the electrons from the hole-electron pair created near the interface with the FTO and the holes created at the surface with the electrolyte to reach the electrolyte and the FTO back contact, respectively, before they recombine. Indeed, LSV measured for CuFe x O ySTRUCT electrodes displayed cathodic photocurrents (Fig. 4B) under visible light irradiation. At 0.4 V vs. RHE photocurrent density reached 85 µA cm -2 (Fig. 4B) which is close to the performances reported for similar delafossite CuFeO 2 samples in the absence of other electron acceptor apart from protons. 45 The energy bands position of the CuFe x O y layers and its behaviour in equilibrium with the electrolyte are represented in Fig. 4C. Assuming that the valence band is located 0.1-0.2 V below the flatband potential (E FB = +0.7 V vs. RHE), [START_REF] Morrison | Electrochemistry at Semiconductor and Oxidized Metal Electrodes[END_REF] we determined E VB = +0.8 to +0.9 V vs. RHE. Considering the indirect band gap (E g ) of 1.6 eV, the conduction band is therefore placed at -0.7 to -0.8 V vs. RHE (E CB = E VB -E g ). Therefore, the conduction band is negative enough for protons reduction, as illustrated in the scheme in Fig. 4C.

Catalyst grafting and photoelectrochemical performance for HER

The CoHEC catalyst is a cobaloxime bearing an axial 4,4′-pyridine ligand functionalized with two carboxylic acid anchoring groups.

Following its use in dye-sensitized photoelectrodes, 25,28 it was selected as a candidate to enhance the efficiency of the photoelectrocatalytic proton reduction at the surface of the CuFe x O y electrodes. The grafting of CoHEC on CuFe x O ySTRUCT electrode was achieved by soaking CuFe x O ySTRUCT electrodes in a methanolic CoHEC (0.7 mM) solution overnight. The presence of CoHEC grafted on CuFe x O y was evidenced by the increase in the absorbance around 450-350 nm in the UV-visible absorption spectrum (Fig. S7 †), as previously reported. [START_REF] Huang | [END_REF] The presence of cobalt in the CuFe x O ySTRUCT -CoHEC samples was confirmed by ICP with Co emission observed at 228.616 eV (Fig. S8 †). The ICP signal is within the detection limit of this technique, indicating a surface concentration of 0.085 nmol cm -2 . XPS analysis of CuFe x O ySTRUCT -CoHEC showed characteristic peaks of Co(II) 2p 3/2 with a multiplet splitting due to the coupling with the unpaired electrons and plasmon losses (779.9, 782.3, and 787.7 eV) likely corresponding to a single Co complex, 48 and two nitrogen atoms (N 1s at 399.0 eV for pyridinic N and 401.1 eV for oxime N), confirming the presence of the cobalt center of the CoHEC as well as nitrogen present in CoHEC ligands (Fig. S9 †). Co : N ratio obtained from the XPS studies is 6.3, which is very close to 6 expected from the stoichiometry of the complex.

Catalytic onset potential for hydrogen evolution mediated by CoHEC at a bare FTO electrode in the dark was measured at -0.26 V vs. RHE in 0.2 M phosphate buffer ( pH 6.7) (Fig. 5 and S10 †), in accordance to the values reported in neutral pH. 25 Sensitization with CuFe x O ySTRUCT shifts this onset potential to +0.7 V vs. RHE, in accordance to the E FB value previously determined for CuFe x O y films. In other words, the CuFe x O ySTRUCT material provides a photovoltage of ∼960 mV to the H 2 evolution reaction, which is definitely much higher than that provided by p-Si or Cu 2 O. Indeed, the onset potentials reported for p-Si, 23,49 Such an improvement was confirmed during chronoamperometric measurements performed at 0.4 V vs. RHE under irradiation (Fig. 6A). The photocurrent for CuFe x O ySTRUCT photocathode reached just -14 µA cm -2 . However, when CoHEC was grafted on the surface of CuFe x O ySTRUCT , photocurrents were up to 4 times higher, varying from -60 to -20 µA cm -2 during the first minute and remaining higher than those measured at CuFe x O ySTRUCT photocathode during the first 8 min.

The amount of hydrogen produced in solution was measured with an hydrogen probe, calibrated following the procedure described in previous publication. 35 The calculated faradaic yields (FY) for both CuFe x O ySTRUCT and CuFe x O ySTRUCT -CoHEC after 20 min were 2% and 6%, respecti-vetely (Table 1). Based on the estimated Co surface concentration, the turnover number (TON) was 57 ± 30 for CuFe x O y -CoHEC, corresponding to an average turnover frequency (TOF) of 0.05 s -1 . However, longer chronoamperometries did not result in higher amount of hydrogen produced. These results clearly show that the efficiency of catalytic hydrogen production is enhanced with CoHEC grafted on the surface of CuFe x O y . However, desorption of the catalyst occurred within minutes of operation as shown in Fig. 6A. The photocurrent decreased along time in the case of CuFe x O y -CoHEC, reaching the values similar to CuFe x O ySTRUCT after 8 minutes, and even lower photocurrents afterwards, indicating some degradation of the photocathode materials. Partial loss of cobalt from the surface is confirmed by ICP (Fig. S8 †) and XPS (Fig. S9 †) measurements after 20 min chronoamperometric measurement. For example, the Co : Cu surface atomic ratio drops from initial 3.7 to 0.14. It is also consistent with a former observation made for similar cobaloxime catalysts immobilized onto surfaces through axial pyridine ligand 27,51 for cathodes operated in the dark and for CoHEC grafted onto dye-sensitized NiO. 25 The CoHEC grafting did not affect apparently the CuFe x O ystruct composition, since the observed Cu 2p and Fe 2p XPS peaks are similar for both samples (Fig. S4 †). However, after applying bias potential to the sample in presence of grafted CoHEC, the Cu visible by XPS was almost completely Optimisation through atomic layer deposition of TiO 2

The instability of semiconductors under photoelectrochemical conditions is a general problem that can be solved by deposition of metal oxides protecting 23,52,53 or passivating layers. 54 A classical protection strategy is the deposition of a few nm of thin amorphous TiO 2 layer by atomic layer deposition (ALD), 23,26,49,55,56 allowing to control the thickness and improve the homogeneity of the surface layer. We therefore deposited an amorphous TiO 2 layer, with a 8.3 nm thickness measured by ellipsometry (Fig. S15 †), onto a flat CuFe x O y film by ALD following the procedure reported elsewhere. 23 Fig. 3Ae shows the top view of amorphous TiO 2 deposited on CuFe x O yFLAT electrode, revealing nanostructured TiO 2 . It has been previously shown that a ∼10 nm layer of amorphous TiO 2 is thick enough to passivate a semiconductor. 56 Since the resistance increases and the photoelectrochemical response drops with the thickness of the TiO 2 layer, 56 we limited the ALD process on CuFe x O yFLAT to 355 cycles, corresponding to a thickness of 8.3 nm measured by ellipsometry.

XRD was performed on CuFe x O y |TiO 2 electrodes. The main phase was CuFe 2 O 4 (Fig. S16 † peaks at 781.8, 786.3, and 789.9 eV, while nitrogen 1 s appears poorly resolved at 400.3 eV (Fig. S9 †). Here again, the amount of cobalt catalyst was estimated by ICP at the detection limit of the technique, i.e. close to 0.085 nmol cm -2 . EDX analysis confirmed the presence of Ti on the photoelectrode surface both before and after the photoelectrochemical experiment.

Photocurrents recorded for CuFe x O y |TiO 2 -CoHEC photocathodes proved to be more stable over time since no decrease in the photocurrent was observed after 20 minutes (Fig. 6B). Within 20 min, CuFe 1.

Additional measurements were carried out under the same conditions but with slightly higher visible-light intensity (up to 100 mW cm -2 ; 1.3 sun). CuFe x O ySTRUCT and CuFe x O ySTRUCT -CoHEC displayed 3% and 6% FY respectively, with similar behaviour regarding the desorption of catalyst and poor stability (Fig. S20 †). However, under these conditions, the protected CuFe x O y |TiO 2 -CoHEC photocathode showed 88% FY and 90 TON achieved within 20 min.

The hybrid photocathodes were evaluated up to two hours without increase in the amount of hydrogen detected, suggesting the photocathodes lost their activity following desorption of the catalyst which was already confirmed by the absence of peaks in the ICP and XPS measurements after 20 min CA (Fig. S8 andS9 †).

Clearly, the presence of TiO 2 significantly and positively affects the FY for H 2 evolution. Although the photocurrent values are significantly lower for TiO 2 -protected photoelectrodes, the amount of hydrogen produced remains similar (4.8 nmol cm -2 for CuFe x O y and 4.3 nmol cm -2 for CuFe x O y | TiO 2 ) or largely enhanced (4.8 nmol cm -2 for CuFe x O ySTRUCT -CoHEC and 7.6 nmol cm -2 for CuFe x O y |TiO 2 -CoHEC). This indicates that TiO 2 has no significant effect in the catalysis for hydrogen evolution but that it prevents at least partially the photocorrosion of the semiconductor. The protected photocathodes showed an average Cu : Fe ratio of 1 : 1.8, unchanged after photoelectrochemical measurement. Any modification in the phase of the material was not observed by XRD after chronoamperometry (Fig. S11 †). Additionally, the presence of TiO 2 significantly limits the decrease in the photocurrent intensity over time, indicating a better stability of the anchoring of the carboxylate functions of CoHEC onto TiO 2 compared to CuFe x O y .

When compared with other hybrid photocathodes from the literature, i.e. the p-GaInP 2 /TiO 2 -cobaloxime architecture described by Turner et al. (100% FY, TON = 1 400 000), 26 p-Si/ mesoTiO 2 -NiP system described by Reisner (80% FY, TON = 646) 49 and the p-Si/mesoTiO 2 -Co C11P photocathode reported recently by us (>80% FY, TON > 200), 23 appear limited. However, previous litterature data were measured at an applied potential of 0-0.1 V vs. RHE, i.e. with a much larger driving force compared to data reported here, which were measured at 0.4 V vs. RHE. Besides stability and selectivity, important metrics for the photocathode performances are indeed its onset photopotential and the maximum photocurrent density. While p-Si and Cu 2 O-based photocathodes display limited onset photocurrent potentials (<0. 4 photocathodes. However, the maximum photocurrent reported here is already one order of magnitude higher compared to dye-sensitized molecular photocathodes and there is still room for improvement through the composition optimization and nanostructuration. The cost and scalability of CuFe x O y materials are similar to those of the above mentioned transparent semiconducting oxides, while there is no need to sensitize them as they naturally absorb in the visible region.

Compared with expensive III-V semiconductors such as InP, GaP or GaInP 2 , such ternary oxide-based semiconductors therefore prove promising for implementation in photoelectrochemical cells. 7,25,27,64

Conclusion

We successfully designed a hybrid photocathode for H 2 evolution exclusively based on Earth-abundant elements. 300 nm thick CuFe x O y films indeed behave as a visible light absorber, perform charge separation and transport electron to the interface with the electrolyte where molecular catalyst can be positioned, for H 2 evolution to occur at a potential 860 mV more positive compared to the dark conditions. Atomic layer deposition of amorphous TiO 2 layer on top of the CuFe x O y film enhances the efficiency and improves the stability of the photoelectrode. Nevertheless, the stability of CuFe x O y material remains challenging and must be further improved for PEC applications, calling for a better understanding of the modifications of CuFe x O y films upon heat, irradiation or electrochemical treatment.

This journal is © The Royal Society of Chemistry 2020

Fig. 1 2 |

 12 Fig. 1 Scheme of electron pathway on CuFe x O y |TiO 2 -CoHEC photocathode and energy potentials.

Fig. 2

 2 Fig. 2 Three-step preparation of the CuFe x O ySTRUCT -CoHEC and CuFe x O y |TiO 2 -CoHEC photocathodes: (1) synthesis of multi-layered CuFe x O y light-harvesting semiconductor samples; (2) protection of the semiconductor with TiO 2 or CuFe x O y nanostructuration, and (3) grafting of the CoHEC molecular catalyst.

Fig. 3 (

 3 Fig. 3 (A) SEM sectional images from (a) clean FTO and (b) CuFe x O yFLAT layer synthetized on FTO (same scale). SEM front images from (c) CuFe x O yFLAT , (d) CuFe x O ySTRUCT and (e) CuFe x O y |TiO 2 (same scale). (B) XRD pattern of CuFe x O yFLAT in blue, CuFe x O ySTRUCT in red and FTO in dark blue. All peaks are referenced to delafossite CuFeO 2 (JCPDS 04-015-2087) (brown); spinel CuFe 2 O 4 (JCPDS 01-074-8585) (black) and Fe 2 O 3 (JCPDS 04-003-2900) (orange).

Fig. 4 ( 4 |

 44 Fig. 4 (A) Tauc plots from CuFe x O yFLAT UV-visible spectrum for direct (r = 1 2 ) and indirect (r = 2) bandgap calculation. (B) Linear sweep voltammogram (LSV) of CuFe x O ySTRUCT films under chopped light irradiation (blue), constant irradiation (red) and in the dark (black) in 0.2 M phosphate buffer pH 6.7 at 10 mV s -1 scan rate, under argon; irradiation with 1 sun visible light (400-780 nm). (C) Scheme of the energy bands of CuFe x O yFLAT films, where E FB , E VB and E CB are the flat-band, valence band and conduction band potentials, respectively, while E H+/H 2 is the redox potential corresponding to the proton reduction: left, the electrode is poised at the flat-band potential; right: the electrode is in equilibrium with the solution at the open circuit potential (OCP).

  Cu 2 O 6 or WSe 2 50 photocathodes are 0.4-0.6 V vs. RHE. As shown in Fig. 5 and Fig. S11, † photocurrents measured at CuFe x O ySTRUCT -CoHEC are twice higher than those measured at unfunctionalized CuFe x O ySTRUCT photocathodes in the absence of CoHEC, showing the benefit of This journal is © The Royal Society of Chemistry 2020 Green Chem., 2020, 00, 1-enhance charge extraction and catalysis.

Fig. 5

 5 Fig. 5 LSV for bare FTO substrate (dashed line in black), bare FTO substrate with 0.7 mM CoHEC in solution (straight red line), CuFe x O ySTRUCT modified FTO substrate (straight blue line) and CuFe x O ySTRUCT decorated with CoHEC (straight brown line). Irradiation of 1 sun visible light (400-780 nm) in 0.2 M phosphate buffer ( pH 6.7) at 10 mV s -1 scan rate under nitrogen.

Fig. 6 (

 6 Fig. 6 (A) Chronoamperometries at +0.4 V vs. RHE on CuFe x O ySTRUCT (black) and CuFe x O ySTRUCT -CoHEC (red) photoelectrodes, irradiated by 1 sun visible light (400-780 nm) under nitrogen. (B) Chronoamperometries at 0.4 V vs. RHE on CuFe x O y |TiO 2 -CoHEC under dark (brown) and light (green) conditions. Measurements in 0.2 M phosphate buffer ( pH 6.7).

Table 1 6 |

 16 Hydrogen production, FY, TON and TOF values for CuFe x O ySTRUCT , CuFe x O ySTRUCT -CoHEC, CuFe x O y |TiO 2 and CuFe x O y | TiO 2 -CoHEC photocathodes. 0.2 M phosphate buffer ( pH 6.7), 1 sun irradiation (65 mW cm -2 , 400-700 nm), at 0.4 V vs. RHE applied bias for 20 Green Chem., 2020, 00, 1-9This journal is © The Royal Society of Chemistry 2020 (I), which can explain the limited faradaic efficiency for H 2 evolution. This reduction phenomenon does not occur in the absence of CoHEC grafting but we observed a significant modification of the CuFe x O y material upon turnover. Indeed, the Cu : Fe atomic ratio measured by EDX changed from 1 : 1.3 before electrochemical treatment to 1 : 5 after treatment, indicating a significant loss of copper during operation (Fig.S12 †). The degradation of the surface of the CuFe x O ySTRUCT electrode was confirmed by SEM (Fig.S13 †). XRD performed on CuFe x O ySTRUCT electrodes after treatment revealed new unknown peaks which correspond neither to CuFe 2 O 4 , CuFeO 2 , Cu 2 O nor to Fe 2 O 3 (Fig.S14 †) . To overcome such a degradation, we decided to protect the CuFe x O y material by depositing a thin TiO 2 layer on top.

  ), indicating a structural change of the bulk materials during ALD deposition. No peaks corresponding to any crystalline phase of TiO 2 were observed in the CuFe x O y |TiO 2 electrodes diffractogram (Fig. S16 †) confirming the amorphous character of the TiO 2 layer, while the presence of Ti was confirmed by EDX (Fig. S17 †) and XPS (Fig. S9B †). LSV under choped light of CuFe x O y |TiO 2 -CoHEC is shown in Fig. S18, † revealing lower photocurrent values and a shift of the onset photocurrent potential to ∼0.6 V vs. RHE. This cathodic shift can be explained by the full oxidation of the CuFe x O y material into CuFe 2 O 4 during the ALD process. Although only approximate bandgap values can be determined for composite films, 57 Tauc plots indicate higher (2 eV) indirect bandgap for CuFe x O y |TiO 2 layers compared to CuFe x O y films (Fig. S19 †), in line with the shift of the onset photocurrent potential. CoHEC catalyst was grafted by immersing CuFe x O y |TiO 2 electrodes in a methanolic solution of 0.7 M CoHEC overnight. The presence of CoHEC grafted on CuFe x O y and CuFe x O y |TiO 2 was confirmed by UV-visible spectroscopy (Fig. S7 †). XPS analysis of the latter reveals a Co species manifested as three 2p 3/2

  the performance of the hybrid CuFe x O y |TiO 2 -CoHEC (54-88% FY, TON = 90) may This journal is © The Royal Society of Chemistry 2020 Green Chem., 2020, 00, 1-9 | 7

The resulting electrodes are referred to as CuFe x O y |TiO 2 . CoHEC grafting on CuFe x O ySTRUCT and CuFe x O y |TiO 2 . CuFe x O ySTRUCT and CuFe x O y |TiO 2 electrodes were immersed in

  0.7 mM CoHEC methanol solution overnight, then rinsed with methanol and dried in air before use.

	Characterization techniques
	UV-visible absorption spectra of CuFe x O y and CuFe x O y |TiO 2
	(before and after CoHEC grafting) were recorded with an
	Agilent Technologies Cary 60 UV-Vis spectrometer. The base-
	line corresponding to the glass|FTO support was recorded and
	automatically subtracted from the spectrum of the electrode.
	Scanning electron microscopy (SEM) images showing the
	electrode morphology were recorded with a scanning electron
	microscope Zeiss Ultra 55 operating at 4 kV.

  x O y |TiO 2 -CoHEC photocathodes produced 7.6 ± 2 nmol H 2 per cm 2 with 54% FY. A control experiment of CuFe x O y |TiO 2 in the absence of the CoHEC catalyst was launched recording 4.3 ± 1 nmol H 2 per cm 2 and 22% FY. The TON based on estimated Co surface concentration was 90 ± 30 for CuFe x O y |TiO 2 -CoHEC. The TOFs for CuFe x O y |TiO 2 -CoHEC were estimated to 0.08 s -1 . All these values are summarized in Table

  V vs. RHE), the CuFe x O y -based ones reported here perform better with 0.7 V vs. RHE onset photocurrent potential. In y ) 1-x Zn 2x S 259 photocathodes, such a positive value vs. RHE compares well with the onset photocurent potentials of dye-sensitized NiO 35,60-62 and CuCrO 263 or CuGaO 2 25

						Although not
	as	efficient	as those	displayed	by	WSe 2	58	or
	(CuGa 1-y

Acknowledgements

This work is supported by the CEA DRF Impulsion 2018 and the Make Our Planet Great Again (MOPGA) programs, as well as the Agence Nationale de la Recherche (Labex ARCANE, CBH-EUR-GS, ANR-17-EURE-0003). We acknowledge Adina Morozan for her help regarding the characterization of the CuFe x O y films.

Conflicts of interest

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