domain: phys sdu

text:
Marian Douspis

Nabila Aghanim

Mathieu Langer

Rajat Léon V E Koopmans

Mani Thomas

Garrelt Mellema

Hiroyuki Tashiro

Vibor Jelić

Saleem Zaroubi

Rajat M Thomas

A cross-correlation study between the cosmological 21 cm signal and the kinetic Sunyaev-Zel'dovich effect

Keywords: cosmic microwave background -cosmology, theory -diffuse radiation -largescale structure of Universe -radio lines, general

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I N T RO D U C T I O N

The Epoch of Reionization (EoR) is one of the least explored periods in the history of the Universe. At present, there are only a few tentative observational constraints on the EoR such as the Gunn-Peterson troughs (Gunn & Peterson 1965;Fan et al. 2006) and the cosmic microwave background (CMB) E-mode polarization (Page et al. 2007) at large scales. Both of these observations provide strong yet limited constraints on the EoR. In the near future, however, a number of observations at various wavelengths [e.g. redshifted 21 cm from H I, Lyman α emitters, high redshift quasi-stellar objects (QSOs) etc.] are expected to probe this pivotal epoch in much greater detail. Among these, the cosmological 21 cm transition line of neutral hydrogen is the most promising probe of the intergalactic medium during reionization (Madau, Meiksin & Rees 1997) E-mail: vjelic@astro.rug.nl A number of radio telescopes [e.g. Low Frequency Array (LOFAR), 1 Murchison Widefield Array (MWA)2 and Square Kilometre Array (SKA)3 ] are currently being constructed/designed that aim at detecting the redshifted 21 cm line to study the EoR. Unfortunately, these experiments will suffer from a high degree of contamination, due to both astrophysical interlopers such as the Galactic and extragalactic foregrounds, and non-astrophysical instrumental effects (e.g. Jelić et al. 2008;Labropoulos et al. 2009). Fortunately, the signal has some characteristics which differentiate it from the foregrounds and noise, and using proper statistics makes it possible to extract signatures of reionization (e.g. Furlanetto, Zaldarriaga & Hernquist 2004;Harker et al. 2009a,b). In order to reliably detect the cosmological signal from the observed data, it is essential to understand in detail all aspects of the data and their influence on the extracted signal.

Given the challenges and uncertainties involved in measuring the redshifted 21 cm signal from the EoR, it is vital to corroborate this result with other probes of the EoR. In this paper, we study the information imprinted on the CMB by the EoR and its cross-correlation with the 21 cm probe. Given the recent launch of the Planck satellite, which will measure the CMB with unprecedented accuracy, it is fit to conduct a rigorous study into the cross-correlation of these data sets.

One of the leading sources of secondary anisotropies in the CMB is due to the scattering of CMB photons off free electrons, created during the reionization process (Zeldovich & Sunyaev 1969). The effect of anisotropies when induced by thermal motions of free electrons is called the thermal Sunyaev-Zel'dovich effect (tSZ) and when due to bulk motion of free electrons, the kinetic Sunyaev-Zel'dovich effect (kSZ). The latter is far more dominant during reionization (for a review of secondary CMB effects, see e.g. Aghanim, Majumdar & Silk 2008).

The kSZ effect from a homogeneously ionized medium, i.e. with ionized fraction only a function of redshift, has been studied both analytically and numerically by a number of authors; the linear regime of this effect was first calculated by Sunyaev & Zeldovich (1970) and subsequently revisited by Ostriker & Vishniac (1986) and Vishniac (1987) -hence also referred to as the Ostriker-Vishniac (OV) effect. In recent years, various groups have calculated this effect in its non-linear regime using semi-analytical models and numerical simulations (Gnedin & Jaffe 2001;Santos et al. 2003;Zhang, Pen & Trac 2004). These studies show the contribution due to non-linear effects being important only at small angular scales (l > 1000), while the OV effect dominates at large angular scales.

The kSZ effect from patchy reionization was first estimated using simplified semi-analytical models (Santos et al. 2003) wherein they concluded that fluctuations caused by patchy reionization dominate over anisotropies induced by homogeneous reionization. However, for a complete picture of the CMB anisotropies induced by the EoR, a more detailed modelling is required. Over and above the underlying density and velocity fields, these details should include the formation history and 'nature' of the first ionizing sources and the radiative transport of ionizing photons to derive the reionization history (sizes and distribution of the ionized bubbles). Some recent numerical simulations of the kSZ effect during the EoR were carried out by Salvaterra et al. (2005), Zahn et al. (2005), Doré et al. (2007) and Iliev et al. (2007).

Cross-correlation between the cosmological 21 cm signal and the secondary CMB anisotropies provide a potentially useful statistic. The cross-correlation has the advantage that the measured statistic is less sensitive to contaminants such as the foregrounds, systematics and noise in comparison to 'auto-correlation' studies. Analytical cross-correlation studies between the CMB temperature anisotropies and the EoR signal on large scales (l ∼ 100) were carried out by Alvarez et al. (2006), Adshead & Furlanetto (2008) and Lee (2009) and on small scales (l > 1000) by Cooray (2004), Salvaterra et al. (2005) and Slosar, Cooray & Silk (2007). Thus far, the only numerical study of the cross-correlation was carried out by Salvaterra et al. (2005). Some additional analytical work on cross-correlation between the E and B modes of CMB polarization with the redshifted 21 cm signal was done by Tashiro et al. (2008) and Dvorkin, Hu & Smith (2009).

In this paper, we first calculate the kSZ anisotropies from homogeneous and patchy reionization based on 100 h -1 Mpc scale nu-merical simulations of reionization. We then cross-correlate them with the expected EoR maps obtained from the same simulations, and we discuss how the large-scale velocities and primary CMB (pCMB) fluctuations influence the cross-correlation. Although similar in some aspects, the work presented here differs from Salvaterra et al. (2005) substantially. First, Salvaterra et al. used a relatively small computational box (20 h -1 Mpc) incapable of capturing relevant large-scale density and velocity perturbations. Secondly, the pCMB fluctuations, which manifest themselves as a large background noise, were not taken into account. And finally, there is a difference in the procedure for calculating the cross-correlation coefficient.

The paper is organized as follows. In Section 2, we discuss the kSZ signal and cosmological 21 cm signal from the EoR. In Section 3, we present the numerical simulations employed to obtain the kSZ and EoR maps for a specific reionization history. Crosscorrelation between the cosmological 21 cm fluctuations (EoR signal) and the kSZ anisotropies, together with the influence of the large-scale velocities and the pCMB fluctuations on the CMB-EoR cross-correlation, is discussed in Section 4. Finally, in Section 5 we present our discussions and conclusions on the topic.

Throughout we assume cold dark matter ( CDM) cosmology with Wilkinson Microwave Anisotropy Probe 3 (WMAP3) parameters (Spergel et al. 2007): h = 0.73, b = 0.0418, m = 0.238 and = 0.762.

T H E O RY

Here we briefly review the theoretical aspects of the kSZ effect and the cosmological 21 cm signal from the EoR. We also present the relevant mathematical forms used to calculate the kSZ and the cosmological 21 cm signals.

Kinetic Sunayev-Zel'dovich effect

The temperature fluctuation of the CMB caused by the Thompson scattering of its photons off populations of free electrons in bulk motion, for a given line of sight (LOS), is

δT T kSZ = -σ T t 0 tr e -τ n e (r • v)dt, ( 1
)
where τ is the optical depth of electrons to the Thomson scattering, v the bulk velocity of free electrons and r the unit vector denoting the direction of the LOS. The integral is performed for each LOS with t r being the time at the epoch of recombination and t 0 the age of the Universe today. Note that all quantities are in physical units.

Temperature fluctuations produced at time t will be attenuated due to multiple scattering along the LOS to the present time and are accounted for by the e -τ term.

The electron density can be written as the product of the total atom density n n and ionization fraction x e . Both n n and x e vary around their average values nn and xe , and thus these fluctuations can be written as δ = n n / nn -1 and δ xe = x e / xe -1, respectively, and consequently the electron density expressed as

n e = nn xe (1 + δ + δ xe + δδ xe ).
(2)

In the first approximation, one can just follow the reionization of hydrogen and assume that the atom density equals the hydrogen density. However, in our simulation we follow both hydrogen and helium. Assuming that both hydrogen and helium follow the underlying dark matter density, the atom density is a sum of the total hydrogen (n H ) and total helium (n He I ) densities: The mean hydrogen and helium densities vary with redshift as nH,He = nH(0),He(0) (1 + z) 3 , where nH(0),He(0) are the mean hydrogen and helium densities at the present time: nH(0) = 1.9 × 10 -7 cm -3 and nHe(0) = 1.5 × 10 -8 cm -3 .

n n = ( nH + nHe )(1+δ
By inserting equation (2) into equation ( 1) and converting equation (1) from an integral in time to one in redshift space4 (z), we get δT T kSZ = -σ T nn(0)

z 0 zr (1 + z) 2 H e -τ xe × (1 + δ + δ xe + δδ xe )v r dz, (4)
where v r is the component of v along the LOS (v r = r • v) and nn(0) = nH I(0) + nHe I(0) . For a CDM universe, the Hubble constant

at redshift z is H = H 0 m (1 + z) 3 +
where H 0 is the present value of the Hubble constant, m is the matter and the dark energy density.

For homogeneous reionization histories, i.e. a uniform change in the ionization fraction as a function of redshift, equation (4) becomes δT T kSZ = -σ T nn(0)

z 0 zr (1 + z) 2 H e -τ xe (1 + δ)v r dz, (5)
which means that the kSZ fluctuations are induced only by spatial variations of the density field. The linear regime of this effect is called the OV effect. The OV effect is of second order and peaks at small angular scales (arcminutes) and has an rms of the order of a few μK.

The cosmological 21 cm signal

In radio astronomy, where the Rayleigh-Jeans law is applicable, the radiation intensity I (ν) is expressed in terms of the brightness temperature T b :

I (ν) = 2ν 2 c 2 kT b , ( 6
)
where ν is the frequency, c is the speed of light and k is Boltzmann's constant. The predicted differential brightness temperature of the cosmological 21 cm signal with the CMB as the background is given by (Field 1958(Field , 1959;;Ciardi & Madau 2003)

δT b = 26 mK x H I (1 + δ) 1 - T CMB T s b h 2 0.02 × 1 + z 10 0.3 m 1/2 . ( 7
)
Here T s is the spin temperature, x H I is the neutral hydrogen fraction, δ is the matter density contrast and h = H 0 /(100 km s -1 Mpc -1 ).

If we express the neutral hydrogen fraction as x H I = xH I (1 + δ x H I ), equation (7) becomes

δT b = 26 mK xH I (1 + δ + δ x H I + δδ x H I ) × 1 - T CMB T s b h 2 0.02 1 + z 10 0.3 m 1/2 . ( 8
)
In his two seminal papers, Field (1958Field ( , 1959) ) calculated the spin temperature, T s , as a weighted average of the CMB, kinetic and colour temperatures:

T s = T CMB + y kin T kin + y α T α 1 + y kin + y α , ( 9
)
where T CMB is the CMB temperature and y kin and y α are the kinetic and Lyman α coupling terms, respectively. We assume that the colour temperature, T α , is equal to T kin (Madau et al. 1997). The kinetic coupling term increases with the kinetic temperature, whereas the y α coupling term depends on Lyman α pumping through the so-called Wouthuysen-Field effect (Wouthuysen 1952;Field 1958).

The two coupling terms are dominant under different conditions and in principle could be used to distinguish between ionization sources, e.g. between first stars, for which Lyman α pumping is dominant, and first mini-quasars for which X-ray photons and therefore heating are dominant (see e.g. Madau et al. 1997;Zaroubi et al. 2007;Thomas & Zaroubi 2008).

S I M U L AT I O N S

The kSZ (δT /T ) and the cosmological 21 cm maps (δT b ) are simulated using the following data cubes: density (δ), radial velocity (v r ) and H I, H II, He I, He II and He III fractions (x H I,H II,He I,He II and He III ). The data cubes are produced using the BEARS algorithm, a fast algorithm to simulate the EoR signal (Thomas et al. 2009).

In the following subsections, we summarize the BEARS algorithm and describe the operations preformed on the output in order to calculate the kSZ and the EoR maps. Furthermore, we show in detail the calculations for obtaining the optical depth and the kSZ signal along a certain LOS. Finally, we present the maps of the kSZ temperature fluctuations for the two patchy reionization models ('stars' and 'mini-quasars') and discuss aspects of their contribution to the signal.

BEARS algorithm: overview

BEARS is a fast algorithm to simulate the underlying cosmological 21 cm signal from the EoR. It is implemented by using an N-body/smoothed particle hydrodynamics (SPH) simulation in conjunction with a 1D radiative transfer code under the assumption of spherical symmetry of the ionized bubbles. The basic steps of the algorithm are as follows. First, a catalogue of 1D ionization profiles of all atomic hydrogen and helium species and the temperature profile that surrounds the source is calculated for different types of ionizing sources with varying masses and luminosities at different redshifts. Subsequently, photon rates emanating from dark matter haloes, identified in the N-body simulation, are calculated semi-analytically. Finally, given the spectrum, luminosity and the density around the source, a spherical ionization bubble is embedded around the source, with a radial profile selected from the catalogue. For more details, we refer to Thomas et al. (2009).

As outputs, we obtain data cubes (2D slices along the frequency/redshift direction) of density (δ), radial velocity (v r ) and hydrogen and helium fractions (x H I,H II,He I,He II and He III ). Each data cube consists of about 850 slices, each representing a certain redshift between 6 and 11.5. This interval is chosen to match the spectral resolution that the frequency-binned LOFAR data will have, i.e. at 0.1 MHz. This implies a δz of about 3 × 10 -4 at the lowest redshift (z = 6) and ≈0.01 at the high redshift end (z = 11.5), which translates to a minimum comoving separation of 0.1 Mpc at low and <2 Mpc at high redshifts. In both cases, ionized bubbles are sampled extremely well because their typical size (in physical units) is ≈6 Mpc in diameter. Slices have a size of 100 h -1 comoving Mpc and are defined on a 512 2 grid. Because these slices are produced to simulate a mock data set for radio-interferometric experiments, they are uniformly spaced in frequency (therefore, not uniform in redshift). Thus, the frequency resolution of the instrument dictates the scales over which structures in the Universe are averaged/smoothed along the redshift direction. The relation between frequency ν and redshift space z is given by

z = ν 21 ν -1, ( 10
)
where ν 21 = 1420 MHz is the rest frequency that corresponds to the 21 cm line.

The final data cubes are produced using approximately 75 snapshots of the cosmological simulations. Since the choice of the redshift direction in each box is arbitrary, three final data cubes can be produced in this manner (x, y and z).

Randomization of the structures

The kSZ effect is an integrated effect and is sensitive to the structure distribution along the LOS. To avoid unnatural amplification of the kSZ fluctuations due to repeating structures in the simulated data cubes, we follow the approach of Iliev et al. (2007) and introduce randomization of the structures along the LOS over a 100 Mpc h -1 scale in two steps. First, each 100 Mpc h -1 chunk of the data cube is randomly shifted (assuming periodic boundary conditions) and rotated in a direction perpendicular to the LOS. The shift can be positive or negative in any direction [x and (or) y] by an integer value between 0 and 512. The rotation can be clockwise or anticlockwise by an nπ/2 angle (n = 0, 1, 2, 3). Secondly, the final data cube is produced by assembling the first 100 Mpc h -1 part from the x data cube, second from the y data cube, third from the z data cube and then back to the x data cube and so on to a distance that spans the comoving radial distance between redshifts 6 and 12.

Optical depth

The Thomson optical depth τ at redshift z is

τ = cσ T z 0 n e (1 + z) 2 H (z) dz, (11)
where c = 2.998 × 10 8 m s -1 is the speed of light, σ T = 6.65 × 10 -29 m 2 the Thomson scattering cross-section for electrons, n e the density of free electrons and H(z) the Hubble constant at redshift z.

In our simulations, we split the integral into two parts. The first part represents the mean Thomson optical depth ( τ06 ) between redshifts 0 and 6 and the second, τ 6z , from redshift 6 to a desired redshift z. This choice is driven by the limited redshift range (z ∼ 6-11.5) of imminent radio astronomical projects designed to map the EoR. Under the assumption that reionization is completed by redshift 6, the mean Thomson optical depth τ06 is 0.0517. Note that our patchy simulations are set to have a mean Thomson optical depth of 0.087, as obtained from the CMB data (τ = 0.087 ± 0.017; Komatsu et al. 2009).

Creating the kSZ and EoR maps

For clarity, we summarize the steps we follow to create the kSZ and EoR maps for a given scenario of the reionization history.

(i) Using the output of BEARS, data cubes for the density, radial velocity, helium and hydrogen fractions are produced.

(ii) Data cubes are randomized over the 100 Mpc h -1 scale along the redshift direction.

(iii) Using equation ( 11) the Thomson optical depth, τ , is calculated to a redshift z.

(iv) Using the integrand of equation ( 4), data cubes with the kSZ signal are produced as a function of redshift.

(v) Integrating along each LOS through the kSZ data cube, the integrated kSZ map is obtained. Note that we assume that the reionization is complete by redshift 6, so the integral in equation ( 4) spans the range z > 6.

(vi) Finally, the brightness temperature fluctuations, δT b , are calculated using equation ( 8).

As examples, Figs 1 and2 show slices through the simulated redshift cube of the cosmological 21 cm signal (δT b ) and the kSZ effect (δT kSZ ) in the case of the 'Stars' and the 'QSOs' patchy reionization models. The angular size of the slices is ∼0. • 6.

In the following sections, we will use the kSZ and EoR maps produced from five different models of reionization.

(i) Homogeneous: reionization history is homogeneous and the ionized fraction follows

x e = 1 1 + e k(z-zreion) , ( 12
)
with z reion being set to 8.5 and k = 2, 4 and 10 which tunes the 'rapidness' of the reionization process. The mean ionization fractions x e (z) for the three different values of k (homogeneous models: HRH1, HRH2 and HRH3) are shown in Fig. 3.

(ii) Patchy stars: reionization history is patchy, gradual and extended with stars as the sources of ionization.

(iii) Patchy QSOS: reionization history is patchy and relatively fast with QSOs as the ionizing sources.

Apart from the difference in the global shape of the reionization histories driven by 'Stars' and 'QSOs' (see Fig. 4), the average sizes of the ionization bubbles are also smaller in 'Stars' compared to those of 'QSOs'. For a detailed description and comparison of reionization histories due to 'Stars' and 'QSOs', see Thomas et al. (2009).

The kSZ anisotropies from patchy reionization are induced by both fluctuations of the density field δ and ionization fraction δ xe (see equation 4). Santos et al. (2003) found that kSZ anisotropies from δ xe fluctuations dominate over the δ modulated fluctuations (OV effect). In order to test this result with our simulations, we split the integral in equation ( 4) into three parts and produce three integrated kSZ maps (for the 'Stars' model, see Fig. 5). The first term '1 + δ' represents the density-induced secondary anisotropies (OV effect). The 'δ xe ' term represents the secondary anisotropies due to patchiness in the reionization and 'δδ xe ' represents a higher order anisotropy.

The mean and rms of the '1 + δ', 'δ xe ' and 'δδ xe ' components of the simulated kSZ maps are given in Table 1 for patchy reionization in the 'Stars' and 'QSOs' models. The rms value of the maps is used as a measure of the fluctuations. We confirm that the 'δ xe ' fluctuations are indeed larger than density-induced anisotropies ('δ') for both patchy reionization models. However, the difference between the 'δ xe ' and 'δ' fluctuations is much larger for the 'Stars' reionization history model than for the 'QSOs' model. Also note that the third-order anisotropy ('δδ xe ') is not negligible in both reionization scenarios. For completeness, we also give the contribution from the pure Doppler term ('1') in equation ( 4).

C RO S S -C O R R E L AT I O N kS Z -E oR M A P S

The kSZ effect from the EoR is expected to be correlated with cosmological 21 cm maps for a homogeneous reionization history and anticorrelated when patchy (Cooray 2004;Salvaterra et al. 2005;Alvarez et al. 2006;Slosar et al. 2007;Adshead & Furlanetto 2008). In this section, the simulations described in Section 3 are used to explore the small angular scale cross-correlation between the kSZ effect and EoR maps for five different reionization histories. Further, we will fold in the influence of (i) the large-scale velocities on the kSZ effect and (ii) the pCMB fluctuations on the crosscorrelation.

Throughout the paper, we will use a normalized cross-correlation in order to be able to compare results from different pairs of maps. The normalized cross-correlation between two images (a i,j and b i,j ) with the same total number of pixels n is defined at zero lag as where ā( b) is the mean and σ a (σ b ) the standard deviation of image a (b). However, the cross-correlation between the kSZ and the EoR map needs to be considered more carefully, as we will explain in the following paragraph.

C 0 = 1 n -1 i,j (a i,j -ā)(b i,j -b) σ a σ b , (13)
The fluctuations of the kSZ effect over the simulated map are both positive and negative, since the radial velocity v r can be both positive and negative (see equation 4). In contrast, the EoR signal fluctuations in our simulations are always positive (see equation 8). When calculating the cross-correlation between these two maps, we are interested in finding the number of points at which both signals are present (homogeneous reionization model) or where one signal is present and the other absent (patchy reionization model). In other words, only the absolute value of the kSZ fluctuation is relevant in our calculation and not its sign.

Homogeneous reionization history

We explore the cross-correlation between the kSZ map and integrated EoR map in the case of three different homogeneous reionization histories (HRH1, HRH2 and HRH3). These histories are given by equation ( 12), with k = 2, 4 and 10 controlling the duration of reionization (see Fig. 3).

The cross-correlation between an integrated kSZ map and an integrated EoR map results in a coefficient C 0,HRH1 = 0.10 ± 0.03 for an extended homogeneous reionization history (HRH1). For HRH2 C 0,HRH2 = 0.21 ± 0.02 and for HRH3 C 0,HRH3 = 0.24 ± 0.02. The errors are estimated by performing a Monte Carlo calculation with 200 independent realizations of the integrated kSZ and EoR maps using the randomization procedure explained in Section 3. Table 1. Note that each map has its own colour scale. As expected, the integrated kSZ and EoR maps are correlated for homogeneous models of reionization. Furthermore, the correlation depends on the duration of reionization with larger values for more 'rapid' reionization. These results are in agreement with Alvarez et al. (2006).

Patchy reionization history

For the patchy reionization models, we first cross-correlate the kSZ and the EoR maps at a given redshift. The resulting zero-lag coefficient (C 0 ), as a function of redshift, is shown in Fig. 6. The solid black line represents the correlation for 'Stars' while the dashed red line the 'QSOs' patchy reionization model. As expected for patchy reionization in both models, the kSZ and the EoR maps anticorrelate at individual redshifts.

The anticorrelation obtained is also evident by visual inspection of the kSZ and EoR slices through the simulated redshift cubes (see Figs 1 and2). One can see that the kSZ signal is present only at the regions where the EoR signal is not. This result is not surprising since the EoR signal is proportional to neutral hydrogen while the kSZ to the ionized, both of which are almost mutually exclusive.

In reality, we are not able to measure the kSZ effect at a certain redshift but only the integrated effect along the entire history. Thus, we can only cross-correlate the integrated kSZ map with the integrated EoR map and/or the EoR maps at different redshifts. 5Fig. 7 shows the integrated EoR and kSZ map for the 'Stars' (first two panels) and 'QSOs' (last two panels) patchy reionization models. The cross-correlation coefficients at zero lag for these two maps are C 0,Stars = -0.17 and C 0,QSOs = -0.02, respectively. In order to determine the error on the kSZ-EoR cross-correlation, we perform a Monte Carlo calculation. After creating 200 independent realizations of the integrated kSZ and EoR maps using the randomization procedure explained in Section 3, we calculate the cross-correlation coefficient for each pair of realizations. Finally, we calculate the mean and standard deviation of the cross-correlations.

For the 'Stars' model we get C 0,Stars = -0.16 ± 0.02, while for the 'QSOs' model C 0,QSOs = -0.05 ± 0.02.

To understand the higher values of the cross-correlation coefficient in 'Stars' compared to the 'QSOs' model, one needs to analyse Fig. 4 and Table 1. It is evident from Fig. 4 that the reionization history is gradual and extended with stars as ionizing sources, compared to a shorter and sharper history with QSOs as ionizing sources. Moreover, the patchy term ('δ xe ') of the kSZ fluctuations is much larger than the homogeneous component in the anisotropy ('δ') in the case of 'Stars' than for the 'QSOs' model (see Table 1). We showed earlier that the kSZ effect correlates with the cosmological 21 cm signal for homogeneous reionization and that the correlation is strongest for an 'instant' reionization history. We also obtain the same result by correlating different kSZ components with the integrated EoR map (see Table 1). Combining these results we see that the cross-correlation is driven by the patchy kSZ anisotropies in the 'Stars' model, while in the 'QSOs' model the homogeneous and patchy kSZ anisotropies tend to cancel each other. As a consequence, the anticorrelation in the 'QSOs' model is much weaker than that in 'Stars'.

In addition to the balance between homogeneous and patchy kSZ anisotropies that governs the (anti)correlation between the kSZ and the EoR maps, the size of the ionized bubbles also plays a key role. Recall that the average size of the ionization bubble is larger for 'QSOs'. As a result, the underlying structure within the ionized bubble will additionally reduce the anticorrelation and might change the scale of (anti)correlation.

From now on, we will concentrate on cross-correlations using 'Stars' since the 'QSOs' model does not show a significant anticorrelation. Fig. 8 shows the correlation coefficient as a function of lag (C(θ )) between the integrated kSZ and the integrated EoR map. The dashed red lines represent the estimated error obtained from Monte Carlo simulations. As in Salvaterra et al. (2005), we find that the two signals are anticorrelated below a characteristic angular scale θ c and this scale indicates the average size of the ionized bubbles which in our case is θ c ≈ 10 arcmin. Salvaterra et al. (2005) also showed that the amplitude of the anticorrelation signal increases with decreasing redshift and that the characteristic angular scale shows a redshift evolution. In order to test this in our simulation, we calculate the redshift evolution of the zero-lag cross-correlation coefficient between the integrated kSZ map and the EoR map at different redshifts (Fig. 9). To calculate the error in the cross-correlation, we generate 200 different realizations of the kSZ and corresponding EoR cubes using the randomization procedure explained in Section 3. Then, around a desired redshift we fix the kSZ effect to zero and integrate along the non-zero part of the kSZ cube. Finally, we cross-correlate the integrated kSZ map with the EoR map at the desired redshift and estimate the error on the cross-correlation between the integrated kSZ map and the EoR map at the certain redshift. Note that the EoR map at a certain redshift is produced by integrating a 100 h -1 Mpc volume around that redshift.

From Fig. 9, we find no coherent redshift evolution of the anticorrelation signal and at a few redshifts the two signals actually correlate instead of anticorrelating. The correlation at a given redshift is caused by the following. (i) The patchy nature of the EoR signal, which implies that there are some redshifts at which the EoR map contains none or only a few small ionized bubbles. If one correlates such an EoR map with the integrated kSZ map, the outcome is a correlation between the two, and because of an insignificant number of the ionization bubbles there is no contribution to the anticorrelation (ii) The patchy nature of the kSZ signal. There are some redshifts where the kSZ signal from a certain ionization bubble does hardly or not at all contribute to the integrated kSZ map. This could happen due to a weak kSZ signal from a certain ionized bubble or due to cancellation of the kSZ signal from another ionization bubble along the LOS.

To illustrate the patchy nature of the kSZ signal and its implications to the cross-correlation, we plot the kSZ signal, δT kSZ (z), and its cumulative integral, z zmax δT kSZ (z)dz, as a function of redshift, for two random LOSs (see Fig. 10). Note from the bottom panel of Fig. 10 that the kSZ signal, as its progress towards the lower redshifts, fluctuates randomly between positive and negative values. Thus, there is no coherent contribution (continuous increase or decrease) to the kSZ signal over the whole redshift range. See also figs 12 and 13 in Iliev et al. (2007) who reached a similar conclusion.

We repeat the analysis of the redshift evolution of the zero-lag cross-correlation coefficient for different bin sizes along redshift (e.g. redshift bins corresponding to 20 h -1 Mpc in comoving coordinates). However, the result does not differ significantly. We also calculate the redshift evolution of the characteristic angular scale (θ c ), but we do not find any coherent evolution. This result is driven by the fact that the contribution of the kSZ signal from a certain redshift to the integrated kSZ map is not significant or even non-existent. As a result, if there is no coherent contribution (continuous increase or decrease) to the integrated kSZ map over the whole redshift range there will be no coherent redshift evolution of the kSZ-EoR cross-correlation signal (Fig. 10).

The discrepancy between our results and those of Salvaterra et al. (2005) is due to (i) the difference in the method to calculate the cross-correlation coefficient and (ii) the different sizes of the computational boxes.

Salvaterra et al. first calculated the cross-correlation coefficient (not normalized with the rms) between a certain kSZ and EoR map. Then, they scrambled both maps without keeping any structural information and calculated the cross-correlation coefficient. They compared the coefficients in the two cases to draw their conclusion. In contrast to Salvaterra et al., we first calculate the normalized cross-correlation coefficient (see equation 13) between a pair of kSZ-EoR maps. And then for comparison, we perform a Monte Carlo simulation to generate different realizations of the kSZ and the EoR maps. However, despite the cross-correlation procedure Bottom panel: for the same LOSs the cumulative integral of the kSZ effect, z zmax δT kSZ (z)dz. Note that there is no coherent contribution (continuous increase or decrease) of the cumulative kSZ effect over the whole redshift range. used, once the pCMB fluctuations are we not able to find any significant kSZ-EoR cross-correlation (see Section 4.4).

In addition, Salvaterra et al. used fairly small computational boxes (4 and 20 h -1 Mpc) compared to our 100 h -1 Mpc box. Since most of the signal comes around the mid-point of reionization, the velocity field and the typical size of reionization bubbles at that redshift put a strong constraint on the size of simulation that one can use. In fig. 2 of Salvaterra et al. (2005), one can see that at 50 per cent reionization, the size of the reionization bubble is about half the simulation box. This means that no matter how one randomizes the box, the bubble will still overlap with the position of the bubble in the next or previous snapshot. Moreover, the small 20 h -1 Mpc simulation box misses ∼90 per cent of the velocity power as given by the linear theory (see section 4.3 in Iliev et al. 2007), and this could lead to velocity coherence. In other words, the redshift 'enhancement' of the kSZ signal is not fully removed.

Large-scale velocity

Our simulation volume is (100 h -1 Mpc) 3 (see Section 3). Thus, large-scale velocities associated with bulk motions, on scales of 100 h -1 Mpc, are missing. The missing velocities represent ∼50 per cent of the total power in the velocity field as given by the linear theory. Iliev et al. (2007) showed that the large-scale velocities on scales of 100 h -1 Mpc increase the kSZ signal. Motivated by this result, we approximately account for the missing large-scale velocities in a similar way as Iliev et al. (2007): first, we assume that every 100 h -1 Mpc chunk of our simulation cube has a random large-scale velocity component v LS . Since our simulation cube is produced using 15 simulation boxes (100 h -1 Mpc), we need in total 15 v LS . We randomly choose a realization of the 15 v LS based on a velocity field power spectrum from linear theory. By doing this, we ensure that the velocities are correlated at large scales. Finally, we add the missing v LS component to each 100 h -1 Mpc chunk of the simulated cube.

Based on 200 realizations of the large-scale velocity field, we have found that the large-scale velocities increase the kSZ signal during the EoR by 10 per cent. But on average we do not find any significant increase or decrease in the kSZ-EoR cross-correlation.

However, for ∼20 per cent of all large-scale velocity realizations we find an increase in the cross-correlation signal by a factor of 2 or larger and for ∼2 per cent a factor of 3 or larger.

Primary CMB

Up to now, our cross-correlation analysis only took into account the secondary CMB anisotropies generated by the kSZ effect. In the actual experiment, the CMB data will comprise not only the kSZ anisotropies which are secondary, but also the primary and other secondary CMB anisotropies (for a recent review, see Aghanim et al. 2008). In this subsection, we will examine the influence of the pCMB fluctuations on the detectability of the kSZ-EoR crosscorrelation.

We simulate the pCMB fluctuations in the following way: first the CMB power spectra are obtained using CMBFAST (developed by U. Seljak and M. Zaldarriaga in 2003) and then the map of the primary anisotropy is produced as a random Gaussian field with this power spectrum. An example of the simulated pCMB map is shown in Fig. 11. The size of the map corresponds to the size of the simulated EoR and kSZ maps. Note the lack of power at small scales due to the Silk damping (Silk 1967).

In order to calculate the noise in the cross-correlation introduced by the pCMB fluctuations, we generate 200 different realizations of the pCMB fluctuations. We then add secondary kSZ anisotropies induced by the 'Stars' (map shown in Fig. 7) and calculate the cross-correlation between the pCMB+kSZ map and the integrated cosmological 21 cm map. The zero-lag cross-correlation coefficient obtained is 0.0 ± 0.3. The noise introduced by the pCMB fluctuations is thus too large to detect any significant kSZ-EoR (anti)correlation. However, one has to remember that the pCMB anisotropies are damped on small angular scales and that on these scales the secondary anisotropies are the dominant component of the CMB power spectra (see Fig. 12). 6 Utilizing this fact, one can do a cross power spectrum and see the correlation as a function of the angular scale. Pursuing this lead, we first calculate the kSZ-EoR cross spectrum without and then with the pCMB added to the kSZ map.

The cross spectrum (C X l ) between the two images of a small angular size is given by

C X l P X k = 1 n k p,q∈k A p,q • B * p,q , ( 14
)
where A p,q is the Fourier transform of the first image, B * p,q the complex conjugate of the Fourier transform of the second image and n k the number of points in the kth bin (k = p 2 + q 2 ). Note that we assume the 'flat-sky' approximation (e.g. White et al. 1999):

k 2 P (k) l(l+1) (2π) 2 C l | l=2πk
which is valid for l 60. Fig. 13 shows the cross power spectrum between the kSZ anisotropies and the integrated cosmological 21 cm map for reion- ization due to 'Stars'. It is evident from the plot that the two images anticorrelate on large scales (l 8000), but that the anticorrelation becomes weaker towards smaller angular scales. At angular scales l 8000, there is no (anti)correlation.

We also calculate the cross power spectrum between the integrated EoR map and integrated kSZ map with pCMB fluctuations included. In this case, the noise introduced by the pCMB is too large to find any significant correlation at scales l 8000.

This result might be driven by the simulation box size and reionization scenarios considered in this study and does not mean that a cross-correlation signal is absent at all scales and reionization histories. In order to test this, one needs to explore the kSZ-EoR cross-correlation using simulations with box sizes larger than 100 h -1 Mpc.

Additional cross-correlation techniques

For better understanding of the properties of the kSZ-EoR crosscorrelation, and with the hope of being able to find the crosscorrelation signal in the presence of the pCMB fluctuation, in this subsection we apply techniques of filtering, wavelet decomposition and relative entropy to our data. We will only use the integrated kSZ map and integrated EoR map from the 'Stars' model of reionization, since as we saw above, this model produces the strongest cross-correlation signal. Note that in the following analysis, we first use the kSZ and the EoR maps and then as a second step include the pCMB fluctuations.

Fig. 14 shows the zero-lag cross-correlation coefficient for the three different filtering procedures. The first one uses a high-pass, the second a low-pass and the third a band-pass filter that passes out only a certain scale. In all three cases, the filter is based on the 'Top hat' function. We filter out the desired scale from both the kSZ map and the EoR map and calculate the cross-correlation coefficient at zero lag. The results are shown for the low-pass and high-pass filters as a function of the FWHM of the filter and for the band-pass filter as a function of scale. The plot on the left-hand panel in Fig. 14 implies that the anticorrelation is strongest on the largest scales of the map. By adding smaller scales, the correlation coefficient decreases meaning that smaller scales introduce noise in the correlation. The middle panel in Fig. 14 suggests the same behaviour. By removing the large scales, the cross-correlation signal becomes very weak. Finally, the third panel of Fig. 14 suggests that the large scales are indeed the dominant component of the anticorrelation signal.

As a next step in our analysis, we include the pCMB fluctuations. However, we obtain the same result as discussed in Section 4.4. On the scales where the kSZ anisotropies dominate over the primary anisotropies, either the anticorrelation signal is too weak or the noise introduced by residuals of the pCMB fluctuations is too large to find any statistically significant kSZ-EoR (anti)correlation.

The wavelet analysis of the maps is preformed using Daubechies and Coiflet wavelet functions. Both the integrated kSZ map with added pCMB fluctuations and the integrated EoR map are decomposed to a certain wavelet mode and then they are cross-correlated. Because the outcome is similar to that of filtering, we will not discuss this further.

The last method applied to the data is the 'relative entropy', also known as the Kullback-Leibler distance. The relative entropy is a measure of the information shared between two variables (two images) by comparing the normalized distribution of the two. This method also did not produce any significant result.

S U M M A RY A N D C O N C L U S I O N S

This paper presents a cross-correlation study between the kSZ effect and cosmological 21 cm signal produced during the EoR. The study uses an N-body/SPH simulation along with a 1D radiative transfer code (the BEARS algorithm; Thomas et al. 2009) to simulate the EoR and to obtain maps of the cosmological 21 cm signal and the kSZ effect. The maps are produced using a 100 h -1 Mpc comoving simulation box for five different (three homogeneous and two patchy) models of reionization history. The homogeneous model with varying degrees of 'rapidness' of the reionization process is given by equation ( 12). The patchy reionization histories include one by 'Stars' (gradual) and the other by 'QSOs' (instant).

For a homogeneous reionization history, we find that the kSZ map and the integrated EoR map are correlated and that the correlation depends on the duration of reionization with larger values for more 'rapid' models. This result agrees with the analyt-ical kSZ-EoR cross-correlation analysis carried out by Alvarez et al. (2006).

For patchy reionization models, we find that the kSZ temperature fluctuations are of a few μK level (see Table 1) and are in agreement with previous results (Salvaterra et al. 2005;Iliev et al. 2007). In addition, we show that the temperature fluctuations induced by the patchiness of the reionization process ('δ xe ' term in equation 4) are larger than the density-induced fluctuations (homogeneous '1 + δ' term in equation 4). The difference between the two is stronger for the extended history ('Stars' model) than for the more rapid reionization history ('QSOs' model) (see Table 1).

As a first step in the kSZ-EoR cross-correlation study of patchy reionization histories, we cross-correlate the kSZ map and EoR map at each redshift (see Figs 1 and2). As expected, the kSZ and the EoR map anticorrelate at certain redshifts (see Fig. 6).

We then cross-correlate the integrated cosmological 21 cm map and the integrated kSZ map for patchy reionization (see Fig. 7). The two signals show significant anticorrelation only in the 'Stars' model (C 0,Stars = -0.16 ± 0.02, C 0,QSOs = -0.05 ± 0.02.). The result is driven by the balance between homogeneous and patchy ('1 + δ' and 'δ xe ' terms in equation 4) kSZ anisotropies and the average size of the ionized bubbles. Since the homogeneous kSZ anisotropies correlate and patchy kSZ anisotropies anticorrelate with the cosmological 21 cm maps, the two effects tend to cancel each other. In addition, the average size of the ionization bubble is larger for 'QSOs' than for the 'Stars' model and the structure of matter within the ionized bubble reduces the cross-correlation. As a consequence, the kSZ-EoR anticorrelation is much stronger for the extended ('Stars' model) reionization history than for a more instant history ('QSOs' model).

For a patchy model of reionization, we estimate the redshift evolution of the correlation coefficient (C 0 ) and characteristic angular scale θ C . This was done by cross-correlating the integrated kSZ maps with the EoR maps at different redshifts (see Fig. 9). In contrast to Salvaterra et al. (2005), we do not find any significant coherent redshift evolution of C 0 and θ C . This discrepancy is caused by the difference in the procedure used for calculating cross-correlation and the different size of the computational boxes.

The influence of the missing large-scale velocities on the kSZ signal and kSZ-EoR cross-correlation was investigated. Although the large-scale velocities increase the kSZ signal by 10 per cent, we do not find, on average, any significant change in the kSZ-EoR cross-correlation. However, for ∼20 per cent of large-scale velocity realizations we find an increase in the cross-correlation signal by a factor of 2 or larger and for ∼2 per cent by a factor of 3 or larger.

The data from CMB experiments contain both the secondary (e.g. kSZ) and primary anisotropies. We calculate the noise in the kSZ-EoR cross-correlation introduced by the pCMB fluctuations and found that its addition reduces the cross-correlation signal to zero (C 0 = 0.0 ± 0.3). The cross-correlation was also performed on scales where the kSZ anisotropies dominate over the pCMB fluctuations (l 4000; see Fig. 12). We have done this by calculating cross-power spectra (Fig. 13), applying different filtering methods (Fig. 14) on the data and by doing wavelet decomposition. However, the outcome of the analysis is that on the scales where the kSZ anisotropies dominate over primary, either the anticorrelation signal is too weak or the noise introduced by residuals of the pCMB fluctuations is still too large to find any statistically significant kSZ-EoR (anti)correlation.

As a further check, we calculate the kSZ-EoR cross-correlation using the simulation from Iliev et al. (2007) ('f250C' 100 h -1 Mpc simulation). The reionization history of this model is similar to our 'QSOs' model. The reionization history is relatively sharp and instant. The cross-correlation coefficient at zero lag for the integrated kSZ map and integrated EoR map is C 0 = -0.04 ± 0.02. The result is in agreement with the result obtained from the 'QSOs' model. We also calculated the redshift evolution of the zero-lag cross-correlation coefficient and have found no coherent redshift evolution.

In view of all the results obtained from our kSZ-EoR crosscorrelation study, we conclude that the kSZ-EoR anticorrelation on scales captured by our simulation box (∼0. • 6) is not a reliable technique for probing the EoR. However, there is still hope that we will be able to find the correlation between the kSZ and EoR signals on scales larger than ∼1 • , where the patchiness of the ionization bubbles should average out (Alvarez et al. 2006;Tashiro et al. 2009). Finally, it is important to note that the kSZ signal induced during the EoR could still be detected in the power spectra of the CMB and used to place some additional constraints on this epoch in the history of our Universe.

Figure 1 .

 1 Figure 1. A slice through the simulated redshift cube of the cosmological 21 cm signal (top panel) and the kSZ effect (bottom panel) in the case of the 'Stars' patchy reionization model. The angular scale of the slices is ∼0. • 6.

Figure 2 .

 2 Figure 2. The same as Fig. 2 but for the 'QSOs' patchy reionization model.

CFigure 3 .

 3 Figure 3. The ionization fraction x e as a function of redshift for three different models of homogeneous reionization (HRH1, HRH2 and HRH3). All three models are defined by equation (12) but have different values of k (different reionization durations).

Figure 4 .

 4 Figure 4. The mean ionization fraction x e as a function of redshift for the 'Stars' and 'QSOs' patchy reionization models.

Figure 5 .

 5 Figure 5. The simulated kSZ anisotropies induced by '1 + δ' (first panel), 'δ xe ' (second panel) and 'δδ xe ' (third panel) terms in equation (4) for the 'Stars' patchy reionization model. The kSZ anisotropies induced by all terms together in equation (4) are shown in the fourth panel ('TOTAL'). The mean and rms of the simulated kSZ maps are given in Table1. Note that each map has its own colour scale.

Figure 6 .

 6 Figure6. The zero-lag cross-correlation coefficient (C 0 ) between the kSZ map and the EoR map at a given redshift. The solid black line corresponds to the 'Stars' and the dashed red line to the 'QSOs' patchy reionization model. For both reionization models, we find an anticorrelation between the maps.

Figure 7 .

 7 Figure 7. The integrated EoR and kSZ map for the 'Stars' (first two panels) and 'QSOs' (second two panels) patchy reionization models. The mean crosscorrelation coefficient at the zero lag between the integrated EoR map and integrated kSZ map is C 0,Stars = -0.16 ± 0.02 for the 'Stars' and C 0,QSOs = -0.05 ± 0.02 for the 'QSOs' model.

Figure 8 .

 8 Figure8. The cross-correlation between the integrated EoR map and integrated kSZ map as a function of lag (C(θ )) for the 'Stars' reionization history scenario (dashed white line). The grey-shaded surface represents the estimated error obtained by the Monte Carlo simulation. Note that the correlation coefficient at the zero lag is C 0 = -0.16 ± 0.02.

Figure 9 .

 9 Figure9. The redshift evolution of the zero-lag correlation coefficient between the integrated kSZ map and the EoR map at the certain redshift. The result is shown for the 'Stars' reionization history model. Note that the EoR map at a certain redshift is produced by integrating 100 h -1 Mpc volume around that redshift.

Figure 10 .

 10 Figure 10. Top panel: two random (solid and dotted) LOs through the 'Stars' kSZ cube, δT kSZ (z), averaged over 10 pixels (∼0.7 arcmin) at each redshift. Bottom panel: for the same LOSs the cumulative integral of the kSZ effect,

Figure 11 .

 11 Figure 11. The map of the pCMB fluctuations generated as a Gaussian random field with the power spectrum obtained from the CMBFAST algorithm.

Figure 12 .

 12 Figure 12. The power spectra of pCMB fluctuations (dotted line) and kSZ anisotropies obtained from the simulated maps (solid line).

Figure 13 .

 13 Figure13. The cross spectrum (see equation14) between the integrated kSZ map and integrated EoR map for the 'Stars' reionization history (dashed white line). The grey-shaded surface represents the estimated error obtained by the Monte Carlo simulation. Note that the pCMB fluctuations are not included.

Figure 14 .

 14 Figure14. The zero-lag cross-correlation coefficient as a function of the three different filtering procedures. The first one uses a high-pass filter, the second one uses a low-pass filter and the third a band-pass filter that passes only a certain scale. In all three cases, the filter is based on the 'Top hat' function. The dashed white line is the mean and the grey-shaded surface represents the estimated error obtained by the Monte Carlo simulation.

  ). Cross-correlating the 21 cm and kSZ from the EoR 2281 Moreover, the electron density can be written as n e = n H x H II + n He x He II + 2n He x He III , (3) where x H II,He II,He III are ionization fractions of H II, HeII and He III, respectively. The ionization fractions are defined as x H II = n H II /n H , x He II = n He II /n He and x He III = n He III /n He , respectively.

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	C 2010 The Authors. Journal compilation C 2010 RAS, MNRAS 402, 2279-2290

Table 1 .

 1 The mean and rms of the '1 + δ', 'δ xe ' and 'δδ xe ' simulated kSZ maps for both the 'Stars' (see Fig.5) and 'QSOs' patchy reionization models. C 0 is a cross-correlation coefficient at a zero lag between corresponding kSZ maps and the integrated EoR map (see Section 4). For completeness, we also list the results for the pure Doppler term ('1') in equation (4).

		δT kSZ	1	1+ δ	δ xe	δδ xe	Total
	Stars	mean (μK)	-0.004	0.03	0.58	0.02	0.63
		rms (μK)	0.14	0.80	1.74	0.40	2.00
		C 0	0.05	-0.003 -0.12 -0.06 -0.11
	QSOs	mean (μK)	-0.002	0.03	0.27	0.01	0.30
		rms (μK)	0.15	0.93	1.28	0.28	1.57
		C 0	0.1	0.04	-0.08 -0.01 -0.06

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In order to make transformation of equation (1) to the redshift space we use dt = -dz H (z)[1+z] , where H(z) is the Hubble constant at redshift z.

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This is because, unlike the kSZ effect, we can potentially obtain redshiftspecific information of neutral hydrogen via upcoming radio telescopes.

Note that a harmonic multipole l translates to degrees as θ [ • ] = 180 • /l. The angular resolution of the simulated maps is ∼5 arcsec, which translates to l max ∼ 1.3 × 10 5 . The maps are expected to convey the physical information for 1.5 × 10 3 l 1.3 × 10 5 . C 2010 The Authors. Journal compilation C 2010 RAS, MNRAS 402, 2279-2290 Downloaded from https://academic.oup.com/mnras/article/402/4/2279/1746946 by guest on 20 November 2022

This paper has been typeset from a T E X/L A T E X file prepared by the author. C 2010 The Authors. Journal compilation C 2010 RAS, MNRAS 402, 2279-2290 Downloaded from https://academic.oup.com/mnras/article/402/4/2279/1746946 by guest on 20 November 2022

AC K N OW L E D G M E N T S

We acknowledge discussion with the LOFAR-EoR key project members. We thank I. Iliev for providing us with the Iliev et al. (2007) ('f250C' 100 h -1 Mpc) simulation, J. Schaye and A. Pawlik for their dark matter simulation and A. Ferrara for useful comments. We are also thankful to the anonymous referee for his illustrative and constructive comments. As LOFAR members VJ, SZ, LVEK and RMT are partly funded by the European Union, European Regional Development Fund and by 'Samenwerkingsverband Noord-Nederland', EZ/KOMPAS.


Etienne Rodriguez
email: etienne.rodriguez@gmail.com

Alireza Mottaghizadeh

Djamal Gacemi

Daniele Palaferri

Zahra Asghari

Mathieu Jeannin

Angela Vasanelli

Azzurra Bigioli

Yanko Todorov

Mattias Beck

Jerome Faist

Qi Jie Wang
email: qjwang@ntu.edu.sg

Carlo Sirtori
email: carlo.sirtori@univ-paris-diderot.fr

Links at 4.9 µm Wavelength

Keywords: Room-Temperature, Wide-Band, Quantum Well Infrared Photodetector for Microwave Optical Quantum devices, Mid-infrared, Photodetectors, Ultrafast devices

come

However, they often operate at low operating temperatures and possess a relatively lowfrequency bandwidth which makes them not suitable for high-speed applications. [START_REF] Varesi | Fabrication of High-Performance Large-Format MWIR Focal Plane Arrays from MBE-Grown HgCdTe on 4" Silicon Substrates[END_REF] Photodetectors based on intersubband transitions, i.e. quantum well infrared photodetector (QWIP) and quantum cascade detectors (QCDs), have also shown remarkable performance in both mid-wavelength infrared (MWIR) (~3-7 µm) and long-wave infrared (LWIR) (~7-14 µm). Due to their high photoconductive gain and high-speed operation, QWIPs have attracted significant attention. [START_REF] Liu | Photoconductive Gain Mechanism of Quantum-Well Intersubband Infrared Detectors[END_REF] This photodetector technology has the advantages of high absorption, [START_REF] Schneider | Quantum Well Infrared Photodetectors[END_REF] wide tunability of the detection wavelength based on band-structure engineering of the quantum structures, [START_REF] Capasso | Band-Gap Engineering: From Physics and Materials to New Semiconductor Devices[END_REF] and high-speed behaviour due to their intrinsic short carrier lifetime. [START_REF] Liu | High-Frequency Quantum-Well Infrared Photodetectors Measured by Microwave-Rectification Technique[END_REF][START_REF] Grant | Room-Temperature Heterodyne Detection up to 110 GHz with a Quantum-Well Infrared Photodetector[END_REF] This physical parameter is an inherent characteristic of the intersubband transitions which leads to highfrequency response exceeding 100 GHz [START_REF] Grant | Room-Temperature Heterodyne Detection up to 110 GHz with a Quantum-Well Infrared Photodetector[END_REF] and a very high saturation intensity [START_REF] Vodopyanov | Intersubband Absorption Saturation Study of Narrow III -V Multiple Quantum Wells in the Spectral Range[END_REF] allowing a wide range of applications including heterodyne detection, [START_REF] Grant | Room-Temperature Heterodyne Detection up to 110 GHz with a Quantum-Well Infrared Photodetector[END_REF][START_REF] Hofstetter | GHz Operation of a Room Temperature Photovoltaic Quantum Cascade Detector at 5.35 Μm[END_REF][START_REF] Palaferri | Room-Temperature Nine-Μm-Wavelength Photodetectors and GHz-Frequency Heterodyne Receivers[END_REF] characterisation of ultrafast laser pulses [START_REF] Hugi | Mid-Infrared Frequency Comb Based on a Quantum Cascade Laser[END_REF] and wideband free space communications. [START_REF] Martini | High-Speed Digital Data Transmission Using Mid-Infrared Quantum Cascade Lasers[END_REF] The development of QWIP has been successful in both the MWIR and LWIR bands. [START_REF] Rogalski | Infrared Detectors: Status and Trends[END_REF][START_REF] Schneider | Dual-Band QWIP Focal Plane Array for the Second and Third Atmospheric Windows[END_REF] Great efforts have been devoted to the development of LWIR QWIPs. It was used to detect the beating signals between two CO2 lasers at a frequency starting from 35 GHz up to 110 GHz at 10 µm. [START_REF] Grant | Room-Temperature Heterodyne Detection up to 110 GHz with a Quantum-Well Infrared Photodetector[END_REF] Recently, ultrahigh-sensitivity heterodyne measurements in QWIP at room temperature have been demonstrated using metamaterials to enhance light-matter interactions and significantly increase signal-to-noise ratio. [START_REF] Palaferri | Room-Temperature Nine-Μm-Wavelength Photodetectors and GHz-Frequency Heterodyne Receivers[END_REF] On the other hand, in the MWIR region, high responsivity of 1 A/W 15 and background-limited detectivity of 2 x 10 11 Jones 16 have been demonstrated but only at cryogenic temperatures. As far as we know no prior work has been devoted to high-speed and high performance QWIPs in the MWIR region at room temperature.

In this work, we have demonstrated a direct modulation of high-speed QWIP up to 26 GHz at room temperature. The active region of the QWIP is made of 40 periods of strain-compensated In0.1Ga0.9As/Al0.4Ga0.6As alloy grown on GaAs substrate and operate at 4.9 µm. We designed a 25 µm square MESA where the top contact is realised with an air-bridge structure for highfrequency modulations. A high-speed QCL [START_REF] Calvar | High Frequency Modulation of Mid-Infrared Quantum Cascade Lasers Embedded into Microstrip Line[END_REF] that can be modulated up to 26 GHz was used to measure the frequency response of our QWIP device and to detect the beatnote signals at 11.5 and 23.1 GHz. Moreover, we also used rectification techniques to measure the frequency response of the QWIP up to 30 GHz, where the signal is attenuated by approximately 10 dBm. The rectified response is consistent with the direct-modulated QCL measurement.

STRUCTURE, SAMPLE DESCRIPTION AND TEMPERATURE CHARACTERISTICS

The QWIP wafer was grown by molecular beam epitaxy on a GaAs substrate. The active layers are composed of 40 periods of the following sequence: 5 Å In0.1Ga0.9As, 25 Å In0.1Ga0.9As (doped: 5 x 10 11 cm -2 ), and 5 Å In0.1Ga0.9As separated by a 400 Å Al0.4Ga0.6As barrier. The active region is sandwiched between two highly-doped GaAs contact layers as it is drawn in Fig. 1. The bandstructure and relevant energy levels of the detector are displayed in Fig. 1 with E1 the lower energy state, E2 the upper energy state and EF, Eact and VB are respectively the Fermi level, the activation energy of the QW and the total barrier height. The upper energy level is in resonance with the top of the barrier, as in the typical bound-to-quasicontinuum design.

The fabrication of the devices has been conceived to achieve high-speed performances. Square mesas were defined using Inductively Coupled Plasma (ICP) etching system on an annealed Ge/Au/Ni/Au top contact up to the bottom contact. Then, we protected the mesas using a thick photoresist before removing the bottom contact where the coplanar waveguide will take place by wet-etching. The air-bridge metal structure is realised by using two different photoresists, one positive photoresist used as a pillow to support the ridge in front of the mesa, and the negative photoresist for the lift-off process, after the Ti/Au deposition by e-beam evaporation technique. The additional bakes of the positives photoresist are essential to soften the angles and to support the development of the negative photoresist. Moreover, the e-beam deposition must be done with three different angles (+30, 0 and -30 °) following the direction of the coplanar waveguide to ensure good adhesion between the bridge and the substrate. The airbridge between the top of the mesa and the microstrip ensures a low time constant of the effective RLC circuit. The microstrip is widened to a width fitted to the high-frequency launcher. The top view of the device is shown Fig. 2 (a), where the inset shows clearly the squared mesa and the gold air-bridge. Finally, we polished the facet of the device at 45° to efficiently couple the light into the active region.

Fig. 2. (a) The top view of the QWIP (The inset: zoom-in of the active region with the air bridge); (b) Cut-off frequency at -3 dB as a function of the MESA dimensions for the conventional wire bonding (dashed line) and the proposed air-bridge (solid line) devices (The inset: electrical circuit of the device).

In order to optimise the electrical circuit for high speed operation, the wire inductance is suppressed by realising a contact with a small air-bridge connected to a 50 Ω microstrip line. The capacitance of the device is directly linked to the size of the mesa. Considering practical implementations, we designed a relatively small square mesa of 25 x 25 µm 2 . This dimension corresponds to an electrical cut-off frequency of 92 GHz calculated using the equivalent electrical circuit model. CD represents the device capacitance, RC is the access resistance of the coplanar line which is designed to be 50 Ohm to match the impedance of the system load RL, and L is the inductance of the wire bonding (in the case of air-bridge the parasite inductance Lp is negligible).

The fabricated QWIP device is mounted on a cryostat and then cooled down at cryogenic temperatures. Photocurrents at different temperatures and bias voltages are measured by 1:1 imaging of the 0.2 inch aperture blackbody at 1000° C. In order to remove the dark current, the photon flux is chopped at 500 Hz, and the measurements are performed using a lock-in amplifier (Stanford Research SR830). The photocurrent amplitude is measured through a shunt resistance in series with the QWIP without the use of any pre-amplifier. We extracted the power on the QWIP by using a calibrated MCT detector in the exact same configuration. Taking into account the dimensions and the spectral responses of the different photodetectors, the impinging power is 160 nW on the 25 µm square QWIP. The spectra are measured using a Bruker Fouriertransform infrared (FTIR) spectrometer with step scan technique. The signal from the detector is pre-amplified by a low-noise current amplifier (DLPCA-200) and sent to the lock-in amplifier. The internal IR source of the FTIR emit about 52 nW only, despite the low input optical power we were able to measure the response spectra at up to 300 K as shown in Fig. 3. (a). The spectra present an asymmetry, which corresponds to the intrinsic characteristic of the bound-toquasicontinuum design. [START_REF] Palaferri | Antenna Resonators for Quantum Infrared Detectors and Fast Heterodyne Receivers[END_REF] With the increase of the temperature, the central wavelength of the QWIP slightly shifts towards the lower energy (from 2050 cm -1 at 80 K to 2010 cm -1 at 250 K due to the depolarization effect. The linewidth starts to increase from 130 K (from 27.03 meV up to 130 K to 33.48 meV at 250 K) due to the electron-phonon interactions. The exponential fits of the evolution of the dark current with the temperature allowed the calculation of the activation energy of the system which corresponds to 263.2 meV at low bias voltages. This energy is the thermal energy required to excite electrons from the Fermi energy to the continuum and is in a very good agreement with the activation energy simulated in Fig. 1.

The background-limited detectivity and the responsivity at 300 K as a function of the bias voltage are, respectively, shown in Figs. 3 (c) and (d). They exhibit a value of 100 mA/W and 1 x 10 7 Jones for a bias voltage of -5 V.

HIGH-SPEED BEHAVIOUR

The first high-speed characterizations of our QWIPs have been done by direct modulation of a QCL. The measurement setup is shown in Fig. 4 (a). A high-speed current modulated QCL at a wavelength of 4.7 µm is mounted in a high-speed cryostat and operates at cryogenic temperatures. A bias-T is used which allows the injection of the DC and the microwave currents simultaneously. The radiation from the laser is collected and then focused on the QWIP using two convex lenses. The output power of the laser corresponds to 110 mW, however, due to the small size of the detector, only a small fraction of the power is coupled with the QWIP. It results with a photocurrent of about 600 µA at room temperature. The QWIP is also connected through a bias-T in order to measure the modulated photocurrent of the QWIP at high-frequency using a spectrum analyser (Agilent E4407B) limited to 26.5 GHz. The injection is tuned from 100 MHz to 26.5 GHz using a signal generator (Anritsu MG3693B) with a step of 300 MHz at a RF power of -10 dBm.

Fig. 4. (a) The schematic illustration of the setup used to investigate the high-speed modulation of the QWIP. A high-speed direct-modulated quantum cascade laser (QCL) is used in the experiment; (b) Photoresponse of the QWIP with the QCL modulated at -10 dBm at 300 K (red line + symbol) at a bias voltage of -4 V. The rectification measurements of QWIP using the cryostat setup (blue line + symbol) and using the coplanar tip configuration (green line + symbol) with in dashed grey line the resonances due to parasitic effects of its electrical circuit. The black dashed line represents the value at -3 dB.

To avoid microwave pick-ups, we separated the emitter and detector by more than a meter, and the power of the injected microwaves remains as low as possible. Furthermore, to remove the remaining antenna effects, we have detected the background by carrying out the same measurements but with laser beam blocked. All these measurements were realised at room temperature with a bias voltage on the QWIP at -4 V, while the laser operate at low temperature with a radio frequency (RF) injection power of -10 dBm as shown in Fig. 4 (b).

This approach is complementary with the detection of the beating signal shown in Fig. 5. In the direct modulation approach, the RF signal injected in the laser will modulate the IV curve of the laser according to the injected power. It is translated by an amplitude modulation of the laser intensity, which is then transmitted to the photodetector. Therefore, the amplitude modulation detected by the QWIP will also be dependent on the quality of the injection of the laser.

We also verified the high speed measurements by microwave rectification techniques utilising an RF launcher for the QWIP characterizations in the cryostat at 300 K. Due to the inherent non-linear I-V characteristics of QWIPs, microwave rectification technique is a way to study the roll-off behaviour of the QWIP by applying an RF signal to the photodetector and measuring the variation of its DC biasing current. This value reflects the frequency roll-off response of the device limited by the intrinsic-transport and the electrical parameters of the circuit. [START_REF] Schneider | Quantum Well Infrared Photodetectors[END_REF] Using this method, high speed characterizations of photodetectors are no longer limited by the spectrum analyser, but only by the synthesiser RF generator. With this, we can extend the high-speed measurement up to 30 GHz. This measurement has also been carried out by operating a lock-in amplifier technique with an electrical modulation of the RF signal using a signal generator. As a result, the RF injection is over modulated by a 10 kHz rectangular square wave with a duty cycle of 50% and sent to the QWIP though the bias-T with the bias voltage of the QWIP applied. The variation of the bias voltage is directly recovered at the terminal of the generator using a BNC-T and sent to the AC input of the lock-in amplifier synchronised with the signal generator.

The rectification measurements, displayed in Fig. 4 (b), exhibit good agreement with the direct modulation using the QCL modulated at -10 dBm, and show a frequency response up to 30 GHz. At 22 GHz the signal of the direct modulation is attenuated by approximately 8 dBm while at 30 GHz the rectification signal shows a diminution by approximately 10 dBm. We do not identify a clear change of the slope in the bode diagram of the rectification measurements or direct modulations which would correspond to the cut-off frequency. The resonances of our system around 10 GHz are due to some parasitic effects in the electrical circuit using the launcher in the cryostat. We conducted the same measurements using a simpler electrical circuit composed of an RF coplanar tip, where these resonances were not shown as seen in green in Fig. 4 (b). However, this setup suffers from another resonance at higher frequencies represented in grey dashed line, also visible in the transmission line of the electrical circuit not displayed here.

APPLICATION ON FERQUENCY COMB QCL AND OPTICAL LINKS

The optical frequency combs have the potential to revolutionise the communication and especially the free-space communication in the atmospheric windows in the mid-IR where, recently, Joerg Pfeifle et al. has achieved a data transfer rate of 1.44 Tbit s -1 over up to 300 km by using frequency combs sources. [START_REF] Pfeifle | Coherent Terabit Communications with Microresonator Kerr Frequency Combs[END_REF] Our wide-band photodetector is designed to work up to RT in the MWIR. Therefore, it can be used in high-speed MWIR communication data and the characterisation of frequency comb laser by analysing the beatnote signal. The detection of the beating signal of the multimode laser is done by mixing the multiple modes in a high-speed mixer, i.e. QWIP. The frequency of this signal is directly related to the length of the cavity but also to the dispersion which makes it a powerful tool to study frequency combs lasers. Moreover, as one of the important applications of mid-IR photonics, photodetectors could be used to characterise frequency comb lasers. We also detected the beatnote arising from the frequency mixing of longitudinal modes of our multimode laser. Fig. 5 (a) and the inset show the detection on the QWIP at 300 K of the first and second harmonic of the beatnote of a 3 mm laser corresponding respectively to a frequency of 11.58 and 23.15 GHz (the width at -3 dBm of the first harmonic of the beatnote is about 4 kHz). We also performed the injection-locking by increasing the injected power and the injected frequency on the laser as respectively displayed in Fig. 5 (b) and (c). We realised these measurements using a coplanar tip at room temperature. The RF signal injected near the round-trip frequency of the laser will act as a master oscillator and pull or inject the beating signal and therefore, impose the intermode frequency. The injection-locking is a power tool to achieve active mode-locking which stabilise the comb emission of the laser and can significantly broaden the emission spectrum by the generation and proliferation of sidebands in the optical range. [START_REF] St-Jean | Injection Locking of Mid-Infrared Quantum Cascade Laser at 14 GHz, by Direct Microwave Modulation[END_REF][START_REF] Li | Dynamics of Ultra-Broadband Terahertz Quantum Cascade Lasers for Comb Operation[END_REF] This method exhibits another direct method of high-frequency detection, same measurements were realised using the launcher configuration at low, and room temperature not shown here.

CONCLUSION

In conclusion, we have demonstrated that high speed QWIP at 4.9 µm with a modulation that goes up to 30 GHz at room temperature. Our high-speed QWIPs have a specific design for very high frequency based on small surface detector and air-bridges which enables the propagation and injection of high-frequency RF signals. It has been tested, up to room temperature, by direct and indirect method up to, respectively, 26 and 30 GHz both limited by the electrical systems.

We also observed on the QWIP at room temperature the two first harmonics of the beating signal and the injection-locking of 3 mm lasers, which correspond to the beating signals at the frequency of 11.57 and 23.15 GHz, respectively. Beyond the high-speed behaviour of our devices, the intrinsic properties of intersubband transitions and their high responsivity at room temperature make them very promising for many applications in the 3 -5 µm atmospheric window in the MWIR, especially in heterodyne detection, free-space communication and frequency comb QCLs characterisations.

Fig. 1 .

 1 Fig. 1. Band-structure of the developed qauntum well infrared photodetector (QWIP) with 3 quantum wells (QWs) active regions at a bias voltage of 2 V. The most relavant physical parameters of the sructure are indicated in the figure.

Fig. 2 (

 2 b) illustrates the electrical cut-off frequency at -3 dB for 40 periods with wire bonding (dashed line) and with air-bridge (solid line) waveguides. The inset of Fig.2 (b)shows the equivalent circuit where the QWIP is considered as a current generator.

Fig. 3 .

 3 Fig. 3. (a) Spectra of the QWIP at several different temperatures up to 300 K; (b) Currentvoltage (IV) characteristics in background condition for different temperature, inset: activation energy as a function of the bias voltage; background-limited detectivity (c) and responsivity (d) of the QWIP device at 300 K as a function of the bias voltage.

Fig. 3 .

 3 (b) shows the evolution of the background current as a function of the voltage up to room temperature.

Fig. 5 .

 5 Fig. 5. (a) The first harmonic of the beatnote, corresponding to the 3 mm laser, is collected on the QWIP at 300 K. The inset: the second harmonic of the beatnote at 23.15 GHz;

Funding.

  Singapore National Research Foundation, Competitive Research Program (NRF-CRP18-2017-02) and Singapore Ministry of Education Tier 2 Program (MOE2016-T2-2-159).

Acknowledgement. We would like to thank Maria Amanti her useful discussions and suggestions on this work.


Fanny Bachelier

Damien Bazin

Benoit Grosselin

Véronique Daële

Development

Development and validation of on-line auto-GC for the analysis of trace level biogenic VOCs for a field measurement campaign

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  Chromatotec, 15 rue d'Artiguelongue, 33240 Saint-Antoine, France fanny.bachelier@cnrs-orleans.fr 1 ICARE -UPR 3021 -CNRS, 1C Av. de la Recherche Scientifique, CS 50060, F-45071, Orléans cedex 2, France Fanny BACHELIER 1,2 , Damien BAZIN 2 , Benoit GROSSELIN 1 , Véronique DAËLE 1 Development and validation of on-line auto-GC for the analysis of trace level biogenic VOCs for a field measurement campaign Context: Volatile Organic compounds (VOCs) play a major role on atmospheric pollution VOCs emitted in the atmosphere from natural or anthropological sources (1) Biogenic VOCs (BVOCs) = hydrocarbons, isoprene, terpenes & oxygenated compounds (OVOCs) (2) These molecules are found in ambient air from ppt to ppb levels that require sensitive analytical systems Problem: Usually laboratory instruments involving one off-sampling and analysis Need for continuous on-site monitoring systems and allowing automatic data reprocessing Goal: Evaluate suitability of continuous and automatic thermodesorption Gas chromatograph with Flame ionization Detector (FID) and Mass Spectrometer (MS) in parallel for BVOCs measurement in ambient air Flow rate: 3 mL/min Sampling flow: 20 mL/min Column: Capillary MXT30CE column (30m x 0,28mm x 1µm) Trap: Carbopack mixture Thermodesorption: 380°C Oven: 36 -202°C Detector: FID at 190°C Single Quadrupole EI: 70eV Transfer line: 200°C Inlet: 250°C Detection: SCAN mode (0 -300 amu) SEM voltage = 1200V Sampling time: 30 min -Cycle time: Non selective Time driftInterest of dual detection FID and MS in continuous and on-site measurements:To validate the system, certified gas mixtures PAMS 58 and TO15 (Takachiho, Japan) at 100 ppb(V) were used Linearity, limit of detection, precision and accuracy calculated according to the Chinese Norm for ambient air measurements(3) for 4 aromatic hydrocarbons, 11OVOCs and 1 terpene 4 OVOCs not quantified by FID → coelutions FID, MS and validated data accuracies at 4 ppb of aromatic hydrocarbons, OVOCs and terpene Calibration curves of Methyl Ethyl Ketone (MEK), Toluene and αAtmospheric ChemistRy Of the Suburban ForeSt (15 June -25 July) Location: 3 sites in Paris including Rambouillet Forest (80 km from Paris) Aim: Understand physico-chemical interactions of urban and biogenically air masses TD-GC-FID/MS objective: : Gu et al., (2019) China, Ecotoxicology and Environmental Safety 169, 797 -806 (2): Dörter et al., (2020), Science of the total Environment 731, 139 -201 (3): HJ 1010-2018 (2019), Specifications and Test procedures for Ambient Air Quality Continuous Monitoring System with Gas Chromatography for Volatile Organic Compounds Calibration curves of Methyl Ethyl Ketone (MEK), Toluene and α-pinene Performance studies of TD-GC-FID/MS were evaluated to quantify 16 VOCs: 4 aromatic hydrocarbons, 11 OVOCs and 1 terpene in ambient air Algorithm optimized to simplify data reprocessing and select accurate result between FID and MS 3 potential BVOCs identified and quantified with the system during the ACROSS campaign in Rambouillet site Perspectives: Performance studies with an internal standard Data analysis and comparison with other instruments deployed during the Rambouillet campaign

Acknowledgements to the Mass Spectrometry French Society (SFSM) for providing me the opportunity to participate to the Analytics 2022 congress This work was carried out as part of a thesis with a CIFRE #2020/1563 agreement from the National Association for Technical Research (ANRT)