Probing patchy reionization through -21cm correlation statistics
We consider the cross-correlation between free electrons and neutral hydrogen during the epoch of reionization. The free electrons are traced by the optical depth to reionization while the neutral hydrogen can be observed through cm photon emission. As expected, this correlation is sensitive to the detailed physics of reionization. Foremost, if reionization occurs through the merger of relatively large halos hosting an ionizing source, the free electrons and neutral hydrogen are anti-correlated for most of the reionization history. A positive contribution to the correlation can occur when the halos that can form an ionizing source are small. A measurement of this sign change in the cross-correlation could help disentangle the bias and the ionization history. We estimate the signal-to-noise of the cross-correlation using the estimator for inhomogeneous reionization proposed by Dvorkin and Smith (2009). We find that with upcoming radio interferometers and CMB experiments, the cross-correlation is measurable going up to multipoles . We also derive parameter constraints and conclude that, despite the foregrounds, the cross-correlation proofs a complementary measurement of the EoR parameters to the 21 cm and CMB polarization auto-correlations expected to be observed in the coming decade.
Subject headings:Reionization, 21 centimeters, Cosmic Microwave Background
The epoch of reionization (EoR) is one of the least understood periods of cosmic history, with only limited observational measurements (see e.g. Loeb and Barkana (2001)). The absence of the Gunn-Peterson trough in the spectra of quasars implies that reionization should have been completed around (Fan et al., 2006a, b). The total optical depth to reionization has been measured to be (Hinshaw et al., 2012). If reionization is assumed to be instantaneous, this would imply a transition redshift of . Besides these constraints, we know very little about the details of reionization, such as the typical halo mass associated with the first ionizing objects as well as the distribution of these objects inside the halos.
The spectral mapping of neutral hydrogen in emission (Hogan and Rees, 1979; Scott and Rees, 1990; McQuinn et al., 2006; Madau et al., 1997; Zaldarriaga et al., 2004; Furlanetto et al., 2004a, b, 2006) promises to be a new probe of the EoR. The spontaneous hyperfine spin flip transition causes the emission of a photon with a wavelength of 21 centimeters in the rest frame. Applying different base filters to the observed emission, it is possible to map the distribution of neutral hydrogen in the Universe as a function of redshift. The auto-correlation of the observed maps is very sensitive to the EoR parameters.
Additionally, cross correlating the observed maps with other observables could provide complementary constraints on the EoR parameters. For example, the 21 cm fluctuations are expected to be correlated with galaxies (Lidz et al., 2009; Wiersma et al., 2012) as well as with the Cosmic Microwave Background (CMB) fluctuations through the Doppler peak and the kinetic SZ effect (see e.g. Alvarez et al. (2006); Salvaterra et al. (2005); Adshead and Furlanetto (2007); Tashiro et al. (2008, 2010, 2011); Holder et al. (2006) and (Jelić et al., 2010; Natarajan et al., 2012) for recent simulations). Unlike the fluctuations in 21 cm, which are a direct representation of the underlying neutral hydrogen, no direct measurement of the electron density at high redshifts exists. The electron density can be measured indirectly through its integrated effect on the CMB, providing us with a number () that tells us the fraction of photons affected by scattering of electrons along the line of sight. One can go further and reconstruct the inhomogeneities in the optical depth field by considering second order effects on the CMB due to the screening mechanism (Dvorkin et al., 2009), Thomson scattering and the kSZ effect (see Dvorkin and Smith (2009), where an estimator of the anisotropic optical depth field is derived and Gluscevic et al. (2012) for an implementation of this estimator to WMAP -year data). In this work we consider the cross-correlation between a reconstructed map of the inhomogeneous optical depth (using CMB polarization observations) and a neutral hydrogen map measured through the redshifted 21 cm lines. Intuitively, these two observables are expected to be anti-correlated on most scales. We would like to stress that this cross-correlation is fundamentally different from direct cross correlations between CMB temperature and polarization and 21 cm maps (Tashiro et al., 2008, 2010); the estimator is a quadratic estimator, making this cross correlation a statistical 3 point correlation function rather than a 2 point function.
Besides providing complementary constraints on the EoR parameters, the cross-correlation between the 21 cm field and the CMB should in principle be less sensitive to the details of the foregrounds. Although current (Masui et al., 2012; Chapman et al., 2012) and upcoming experiments (Harker et al., 2010; Mellema et al., 2012) are expected to be capable of measuring the auto-correlation of 21 cm maps, one very persistent nuisance in extracting the signal from reionization are the foregrounds. Inhomogeneities in the 21 cm signal due to patchy reionization must be separated from fluctuations in foreground sources. Typical foreground sources are faint radio galaxies, starburst galaxies and galaxies responsible for reionization. In addition, our own Galaxy is very bright at the frequencies one aims at for mapping the 21 cm signal from reionization, exceeding the 21 cm reionization signal by several orders of magnitude. Attempts have been made to characterize these foregrounds (Liu et al., 2009; Jelić et al., 2008; Bernardi et al., 2010, 2009; Liu et al., 2012). Despite these efforts, foregrounds can never be fully removed, simply because we do not know their exact origin.
This paper is organized as follows. We review the physics of reionization and derive the expressions for the fluctuations in 21 cm and fluctuations in the optical depth in §2. The former is proportional to the neutral hydrogen fraction, while the latter is proportional to the free electron fraction. Using a simple reionization model (Furlanetto et al., 2004b; Wang and Hu, 2006) where regions of HII are represented by spherical bubbles of typical size we derive an expression for the cross-correlation in section §3. In §5 and §6 we study the one-bubble and two-bubble contributions to the power spectrum. We compute the angular power spectrum of the cross-correlation in §7. We assess the observability of the cross correlation by using a redshift weighting to maximize the signal to noise. We end this section with an estimate of the EoR parameter constraints, when considering LOFAR and SKA noise levels. We present our conclusions in §9. In the Appendix we discuss the dependence of the cross-correlation on the parameters of the reionization model.
Unless specifically mentioned, we use the following set of parameter values throughout the paper: and . Lensing is included and we use the non-linear halo fit model to determine the power spectrum of density fluctuations. We use the WMAP pivot scale Mpc and (Hinshaw et al., 2012).
2. 21 cm brightness and the optical depth to reionization
In this section we will review the standard results for fluctuations in the 21 cm temperature brightness and relate those to fluctuations in the neutral hydrogen fraction. In the second half of this section, we will derive the fluctuations in the optical depth caused by fluctuations in the free electron fraction along the line of sight, confirming the results first obtained in (Holder et al., 2006).
The optical depth of a region of the IGM in the hyperfine transition is given by (Field, 1958)
where MHz is the rest frame hyperfine 21 cm () transition frequency, is the spontaneous emission coefficient for this transition, is the spin temperature of the IGM, weighting the relative population of the atoms in the singlet state to atoms in the triplet state (Field, 1958), is the neutral hydrogen density, and the proper velocity along the line of sight. At high redshifts, where peculiar motions along the line of sight are small compared to the Hubble flow, . At dark energy and radiation are both unimportant and we can solve for in a matter dominated Universe, .
We can now write the following expression for the optical depth:
Here we used , and
where is the neutral hydrogen fraction, i.e. , is the matter energy density and is the proton mass. The factor addresses the fact that not all protons are in hydrogen, but a fraction is in Helium.
The intensity along the line of sight from a thermal source is given by
with the absorption coefficient and the emissivity of photons. Using in the Rayleigh-Jeans limit, we have , while and . Hence, we can write the 21 cm brightness temperature as:
The brightness temperature increment is defined at an observed frequency corresponding to a redshift as
There are usually two types of filters associated with the resolution of the experiment. First, there is a finite angular resolution, which will affect all modes perpendicular to the line of sight. Second, since the brightness temperature of the 21 cm emission is a 3-dimensional field, we are confined to a frequency resolution or bandwidth, which affects the modes along the line of sight (or, equivalently, a redshift resolution).
The total integrated 21 cm surface brightness is given by
Here is an experimental band filter that is due to the finite frequency resolution of the instrument, which is centered around (comoving distance). We define the dimensionless brightness temperature as
In the limit of , .
Now can be written as
We will now consider fluctuations in the free electron density, which in turn will induce fluctuations in the optical depth. The optical depth to distance along the line of sight is given by
where is the Thomson cross-section, is the electron number density and is the scale factor. Relating the free electron density to the free electron fraction , we can write
assuming that Helium is singly ionized.
The average baryon density diffuses in an expanding background as , and the free electron density becomes
The optical depth can in turn be written as
Therefore, we can relate fluctuations in the optical depth to fluctuations in the 21 cm brightness temperature (Holder et al., 2006) as
where we have assumed a delta window function.
It is worth noticing that the above expression for the fluctuations in the 21 cm brightness temperature is only valid for . Early on, when the number of ionizing sources are rare and the temperature of the IGM close to these sources is coupled to the kinetic temperature by Ly photons associated with these local sources, this assumption breaks down, and the 21 cm signal can appear in absorption. We neglect this effect in this paper.
3. Correlating and
We will now cross-correlate the optical depth fluctuations with the temperature brightness. As we saw before, the CMB optical depth is proportional to the free electron density. If reionization is inhomogeneous, the free electron density is a function of position in the sky. Anisotropies in the optical depth produce three effects in the CMB: (i) screening of the temperature and polarization fluctuations that we observe today by an overall factor of . This effect generates CMB B-mode polarization; (ii) Thomson scattering: new polarization is generated by scattering of the local temperature quadrupole that each electron sees along the line of sight. This effect also produces B-modes; and (iii) new temperature anisotropy is generated from the radial motion of ionized bubbles relative to the observer (the kinetic Sunyaev Zel’dovich effect).
The two-point correlation function between the E-modes and the B-modes generated from patchy reionization is proportional to the anisotropic part of the optical depth. This fact allowed the authors in Ref. (Dvorkin and Smith, 2009) to write a minimum variance quadratic estimator for the field . In this work, we will use the CMB polarization fluctuations to reconstruct a map of and cross-correlate it with the 21 cm field.
We will use the shorthand notation . We can write the cross-correlation between the field X (measured through the CMB) and the field (measured through 21 cm) as
Here we defined , and . We make the simplistic assumption that the connected part of vanishes. Here corresponds to fluctuations in the neutral hydrogen fraction, which is given by .
Before we can compute we need to specify our model of reionization. We will assume that the Universe reionized through the growth of ionized bubbles associated with massive halos. The bubbles themselves contain a single source and we assume their size to be larger than the non-linear scale.
We will adopt the following average reionization fraction as a function of redshift
which is the one used in the code CAMB (Lewis et al., 2000). Here , and are free parameters that satisfy our integrated optical depth along the line of sight .
The ionized bubble around a given source is assumed to be spherical with an average radius . We will assume that the typical ionized bubble radii are log-normal distributed (Zahn et al., 2006), i.e. there is a skewness towards smaller bubble sizes,
where is the variance of the distribution.
The average bubble volume is then given by
Hence, we can define a volume weighted radius, such that , which can be written as
If we assume that a given point in space is ionized with Poisson probability, we can write the ionization fraction as
with the number density of bubbles. The brackets around are placed to remind us that we are considering a Poisson distribution of sources, and the result is averaged over the Poisson process. We further assume that the number density of bubbles traces the large-scale structure with some bias :
while the average bubble number density is related to the mean ionization fraction as
Here is the matter over-density smoothed by a top hat window of radius ,
In momentum space,
which is the Fourier transform of for and otherwise.
We further define
4. Reionization parameters
Reionization can only proceed if the seed halo is massive enough for cooling. In particular, line cooling and atomic cooling are important for collapse. One can relate the virial temperature of the halo to the mass of the halo
Therefore, setting a condition on the amount of cooling necessary to form ionizing objects sets a typical mass of the halo, which will be redshift dependent. We can relate the mean number density of bubbles, the average reionization fraction and the typical bubble volume by inverting Eq. (22):
The mean bubble number density introduced in Eq. (22) is derived through an integral over the halo mass function, with a mass threshold which can roughly be set by the viral temperature in Eq. (27) 111Although the model we are using here is self consistent, relating our toy-reinoization model to all other relevant parameters, we find that the resulting bubble radius as a function of redshift is too small compared to simulations (Shin et al., 2008). One can alleviate this discrepancy somewhat by raising . For that purpose we assume the critical temperature to form an ionizing object to be five times the virial temperature.:
We use the Sheth and Tormen (Sheth and Tormen, 2002) halo mass function
where and is the variance of the density field smoothed with the top-hat window function enclosing a mass :
It is straightforward to show that
The parameters , and can be fitted from simulations. is then derived through the constraint . Consequently, by setting we can find an expression for the average bubble volume, and we can infer a function of the average bubble radius as a function of redshift (in the assumption of a log normal distribution of radii at any given redshift).
Likewise, the bubble bias can be related to the halo bias (see e.g. Wang and Hu (2006)) as:
The integral runs over all masses with a threshold mass scale . Sheth and Tormen can be used for the halo bias
Therefore, for any given model of we can compute , and . In this paper we will use , and (Reed et al., 2007).
In Fig. 1 we have plotted the model we use for our average ionization fraction as a function of redshift. Our choice of parameters corresponds to a scenario with a neutral Universe at and a completely ionized Universe at .
5. Two-bubble correlations
The cross-correlation between the neutral hydrogen and free electron fraction has two main contributions. First, the correlation is set by the Poisson distribution of ionizing sources inside the bubbles. This term is referred to as the one-bubble term, and it is dominated by the shot noise. Second, cross correlations can also be induced by the enhanced probability of bubble formation (or ionizing sources) inside overdense regions (with probability ). This term is referred to as the two-bubble term, and is relevant for scales much larger than the average size of a bubble.
with the bubble bias introduced earlier.
Taking the Fourier transform of and , we obtain
Note that here we are considering correlations of the neutral hydrogen fraction (perturbing the free electron fraction accounts for a minus sign in Eq. (38)). The superscript “” denotes the two-bubble contribution. Also, we assume that the baryon fluctuations (the gas) trace the dark matter fluctuations. The total two-bubble contribution to the power spectrum of results in
Let us write , with . We can distinguish two limiting cases: for the power spectrum effective bias is negative, representing an anti-correlation, while for , is positive, and the two-bubble term is positively correlated.
In the large scale limit, when , . This function changes sign when . The bubble bias on average grows towards larger , despite the bubbles being smaller, the bubbles become rare (larger ) and highly correlated. For our choice of parameters we can solve this equation for the redshift and find as the redshift at which the two-bubble term turns negative on large scales. Fig. 2 shows the two-bubble term for various redshifts.
At scales that are smaller than the radius of the bubbles, the two-bubble term is ill-defined (Baldauf et al., 2013): the correlation length becomes shorter than the size of the bubbles, effectively rendering them to one bubble. Therefore, we will neglect the two-bubble correlation term at those scales. In practice, we apply a smoothing filter that effectively cuts the correlation for . We show the two-bubble contribution to the cross-correlation in Fig. 3.
6. One-bubble correlations
For scales much smaller than the average bubble radius, the correlation is dominated by the presence (or absence) of a single bubble (Wang and Hu, 2006). The correlation between two (ionized) points separated by can be written as (Zaldarriaga et al., 2004; Furlanetto et al., 2004a)
where is a function with the following limits: for and for . If the probability for finding one point inside an ionized bubble is , then when the probability of finding the second point in the same bubble is , hence their joint correlation probability is . For large separations, the probability of finding two points in separate bubbles is the product of both probabilities, i.e., . Eq. (40) effectively encodes the smooth transition between these two regimes. The one-bubble correlation for free electrons (or equivalently neutral hydrogen) then becomes:
As long as the bubbles do not overlap, the function can be described by the convolution of two top hat window functions . The one-bubble contribution to the power spectrum can then be written as
The first term in Eq. (42) is the shot noise of the bubbles, which is a direct consequence of randomly placing ionizing galaxies in the Universe. We will later see that this term typically dominates the total correlation function at late times. This can be understood by realizing that the bubbles tend to be larger at late times, assuming that the bubble size increases over time through bubble merging.
Note that when correlating the free electrons with the neutral hydrogen, the one-bubble contribution is always negative, i.e. these are fully anti-correlated when considering just single bubbles in the Universe. In the previous section we have also shown that the two-bubble cross-correlation at early times. As reionization proceeds and the neutral hydrogen fraction decreases, the correlation on all scales will become anti-correlated. More importantly, the signal is proportional to the matter power spectrum, which grows as during matter domination. Within the model applied in this work, we find that at redshift , the two-bubble term is negligible compared to the one-bubble term on most scales.
It was shown by Ref. (Wang and Hu, 2006) that can be approximated as
which is derived by equating the small and large scale limits of Eq. (43) with
We have found the simple fitting solution of Eq. (44) to be accurate to the percent level for most values of . In the Appendix we will show that the contribution from to the one-bubble peak is relatively small for all parameter values in the range of interest for the cm cross-correlation, but is non-negligible and relevant at small scales.
We show the shot noise and in Figs. 4 and 5. The sum of these terms gives the one-bubble power spectrum shown in Fig. 6. Note that the one-bubble term from all three possible correlations (, and ) is equivalent up to a sign.
In the small scale limit, , and the one-bubble term becomes: , rendering the total cross-correlation negative at these scales (at these scales we are applying a smoothing filter to the two-bubble term, so it effectively does not contribute to the total cross-correlation). In Fig. 7 we show the total cross-correlation for at (changing the redshift of the cross-correlation will predominantly affect the average ionization fraction ).
7. Projected cross-correlation
Fourier transforming the dimensionless brightness temperature , we can write the spherical harmonic coefficient for the 21 cm fluctuation as
Note that the response function is centered around , and in practice we take this distance to be somewhere between and .
We can do the same for the optical depth to reionization, i.e. :
where corresponds to the distance to last scattering.
Cross-correlating the two maps yields
Let us consider the cross-correlation in the Limber approximation. Under this approximation, we can assume that the Bessel functions are small, , for and peak when . The integral over comoving momentum will get most of its contribution from modes . Therefore we can make the approximation that and re-write Eq. (LABEL:eq:Clclean) as
Again, in practice we take the window function to be centered around .
In the Limber approximation we can perform the integral over the product of Bessel functions as:
Thus, we can write the angular cross spectrum as
We will assume that the window function is a Gaussian centered around redshift with width given by
where is the bandwidth frequency of the instrument. We have taken into account that the power spectrum explicitly depends on redshift.
We show the angular cross-correlation for MHz and several values of in Fig. 8.
In the previous sections we have shown that the cross-correlation between free electrons and neutral hydrogen has a strong dependence on the parameters that determine the bubble distribution as well as its bias. The location of the peak is set by an effective scale, which we derive in the Appendix. We will see that for a log-normal bubble distribution, this effective scale is exponential in the width of the distribution and inversely proportional to the average bubble radius. Therefore, a small change in the width of the distribution can produce a large change in the location of the peak. We have also shown that at early times the two-bubble term can be positively correlated at large scales. The positive contribution to the correlation function at large scales eventually vanishes when the universe further reionizes. However, if the bubbles are small enough, a positive contribution to the correlation function could persist until late times. Oppositely, if bubbles are relatively large (a few Mpc), the shot noise, which is negative for all scales, will dominate the correlation function. For the reionization parameterization in this paper, the shot noise is the dominated term at the most relevant redshifts (peaking around ).
8. Is the cross-correlation detectable?
In this section we will determine if the cross-correlation is detectable. An important issue that we will address here are the foregrounds. As previously mentioned, the cm emission should be swamped by foregrounds, dominated on large scales by polarized Galactic synchrotron, with a total intensity of 3-4 orders of magnitude larger than the 21 cm brightness from reionization. On small scales, the redshifted 21 cm brightness is obscured by extragalactic sources (Shaver et al., 1999; Jelić et al., 2008). We do not know the spectral dependence of all these foregrounds, but in general we can assume that they are relatively smooth in frequency along the line of sight, as they are associated with same source (e.g. our own Galaxy). In principle, one can therefore remove a large part of the (large scale) foregrounds by removing the largest modes along the line of sight (see e.g. Liu and Tegmark (2012) for a recent discussion).
However, when cross-correlating the 21 cm field with the optical depth, we want to keep the largest modes, to which the integrated optical depth is most sensitive. Hence, we will keep the foregrounds in the observed maps and show that the cross-correlation between foregrounds in the 21 cm field and in the CMB should be small. In order to neglect the cross-correlation of the foregrounds between and cm, we typically need the foreground of the CMB to be times the signal (Liu et al., 2009) (given that ). The synchrotron emission is roughly equal to the CMB signal at GHz. Therefore, if we assume that the synchrotron scales as (Kogut et al., 2007), at GHz ( band) we have . Thus, we estimate that the signal will be larger than the remaining foregrounds after cross-correlating the two maps.
Additionally, by not removing the foregrounds, the 21cm foregrounds will effectively act as noise term in the cross-correlation. In other words, even in the absence of correlation between foregrounds, there is still a finite probability that any given data point in the map will correlate with a foreground measurement from 21 cm, i.e. the induced noise contains a term , where the latter is the spherical harmonic coefficient of the 21 cm foreground map.
Unfortunately, we do not know exactly what the level of synchrotron foreground is, but typically , with . We will assume that the synchrotron emission scales as . La Porta et al. (2008) showed synchrotron emission at 480 MHz has a normalized amplitude of , with the actual amplitude and slope depending on the position in the sky.
Although we cannot remove the foregrounds through implementing a large scale cutoff, we can alternatively try to remove a substantial part of galactic foreground emission. If there is a large correlation between different frequencies of the foreground maps, one could measure the foregrounds at high frequency (corresponding to a completely ionized universe and, hence, with no signal in the cross-correlation), extrapolate with an appropriate scaling and subtract those from the high redshift maps 222 This is scaling is approximate and simplistic. In reality one would probably have to consider a slope that changes as a function of scale and frequency. We are assuming the scaling will be further understood as a function of frequency once we are capable of performing this cross correlation. (Shaver et al., 1999). If the correlation between different maps at high frequencies is of order , one could reduce the overall amplitude of the foreground by a factor of and the power by a factor of a 333Note that this is a very crude estimate. For example, it might be relevant to consider 21 cm signals after reionization (at low ) due to residual neutral hydrogen, primarily in Damped Ly absorbers (DLA’s). . In addition, Liu et al. (2012) showed that down weighting the most heavily contaminated regions in the sky can reduce the effective foreground as much as a factor of . Note that this approach is different from the usual spectral fitting techniques (Shaver et al., 1999; Santos et al., 2005; Wang et al., 2006; McQuinn et al., 2006; Harker et al., 2009; Jelić et al., 2010; Chapman et al., 2012, 2013).
At small scales we expect extra galactic radio sources to dominate the foregrounds. However, there are several strategies that will likely suppress the noise term due to correlations between millimeter and radio emission: Ê (i) since bright sources are expected to dominate the variance at radio frequency (de Zotti et al., 2010; Planck Collaboration et al., 2011; Heywood et al., 2013), Êthese sources can be masked at 5 . ÊThis will suppress the radio source contribution without removing very much of the sky;Ê(ii) at a given location in the map, the radio data short wards and long-wards of the 21 cm radio emission can be used to remove both galactic and extragalactic foreground by assuming that the sources at a given location can be fit by a power lawÊ; (iii) at millimeter and sub millimeter wavelengths, multi-frequency data can be used to separate CMB signal from dusty galaxy foregrounds.ÊThe Planck data shows that the 353 GHz data can be used to remove of the dusty galaxy foreground at 220 GHz (Planck Collaboration et al., 2013). All of these strategies will likely be employed to remove foregrounds in both maps.
We will consider a case in which the angular power spectrum of the foreground is given by:
where corresponds to the frequency of the redshift considered and is the foreground reduction factor that we can hope to achieve through a measurement at low redshift. As we will show later, the signal-to-noise of the cross-correlation does not vary substantially for different values of in this range.
We will do forecasts for a total integration time of 1000 hours, and a beam with an angular diameter of arcmin. We set the bandwidth to MHz. For a LOFAR (van Haarlem et al., 2013) type experiment, we use m and for a Square Kilometer Array (SKA) type experiment we use m.
On the CMB side, experiments are rapidly improving (Pla, 2006; Niemack et al., 2010; Austermann et al., 2012; Zaldarriaga et al., 2008), with high sensitivity experiments coming soon (Planck, ACTPol, SPTPol, CMBPol) and we should have observations of the E- and B-modes polarization spectra up to small scales in the near future. We will now consider a next generation polarization experiment that allows us to reconstruct a map of the optical depth with the estimator proposed by Dvorkin and Smith (2009). This estimator was built to extract the inhomogeneous reionization signal from future high-sensitivity measurements of the cosmic microwave background temperature and polarization fields. Dvorkin and Smith (2009) wrote a minimum variance quadratic estimator for the modes of the optical depth field given by:
where and are the - and -mode polarization power spectra. and correspond to the CMB noise power spectra, and are given by:
where is the detector noise and is the beam size.
The coupling can be written as
Here is the cross-power spectrum between the CMB -mode polarization without patchy reionization and the response field to fluctuations . ( is positive at large scales due to Thomson scattering, and negative at small scales due to the screening).
Furthermore, the reconstruction noise power spectrum is given by: