Reconstruction of Cosmological Initial Density Field with Observations from the Epoch of Reionization

The initial density distribution provides a basis for understanding the complete evolution of cosmological density fluctuations. While reconstruction in our local Universe exploits the observations of galaxy surveys with large volumes, observations of high-redshift galaxies are performed with a small field of view and therefore can hardly be used for reconstruction. Here, we propose reconstructing the initial density field using the H i 21 cm and CO line intensity maps from the epoch of reionization. Observations of these two intensity maps provide complementary information on the density field—the H i 21 cm field is a proxy of matter distributions in the neutral regions, while the CO line intensity maps are sensitive to the high-density, star-forming regions that host the sources for reionization. Technically, we employ the conjugate gradient method and develop the machinery for minimizing the cost function for the intensity mapping observations. Analytical expressions for the gradient of cost function are derived explicitly. We show that the resimulated intensity maps match the input maps of mock observations using semi-numerical simulations of reionization with an rms error ≲7% at all stages of reionization. This reconstruction is also robust with an rms error of ∼10% when an optimistic level of shot noise is applied to the CO map or white noise at the level of ≲10% of the standard deviation is applied to each map. Our proof-of-concept work demonstrates the robustness of the reconstruction method, thereby providing an effective technique for reconstructing the cosmological initial density distribution from high-redshift observations.


INTRODUCTION
In the standard theory of cosmology (see, e.g.Mo et al. 2010), initial density perturbations of our Universe were generated by quantum fluctuations and then amplified during the inflationary era.A hierarchy of inhomogeneous structures was formed from small scales (e.g.galaxies and halos) to large scales (e.g.filaments and voids) thanks to gravitational instability.Bridging theories and observations, statistical observables are usually employed to quantify the measurements, which nevertheless are subject to cosmic variance unavoidably.Given observational data of large-volume galaxy surveys in our local Universe, reconstruction of cosmological initial density field is a solution to avoid the cosmic variance, in that galaxies and halos from simulations can be compared Corresponding author: Meng Zhou, Yi Mao zhoum18@mails.tsinghua.edu.cn(MZ), ymao@tsinghua.edu.cn(YM) directly to their counterparts in observations if the initial conditions are all known.Given the input observations of galaxy surveys, Jasche & Wandelt (2013); Wang et al. (2013) sampled the posterior distribution function of the initial density field with the Hamiltonian Monte Carlo (HMC) method and then reconstructed the initial density field accurately in real space (Jasche & Wandelt 2013) or in Fourier space (Wang et al. 2013).
Looking deeper into the Universe, observations of highredshift galaxies by, e.g. the James Webb Space Telescope, are performed with a small field of view (Chen et al. 2023) and therefore can hardly be used for reconstruction.However, line intensity mappings (Bernal & Kovetz 2022) emerge as a promising cosmological probe with a large field of view, generically.In particular, during the epoch of reionization (EoR), the tomographic mapping of the 21 cm line intensity, due to the hyperfine transition of atomic hydrogen, promises to make a 3D image that reveals the evolutionary history of the

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Universe from being neutral to ionized because H i 21 cm line is a proxy of matter density in the neutral regions.Nevertheless, almost no 21 cm signal can be emitted from the ionized regions (aka H ii bubbles).Thus the matter densities in the ionized regions can only be inferred indirectly from their impact on the morphology of H ii bubbles in the 21 cm maps.In the inside-out scenario of reionization, the ionized regions started to form surrounding the first galaxies at high-density regions on average.To fill the gap of ionized regions in the 21 cm maps, therefore, the intensity mappings of molecular lines, e.g.CO(1-0) (Gong et al. 2011;Lidz et al. 2011) and C ii line (Gong et al. 2012;Silva et al. 2015;Dumitru et al. 2019) that are good tracers of high-density, star-forming regions that host the sources of reionization, serve as complementary cosmological probes during the EoR, e.g. they are anti-correlated with the 21 cm signal statistically (Chang et al. 2015).
In this paper, we propose for the first time to reconstruct the cosmological initial density field using future intensity mappings of the H i 21 cm line and CO(1-0) line from the EoR.Technically, we will build new machinery for minimizing the cost function for the intensity mapping observations, and apply it to the EoR observations with sophisticated derivations of the analytical expressions for the gradient of cost function.
The rest of this paper is organized as follows.We introduce our method in Section 2, show the main results in Section 3, and make concluding remarks in Section 4. Some technical details are left to Appendix A (on the optimization of reconstruction) and Appendix B (on the effect of subtracting the mean from signals).

METHODOLOGY
The initial density reconstruction involves two ingredients: (1) a framework that finds the most likely configuration of the initial density field from which the resimulated maps match the input maps of observation, and (2) an underlying theoretical model or simulation tool that connects the initial density field with the simulated maps.For instance, the reconstruction from the galaxy survey observation employs the HMC method as the matching algorithm and the second-order Lagrangian perturbation theory (in Jasche & Wandelt 2013) or modified Zel'dovich approximation (in Wang et al. 2013) for evolving the initial density field to the matter distribution at  = 0.In this paper, for the reconstruction at the time of EoR, we employ the conjugate gradient method1 to minimize the cost function directly.Also, as a first attempt for such a reconstruction, we employ the excursion set model of reionization (ESMR;Furlanetto et al. 2004) as the underlying reionization theory to evolve the initial density field to the ionization field and the Zel'dovich approximation for the evolution of density fluctuations.Throughout this paper, we adopt a ΛCDM cosmology model with Ω m = 0.31, Ω b = 0.048, ℎ = 0.68, and  8 = 0.81.
In §2.1, a general framework for reconstruction will be constructed, assuming that an underlying theory for reionization and density fluctuations is available, and a specific implementation will be made in §2.2 (using the Zel'dovich approximation), §2.3 (modeling of the 21 cm and CO line intensity maps) and §2.4 (using the ESMR), with a description of our simulation setup in §2.5.

Reconstruction Framework
Consider an initial density field  ini  , where the subscript  is the index of comoving cells in 3D real space.In this section, we also represent the initial density field as an -dimensional vector  ini , where  is the total number of cells.
Assume that the initial density field is evolved to the simulated maps,  mod, coev ,  in a coeval simulation box at a later observation time.Here, the superscript "mod" refers to the map from simulations or models, "coev" refers to the coeval box, and the subscript  refers to the type of observation maps, e.g. in §2.3,  = "21cm" and "CO", which denote the 21 cm and CO(1-0) intensity maps, respectively.
Note that the simulated maps usually have higher resolutions, in order to resolve nonlinear structures, than the maps of observation that are subject to instrumental effects.Thus we use  mod ,  to denote the simulated maps that are smoothed to match the resolution in observation, where  is the index of comoving coarse-grained cells.
Here, " ∈ " refers to the summation over the simulation cells (with the dummy index ) that are inside the coarsegrained cell with the index , and  trans is a normalization factor that parametrizes the contribution of each simulation cell to the coarse-grained cell.We adopt a simple treatment of smoothing in Equation (1) for the purpose of the proof of concept.In principle, the smoothed simulated maps should be the convolution of the simulated maps with some window function or filters.More instrumental and/or observational effects can also be included in the forward simulation, which will be the focus of a follow-up paper.
Our goal is to match the smoothed simulated maps  mod ,  to the input maps of observation  inp ,  as accurately as possible.For this purpose, we define the cost function, which is the negative logarithm of the likelihood, as a functional of the initial density field (Wang et al. 2013), where  cost is an overall normalization factor and   is the weight between different types of observations that can be adjusted for optimization,    = 1.Here we assume a white noise  N  that sums up all noises in observations.In the ideal case with only cosmic signals and no noise, we take  N  to be 0.1% of the standard deviation in each mock map .In principle, the framework in this paper can be readily extended to include more realistic effects of noises and systematics, which we will investigate in a follow-up paper.
Reconstruction of the initial density is the process of minimizing the cost function (or equivalently maximizing the likelihood).If we regard the initial density field at each point  as a free parameter, then this is a typical problem of multidimensional optimization.Multidimensional optimization is usually realized by performing one-dimensional optimization (i.e.line minimization) iteratively along different line directions.Here we introduce the conjugate gradient method (Press et al. 2007) that is widely applied in multidimensional optimization.The conjugate gradient method is particularly computationally efficient by constructing a set of directions that are mutually conjugate, which speeds up the convergence.
We start from a guess of initial density field  ini 0 that is randomly picked from a multi-dimensional Gaussian distribution with the covariance matrix given by a modified linear power spectrum2.Suppose that the underlying theory of reionization and density fluctuations is differentiable, i.e. an analytical expression for the gradient  = / ini can be derived from the theory, which is the focus of §2.2 and §2.4.For the initial guess, we evaluate the initial gradient  0 = / ini 0 and set  0 = − 0 as the initial direction of line minimization.We then perform the iterations of the conjugate gradient until convergence is achieved.Assuming that  ini  ,   and   are given from the previous iteration , the algorithm in the ( + 1)-th iteration is as follows.
(1) Perform the line minimization along the direction   and find the minimum point  min, (see the detail below).
(2) Update the field (5) Go to the next iteration.
To perform the line minimization, we follow the Dbrent method (Press et al. 2007), a modified version of Brent's method (Brent 1973).Given the field  ini  and the direction   , the cost function ( ini  +   ) is a scalar function of , where  is a free parameter.Its derivative is The Dbrent method finds the value of  at the minimum point,  min, , by tightening a bracket of  values using the sign of the derivative evaluated with Equation (3) at the central point of the bracket.

Analytical Expression for the Gradient with respect to the Initial Density
This subsection focuses on deriving an analytical expression for the gradient of the cost function with respect to the initial density field, / ini  .The cost function is generally a function of the evolved density field  evol that is in turn a function of the initial density field  ini .Both  evol and  ini are -dimensional vectors.The superscripts "evol" and "ini" imply the evolved and initial density field, respectively.Thus the gradient of the cost function is written using the chain rule as In this paper, we employ the Zel'dovich approximation for the evolution of density fluctuations, i.e. for connecting the initial density field  ini with the evolved density field  evol .Zel'dovich approximation assumes that the separation between the comoving coordinates of a particle and its Lagrangian coordinates is linear to the time-independent displacement.Given the initial position  ini  of the -th particle, its final position  evol  is given by where  evol and  ini are growth factors at the redshifts of the evolved and initial density fields, respectively.The displacement   can be written as where   is the -th wave vector.3

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We smooth the particles to mesh with the Cloud-in-Cell (CIC) algorithm.We assume a cubic particle as the mass of (1 +  ini ) with the same volume of a mesh cell.Therefore, the evolved density is where  CIC () is the CIC window function,   is the position of the -th mesh cell for the evolved density field at the final redshift,  evol  is the final position of the -th particle at the final redshift.The -th particle is the -th mesh cell for the initial density field at the initial redshift.
Thus the partial derivative  evol  / ini  can be written as where Note that  ′ CIC () -the derivative of the CIC window function with respect to  -is a vector.
Suppose that the underlying theory of reionization is differentiable, i.e. an analytical expression for the gradient / evol can be derived from the theory, which is the focus of §2.4.For simplicity, we define   = / evol  and Putting all together, the gradient of the cost function can be written as Equation ( 10) can be computed efficiently with the FFT.Basically,   and the first term in the RHS of Equation ( 10) are just convolutions of   and some kernels, so they can be calculated in Fourier space using the convolution theorem.Also, the second term in the RHS of Equation ( 10) can be rewritten as follows, so it can calculated easily with the FFT. 1

The 21 cm and CO Intensity Mappings
The EoR 21 cm brightness temperature  mod, coev 21cm,  at the comoving cell  is given by the local neutral fraction  HI,  and the local overdensity  evol  , where  21cm = 27 √︁ [(1 + )/10] (0.15/Ω m ℎ 2 ) (Ω b ℎ 2 /0.023) in units of millikelvins.Here, all quantities are implicitly evaluated at a time during the EoR.In this paper, we focus on the limit where the spin temperature is much higher than the cosmic microwave background temperature, which is valid soon after reionization begins.As such, the dependence on spin temperature is neglected as in Equation ( 11).
We employ the ESMR to identify the ionized regions.Specifically, cells inside a spherical region are identified as ionized, if the number of ionizing photons in that region is larger than that of neutral hydrogen atoms, or  coll ( evol ,  ,  vir ) ≥  −1 .Here,  is the ionizing efficiency,  vir is the minimum virial mass of haloes that host ionizing sources,  evol ,  is the local, evolved overdensity that is smoothed over a sphere with the radius  and the center at the cell , and  coll ( evol ,  ,  vir ) is the collapsed fraction smoothed over that sphere.The smoothing scale  proceeds from the large to small radius until the above condition is satisfied.If this does not happen with  down to the cell size, then the cell at  is considered as partially ionized with the ionized fraction of   coll ( evol ,  ,  vir ).Without loss of clarity, we use  to denote the scale of cell size4 in the rest of this paper.In other words, the neutral fraction field is given by For the CO(1-0) line, we assume that the CO brightness temperature is proportional to the local star formation rate density that is proportional to the collapse fraction on the scale of cell size, .

Analytical Expression for the Gradient with respect to the Evolved Density
This subsection focuses on deriving an analytical expression for the gradient of the cost function with respect to the evolved density field, / evol  , based on the ESMR.It can be written as where The overall normalization factor  cost is a free parameter, so we set  trans  cost = 1 for convenience.
For the 21 cm map, the derivative is Both 21 cm and CO maps are determined by the local collapse fraction  coll (see Equations 12,13,18,19).We employ the Press-Schecter (PS) halo mass function (Press & Schechter 1974) to calculate the collapse fraction and its derivative, and normalize them with the Sheth-Torman (ST) correction (Sheth & Tormen 1999).
where fST and fPS are the mean ST and PS collapse fractions, respectively.  () is the critical overdensity for collapse at redshift .  is the variance of density fluctuations at the smoothing scale , and  min = ( vir ) is the variance at the scale corresponding to the minimum halo mass  vir .The last ingredient is the derivative  evol ,  / evol  that is involved in the calculations for both derivatives of the 21 cm and CO maps (see Equations 18 and 19).The smoothed overdensity  evol ,  is the convolution of the overdensity  evol  and a window function.If we rewrite it in the form of matrix multiplication as  evol ,  =  F    evol  , then  evol ,  / evol  = F   , where the matrix F   is given by the smoothing kernel and the smoothing scale .In practice, we exploit the symmetry We compare the gradients calculated analytically ("/") and numerically("Δ/Δ") at 50 random locations, respectively.The diagonal dashed line indicates the perfect matching.
F   = F   to simplify the calculation when summations over  and  are put together in Equation ( 14).Now we can calculate the derivative / ini for the 21 cm and CO intensity maps with an analytical formalism.To test the accuracy of this formalism, we choose 50 locations randomly and compare the derivatives / ini  that are calculated analytically and numerically in Figure 1, respectively.For the numerical implementation, we shift the initial density at the chosen location (with the cell index ) with a small variation Δ ini  = 10 −4  ini  and calculate the resulting difference of the cost function Δ, and then use the finite difference method to estimate the gradient, Δ/Δ ini  .The comparison shows that the analytical results are in good agreement with the numerical results.

Simulations
We perform semi-numerical simulations of reionization with a modified5 version of the code 21cmFAST (Mesinger et al. 2011).It is based on the semi-numerical treatment of cosmic reionization with the excursion-set approach (Furlanetto et al. 2004) to identify ionized regions.Our simulations were performed on a cubic box of 368 comoving Mpc on each side, with 128 3 grid cells.We choose a reference simulation with the parameter value  = 25 and  vir = 5 × 10 8  ⊙ .Note that these reionization parameters are fixed in the initial density reconstruction.The reconstruction of initial density fields considered in this paper is made in the comoving cubic volume where the mock 21 cm and CO brightness temperature fields are measured at three different redshifts  = 7.56, 8.20 and 9.54 (corresponding to three stages of reionization, xHI = 0.25, 0.50 and 0.75), independently.We set the initial density field of the mock observations at the redshift  ini = 20.The mock observations are constructed from the reference simulation in a coeval manner in the sense that we neglect the lightcone effect (Datta et al. 2012(Datta et al. , 2014) ) across the simulation box along the line-of-sight.We list the mean 21 cm and CO brightness temperature and their standard deviations in Table 1.We also assume the pixel resolution of observations corresponds to 5.75 comoving Mpc in the coarse-grained cells.

RESULTS
We present the optimal performance of the reconstruction in this section while leaving the optimization technique to Appendix A.

Mock with Cosmic Signals
We first consider the reconstruction in the ideal case with only cosmic signals and no noise.From the reconstructed initial density, we resimulate the 21 cm and CO intensity maps at given redshifts.In Table 1, we show the normalized rms difference,  tot , as a metric for evaluating the goodness of reconstruction, which is defined as where   is the number of coarse-grained cells in mock observations, and  S  is the standard deviation of the "signal maps", i.e. the mock maps that include only cosmic signals and no noise.Note that we use  S  , instead of the amplitude of the input temperature itself, as the denominator in Equation ( 21), because the temperature can be nearly zero in some points but the field amplitude can be represented by its standard deviation  S  statistically.We find that the reconstruction is accurate in the sense that the errors in the resimulated 21 cm or CO maps are within 7% on average with respect to the input maps.
For visualization purposes, we show a slice of the resimulated 21 cm intensity maps and the input, mock observation in Figure 2, and the same comparison for the CO intensity maps in Figure 3.Comparison by eye finds almost no difference between the resimulated map and the input map.In Figure 4, we compare a slice of the input, true initial overdensity with the reconstructed initial overdensity field using the mock observations from three different redshifts.Their difference is small as seen in the bottom panel, too.
To evaluate the difference in a quantitative manner, we plot the comparison of overdensity in the top panel of Figure 5, and the probability distribution function (PDF) of the residual overdensity in the bottom panel of Figure 5.We find that the distribution of the residual overdensity is Gaussian.At high redshift  = 9.54 ( xHI = 0.75), the distribution has a zero mean, which implies that the reconstruction is unbiased.However, at lower redshifts  = 8.20 and 7.56 ( xHI = 0.50 and 0.25 respectively), the mean of the distribution is on the positive side, which indicates that the reconstructed initial overdensity is overestimated with respect to the true initial overdensity.Also, the top panel shows that this overestimation mostly takes place in the overdense regions ( > 0).This is likely due to the fact that at lower redshifts, the ionized bubbles have large sizes (∼ tens of Mpc), so the impact of the initial density field on the 21 cm maps at the later stage of reionization is not as local as that in the early stage of reionization.
To further evaluate the clustering of the reconstructed initial density field, we plot its power spectrum in the first panel of Figure 6, and its fractional difference with respect to the input, true initial power spectrum in the second panel of Figure 6.While we find their agreement at large scales, the reconstructed overdensity power spectrum is overestimated at small scales, particularly at the middle stage of reionization ( xHI = 0.50) at the level of error ∼ tens of percent.
Regarding the power spectrum of the 21 cm and CO maps, the power spectra of the resimulated maps and the input maps are almost indistinguishable, so we only plot their fractional difference in the third and fourth panel of Figure 6.We find that the power spectra of the resimulated map have an error of ≲ 4% with respect to the true power spectra in most cases.

Mock with Noises
We estimate the effect of noise on the reconstruction in this subsection.Again, as a proof of concept, here we assume a white noise that sums up all noises in observations and leave it to follow-up work to include more realistic modeling of      as a function of wavenumber .We show the input, true overdensity power spectrum (black) and that reconstructed from the mock observations of the 21 cm and CO maps at different stages of reionization xHI = 0.25 (blue), 0.50 (red) and 0.75 (yellow), respectively.(Second panel) the fractional difference of the reconstructed overdensity power spectrum with respect to the true power spectrum.(Third panel) the fractional difference of the power spectrum of the resimulated 21 cm map with respect to that of the input 21 cm map.(Fourth panel) same as the third panel but for the CO map.noises and systematics.We include a noise  N  in the input mock map, with three scenarios in terms of its ratio to the standard deviation of the "signal map"  S  :  N  / S  = 10 −3 , 10 −2 , and 10 −1 .In Figure 7, we plot the goodness of reconstruction as a function of  N  / S  .We find that the goodness of reconstruction is as good as the ideal case without noise if  N  / S  ≲ 0.01.However, once the noise is at the level of 0.1  S  , the reconstruction is significantly worse (with rms error ≳ 0.1) than the ideal case.This can be considered as a rough estimation of the noise level required for future observations for the purpose of the initial density reconstruction.

CONCLUSION
In this paper, we propose to reconstruct the cosmological initial density field using the H i 21 cm and CO line intensity maps from the EoR.We employ the conjugate gradient method and develop the machinery for minimizing the cost function for the intensity mapping observations and apply this framework to the reconstruction from the EoR observations.Specifically, an analytical formalism for the gradient of the cost function is derived using the ESMR and the Zel'dovich approximation as the underlying theory for reionization and density fluctuations.Our results demonstrate that the resim- We show the reconstruction from the mock observations of the 21 cm and CO maps at different stages of reionization xHI = 0.25 (blue), 0.50 (red) and 0.75 (yellow), respectively.The dashed lines indicate the results for the ideal case without noise.Note that each case has only one realization, so the cases with small noise  N  / S  ≲ 10 −2 may have even better results than the ideal case.

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ulated intensity maps match the input maps of mock observations with an rms error ≲ 7% at all stages of reionization.This reconstruction is also robust at the same level of accuracy when a noise at the level of ≲ 1% of the standard deviation of the signal map is applied to each map.This suggests that our work provides an effective technique for reconstructing the cosmological initial density distribution from high-redshift observations.Nevertheless, our proof-of-concept work has a few limitations.We only adopt a simple treatment of smoothing kernels from the simulation cells to the observation pixels, and assume a white noise that sums up all noises in observations.In principle, our work can be extended to include more realistic modeling of smoothing, noises, and observational effects (e.g.redshift-space distortions) as parts of forward simulations, which will be further developed in a follow-up paper.Note.-For each strategy at a given stage of reionization (labeled by xHI ), we show the optimum weight  21cm in the step of full construction, the number of iterations in the preprocessing  pre , the number of iterations in the full reconstruction  full , the total wall-clock time  tot based on a test using eight CPU cores, and the goodness of reconstruction  tot .The star ('*') marks the optimal choice of strategy and weight for each mock observation at a given xHI , which is used in the main text of this paper. .(B1) The third term on the RHS of Equation ( B1) is zero statistically in coeval boxes and in a lightcone as it is the mean of subtracted signals at each frequency channel.The second term on the RHS of Equation (B1) is the difference of mean signals.It may be different from the full signal case at the first few steps if fluctuations of our initial guess are smaller than the true field.Nevertheless, this effect becomes small once beyond the burn-in phase.Therefore, we conclude that for the reconstruction of initial density, the effect of subtracting the mean is negligible.

Figure 2 .
Figure 2. The 21 cm brightness temperature maps of the input field (top) and the resimulated field (bottom) in units of millikelvins.We show the maps in a slice of simulated coeval box with 368 comoving Mpc on each side, (from left to right) at redshift  = 7.56, 8.20 and 9.54, corresponding to global neutral fraction xHI = 0.25, 0.50 and 0.75, respectively.

Figure 3 .
Figure 3. Same as Figure 2 but for the CO brightness temperature maps in units of microkelvins.

Figure 4 .
Figure 4.The initial overdensity field  ini in a slice of comoving volume with 368 Mpc on each side.We show the input, true field (top left), and the reconstructed field (middle) using the mock observations of the 21 cm and CO maps in a coeval box (from left to right) at redshift  = 7.56, 8.20, and 9.54, respectively.For the purpose of comparison, we also show the residual between the reconstructed and the true initial overdensity (bottom).

Figure 5 .
Figure 5. Calibration of the initial density reconstruction.(Top) we show the reconstructed initial overdensity (" recon ") vs the input, true one (" original ").(Bottom) the PDF of the residual  recon −  original .The reconstruction is made using the mock observations of the 21 cm and CO maps in a coeval box (from left to right) at redshift  = 7.56, 8.20, and 9.54, respectively.The dashed lines indicate the perfect matching.

Figure 6 .
Figure 6.(First panel from the left) the dimensionless overdensity power spectrum Δ 2 as a function of wavenumber .We show the input, true overdensity power spectrum (black) and that reconstructed from the mock observations of the 21 cm and CO maps at different stages of reionization xHI = 0.25 (blue), 0.50 (red) and 0.75 (yellow), respectively.(Second panel) the fractional difference of the reconstructed overdensity power spectrum with respect to the true power spectrum.(Third panel) the fractional difference of the power spectrum of the resimulated 21 cm map with respect to that of the input 21 cm map.(Fourth panel) same as the third panel but for the CO map.

Figure 7 .
Figure7.The goodness of reconstruction  tot as a function of the ratio between the noise and the standard deviation of the signal map,  N  / S  .We show the reconstruction from the mock observations of the 21 cm and CO maps at different stages of reionization xHI = 0.25 (blue), 0.50 (red) and 0.75 (yellow), respectively.The dashed lines indicate the results for the ideal case without noise.Note that each case has only one realization, so the cases with small noise  N  / S  ≲ 10 −2 may have even better results than the ideal case.

Table 1 .
Mock Observations of the 21 cm and CO Brightness Temperature Fields -the Redshift, the Mean Neutral Faction, the Global Average, and the Standard Deviation of the 21cm Field and the CO Field.We Also Show the Goodness of Reconstruction for the Resimulated Fields.