--- title: 21cm One-Point Statistics Overview url: https://www.emergentmind.com/topics/21cm-one-point-statistics type: topic --- # 21cm One-Point Statistics Overview Searching arXiv for recent and foundational papers on 21 cm one-point statistics, PDFs, counts-in-cells, and wavelet-based non-Gaussian summaries. 21 cm one-point statistics are statistics of the distribution of the 21 cm signal evaluated at single spatial locations, voxels, spherical cells, or line-of-sight samples, rather than explicit correlations between distinct points. In practice, the term covers the probability distribution function (PDF) of the brightness-temperature field, its moments such as variance and skewness, voxel intensity distributions, counts-in-cells PDFs, conditional one-point estimators such as stacking, and multiscale summaries defined in wavelet space. They are used in three closely related settings: Cosmic Dawn and the Epoch of Reionization, where the 21 cm field is highly non-Gaussian because of patchy ionization and heating; post-reionization HI intensity mapping, where nonlinear structure formation and tracer bias generate non-Gaussian voxel statistics; and the 21 cm forest, where narrow absorption features yield rare-event tails and strong scale coupling along the line of sight [1412.3332][2606.30200][2504.14656]. ## 1. Definitions and field variables The underlying observable depends on the regime. During Cosmic Dawn and the Epoch of Reionization, the differential brightness temperature is written as $$ T_b(\mathbf{x}, z) \simeq 27\,x_{\rm HI}(\mathbf{x}, z)\,\big(1 + \delta_m(\mathbf{x}, z)\big)\, \left(1 - \frac{T_{\rm CMB}(z)}{T_S(\mathbf{x}, z)}\right) \left(\frac{1+z}{10}\right)^{1/2} \left(\frac{0.15}{\Omega_m h^2}\right)^{1/2} \left(\frac{\Omega_b h^2}{0.023}\right)\,{\rm mK}, $$ up to the approximations stated in the literature, so one-point statistics probe the joint distribution of neutral fraction, density, and spin-temperature fluctuations [1412.3332]. In the post-reionization regime, the brightness temperature is commonly written as $$ T_b(z) = \bar{T}_b(z)\,[1+\delta_{\rm HI}(z)], $$ with $$ \bar{T}_b(z) = 0.566\,h\,\frac{H_0}{H(z)}\!\left(\frac{\Omega_{\rm HI}}{0.003}\right)\!(1+z)^2~{\rm mK}, $$ so one-point observables become statistics of the HI overdensity field and its tracer mapping from matter [2606.30200]. For the 21 cm forest, the observable is a one-dimensional brightness-temperature or optical-depth spectrum along a pencil beam, with $$ \delta T_b(s,\nu) \approx \frac{T_S(s,z) - T_\gamma(s,\nu_0,z)}{1+z}\,\tau_{\nu_0}(s,z), $$ and the optical depth itself depends on density, neutral fraction, spin temperature, and velocity gradients [2504.14656]. In all three settings, one-point statistics are defined from the empirical distribution of field values after smoothing, voxelization, or filtering. The common feature is that they compress the field in value space rather than configuration space. This makes them natural probes of non-Gaussianity, rare events, and environmental dependence, but also makes them especially sensitive to the mapping between the underlying cosmological field and the measured observable. ## 2. Canonical one-point observables The most basic one-point observable is the PDF itself. For post-reionization intensity mapping, the voxel intensity distribution is defined as $$ \mathcal{P}(T) = \sum_N \mathcal{P}(T|N)\mathcal{P}(N), $$ where \(N\) is the number of sources in a voxel, \(\mathcal{P}(N)\) is the source-count distribution, and \(\mathcal{P}(T|N)\) is the temperature distribution conditional on \(N\) [2606.30200]. Closely related are counts-in-cells PDFs for the density in spheres. In large-deviation descriptions, $$ \mathcal{P}_R(\rho) \propto \exp\left[-\Psi_R(\rho)\right], \quad \Psi_R (\rho)= \frac{1}{2} \frac{\delta_{\rm L}^2}{\sigma_{\rm L}^2(R\rho^{1/3})} \frac{\sigma_{\rm L}^2(R)}{\sigma_{\rm NL}^2(R)}, $$ with \(\rho=1+\delta\) and spherical-collapse variables defined in the usual way [2606.30200]. Moments of the one-point distribution are the standard low-dimensional summaries. For a pixelized field \(X\), the variance and skewness are estimated as $$ \sigma_X^2 = \frac{1}{N_{\rm pix}} \sum_{i=1}^{N_{\rm pix}} \left( X_i - \bar{X} \right)^2, $$ and $$ \gamma_X = \frac{1}{N_{\rm pix}\,\sigma_X^3} \sum_{i=1}^{N_{\rm pix}} \left( X_i - \bar{X} \right)^3, $$ with negative skewness corresponding to a longer tail toward lower values and positive skewness to a tail toward higher values [1412.3332]. In continuous form, the variance is the integral of the power spectrum, $$ \sigma^2 = \int \frac{d^3 k}{(2\pi)^3} P(\mathbf{k}), $$ while skewness is related to the bispectrum integral, so one-point moments are integrated higher-order statistics rather than independent from them [2509.14751]. A distinct but related construction is the local variance, defined from the distribution of patch means rather than the distribution of pixels themselves. For the ionization field in slices, $$ \sigma_\mathrm{loc,x_e}^2(z) = \frac{1}{N}\sum_{k=1}^{N} \mu(k,z)^2 - \bar{x}_e(z)^2, $$ with \(\mu(k,z)\) the slice mean and \(\bar{x}_e\) the global mean ionized fraction [2102.04352]. This quantity was introduced specifically to turn what is usually regarded as sample variance into a morphology-sensitive one-point observable. ## 3. Major statistical constructions A large part of the modern literature consists of generalizations of the PDF and low-order moments that retain one-point character while adding scale information. In post-reionization intensity mapping, the principal examples are the voxel intensity distribution, counts-in-cells PDFs, emission line stacking, and the starlet \(\ell_1\)-norm. Stacking is defined as $$ I(\Delta\alpha,\Delta\phi,\Delta\nu) = \frac{\sum_i I(\alpha_i{+}\Delta\alpha,\,\phi_i{+}\Delta\phi,\,\nu_i{+}\Delta\nu) w_i}{\sum_i w_i}, $$ and functions as a conditional one-point estimator around selected tracers [2606.30200]. The starlet \(\ell_1\)-norm is built from a spherical starlet decomposition, $$ M(\hat{n}) = \sum_{j=1}^J w_j(\hat{n}) + c_J(\hat{n}), $$ with binned coefficient amplitudes $$ \ell_1^{(j,k)} = \sum_{\hat{n} \in \mathcal{B}_k} \left| \frac{w_j(\hat{n})}{\mathcal{N}_j} \right|, $$ and is explicitly described as a non-Gaussian one-point statistic in wavelet space [2606.30200]. During reionization, wavelet-domain summaries have been developed as structured multiscale one-point statistics. Wavelet moments are $$ M_1(i) = \int |I * \psi_{w,i}(\vec{x})|\, d^2 x,\qquad M_2(i) = \int |I * \psi_{w,i}(\vec{x})|^2\, d^2 x, $$ with \(M_2(i)\) exactly equal to a power-spectrum bin and \(M_1(i)\) sensitive to sparsity and non-Gaussian tails [2311.00036]. The Wavelet Scattering Transform extends this hierarchy. For a one-dimensional forest spectrum \(\delta T_b(x)\), $$ S_0 = \langle \delta T_b \rangle, $$ $$ S_1^{j_1} = \big\langle \, |\, \delta T_b * \psi_{j_1}(x)\,| \,\big\rangle, $$ and $$ S_2^{j_1,j_2} = \big\langle \,\big|\, \big| \delta T_b * \psi_{j_1}(x) \big| * \psi_{j_2}(x) \big| \,\big\rangle, $$ where first-order coefficients measure scale-dependent local amplitudes and second-order coefficients measure scale-scale coupling [2504.14656]. The same paper explicitly interprets \(S_1(j)\) as a scale-dependent one-point statistic of filtered amplitudes and \(S_2(j_1,j_2)\) as a non-Gaussian statistic of interactions across scales. A further generalization is the Deconvolved Distribution Estimator, defined in characteristic-function space as $$ \mathcal{R}(t_1,t_2) \equiv \frac{\widetilde{P}_{\rm 2D}^{\rm obs}(t_1,t_2)}{\widetilde{P}_1^{\rm obs}(t_1)\, \widetilde{P}_2^{\rm obs}(t_2)} - 1, $$ with an observable estimator $$ R_{ij} = N_{\rm vox}\; \frac{\widetilde{B}_{ij}}{\widetilde{B}_{1,i}\,\widetilde{B}_{2,j}} - 1. $$ It was introduced as an unbiased, cross-correlation estimator for one-point statistics and is designed so that uncorrelated noise and systematics cancel in expectation [2210.14902]. This suggests a natural role for cross-one-point statistics in 21 cm analyses with overlapping tracers. ## 4. Physical interpretation across cosmic epochs In Cosmic Dawn and the Epoch of Reionization, one-point moments are primarily timing diagnostics. The redshift evolution of the 21 cm power spectrum can be interpreted using the variance and skewness of the brightness-temperature and spin-temperature fields, and the peaks and dips are described as being deeply related to X-ray heating of the intergalactic gas [1412.3332]. In particular, the skewness of the spin-temperature-related field changes sign when X-ray heating becomes effective, and the skewness of the brightness-temperature distribution is identified as a key observable to identify the onset of X-ray heating [1412.3332]. In isocurvature analyses, the variance is highly sensitive to the timing of cosmic events and provides tight constraints on isocurvature parameters, whereas skewness is more strongly affected by astrophysical uncertainties and observational noise [2509.14751]. The same logic extends to morphology. For binary or nearly binary ionization fields, the whole-volume variance mainly traces \(\bar{x}_e(1-\bar{x}_e)\), whereas the local variance traces the distribution of patch means and therefore the ionization morphology. Larger and more clustered bubbles increase the spread of patch means and hence increase \(\sigma_\mathrm{loc}\); the maximum local variance typically appears near the stage when both neutral and ionized regions are large [2102.04352]. This is why local variance was proposed as a tracer of both reionization history and ionization topology rather than only filling fraction. In post-reionization intensity mapping, one-point statistics probe non-Gaussianity from nonlinear structure formation, complex galaxy biasing, and possible primordial non-Gaussianity [2606.30200]. The VID is sensitive to the line-luminosity function, halo occupation, and the HI mass function, and counts-in-cells PDFs are sensitive to \(\sigma_8\), \(\Omega_M\), HI bias, and shot noise through the mapping from matter to HI [2606.30200][1808.09968]. This is why the PDF is repeatedly described as complementary to the power spectrum and particularly useful for breaking bias–amplitude degeneracies [2607.06526]. For the 21 cm forest, the field is intrinsically highly non-Gaussian because of deep, rare absorption features from minihalos or filaments, asymmetries due to peculiar velocities and temperature gradients, and clustered small-scale structures modulated by large-scale heating and ionization [2504.14656]. In that setting, first-order scattering coefficients distinguish CDM, WDM, and X-ray-heated scenarios through their scale-dependent amplitudes, while second-order coefficients expose scale coupling that is suppressed in WDM and in strong X-ray heating scenarios [2504.14656]. The physical interpretation is direct: rare, narrow lines populate the tails of one-point amplitude distributions, and their environmental modulation produces higher-order structure. ## 5. Modeling and inference The main analytic backbone for post-reionization one-point statistics is large-deviation statistics plus spherical collapse. In the HI context, one starts from a matter PDF in spherical cells, models the tracer relation between HI and matter, and then marginalizes over matter to obtain the HI PDF [1808.09968][2607.06526]. A representative tracer model writes $$ \langle \delta_{\rm HI}|\delta_m\rangle = b_1\,\delta_m + \frac{b_2}{2}\,\left(\delta_m^2 - \sigma_m^2\right), $$ together with a density-dependent stochasticity parameterization $$ \alpha(\delta_m) = \alpha_0 + \alpha_1\,\delta_m + \alpha_2\,\delta_m^2, $$ and then constructs $$ \mathcal{P}(\delta_{\rm HI}) = \int \mathcal{P}(\delta_{\rm HI}|\delta_m)\,\mathcal{P}(\delta_m)\,d\delta_m $$ [2607.06526]. This is the same conceptual structure used in broader tracer-PDF work and provides a direct route from matter statistics to HI intensity statistics. Inference studies consistently find complementarity with two-point observables. For an SKA-Mid forecast in \(1.1