21-cm Forest: Probing Early Universe Structures

• The 21-cm forest is a collection of narrow absorption features imprinted by neutral hydrogen on high-redshift radio source spectra, offering a direct probe of early universe conditions.
• It utilizes advanced statistical methods such as 1D power spectra and topological analysis to resolve small-scale structures and distinguish astrophysical parameters like IGM temperature and minihalo abundance.
• High-resolution radio spectroscopy is employed to detect subtle absorption signals, enabling constraints on dark matter properties, X-ray heating effects, and other non-standard physics influencing cosmic dawn.

The 21-cm forest is the ensemble of narrow absorption features produced when neutral hydrogen along a line of sight imprints the redshifted hyperfine transition on the spectra of distant, compact, radio-bright background sources during Cosmic Dawn and the Epoch of Reionization. In contrast to the global 21-cm signal and 3D 21-cm tomography, which probe diffuse brightness against the CMB, the forest is a pencil-beam observable that resolves small-scale neutral structure in frequency space and is therefore acutely sensitive to the thermal state of the neutral intergalactic medium, the abundance of minihalos and filamentary absorbers, and any physics that modifies structure formation below galactic scales (Cang et al., 23 Jun 2026, Sun et al., 2024, Thyagarajan, 2020).

1. Physical basis and radiative transfer

The forest arises because the observed continuum from a high-redshift radio source is attenuated at frequencies corresponding to intervening neutral hydrogen. A standard thermally broadened line-of-sight optical depth used in recent work is

$$\tau_\nu = \frac{3 h_P c^3 A_{10}}{32 \pi^{3/2} k_B \nu_{21}^2} \int \frac{n_{\mathrm{HI}}(x)}{b(x)\,T_S(x)} \exp\!\left[-\frac{(u(\nu)-v(x))^2}{b^2(x)}\right] dx,$$

with $b=(2k_B T_K/m_H)^{1/2}$, where $n_{\mathrm{HI}}$ is the neutral hydrogen density, $T_S$ the spin temperature, $T_K$ the kinetic temperature, and $v(x)$ the line-of-sight peculiar velocity. In the optically thin limit, the differential brightness temperature against the background radiation is

$$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$

with $z=\nu_{21}/\nu-1$ (Shimabukuro, 17 Nov 2025). A widely used intuitive approximation for the line optical depth is

$$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$

which makes explicit the dependence on density, spin temperature, redshift, and velocity gradients (Shimabukuro, 17 Nov 2025, Ciardi et al., 2015).

Absorption requires $T_S<T_\gamma$. The spin temperature is set by the balance between CMB coupling, collisions, and Wouthuysen–Field coupling,

$b=(2k_B T_K/m_H)^{1/2}$0

so the forest strengthens when Ly$b=(2k_B T_K/m_H)^{1/2}$1 coupling drives $b=(2k_B T_K/m_H)^{1/2}$2 while the gas remains cold, and it fades once X-ray heating raises $b=(2k_B T_K/m_H)^{1/2}$3 and hence $b=(2k_B T_K/m_H)^{1/2}$4 well above the background temperature (Ewall-Wice et al., 2013, Mack et al., 2011). This dependence makes the forest simultaneously a probe of thermodynamics and of the local gas distribution.

A persistent misconception is that the forest is synonymous with isolated minihalo lines. Recent simulation-based studies treat it more broadly: diffuse neutral IGM structures, filaments, infall regions around halos, and collapsed starless systems can all contribute. In detailed radiative-hydrodynamic modeling, the strongest features can arise in moderately overdense filaments and halo outskirts, while unresolved minihalos would add an even narrower, deeper component (Semelin, 2015, Shao et al., 2023).

2. Absorbers, thermal history, and small-scale structure

The forest is unusually sensitive to small-scale cosmological structure because the relevant absorbers live on mass and length scales far below those probed by standard 21-cm tomography. In one minihalo-based treatment, the dominant absorber masses are $b=(2k_B T_K/m_H)^{1/2}$5–$b=(2k_B T_K/m_H)^{1/2}$6 just above the Jeans scale, corresponding roughly to comoving radii $b=(2k_B T_K/m_H)^{1/2}$7–$b=(2k_B T_K/m_H)^{1/2}$8 and hence to modes $b=(2k_B T_K/m_H)^{1/2}$9–$n_{\mathrm{HI}}$0 (Shimabukuro et al., 2014). This is the regime where warm dark matter free streaming, neutrino mass, primordial running, primordial black holes, and baryon–dark-matter relative velocity can all leave detectable imprints (Shimabukuro et al., 2014, Shao et al., 1 Jan 2025, Cang et al., 23 Jun 2026).

Warm dark matter is a canonical example. Its finite free-streaming length suppresses the linear matter power spectrum on small scales, lowers the abundance of low-mass halos, and erodes the fine-grained absorption network. A standard transfer function used for intuition is

$n_{\mathrm{HI}}$1

with $n_{\mathrm{HI}}$2 decreasing as $n_{\mathrm{HI}}$3 increases (Shimabukuro, 17 Nov 2025). In forest observables, this appears as a depletion of narrow absorbers, earlier mergers of neighboring troughs, and suppression of high-$n_{\mathrm{HI}}$4 power in the 1D line-of-sight spectrum (Shao et al., 2023, Shao et al., 2024).

Heating acts differently. X-ray emissivity is commonly parameterized by $n_{\mathrm{HI}}$5, which rescales the X-ray output per unit star-formation rate. Larger $n_{\mathrm{HI}}$6 raises the IGM kinetic temperature, pushes $n_{\mathrm{HI}}$7 upward, and compresses absorption contrast. In semi-numerical models this suppresses the forest more uniformly across scales than warm dark matter does, because heating modifies optical depth amplitudes even when the underlying small-scale density ranking is approximately preserved (Shimabukuro, 17 Nov 2025, Shao et al., 2023). This distinction underlies much of the modern statistical analysis of the forest.

The same logic extends beyond warm dark matter. Massive neutrinos suppress the matter power spectrum below the free-streaming scale and can, in favorable scenarios, be constrained by the forest 1D power spectrum to around $n_{\mathrm{HI}}$8 when the IGM temperature is externally constrained (Shao et al., 1 Jan 2025). Primordial black holes have a two-sided effect: PBH shot-noise isocurvature enhances low-mass halo formation, increasing line counts, whereas PBH accretion heating raises the IGM temperature and suppresses them. Forecasts indicate competitive upper limits as low as $n_{\mathrm{HI}}$9 at $T_S$0 under suitable assumptions (Villanueva-Domingo et al., 2021). Subhalos within host minihalos can also matter: one study found that although the boost is negligible for $T_S$1 hosts, substructure can enhance optical depth by an order of magnitude for $T_S$2 hosts and increase the integrated absorber abundance by up to order $T_S$3 (Kadota et al., 2022).

3. Statistical descriptions of the forest

Direct line-by-line spectroscopy is only one way to use the forest. A central development has been the move toward statistical observables that aggregate information from many weak features. Along a sightline, the standard summary is the one-dimensional power spectrum,

$T_S$4

with $T_S$5 obtained by averaging over segments or sources (Sun et al., 2024, Thyagarajan, 2020). This statistic is especially effective because heating mainly changes the overall amplitude, whereas small-scale structure suppression changes the scale dependence. In forecasts for the forest contribution to the 21-cm power spectrum, the absorption signal was found to dominate a distinctive high-$T_S$6 region, $T_S$7, when the IGM is cool (Ewall-Wice et al., 2013).

The 1D power spectrum also has practical advantages. It reduces cosmic variance through averaging over independent segments and can detect the forest even when individual absorbers are too weak to identify. A dedicated z=6 detectability study showed that with 10 radio-loud sources the 1D forest power spectrum is detectable for $T_S$8 with 500 hr on the uGMRT and for $T_S$9 with 50 hr on SKA1-low if the IGM is $T_K$0 neutral and the neutral regions have spin temperature $T_K$1 (Šoltinský et al., 2024). Earlier work on narrow-sightline statistics similarly concluded that a $T_K$2 hr campaign targeting $T_K$3 narrow sightlines to $T_K$4–10 mJy sources could detect the LOS power spectrum and discriminate reionization scenarios (Thyagarajan, 2020).

A more recent development is the use of explicitly non-Gaussian and topological summaries. In “Topological Signatures of Heating and Dark Matter in the 21 cm Forest” the forest is converted into a standardized 1D field $T_K$5, a sublevel filtration

$T_K$6

and a Betti-0 curve

$T_K$7

where each trough $T_K$8 has birth $T_K$9, death $v(x)$0, and persistence lifetime $v(x)$1 (Shimabukuro, 17 Nov 2025). The resulting descriptors—trough-line density $v(x)$2, lifetime variance $v(x)$3, and lifetime skewness $v(x)$4—were shown to respond in nearly orthogonal directions across the $v(x)$5 parameter space. In that analysis, a common persistence cut $v(x)$6 and a common threshold $v(x)$7 fixed from the baseline noiseless CDM model stabilized the statistics, and the topological signatures remained detectable under SKA1-Low-like thermal noise. This suggests that the forest contains merger-hierarchy and connectivity information not accessible to amplitude-only summaries.

4. Background sources, instrumental requirements, and detectability

The limiting resource for forest work is not only telescope sensitivity but also the availability of sufficiently bright, compact background sources. Radio-loud quasars are the most studied candidates. A recent physics-driven source-population forecast found that if the radio-loud fraction remains roughly constant with redshift, a one-year SKA-LOW survey could detect approximately 20 radio-loud quasars at $v(x)$8 bright enough to resolve individual forest lines; if the radio-loud fraction declines strongly with redshift, the yield drops sharply and line spectroscopy becomes much harder (Niu et al., 2024). At lower redshift, prospects are already improving: more than 30 radio-loud quasars are now known at $v(x)$9, which materially changes the observational outlook for late-reionization forest searches (Šoltinský et al., 2024).

Fine spectral resolution is indispensable. Several studies adopt $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$0 kHz channelization for direct or statistical analyses because the narrowest features are kHz-scale. In the topological study, degrading the resolution from 1 kHz to 5–10 kHz merged neighboring troughs and lowered and narrowed the Betti-0 peak (Shimabukuro, 17 Nov 2025). In direct-detection forecasts with SKA1-Low, line widths of order 1–5 kHz are explicitly resolved, and targeted observations at $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$1–20 kHz over 1000 hr were judged capable of detecting strong features for favorable source fluxes in the $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$2–15 range (Ciardi et al., 2015).

Individual-line detection remains difficult. One early simulation study concluded that statistically significant 21-cm absorption against a Cygnus A–type source at $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$3 could be detected by SKA in less than a year, whereas significant detection of the detailed forest features would require nearly a decade under per-channel methods (Mack et al., 2011). In detailed radiative-hydrodynamic modeling, LOFAR was estimated to detect a few strong absorption features over a few tens of MHz for a 20 mJy source at $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$4, while the SKA would recover a much larger fraction of the absorption information for the same source (Semelin, 2015). The practical implication is that direct line spectroscopy is possible but selective, whereas statistical observables are likely to dominate the first detections.

Calibration and continuum control remain central. The forest is less vulnerable than diffuse 21-cm tomography to wide-field foreground confusion because it uses compact sources as backlights, but it is not immune to bandpass structure, chromatic synthesized PSF leakage, radio-frequency interference, and continuum-model residuals (Thyagarajan, 2020, Šoltinský et al., 2024). Because many forest statistics depend primarily on rank ordering or on differential fluctuations after continuum subtraction, they can be more robust than absolute-amplitude observables, but only if spectral calibration is stable on kHz scales (Shimabukuro, 17 Nov 2025).

5. Parameter inference and constraints on astrophysics and fundamental physics

The forest is now used not only for detection but for quantitative inference. In a Fisher-forecast framework based on the 1D power spectrum, simulated SKA1-LOW observations at $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$5 with 10 sources of $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$6, 100 segments of 10 cMpc, and 100 hr per source yielded marginalized uncertainties of $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$7 and $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$8 for a mildly heated fiducial model with $$\delta T_b(\hat{s},\nu) \approx \frac{[T_S(\hat{s},z)-T_\gamma(\hat{s},z)]\,\tau(\hat{s},z)}{1+z},$$9 and $z=\nu_{21}/\nu-1$0; the corresponding SKA2-LOW forecast gave $z=\nu_{21}/\nu-1$1 and $z=\nu_{21}/\nu-1$2 (Shao et al., 2023). Even for a strongly heated case with $z=\nu_{21}/\nu-1$3, the same study found $z=\nu_{21}/\nu-1$4 and $z=\nu_{21}/\nu-1$5 for SKA2-LOW.

Likelihood-free inference has pushed this further. A normalizing-flow pipeline trained on simulated 21-cm forest power spectra recovered, for a low-heating SKA1-LOW 100 hr scenario with true $z=\nu_{21}/\nu-1$6 and $z=\nu_{21}/\nu-1$7,

$z=\nu_{21}/\nu-1$8

and for a high-heating SKA2-LOW 200 hr case,

$z=\nu_{21}/\nu-1$9

while explicitly modeling the non-Gaussian distribution of the 1D power spectrum (Sun et al., 2024). A later z=6 study compared five inference pipelines and found that the most effective method bypassed the power spectrum entirely: a 1D U-Net produced a 256-dimensional latent representation of the noisy spectrum, and XGBoost regression on that latent space yielded meaningful constraints on $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$0 and $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$1 even from a single 50 hr uGMRT sightline, corresponding to an approximately one-order-of-magnitude reduction in integration time relative to earlier power-spectrum-based techniques (Patil et al., 15 Jul 2025).

The forest also supports broader fundamental-physics constraints. Under weak astrophysical heating, one forecast based on the forest 1D power spectrum found sensitivity to dark-matter annihilation at $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$2 and to decay lifetimes $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$3 for $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$4 particles, together with sensitivity to primordial black holes at $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$5 and abundance $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$6 (Zhao et al., 6 Sep 2025). For neutrino mass, another forecast argued that, in an ideal scenario with external temperature information, the forest could constrain the summed mass to around $$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$7 (Shao et al., 1 Jan 2025). These numbers are highly model-dependent, but they illustrate the parameter space that becomes available once the forest is treated as a high-dimensional statistical field rather than as a set of isolated lines.

6. Limitations, misconceptions, and outlook

The dominant limitations are astrophysical degeneracy, source scarcity, and realism of small-scale modeling. Heating is the most persistent degeneracy: X-ray preheating can suppress the forest so efficiently that warm-dark-matter suppression, PBH heating, or neutrino-induced small-scale damping become difficult to distinguish unless the thermal history is constrained independently (Shao et al., 2023, Zhao et al., 6 Sep 2025). Source scarcity remains a practical bottleneck at the highest redshifts, especially if the radio-loud fraction evolves downward (Niu et al., 2024). On the modeling side, unresolved minihalo gas physics, self-shielding, radiative transfer, shock heating, peculiar velocities, and reionization patchiness all affect line statistics and high-$$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$8 power (Semelin, 2015, Shao et al., 2024).

Several common misconceptions can therefore be stated precisely. The first is that the forest is only a direct-detection problem. In fact, a large fraction of current progress relies on statistical observables—LOS power spectra, fluctuation variance, topology, and machine-learning summaries—which remain informative when no individual line is significant (Mack et al., 2011, Šoltinský et al., 2024, Shimabukuro, 17 Nov 2025). The second is that the forest is just a high-redshift analog of the Ly$$\tau_{21} \approx \frac{3 c^3 h_P A_{10} n_{\mathrm{HI}}}{32 \pi k_B \nu_{21}^2 T_S (1+z) (dv_{\parallel}/dr_{\parallel})},$$9 forest. The analogy is useful, but the 21-cm forest is sensitive to spin temperature and to the radio background in a way that makes early heating a central part of the signal model (Cang et al., 23 Jun 2026). The third is that foregrounds cease to matter because the background source is compact. Smooth foregrounds are less central than in diffuse tomography, but bandpass structure, continuum subtraction, chromatic PSF response, and RFI remain decisive systematics (Thyagarajan, 2020, Šoltinský et al., 2024).

The near-term outlook is nonetheless substantially stronger than it was a decade ago. Source catalogs are expanding, late-end reionization models leave open a z≈6 forest window, SKA-Low-like sensitivities make statistical detection plausible, and analysis frameworks now include halo-model predictions, simulation-based inference, deep latent-space compression, and topological data analysis (Šoltinský et al., 2024, Niu et al., 2024, Patil et al., 15 Jul 2025, Shimabukuro, 17 Nov 2025). A plausible implication is that the first robust forest measurements will be hybrid: a small number of deep spectra from bright radio-loud quasars analyzed jointly with power-spectrum, non-Gaussian, and topology-aware summaries, then interpreted in combination with external constraints from global 21-cm experiments, 3D tomography, Ly$T_S<T_\gamma$0 forest data, and high-redshift source surveys (Cang et al., 23 Jun 2026).