-
Structure suppression is modeled as an *infinitely sharp* halo-mass cutoff $M_{\rm cut}$ (Sec. 2.2), i.e. all halos with $M<M_{\rm cut}$ are removed. For DM–baryon scattering (drag/heat exchange), realistic suppression is typically smooth and redshift/scale dependent (via a transfer function), and can also alter collapse thresholds, concentrations/formation times, and gas content. A step-function truncation risks (i) exaggerating or distorting the low-$\tau$ shape change in $dN/d\tau$ (the central “fingerprint”), and (ii) producing over-optimistic or mis-centered Fisher constraints on $M_{\rm cut}$ (Sec. 3.1–3.3), which then propagate directly to $\sigma_0/m_\chi$ (Sec. 3.4, Conclusions).
Recommendation: In Sec. 2.2, replace the step cutoff with a physically motivated smooth suppression, or at minimum bracket the step model with a smoothed cutoff (e.g., logistic/erf in $\log M$ with tunable width) calibrated to published transfer-function/HMF results for DM–baryon scattering. In Sec. 3.1–3.2, add a robustness test showing how $dN/d\tau$, $N(>\tau)$, and $D_{\rm KL}$ change when the cutoff is smoothed and/or when a standard “half-mode mass” style mapping is used. Quantify how the inferred $\sigma(M_{\rm cut})$ (Sec. 3.3) shifts under these alternatives and temper claims that depend sensitively on the sharpness of the cutoff.
-
The mapping between the phenomenological cutoff $M_{\rm cut}$ and the microscopic scattering parameter $\sigma_0/m_\chi$ (for $n=0$ and $n=-4$) is asserted but not transparently derived or reproducible (Sec. 2.2, Sec. 3.4). The manuscript does not specify: the perturbation calculation/transfer function used, how the suppression scale is defined (e.g., 50% suppression in $T(k)$, filtering mass, half-mode), how it is translated to a halo-mass scale at relevant redshifts, dependence on $m_\chi$ and cosmology, or what baryonic target (protons/electrons/neutral H) is assumed. Since the headline result is the $\sigma_0/m_\chi$ reach and the “4–5 orders of magnitude over CMB” comparison (Sec. 3.4, Conclusions), this opacity undermines credibility and prevents verification.
Recommendation: Add a dedicated subsection (end of Sec. 2.2 or new Sec. 2.5 / Appendix) detailing the full pipeline $\sigma_0/m_\chi \rightarrow$ transfer-function suppression $\rightarrow$ halo-mass-function modification $\rightarrow$ an effective $M_{\rm cut}$ used in HAYASHI. Explicitly state: assumed DM mass $m_\chi$, scattering targets, cosmological parameters, the definition of the cutoff scale (e.g., $k_{1/2}$ where $T(k)=1/2$), and the conversion to a characteristic mass (e.g., half-mode mass). Provide a table/figure of $M_{\rm cut}$ versus $\sigma_0/m_\chi$ for representative redshifts (e.g., $z=8,10$) and both $n=0$ and $n=-4$. In Sec. 3.4, overlay or tabulate the corresponding $M_{\rm cut}$ values for the quoted $\sigma_0/m_\chi$ sensitivities and include an estimate of systematic uncertainty from the mapping.
-
The thermal/spin-temperature and minihalo gas modeling is too compressed for the paper’s strong conclusion that even extreme $f_{\rm cool}$ has negligible impact and that the $M_{\rm cut}$-induced *shape* change robustly breaks astrophysical degeneracies (Sec. 2.1–2.2, Sec. 3.1–3.2). It is unclear how HAYASHI computes $T_s$ (collisional vs. Wouthuysen–Field coupling, radiation backgrounds), whether the regime $T_s\approx T_k$ is always realized over $z=7$–$15$, whether absorption is in the linear $\tau\ll1$ regime, and whether $f_{\rm cool}$ is applied to IGM only or also to minihalo gas (where line profiles/central densities set $\tau$). The current parameter set $\{M_{\rm cut}, f_{\rm cool}, \bar{x}_{\rm HI}\}$ may not capture astrophysical effects that also change the *shape* of $dN/d\tau$ (e.g., photoheating feedback reducing gas in low-mass halos, mass-dependent gas fractions, X-ray/Ly$\alpha$ fluctuations).
Recommendation: Expand Sec. 2.1 with an explicit (but concise) summary of the ingredients that determine $\tau$ in the IGM and minihalos: gas density/temperature profiles, how $T_s$ is computed (collisions/Ly$\alpha$ coupling), what backgrounds are assumed, and the halo mass/redshift range driving the statistics. In Sec. 2.2 and Sec. 3.1, clarify exactly where $f_{\rm cool}$ is applied (IGM only vs also halos), whether it is redshift dependent, and report representative $T_k$ and $T_s$ ranges to support the “saturation” argument. To support degeneracy-breaking claims (Sec. 3.2), add at least one additional nuisance parameter that can preferentially suppress low-$\tau$ absorbers (e.g., a feedback/photoheating suppression mass, a low-mass gas-fraction suppression parameter, or a temperature floor) and show how it compares to changing $M_{\rm cut}$ in $dN/d\tau$ and in the Fisher constraints.
-
Forecast/statistical methodology is under-specified and likely optimistic: the Fisher analysis (Sec. 2.4) appears to assume independent Poisson counts per $\tau$-bin with ${\rm Var}(N_k)=N_k$, but the paper does not fully define the data vector (counts vs normalized PDF), $\tau$-binning and $\tau$-range, redshift binning/path length, or how $N_{\rm los}(z)$ enters $N_k$. Instrumental effects and analysis systematics that are central for low-$\tau$ features—finite spectral resolution, thermal noise, continuum fitting, line blending/confusion, completeness/detection thresholds in $\tau$—are neglected, yet much of the claimed information is in the low-$\tau$ regime (Sec. 3.1–3.3). This makes $\sigma(M_{\rm cut})$ and the optimal $z\approx8$–$10$ window potentially fragile (Sec. 3.3), and the resulting $\sigma_0/m_\chi$ reach likely too optimistic (Sec. 3.4, Conclusions).
Recommendation: In Sec. 2.3–2.4, explicitly define the observable: whether $N_k$ are total absorber counts across all sightlines in $(z,\tau)$ bins (preferred for Poisson likelihood), or a normalized shape-only statistic; provide the $\tau$-bin edges, $\tau$-range, and any redshift binning/path length per sightline. Write $N_k(z) = N_{\rm los}(z)\times\lambda_k(z,\theta)$ (or equivalent) so the $N_{\rm los}$ scaling is explicit. Then incorporate a minimal observational realism layer: e.g., (i) a $\tau$ detection threshold/completeness curve, (ii) an effective reduction in usable $N_{\rm los}$, and/or (iii) an added fractional variance term per bin to represent non-Poisson systematics. Recompute (or at least bracket) $\sigma(M_{\rm cut})$ and the preferred redshift window under optimistic vs pessimistic thresholds, and soften conclusions where they depend on Poisson-limited low-$\tau$ performance.
-
KL divergence is used as a central “intrinsic distinguishability” metric (Sec. 2.3, Fig. 3, Sec. 3.2) but the manuscript does not define the normalized distributions entering $D_{\rm KL}$. KL divergence requires proper probability distributions $p(\tau)$, $q(\tau)$ (or discrete probabilities $p_k$, $q_k$). If computed from $dN/d\tau$ or raw $N_k$ without normalization, the interpretation changes (shape-only vs amplitude+shape), and the connection to detectability with finite counts is unclear.
Recommendation: In Sec. 2.3, add the explicit definition of $D_{\rm KL}$ and specify precisely how $p$ and $q$ are constructed from $dN/d\tau$ or binned $N_k$ (including normalization over the chosen $\tau$-range). If the intent is “shape-only,” state that explicitly and show how amplitude information is treated separately. In Sec. 3.2, add a brief interpretive bridge: for representative $N_{\rm los}$ and total counts, what $D_{\rm KL}$ values correspond to a meaningful likelihood-ratio/$\Delta\chi^2$ separation? (A simple approximation or illustrative conversion is sufficient.)
-
The assumed redshift evolution of background radio-loud sources $N_{\rm los}(z)=10[(1+z)/8]^{-2.5}$ (Sec. 2.4) is a key driver of the forecast and the claimed optimal $z\approx8$–$10$ window (Sec. 3.3), but is currently schematic and not clearly justified by a cited luminosity-function model, flux limit, or survey strategy. Likewise, the “4–5 orders of magnitude better than CMB” statement (Sec. 3.4, Conclusions/Abstract) is presented without a like-for-like comparison (model assumptions, $m_\chi$, $n$, ionization history) and without plotting the referenced CMB limits alongside the forecast.
Recommendation: In Sec. 2.4, cite the provenance of the $N_{\rm los}(z)$ scaling (data or simulations) and add at least one optimistic and one pessimistic alternative (varying normalization and exponent, or tying $N_{\rm los}$ to a flux limit and a luminosity function). In Sec. 3.3, show how $\sigma(M_{\rm cut})$ and the optimal redshift shift under these alternatives. In Sec. 3.4, state explicitly which CMB limits are used (citations, $m_\chi$, targets, $n$, and assumptions) and overlay them on the same $\sigma_0/m_\chi$ plot; then restate the improvement factor conditional on those assumptions, and temper the language in the Abstract/Conclusions if it only holds in optimistic source/noise scenarios.