{"id":"22aa0828-8141-4640-9f31-cfbb535511b2","arxiv_id":"2506.03768","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A two-moment cosmic ray MHD simulation suite shows that cosmic rays couple to isothermal supersonic turbulence only when both diffusion and streaming are slow relative to turbulent transport, imprints a mixed Pc ∝ ρ^(2/3)–ρ^(4/3) equation of state, and leaves solenoidal velocity modes unchanged.","lead":"Cosmic rays in the interstellar medium can either be locked to the gas or pass through it freely, depending on how fast they diffuse and stream compared to the turbulence. This paper runs magnetohydrodynamic simulations with a two-moment cosmic ray fluid and maps out exactly which values of diffusion and streaming place a plasma in the coupled regime, where cosmic rays smooth density fluctuations and heat themselves.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The regime map's streaming Prandtl number assumes χ=1; for realistic ISM ionization fractions the coupled window shrinks by orders of magnitude, so the cold-ISM relevance claim is the least secure link.","rationale":"The reader's weakest_assumption identifies exactly the assumption I find most load-bearing: Pms is computed with vA rather than vA,ion = vA/√χ. This assumption controls whether the paper's headline coupled regime has any overlap with the cold ISM it claims relevance to. The attack is not an internal inconsistency in the MHD/CRMHD solver; rather, it is an external physical premise that the paper explicitly flags in §6.1.1 but never tests. Because the manuscript is transparent about the assumption and because the internal parameter study is executed competently, I do not think the verdict should be hardened to REJECT or softened to ACCEPT. The conditional verdict remains appropriate: the central claim should be adopted only with the caveat that the regime map has been demonstrated for fully ionized plasma, and its cold-ISM applicability is an unverified extrapolation. I agree with the reader that this is the weakest assumption; no separate concern about the numerical scheme or the spectral measurements outweighs it, because those are secondary to the physical interpretation of the regime map.","tokens_in":30436,"tokens_out":12934,"duration_ms":141977,"concrete_test":"Rerun the super-Alfvénic coupled case M4MA10D22 (Pm_c = 0.0045, Pms = 0.09) with the streaming speed in Eq. 4 multiplied by 10 and by 100, representing χ = 10^-2 and 10^-4, while keeping Pm_c, forcing, and resolution fixed. Over the same 5τ averaging window, measure the density-PDF width σ_s^2, the Ec–ρ joint-PDF envelope against the ρ^{2/3} and ρ^{4/3} lines, and the heating rate Qc. If the coupled signatures disappear at the higher streaming speeds, the central regime map is confined to χ≈1 plasma and the cold-ISM relevance claim fails; if the signatures persist, the concern is non-load-bearing.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central regime map is built on Pms = vA0ℓ0/Dcrit = 1/MA0, where vA0 is the total-density Alfvén speed. But the physical streaming speed in Eq. 4 is vA,ion = B/√(χρ), and Section 6.1.1 concedes that χ = 10^-2 to 10^-8 in cold ISM phases, making the effective streaming speed 10 to 10^4 times larger. Using the authors' own mapping, Pms_real = 1/(MA0√χ): a nominally super-Alfvénic coupled run (MA0 = 10, Pms = 0.09) becomes Pms_real = 10 for χ = 10^-4, i.e. decoupled by the same criterion. This is not a small quantitative shift: it moves every coupled run in Table 2 into the streaming-dominated decoupled regime. The abstract and conclusions state the coupled regime is 'characteristic of slow streaming (Pms < 1)' and frame the study as 'relevant to the cold ISM,' but the demonstrated coupled window applies only to χ ≈ 1 plasma. The acknowledgment in §6.1.1 is honest, but it is a limitation, not a resolution: no simulation or analytic test in the paper shows that any coupled signature survives at realistic χ. The internal simulation results are plausible and the dimensionless framework is coherent; the load-bearing external assumption is that the fully-ionized streaming speed is representative of the ISM phases the paper targets.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies the dynamical coupling between cosmic rays (CRs) and a turbulent, isothermal, magnetized plasma using the two-moment CRMHD framework of Jiang & Oh (2018). The authors introduce three dimensionless numbers—Pm_c, Pm_s, and Pm_f—based on the non-dimensionalized interaction coefficient, and propose that the CR fluid and plasma are strongly coupled only when Pm_c < 1 and Pm_s < 1. They run a suite of 256^3 simulations spanning sub-, trans-, and super-Alfvénic regimes, with diffusion coefficients varying over four orders of magnitude. In the coupled regime (weak diffusion and weak mean field), they report that the CR pressure imprints a mixed equation of state between P_c ∝ ρ^(4/3) and P_c ∝ ρ^(2/3), which raises the effective sound speed, narrows the density PDF, and produces secular CR heating through compressible compressions. They further claim that decoupling occurs at Pm_c ≥ 1 in all plasma regimes and that the solenoidal velocity spectrum remains invariant to CR coupling. The paper concludes that these results are relevant to the cold ISM.","tokens_in":30740,"tokens_out":9067,"duration_ms":90765,"significance":"If the claims are correct, the paper provides a useful dimensionless framework for interpreting CRMHD turbulence simulations and offers specific, testable predictions: the density-PDF width is bounded by the M and Meff lognormal models, the CR flux correlation length transitions from ℓv to ℓB across Pm_c = 1, and the CR heating rate is controlled by compressible structures. The statistical evidence (2D PDFs, power spectra, correlation lengths, heating rates) is internally consistent and supports the proposed regime classification for the fully ionized (χ = 1) case. The study is a valuable contribution for understanding CR-plasma coupling in highly ionized media and for comparing two-moment CRMHD results with earlier one-moment studies. However, the direct applicability to the cold ISM is not established, because the streaming speed is set to the full Alfvén speed rather than the ion Alfvén speed, and the paper's own limitation section concedes that realistic ionization fractions would move essentially all of the simulated coupled runs into the decoupled regime.","major_comments":[{"comment":"The claim that the coupled regime is 'relevant to the cold ISM' is not supported by the simulations, because the streaming speed in Eq. (4) is set to the full Alfvén speed vA, which implicitly assumes a fully ionized plasma (χ = 1). The manuscript itself states in Section 6.1.1 that cold ISM phases have mass ionization fractions χ = 10^-2 to 10^-8, increasing the streaming speed by 10 to 10^4 times. Using the authors' own mapping, the streaming Prandtl number becomes Pms_real = vA,ionℓ0/Dcrit = (1/MA0)/√χ. For the nominal coupled super-Alfvénic run M4MA10D22 (MA0 ≈ 11, Pms = 0.09), χ = 10^-4 gives Pms_real ≈ 10, which falls in the streaming-dominated decoupled regime by the paper's own criterion. Thus every coupled run in Table 2 would be decoupled at realistic ISM ionization fractions. The acknowledgment in Section 6.1.1 is honest, but no simulation or analytic argument in the paper demonstrates that any coupled signature survives at χ < 1; the abstract and title therefore overstate the applicability of the regime map to the cold ISM.","section":"Section 6.1.1; Abstract"},{"comment":"The abstract and conclusions repeatedly claim that CR coupling 'reduces the turbulent Mach number.' However, Table 2 shows that the measured sonic Mach number M = σv/cs is essentially constant across coupling strengths (e.g., the super-Alfvénic runs all have M = 3.3–3.5 regardless of Pmc). What actually decreases is the effective Mach number Meff = σv/ceff, because ceff increases through the CR-modified sound speed in Eq. (16). The density-PDF variance in Eqs. (17)–(18) is controlled by Meff, not M, so the physical statement should be that coupling reduces the effective Mach number Meff, not the turbulent Mach number M. The current wording conflates the two quantities and should be corrected throughout.","section":"Section 3.1, Eq. (16), Table 2"},{"comment":"The conclusions bullet states 'we show that decoupling occurs at Pm c ≥ 1 in all plasma regimes.' This is internally inconsistent with the paper's own results: in the sub-Alfvénic regime (e.g., M4MA05D22, Pms = 1.30), the CR fluid is decoupled even at Pmc = 0.0045 because Pms > 1, as the paper itself explains in Section 3.1. The correct statement is that decoupling occurs when either Pmc ≥ 1 or Pms ≥ 1; the bullet as written overstates the role of Pmc and should be revised to reflect the joint condition.","section":"Section 6, conclusions bullet"},{"comment":"The coupling boundary is largely built into the non-dimensionalization of the interaction coefficient. In Eq. (14) and Eq. (A4), the inverse interaction coefficient σc^{-1} is decomposed as Pmc + Pms times a streaming operator, so the statement that strong coupling occurs when Pmc < 1 and Pms < 1 is a direct restatement of the model's interaction parameter rather than an independently inferred threshold. The supporting statistics (PDFs, spectra, correlation lengths, heating rates) do show behavioral changes across this boundary, so this is not circular in a damaging sense, but the paper should explicitly acknowledge that Pmc and Pms are the natural control parameters of the model, not empirical quantities measured from the simulations. This framing would prevent readers from overinterpreting the regime map as a prediction rather than a convenient parameterization.","section":"Section 2.2.2, Eq. (14), Appendix A"}],"minor_comments":[{"comment":"The table note defines Pmf = Pms/Pmc = vA0ℓ0/Dc, but Eq. (15) in the text defines Pmf = Pmc/Pms = Dc/(vA0ℓ0). The listed values in Table 2 (e.g., 0.003 for M4MA05D22) correspond to Pmc/Pms, not Pms/Pmc; the note should be corrected to match the definition in Eq. (15).","section":"Table 2, column (7)"},{"comment":"The caption contains several typos: 'T op row' and 'F ourth row' should be 'Top row' and 'Fourth row', and 'super-Alfve´ nic' should be 'super-Alfvénic'.","section":"Figure 3 caption"},{"comment":"The phrase 'strong enough to decoupling the CR fluid' should be 'strong enough to decouple the CR fluid'.","section":"Section 3.1, paragraph 4"},{"comment":"Calling Pm_c, Pm_s, and Pm_f 'Prandtl numbers' may be confusing because they are ratios of transport coefficients to turbulent diffusivity, not ratios of momentum diffusivity to thermal diffusivity as in the classical Prandtl number. A short remark that these are 'Prandtl-like coupling parameters' would improve clarity.","section":"Section 2.2.2"},{"comment":"The statement that 'we only properly resolve a handful of large-scale, supersonic modes' is an important caveat for the spectral claims, particularly the invariance of the solenoidal spectrum; it would be helpful to state this caveat in the abstract or conclusions, since the power-spectrum analysis spans only a narrow k range.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically sound in its internal logic and the simulations appear to support the regime classification for the χ = 1, fully ionized case. The main issue is an overclaim of cold ISM applicability: the streaming speed is set to vA rather than vA,ion = B/√(χρ), and the paper's own limitation section admits that realistic ionization fractions would move all coupled runs into the decoupled regime. This is a load-bearing point for the title and abstract. If the authors reframe the claims to be specific to highly ionized plasmas, or add a clear quantitative caveat (e.g., a rescaled Pms formula for finite χ), the paper would be acceptable. The inconsistency in the conclusions bullet about decoupling at Pmc ≥ 1 should also be fixed. A minor issue: the table definition of Pmf is inconsistent with Eq. (15). Overall, I support publication after major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Sampson et al. deliver a genuinely useful regime map for when CR pressure backreacts on supersonic MHD turbulence. The two-moment treatment with both anisotropic diffusion and Alfvenic streaming, scanned across Pm_c and MA0, is new, and the paper earns the headline results: a coupled window at Pm_c<1 and Pm_s<1, a mixed Pc proportional to rho^4/3 / rho^2/3 equation of state in that window, and a non-monotonic CR heating rate peaking at Pm_c about 0.45 rather than at the strongest coupling. The solenoidal spectral invariance is a real point of contrast with Bustard and Oh, and the authors sensibly attribute it to their mixed compressive/solenoidal driving. These are solid, falsifiable claims backed by multiple independent statistics: PDFs, 2D joint distributions, correlation lengths, and spectra.\n\nThe soft spots are real but mostly addressable. The biggest one concerns the cold-ISM framing. The streaming speed is set to the full vA, i.e., chi=1; Section 6.1.1 acknowledges that realistic cold neutral phases have chi=10^-2 to 10^-8, which makes the effective streaming speed 10 to 10^4 times larger and pushes Pms above unity. That is not a small correction. Under the paper's own criterion, every coupled super-Alfvenic run becomes decoupled once chi drops below about 10^-2. The acknowledgment is honest, but the abstract still promises relevance to the cold ISM without qualification. The regime map as presented is for chi about 1 plasma; its extension to neutral-rich gas is speculative. That should be a prominent caveat, not a footnote.\n\nSecond, the abstract overstates the Mach number reduction. Table 2 shows M roughly constant at 3 to 5 across all Pmc; what changes is Meff. The distinction matters: the driving is momentum-fixed, so the density-PDF narrowing is a consequence of the increased effective sound speed, not a reduction in the turbulent Mach number.\n\nThird, the density-variance and mixed-EOS claims lack quantitative error bars; sigma_s^2 versus Pmc would be easy to plot. Fourth, the RAMSES implementation is unreleased and cited as Rosdahl et al. in prep., which makes the central figures hard to check.\n\nThere is a mild circularity in the Pm<1 boundary, since it is built into the non-dimensionalized interaction coefficient, but the paper does not rely on that alone; the behavioral diagnostics are independent confirmations. I do not see a load-bearing flaw in the internal simulation logic.\n\nWho gets value: anyone running galaxy or ISM simulations with two-moment CR schemes, and CR transport theorists. It deserves a serious referee; I would recommend accept after major revisions that fix the M/Meff language, quantify the density variances, and move the chi caveat into the abstract. The regime map itself is worth having even if its cold-ISM reach is narrower than claimed.","headline":"A useful but overreaching regime map for CR-plasma coupling: solid for chi=1 plasmas, and the cold-ISM relevance claim dies with realistic ionization fractions.","tokens_in":31382,"tokens_out":3767,"would_cite":true,"duration_ms":39082,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Two numbers decide when cosmic rays reshape interstellar gas.","keywords":["cosmic rays","interstellar medium","magnetohydrodynamic turbulence","CR streaming","CR diffusion","density PDF","two-moment CRMHD","Alfvén Mach number"],"falsifier":"Measure the joint distribution of cosmic-ray energy density and gas density in a super-Alfv\\'enic, low-diffusion interstellar region: the paper predicts a tight correlation along the band $P_{\\rm c} \\propto \\rho^{4/3}$ to $P_{\\rm c} \\propto \\rho^{2/3}$ when both $\\mathrm{Pm}_{\\rm c}$ and $\\mathrm{Pm}_{\\rm s}$ are below 1, and a flat, uncorrelated distribution when either exceeds 1.","tokens_in":30195,"feed_emoji":"🌌","tokens_out":8622,"duration_ms":78300,"temperature":0.7,"pith_summary":"This paper tries to establish when cosmic rays dynamically affect the turbulent, magnetized gas of the interstellar medium rather than passively diffusing through it. The authors define two dimensionless numbers, $\\mathrm{Pm}_{\\rm c} = D_{\\rm c}/D_{\\rm crit}$ and $\\mathrm{Pm}_{\\rm s} = v_{A0}/\\sigma_v$, comparing CR diffusion and CR streaming with the turbulent diffusion of the plasma, and argue that the two fluids couple only when both are below 1. In the coupled regime, CRs imprint on the gas a mixed equation of state between $P_{\\rm c} \\propto \\rho^{4/3}$ and $P_{\\rm c} \\propto \\rho^{2/3}$, effectively raising the sound speed and suppressing density fluctuations. Decoupling occurs at $\\mathrm{Pm}_{\\rm c} \\ge 1$ in all plasma regimes tested, which would sharply narrow the region of parameter space where galaxy simulations must treat cosmic-ray pressure as dynamically active.","feed_headline":"Two numbers decide when cosmic rays reshape interstellar gas","feed_subtitle":"Below those speeds, cosmic-ray pressure stiffens the gas and heats it; above them, the two fluids decouple.","key_machinery":"The load-bearing object is the interaction coefficient $\\sigma_{\\rm c}$ in the two-moment cosmic-ray flux equation, whose inverse combines the microphysical diffusion tensor and the Alfv\\'enic streaming flux. Rescaling that inverse by the turbulent correlation length and velocity yields the three dimensionless Prandtl numbers $\\mathrm{Pm}_{\\rm c}$, $\\mathrm{Pm}_{\\rm s}$, and $\\mathrm{Pm}_{\\rm f} = \\mathrm{Pm}_{\\rm c}/\\mathrm{Pm}_{\\rm s}$, which decide which transport term dominates and mark $\\mathrm{Pm}_{\\rm c} = 1$ as the coupled-to-decoupled boundary. In the coupled state the cosmic-ray pressure behaves as a mixture of the relativistic equation of state $P_{\\rm c} \\propto \\rho^{4/3}$ and the streaming equation of state $P_{\\rm c} \\propto \\rho^{2/3}$, a signature the paper reads directly from two-dimensional $E_{\\rm c}$--$\\rho$ probability distributions.","core_discovery":"Using a two-moment cosmic-ray magnetohydrodynamic model with Alfv\\'enic streaming and anisotropic diffusion, the paper identifies coupled and decoupled states of the cosmic-ray fluid and the plasma, controlled by dimensionless Prandtl numbers $\\mathrm{Pm}_{\\rm c} = D_{\\rm c}/(\\sigma_v \\ell_0)$ and $\\mathrm{Pm}_{\\rm s} = v_{A0}/\\sigma_v$. The central claim is that the fluids interact only for $\\mathrm{Pm}_{\\rm c} < 1$ and $\\mathrm{Pm}_{\\rm s} < 1$; whenever $\\mathrm{Pm}_{\\rm c} \\ge 1$, they decouple regardless of magnetization. In the coupled regime the joint distribution of cosmic-ray energy density and gas density lies in the band between the relativistic equation of state $P_{\\rm c} \\propto \\rho^{4/3}$ and the streaming-modified $P_{\\rm c} \\propto \\rho^{2/3}$. There the cosmic rays raise the effective sound speed, lower the effective turbulent Mach number, narrow the log-density PDF without changing its correlation length, and secularly heat the cosmic-ray fluid through converging flows. The solenoidal velocity spectrum is left unchanged, while compressible-mode power drops on all resolved scales.","pith_inferences":["Inference: because the paper's own limitation section notes that real interstellar phases have ionization fractions $\\chi = 10^{-2}$ to $10^{-8}$, the effective streaming speed would be $10$ to $10^4$ times larger, pushing most neutral-dominated regions to $\\mathrm{Pm}_{\\rm s} > 1$; the coupled window would then shrink to highly ionized, weakly magnetized gas.","Inference: the measured $E_{\\rm c}$--$\\rho$ correlation suggests an observational test: mapping gamma-ray or synchrotron emissivity against column density in a super-Alfv\\'enic cloud should show the $\\rho^{4/3}$ to $\\rho^{2/3}$ band only where both Prandtl numbers are below 1.","Inference: a natural extension is to make the streaming speed depend on the local ionization fraction and include neutral damping of the streaming instability, which could turn the sharp $\\mathrm{Pm}_{\\rm c} = 1$ boundary into a phase-dependent surface.","Inference: because the simulations inject solenoidal and compressible modes in equal proportion, the reported invariance of the solenoidal spectrum may not carry over to purely compressive driving, the configuration used in some earlier subsonic studies."],"forward_implications":["Galaxy simulations can use $\\mathrm{Pm}_{\\rm c}$ and $\\mathrm{Pm}_{\\rm s}$ as a pre-run diagnostic: only plasmas with $D_{\\rm c} < D_{\\rm crit}$ and $v_{A0} < \\sigma_v$ need to resolve cosmic-ray backreaction on the gas dynamics.","In coupled super-Alfv\\'enic gas, cosmic-ray pressure lowers the amplitude of density fluctuations at fixed momentum injection, so density-variance based star formation or column-density diagnostics will imply a smaller Mach number than the velocity dispersion alone suggests.","Cosmic-ray heating in the coupled regime comes from coherent converging structures in the compressible velocity field, so reacceleration is tied to compressive motions and does not drain the solenoidal cascade.","In decoupled regimes with $\\mathrm{Pm}_{\\rm c} \\ge 1$ or $\\mathrm{Pm}_{\\rm s} \\ge 1$, cosmic rays behave as a nearly uniform, passively distributed pressure and have negligible dynamical effect on the plasma."],"supporting_citations":[{"why":"Defines the critical diffusion coefficient $D_{\\rm crit} = \\sigma_v \\ell_0$ and supplies the one-moment coupling results that this study directly extends and qualifies.","marker":"Commerçon et al. (2019)"},{"why":"Provides the two-moment cosmic-ray transport equations and the interaction coefficient $\\sigma_{\\rm c}$ on which the dimensionless Prandtl-number analysis is built.","marker":"Jiang & Oh (2018)"},{"why":"Supplies the comparison CRMHD turbulence results for cosmic-ray heating and spectral changes, including the solenoidal-spectrum contrast.","marker":"Bustard & Oh (2023)"},{"why":"Shows that streaming redistributes cosmic rays and homogenizes $E_{\\rm c}$, the decoupling effect the paper attributes to $\\mathrm{Pm}_{\\rm s} > 1$.","marker":"Dubois et al. (2019)"},{"why":"Gives the streaming-dominated equation of state $P_{\\rm c} \\propto \\rho^{2/3}$ used as one boundary of the coupled regime.","marker":"Wiener et al. (2017a)"},{"why":"Provides the high-density streaming scaling that defines the lower edge of the measured $E_{\\rm c}$--$\\rho$ correlation.","marker":"Quataert et al. (2022)"},{"why":"Supplies the turbulent-driving scheme and the log-density variance scaling used to build the $M$ and $M_{\\rm eff}$ theoretical PDFs.","marker":"Federrath et al. (2010)"},{"why":"Establishes that cosmic-ray streaming proceeds near the Alfv\\'en speed, the physical basis for $\\mathrm{Pm}_{\\rm s}$.","marker":"Kulsrud & Pearce (1969)"},{"why":"Introduces the effective sound speed from cosmic-ray pressure that explains the narrowing of the density PDF.","marker":"Salem et al. (2014)"}],"fun_headline_variants":["Two numbers set when cosmic rays couple with gas","Cosmic ray-gas coupling hinges on two Prandtl numbers","Cosmic rays reshape gas when both Prandtl numbers drop below 1","The two Prandtl numbers that unlock cosmic ray-gas coupling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that cosmic-ray streaming runs at the full Alfv\\'en speed, which holds only for a fully ionized plasma; with realistic interstellar ionization fractions of $10^{-2}$ to $10^{-8}$, the streaming speed would be $10$ to $10^4$ times larger, pushing most of the coupled runs into the decoupled regime.","fun_headline_variants_meta":{"raw":{"variants":["Two numbers set when cosmic rays couple with gas","Cosmic ray-gas coupling hinges on two Prandtl numbers","Cosmic rays reshape gas when both Prandtl numbers drop below 1","The two Prandtl numbers that unlock cosmic ray-gas coupling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000753,"raw_usage":{"total_tokens":3456,"prompt_tokens":1158,"completion_tokens":2298,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":774,"completion_tokens_details":{"reasoning_tokens":2227}},"tokens_in":774,"tokens_out":2298,"duration_ms":16716,"temperature":1.0,"reasoning_tokens":2227,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:56:00.161070+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the joint distribution of cosmic-ray energy density and gas density in a super-Alfv\\'enic, low-diffusion interstellar region: the paper predicts a tight correlation along the band $P_{\\rm c} \\propto \\rho^{4/3}$ to $P_{\\rm c} \\propto \\rho^{2/3}$ when both $\\mathrm{Pm}_{\\rm c}$ and $\\mathrm{Pm}_{\\rm s}$ are below 1, and a flat, uncorrelated distribution when either exceeds 1.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the two-moment cosmic-ray transport equations and the interaction coefficient $\\sigma_{\\rm c}$ on which the dimensionless Prandtl-number analysis is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the high-density streaming scaling that defines the lower edge of the measured $E_{\\rm c}$--$\\rho$ correlation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes that cosmic-ray streaming proceeds near the Alfv\\'en speed, the physical basis for $\\mathrm{Pm}_{\\rm s}$."},{"cited_title":"L., & Hummels , C","cited_arxiv_id":null,"evidence_quote":"Introduces the effective sound speed from cosmic-ray pressure that explains the narrowing of the density PDF."}],"review_version":1}