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REVIEW 5 major objections 4 minor 186 references

Explaining JWST star formation history at $z \sim 17$ by modifying $\Lambda$CDM

T0 review · 5 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that k-mouflage gravity, a modified theory with a screening scalar field, can simultaneously explain JWST's surprisingly massive early galaxies and the completion of reionization, while six other cosmologies tested fail…

desk verdict A serious but statistically unfinished attempt to constrain modified gravity with JWST high-z data; the headline k-mouflage claim rests on visual inspection and LCDM-calibrated baryonic physics. read the letter →

arxiv 2501.11103 v1 pith:QTIOJIY7 submitted 2025-01-19 astro-ph.CO

Oleksii Sokoliuk This is my paper · ORCID
classification astro-ph.CO
keywords modifiedgravityJWSThigh-redshiftgalaxiesstarformationhistoryhalomassfunctionreionizationk-mouflagenDGPbraneworldcosmologicaltensions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that JWST's surprisingly massive, early galaxies and the requirement that reionization finish on schedule can be explained together if gravity is modified in a specific way. It builds galaxy observables—stellar mass functions, stellar mass densities, star formation rate densities, UV luminosity functions, and reionization histories—from halo mass functions computed in several beyond-ΛCDM theories, then asks which parameter choices survive. The answer it reaches is that only k-mouflage gravity, a scalar-field theory with a screening mechanism, combined with a double power-law stellar mass-to-halo relation, satisfies both the high-redshift JWST stellar mass density and the reionization constraints up to stellar masses around $10^{11}\,M_\odot$. The other tested theories—phenomenological modifications, varying-growth-index cosmologies, and the normal-branch DGP braneworld—fail at least one of these probes. If true, this narrows the modified-gravity landscape and suggests the early-galaxy excess is a gravitational, not a baryonic, effect.

What carries the argument

The machinery is an analytic chain that converts a theory of gravity into galaxy observables. Modified gravity enters through the functions $\mu(a,k)$ and $\gamma(a,k)$ that rescale the gravitational coupling and the Bardeen potentials; these feed a modified Einstein-Boltzmann solver that produces linear matter power spectra. From those spectra, spherical collapse gives the linear density threshold $\delta_c$ and virial quantities, and the extended Press-Schechter formalism yields the halo mass function. Abundance matching with two stellar mass-to-halo relations converts halos into stars, producing the stellar mass function, stellar mass density, and—through a star-formation-rate-to-UV luminosity conversion with dust attenuation—the UV luminosity function and star formation rate density. Reionization is modeled with $Q_{\rm HII}$ and $\tau_{\rm reion}$, using escape fraction $f_{\rm esc}=0.25$, clumping factor $C_{\rm HII}=3.0$, and photon production rate $\log_{10} N_{\rm ion}=53.14$. The load-bearing identity is that only the halo mass function is changed by gravity; all baryonic conversion recipes are carried over from ΛCDM unchanged.

What would settle it

Re-calibrate the double power-law stellar mass-to-halo relation and the UV scatter from hydrodynamic simulations run inside k-mouflage gravity and repeat the fits: if the JWST stellar mass density and reionization constraints no longer overlap for $\beta \approx 0.1$ and $K_0 \gtrsim 0.9$, the central claim is falsified. A simpler observational check is new JWST spectroscopy showing that the $z \gtrsim 10$ galaxies assigned $M_\star \sim 10^{11}\,M_\odot$ actually have lower stellar masses, which would relax the tension that k-mouflage is invoked to resolve.

Watch

Extended reading notes

Core claim

The central claim is that k-mouflage gravity with scalar-field parameters $\beta \approx 0.1$, $K_0 \gtrsim 0.9$ and the double power-law stellar mass-to-halo relation can simultaneously account for the JWST stellar mass density at $z \sim 8$–$16$ and the epoch of reionization constraints on the ionized hydrogen filling fraction $Q_{\rm HII}$ and the CMB optical depth $\tau_{\rm reion}$, up to stellar masses of roughly $10^{11}\,M_\odot$. No other model considered—ΛCDM, the phenomenological parameterizations, the varying-growth-index $w\gamma$CDM model, or the nDGP braneworld—passes both sets of constraints. The paper also derives new parameter preferences: nDGP favors a crossover scale $r_c \gtrsim 10^{3.5}\,{\rm Mpc}$, and in the $w\gamma$CDM case phantom-like dark energy with $w_\Lambda \lesssim -1$ is preferred over quintessence. The result is presented as using JWST to narrow the landscape of viable modified gravity theories.

Load-bearing premise

The star-formation efficiencies, UV scatter and dust corrections, and reionization parameters ($f_{\rm esc}=0.25$, $C_{\rm HII}=3.0$, $\log_{10} N_{\rm ion}=53.14$) were calibrated inside ordinary ΛCDM and are assumed to remain correct when gravity is modified, so only the halo mass function changes.

Editorial extensions

If this is right

  • If k-mouflage gravity is correct, JWST's excess of massive galaxies at $z > 10$ is a prediction of the modified halo mass function rather than an anomaly requiring extreme star-formation efficiency.
  • The combined reionization and stellar mass density constraints single out $\beta \approx 0.1$, $K_0 \gtrsim 0.9$ for k-mouflage, a parameter region that future galaxy surveys can either confirm or exclude.
  • nDGP is driven to $r_c \gtrsim 10^{3.5}\,{\rm Mpc}$, meaning a viable braneworld must have a very large crossover scale, nearly returning to ΛCDM on observable scales.
  • Phenomenological modified gravity and $w\gamma$CDM cannot fix the JWST high-redshift tension because their halo mass functions barely deviate from ΛCDM at $z \gtrsim 4$; only screened theories with an enhanced small-scale gravitational force succeed.
  • The strong dependence of the constraints on the choice of stellar mass-to-halo relation means modified gravity parameters are degenerate with baryonic feedback assumptions, so joint fits are needed before concluding that gravity is modified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same pipeline could test other screened theories, such as symmetron or chameleon models, because the paper shows the discriminating power comes from the high-mass end of the halo mass function.
  • If the gravity-dependence of baryonic recipes is small, the JWST excess is evidence for enhanced small-scale structure formation; if it is large, the k-mouflage preference may be an artifact of using ΛCDM-calibrated star-formation physics, and a hydrodynamic simulation in k-mouflage gravity would settle which.
  • The reionization analysis fixes $f_{\rm esc}$ and $N_{\rm ion}$; treating them as free parameters would likely widen the allowed $\{\beta, K_0\}$ region and could remove the single-model victory, so the claimed success is conditional on those fiducial values.
  • A direct observational extension is to measure the UV luminosity function at $z \sim 14$–$17$ with future JWST observations: k-mouflage predicts a specific bright-end excess relative to ΛCDM that the current UVLF data constrain only weakly.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper explores several modified gravity (MG) models—phenomenological gravity, wγCDM, the normal branch of DGP (nDGP), and k-mouflage—as potential explanations of JWST observations at z ≲ 17. The author implements the halo mass function and spherical collapse in MG, then applies fixed baryonic prescriptions to compute the stellar mass function (SMF), stellar mass density (SMD), UV luminosity function (UVLF), star formation rate density (SFRD), and reionization quantities Q_HII and τ_reion. The main claimed results are that nDGP prefers r_c ≳ 10^3.5 Mpc, k-mouflage prefers β ~ 0.1 with K_0 ≳ 0.9, and phantom-like w_Λ is preferred; the central conclusion is that k-mouflage with a double-power-law SMHR can simultaneously satisfy both reionization and high-redshift JWST SMD constraints up to M_* ~ 10^11 M_sun.

Significance. If the central claim were quantitatively established, the paper would provide a concrete modified-gravity candidate that explains JWST's early massive galaxies and the epoch of reionization, with falsifiable predictions for future surveys. The manuscript has strengths: it covers a broad landscape of MG models, uses a consistent MG-CLASS-based pipeline, includes recent JWST data, and states that the code is available. However, the analysis as presented is exploratory: constraints are drawn by visual inspection of overlaid model curves, baryonic prescriptions are calibrated within ΛCDM and assumed unchanged in MG, and the paper itself acknowledges that several key parameters are unconstrained. These issues are load-bearing for the paper's main conclusion, so the significance is currently limited to a proof-of-concept under strong assumptions.

major comments (5)
  1. [Section 7 and Figure 23] The parameter constraints and the central conclusion that k-mouflage 'can satisfy' the JWST constraints are based on visual inspection of model curves overlaid on data, with no likelihood, no MCMC exploration, no parameter uncertainties, and no goodness-of-fit statistic. Statements such as 'preferred values', 'excluded', and 'best-fit parameters' in Sections 7.1–7.5 and Figure 23 are therefore not quantitatively supported. This is not a minor presentational issue: the paper's main claim depends on distinguishing models that 'satisfy' the data from those that 'fail', and that distinction cannot be made reliably without a statistical comparison that accounts for the error bars on the data points and the scatter prescriptions.
  2. [Section 5.2 and Section 8] The baryonic prescriptions are calibrated within ΛCDM and then assumed to remain exactly valid in modified gravity. This applies to the double-power-law SMHR of Eq. (68), the UV scatter parameters in Eq. (88), the reionization inputs f_esc = 0.25, C_HII = 3.0, and log10 N_ion = 53.14 from Section 6, and the peak star formation efficiency ϵ_*,0. The paper itself notes in Section 5.2 that the UV scatter parameters 'were derived on the basis of ΛCDM cosmology. Clearly, this is an approximation', and in Section 8 that ϵ_*,0 is 'not bounded at all' and an ΛCDM value was assumed. Since SMD, UVLF, and Q_HII depend sensitively on these parameters, a modest gravity-induced change in star formation efficiency or escape fraction would be degenerate with the fitted MG parameters (β, K_0, r_c). The preferred regions in Figure 23 could therefore shift substantially if the baryonic prescriptions were re-derived or marginalized within each MG model.
  3. [Sections 4.1, 5.2, 6 and 7] There is a potential circularity in testing the MG models against observables that were used, at least in part, to calibrate the baryonic ingredients of the same pipeline. The Rodríguez-Puebla SMHR (Eqs. 64–70) is calibrated on SMF and SMD data, the double-power-law SMHR is calibrated on UVLF data, the UV scatter model (Eq. 88) is calibrated on UVLF data, and log10 N_ion is derived from HUDF data. The paper then uses SMF, SMD, UVLF, and reionization observations from Section 7 as constraints. If the calibration data overlap with the 'test' datasets, then the claim that k-mouflage 'satisfies' the JWST constraints is not an independent validation; at minimum, the effective number of degrees of freedom is decreased and the comparison is biased. This should be quantified or the analysis reframed as a consistency test under a stated calibration scheme.
  4. [Abstract, Section 7.1, Section 7.4, Section 8] The paper's statements about the preferred dark energy equation of state are internally inconsistent. The abstract states that 'phantom-like dark energy EoS w_Λ ≲ −1 is preferred over the quintessence', but Section 7.1 reports that Model II prefers the range −1 ≲ w_Λ < ∞, Section 7.4 says that 'w_Λ ≤ 1 ... makes quintessence cosmology a viable choice', and Section 8 states that wγCDM 'prefers w_Λ ≳ −1 instead of quintessence'. These statements cannot all be correct, and the abstract's strong phantom preference is not supported by the body text. This needs to be corrected and the conclusion about w_Λ made consistent with the actual results.
  5. [Section 7.4 and Section 8] The central claim in Section 8—that k-mouflage with the double-power-law SMHR can satisfy both reionization and high-redshift SMD constraints up to M_* ~ 10^11 M_sun—is qualified by the paper's own finding in Section 7.4 that 'for any model and any scatter value, it is still not possible to produce the nearly constant SFRD, required by JWST at z ≳ 12'. The title and abstract claim to 'explain JWST star formation history at z ~ 17', but the SFRD at z ≳ 12 is not reproduced. The paper should either soften the claim to the specific observables and mass range that are actually reproduced, or provide quantitative evidence that the remaining SFRD discrepancy is within the observational and modeling uncertainties.
minor comments (4)
  1. [Equation (43)] The Christoffel symbol in Eq. (43) has a typo: the first two terms are both written as ∂_ν g_{βμ}; the standard expression is Γ^α_{μν} = (1/2)g^{αβ}(∂_μ g_{βν} + ∂_ν g_{βμ} − ∂_β g_{μν}).
  2. [Section 7.1 and Section 8] The text in Section 7.1 says 'first model suggests higher deviation from the fiducial cosmology than the second one', and Section 8 says 'the double power-law best-fit values are noticeably closer to the ΛCDM than Rodriguez-Puebla'. These are consistent, but the phrasing 'closer to the ΛCDM than Rodriguez-Puebla' is missing an explicit comparison object and should be reworded.
  3. [Throughout] There are repeated typographical issues, e.g. 'di fferent', 'it’s power spectrum', 'can easily be implement', and 'JDB' instead of 'JBD' in Section 2.4. A careful proofread is needed.
  4. [Figure 23] The caption of Figure 23 says 'Arrow signs signify an upper limit' and uses circles/squares for degenerate solutions, but the marker codes are not explained in the figure caption; please define the symbols explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction: the k-mouflage result is a forward-model fit, with explicitly flagged baryonic assumptions.

full rationale

I walked the derivation chain: the modified-gravity parameters (rc, beta, K0) enter only through the halo mass function, delta_c, Mmin, and the resulting SMF/SMD/UVLF/SFRD/QHII are computed by explicit integrals (Eqs. 71, 76-77, 78, 89, 90-93). None of these equations is defined in terms of the JWST target quantities; the baryonic recipes (SMHR, UV scatter, fesc, CHII, Nion) are fixed inputs taken from external, non-self-cited literature. The paper explicitly flags the two places where the argument is an approximation rather than a derivation: Sec 5.2 notes sigma_UV parameters 'were derived on the basis of LCDM cosmology. Clearly, this is an approximation', and Sec 8 admits epsilon_star,0 'are not bounded at all' and that an LCDM value was assumed. The ad hoc SFRD scatter (0.2/0.05 dex, Sec 7.4-7.5) is an adjustable input that affects QHII, but it is disclosed and is not defined in terms of QHII, so it does not make the reionization 'prediction' equal to its input by construction. The absence of a likelihood and the use of visual inspection (Sec 7, Fig 23) are statistical/reproducibility weaknesses, not circularity. There are no self-citations and no imported uniqueness theorems. The central claim is therefore self-contained as a forward-model constraint, conditional on the stated LCDM-calibrated baryonic assumptions.

Assumptions & free parameters 13 free parameters · 5 assumptions · 0 invented entities

The central claim depends on a large set of baryonic parameters calibrated within Lambda-CDM, plus the MG model parameters themselves. The MG parameters are effectively free and are constrained only by visual inspection, while the baryonic parameters are assumed and not marginalized over.

free parameters (13)
  • log10 rc (nDGP crossover scale) = 3.5 to 3.9
    Preferred values from visual inspection of SMF, SMD, UVLF, SFRD and EoR data (Section 7).
  • beta (k-mouflage coupling) = 0.1 (double power-law SMHR) or 0.3 (Rodriguez-Puebla SMHR)
    Preferred values from the same visual inspection (Sections 7.1-7.5).
  • K0 (k-mouflage kinetic term coefficient) = >=0.9 (double power-law SMHR) or ~0.3-1 (Rodriguez-Puebla SMHR)
    Preferred values from visual inspection (Sections 7.1-7.5).
  • wLambda (dark energy equation of state) = <= -1 per abstract; >= -1 per Section 8
    Contradictory statements; no statistical fit performed.
  • gamma (growth index in wgammaCDM) = 0.4 to 0.6
    Range constrained by SMF and SFRD visual comparisons (Sections 7.1, 7.4).
  • E11, E22, g_mu, g_gamma, T1, T2 (phenomenological MG parameters) = not strongly constrained; qualitative preferences only
    Visual inspection of power spectra and SMF/SFRD curves (Sections 2, 7.1, 7.4).
  • epsilon_star_0 (peak star formation efficiency) = 0.21 (assumed)
    Fixed to Lambda-CDM value from literature, not fitted (Section 4).
  • fesc (escape fraction) = 0.25 (assumed)
    Taken from SIMBA simulations; not varied (Section 6).
  • CHII (clumping factor) = 3.0 (assumed)
    Fixed from Kaurov & Gnedin 2015 (Section 6).
  • THII (ionized hydrogen temperature) = 2e4 K (assumed)
    Fixed following Gong et al. 2023 (Section 6).
  • log10 Nion (ionizing photon production rate) = 53.14 (derived from HUDF data)
    Fixed from Robertson et al. 2013, 2015 (Section 6).
  • sigma_UV parameters (A, B, C, sigma_min) = A=1.1, B=0.34, C(z), sigma_min=0.2
    Derived from Lambda-CDM simulations and applied unchanged to MG (Section 5.2).
  • Mh_cut (minimum halo mass) = 10^6.5 solar masses (assumed)
    Assumed cut-off for star formation efficiency (Section 4).
assumptions (5)
  • standard math Press-Schechter and Sheth-Tormen halo mass function formalisms are assumed applicable to modified gravity.
    Used in Section 3 without re-calibration.
  • standard math The spherical collapse equations with mu(a) inserted for the gravitational constant give the correct critical overdensity.
    Section 3.1 assumes this form of collapse for modified gravity.
  • domain assumption Stellar mass-to-halo relation, dust attenuation, and UV scatter calibrations derived in Lambda-CDM are unchanged in modified gravity.
    Sections 4 and 5.2 explicitly assume these prescriptions hold, despite noting they were calibrated on Lambda-CDM.
  • domain assumption Reionization inputs fesc=0.25, CHII=3.0, THII=2e4 K, log10 Nion=53.14 are fixed.
    Section 6 adopts these values without marginalization.
  • domain assumption Planck2018 cosmological parameters are held fixed; MG affects only the specified mu and gamma functions.
    Section 2 states this, noting it is an approximation.

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Cite this review

Pith. "Pith review of Explaining JWST star formation history at $z \sim 17$ by modifying $\Lambda$CDM." pith.science (2026). https://pith.science/paper/QTIOJIY7

@misc{pith2026250111103,
  author       = {Pith},
  title        = {Pith review of: Explaining JWST star formation history at $z \sim 17$ by modifying $\Lambda$CDM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QTIOJIY7}},
  note         = {Machine review of arXiv:2501.11103}
}
abstract

Recent cosmological observations indicate a $5\sigma$ discrepancy between the values of the Hubble constant $H_0$ derived from late and early universe probes. A further possible tension at the $\sim 3\sigma$ level arises from different measurements of $\sigma_8$. These measurements suggest the existence of new physics. Here, we explore several theories of modified gravity that may help to resolve these cosmological tensions. These include a family of phenomenological modified theories, where only Newton's gravitational constant and the Einstein-Boltzmann equations are affected. We consider one particular class of these theories: cosmologies with varying growth index $\gamma$ and varying dark energy Equation of State (EoS) $w_\Lambda$. We also consider the normal branch of the Dvali-Gabadadze-Porrati (nDGP) model as well as $k$-mouflage gravity, which involves a non-trivially coupled scalar field. Our main aim is to narrow down the modified gravity landscape by constraining each model using high-redshift JWST data. Several probes are considered in this work: Stellar Mass Function (SMF), Stellar Mass Density (SMD), Star Formation Rate Density (SFRD) and Ultra-Violet Luminosity Function (UVLF) along with the Epoch of Reionization (EoR). We find that generally, the choice of $r_c\gtrsim 10^{3.5}$ Mpc is preferred for nDGP, while $\beta\sim0.1$, $K_0\gtrsim 0.9$ is favored for $k$-mouflage. Moreover, in the context of phenomenological gravity, phantom-like dark energy EoS $w_\Lambda\lesssim -1$ is preferred over the quintessence.

Figures

Figures reproduced from arXiv: 2501.11103 by the authors.

Figure 1
Figure 1. Structure of this paper propriate form of the non-canonical scalar field potential K(χ), where χ is the scalar field. However, at smaller scales, such as within the Solar System, k-mouflage becomes highly non-linear and unstable due to the presence of higher-order modes in the field equation. This makes it challenging to decide on the valid￾ity of a theory on such scales (Brax & Valageas 2014). Aside from the proble… view at source ↗
Figure 2
Figure 2. Matter power spectrum predictions for phenomenological modified gravity versus observational data with varying values of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure (2) continued, but for the nDGP gravity with varying rc (first subplot) and k-mouflage gravity with varying β, K0 (second subplot) The only unknown here is the Hubble parameter, which can be obtained using the modified Friedmann equation (22) and rewrit￾ten in a more familiar form: H(a) = H0 p Ωm0a −3 + Ωr0a −4 + ΩΛ + Ωrc − H0 p Ωrc. (24) where Ωrc = 1/(4H 2 0 r 2 c ) with normalization condition √ Ωm0 + Ωr0 … view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: Linear density contrast for the spherical collapse occur [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Linear density contrast for the spherical collapse occur [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Virial overdensity for phenomenological models [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Virial overdensity for the theories of modified gravitation [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Stellar mass function for the Planck parameterization under both Rodriguez-Puebla (top row) and double power-law (bottom [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Figure (8) continued, but for the z_flex parameterization phenomenological theories have a substantial deviation from the fiducial model at Mh ∼ 1015M⊙ and above, which translates to M⋆ ≳ 1012M⊙ (this value strongly depends on the SMHR). There are only several data poi…
Figure 10
Figure 10. Figure 10: Figure (8) continued, but for the DES parameterization 102.7 Mpc. On the other hand, double power-law SMHR shows strong evidence for rc ≳ 103.5 Mpc at 0 < z < 4 but can allow for rc ∼ 103 Mpc at the higher redshift range, the beginning of Epoch of Reionization, i.e. 5…
Figure 11
Figure 11. Figure 11: Figure (8) continued, but for the wγCDM modified gravity results being displayed in the Figures (15) and (16) respectively. We did not consider phenomenological modifications of grav￾ity such as Planck/DES/wγCDM, as they do not affect high-z structure formation, and t…
Figure 12
Figure 12. Figure 12: Stellar mass function for the nDGP gravity under both Rodriguez-Puebla (orange lines) and double power-law (blue lines) [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Figure (12) continued, but for the k-mouflage gravity under the Rodriguez-Puebla SMHR Article number, page 23 of 35 [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Figure (12) continued, but for the k-mouflage gravity under the double power-law SMHR Article number, page 24 of 35 [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]
Figure 15
Figure 15. Figure 15: Stellar mass density for nDGP modified gravity theory versus recent observational data using Rodriguez-Puebla SMHR [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]
Figure 16
Figure 16. Figure 16: Figure (15) continued, but for the k-mouflage gravity EAGLE simulation output (Furlong et al. 2015), Horizon Run 5 simulation output (Lee et al. 2021) and Illustris-TNG simu￾lation output (Pakmor et al. 2023) respectively. The solution was to introduce an additional s…
Figure 17
Figure 17. Figure 17: Ultra-violet luminosity function for the nDGP gravity under both Rodriguez-Puebla (orange lines) and double power [PITH_FULL_IMAGE:figures/full_fig_p027_17.png]
Figure 18
Figure 18. Figure 18: Figure (17) continued, but for the k-mouflage gravity under the Rodriguez-Puebla SMHR Article number, page 28 of 35 [PITH_FULL_IMAGE:figures/full_fig_p028_18.png]
Figure 19
Figure 19. Figure 19: Figure (17) continued, but for the k-mouflage gravity under the double power-law SMHR Article number, page 29 of 35 [PITH_FULL_IMAGE:figures/full_fig_p029_19.png]
Figure 20
Figure 20. Figure 20: Star formation rate density for Planck (first row) and z_flex (second row), DES (third row) and [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]
Figure 21
Figure 21. Figure 21: Figure (20) continued, but for nDGP (first row) and k-mouflage (second row) modified gravity theories gen predicted by nDGP cannot satisfy observational constraints for any value of rc and in general, the variation of QHII(z) with rc is only marginal. On the other han…
Figure 22
Figure 22. Figure 22: Volume filling fraction of HII (left column) and optical depth (right column) for nDGP (first row), [PITH_FULL_IMAGE:figures/full_fig_p032_22.png]
Figure 23
Figure 23. Figure 23: Compilation of the best-fit parameters determined by the visual inspection of various observables for nDGP (first column) [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]

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