arXiv subjects

Tirthankar Roy Choudhury

Publications and source records attributed to Tirthankar Roy Choudhury.

At least 19 recordsLinked to original sources

arXiv2026

Reconciling large-scale Lyman-$α$ correlations with the SCRIPT Semi-numerical Model

Recent analyses of high-redshift Lyman-$α$ forest observations have revealed strong correlations on scales exceeding 200 cMpc at redshift z = 6. Reproducing these large-scale correlations has proven challenging for current large-volume reionization simulations. In this work, we investigate these large-scale correlations using mock spectra generated from the extended SCRIPT semi-numerical reionization model. We find that while the fiducial model ensemble systematically predicts smaller correlation lengths than those inferred from the 67 sightlines in the extended XQR-30 sample, 17.5% of individual mock realizations can naturally reproduce the observed signal. Using a delete-2 jackknife analysis, we demonstrate that the observed large-scale correlation length is disproportionately driven by a rare pair of highly transmissive sightlines associated with high-redshift transmission spikes. By inserting two such highly transmissive sightlines into our mock realizations, the fraction of realizations consistent with the observed redshift evolution and correlation length increases significantly from 17.5% to 74.1%. Furthermore, we show that spatial fluctuations in the ionizing mean free path remain an essential physical ingredient for reproducing the observed correlation structure. Our results suggest that the unexpectedly large Lyman-$α$ correlations can be reconciled with existing reionization models when accounting for cosmic variance and the outsized statistical impact of highly transmissive sightlines that occur as prominent outliers within the observational sample.

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arXiv2026

SCRIPT in the Cosmic Dawn: Distinguishing Redshift-Evolving Galaxy Populations with 21-cm Fluctuations

Recent James Webb Space Telescope (JWST) observations suggest an unexpectedly high abundance of luminous galaxies at $z \gtrsim 10$, challenging traditional models of star formation. The 21-cm signal of neutral hydrogen offers a complementary, volume-averaged probe of these early epochs. In this work, we present a major extension to the explicitly photon-conserving semi-numerical framework SCRIPT, enabling the self-consistent calculation of spin temperature fluctuations driven by inhomogeneous Lyman-$α$ coupling and X-ray heating during the Cosmic Dawn. Using this framework, we compare a JWST-informed fiducial galaxy model, featuring a redshift-evolving star-formation efficiency, against a static baseline model. We demonstrate that the global 21-cm signal is highly degenerate: a static galaxy population can replicate the absorption trough of an evolving population if a steep, mass-dependent X-ray efficiency is invoked to shift intergalactic medium heating toward highly abundant, low-mass halos. However, this degeneracy is definitively broken by the 21-cm power spectrum. Because spatial fluctuations retain a structural memory of the clustering bias of the source halos, the tuned static model predicts a substantially lower power spectrum amplitude across all characteristic peaks. We conclude that combining global and fluctuation measurements is essential to robustly constrain the redshift evolution of high-redshift galaxies.

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arXiv2026

Overview of 21cm Experiments at high redshift with SKAO

We provide an overview of the eight SKAO Science Book chapters that motivate the Epoch of Reionisation and Cosmic Dawn experiments with SKA-Low. We describe the individual SKA-Low experiments and expected sensitivity - power spectrum, tomography, 21-cm forest, cross-correlations, building on the broad observational plan laid out in the 2015 SKA Science Book. Finally, we outline features of the telescope that will be critical for the success of EoR/CD science, e.g., beam apodization, substations, and multi-beaming.

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arXiv2026

Gravitational Wave Informed Inference of 21-cm Global Signal Parameters

Understanding how and when the first stars and galaxies formed remains one of the central challenges in modern cosmology. These structures emerged during the transition from the Dark Ages to the Cosmic Dawn, a period that remains observationally unconstrained despite strong theoretical progress. During this epoch, neutral hydrogen absorbed a fraction of cosmic microwave background photons through its 21-cm hyperfine transition, producing a 21-cm absorption signal whose evolution encodes the early Universe's thermal and ionization history. However, extracting the underlying astrophysical parameters from this signal is limited by severe parameter degeneracies, which cannot be resolved without independent observational probes. The next-generation gravitational wave (GW) detectors, such as Cosmic Explorer (CE), will observe binary black hole (BBH) mergers up to very large redshifts and hence will detect a fraction of them formed within the redshift range $\sim 13-25$. The merger rate of these BBHs will depend on the star formation rate density (SFRD) at these redshifts, together with the BBH formation efficiency and a time delay distribution. Therefore, the merger rate of these BBHs can work as a tracer of the SFRD in the redshift range $\sim 13-25$. In this Letter, we establish a novel multi-messenger framework and present a proof-of-principle concept of how the observations of BBH mergers form next-generation GW detectors can improve the inference of parameters generating the 21-cm cosmic hydrogen signal, and help break degeneracies between them.

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arXiv2026

Nearest Neighbour-Based Statistics for 21cm-Galaxy Cross-Correlations in the Epoch of Reionization

21cm radiation from neutral hydrogen serves as a direct probe of the Epoch of Reionization. However, both its detection and physical interpretation are severely hindered by contamination from astrophysical foreground emission and instrumental noise that are several orders of magnitude brighter than the signal of interest. A promising way to tackle these challenges is to cross-correlate the 21cm signal with other independent tracers of large-scale structure, most notably high-redshift galaxies. Besides validating putative 21cm detections, such joint analyses are expected to provide independent insights into the properties of ionizing sources and the evolving morphology of ionized regions during reionization. The 21cm signal, however, is intrinsically highly non-Gaussian, limiting the effectiveness of conventional two-point cross-correlation statistics, which capture information only up to the second order. In this work, we therefore investigate the utility of k-nearest-neighbour cumulative distribution functions (kNN CDF), which encode information from the joint clustering at all orders, as an alternative framework for probing 21cm-galaxy cross-correlations. Using self-consistently simulated mock 21cm fields and a catalog of line-emitting galaxies at z = 7, we conducted a proof-of-concept study comparing the kNN CDF formalism and the two-point cross-correlation approach. We find that the kNN CDF statistics outperform the two-point statistics in detecting 21cm-galaxy cross-correlations, even in the presence of instrumental noise and aggressive foreground filtering. Moreover, at a fixed global ionized fraction, it is even able to differentiate between reionization models that remain indistinguishable using two-point statistics. These results demonstrate the power and unexplored potential of exploiting higher-order statistics for extracting maximal information from 21cm-galaxy synergies.

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arXiv2026

Conditioning halos on the tidal environment for fast and accurate HI power spectra during reionization

Predicting the statistical properties of the neutral hydrogen (HI) density field during reionization is an important step in using upcoming 21 cm observations to constrain models of reionization. Semi-numerical models of reionization are often coupled with the collapse fraction field $f_{\text{coll}}(\mathbf{x})$, which determines the fraction of dark matter within halos. In this work, we improve upon earlier prescriptions that compute $f_{\text{coll}}$ based on the dark matter overdensity $δ(\mathbf{x})$ alone, to include more information about the environment in the form of eigenvalues of the tidal tensor. We compute the mean of the $f_{\text{coll}}$ conditioned on these eigenvalues from a set of high-resolution, small-volume simulations and use them to sample the $f_{\text{coll}}$ field of a low-resolution, large-volume simulation. We subsequently use a semi-numerical code for reionization to compute the HI density field and its power spectrum, and benchmark our results against a reference high-resolution, large-volume simulation. Across variations in redshift, ionized fraction, grid resolution, and minimum halo mass, our method recovers the large-scale HI power spectrum with errors at the $\lesssim 2\%-5\%$ level for $k \lesssim 0.5~ h~ \text{Mpc}^{-1}$, providing a substantial improvement over the $\sim 10\%$ results previously obtained using density-only conditioning. Overall, this makes our method a simple yet efficient tool for forward modeling HI maps during reionization.

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arXiv2026

Accelerating reionization constraints: An ANN-emulator framework for the SCRIPT Semi-numerical Model

Constraining the Epoch of Reionization (EoR) with physically motivated simulations is hampered by the high cost of conventional parameter inference. We present an efficient emulator-based framework that dramatically reduces this bottleneck for the photon-conserving semi-numerical code SCRIPT. Our approach combines (i) a reliable coarse-resolution MCMC to locate the high-likelihood region (exploiting the large-scale convergence of SCRIPT) with (ii) an adaptive, targeted sampling strategy to build a compact high-resolution training set for an artificial neural network based emulator of the model likelihood. With only $\approx 10^3$ high-resolution simulations, the trained emulators achieve excellent predictive accuracy ($R^2 \approx 0.97-0.99$) and, when embedded within an MCMC framework, reproduce posterior distributions from full high-resolution runs. Compared to conventional MCMC, our pipeline reduces the number of expensive simulations by a factor of $\sim 100$ and lowers total CPU cost by up to a factor of $\sim 70$, while retaining statistical fidelity. This computational speedup makes inference in much higher-dimensional models tractable (e.g., those needed to incorporate JWST and upcoming 21 cm datasets) and provides a general strategy for building efficient emulators for next generation of EoR constraints.

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arXiv2026

Can an Anti-de Sitter Vacuum in the Dark Energy Sector Explain JWST High-Redshift Galaxy and Reionization Observations?

The unexpectedly large abundance of UV-bright galaxies at $z>10$ discovered by the James Webb Space Telescope poses a significant challenge to the standard $Λ$CDM cosmology. This work tests whether modifying the cosmological background, and thereby the growth of structures, can resolve this tension without invoking significant evolution in the astrophysical properties of early galaxies. We investigate an alternative framework featuring an anti-de Sitter vacuum in the dark energy sector, which naturally arises in quantum gravity models like string theory and can enhance early structure formation. Using a self-consistent semi-analytical model that couples galaxy evolution with reionization, we confront this scenario with a wide range of observations. We show that while a model tailored to fit the high-$z$ UV luminosity functions (UVLFs) appears promising, it is in strong tension with cosmological constraints from the CMB and other low-redshift probes. Conversely, models within this framework that satisfy these constraints provide only a modest boost to structure formation and fail to reproduce the observed galaxy abundances at $z>10$. Although these models remain consistent with the cosmic reionization history, we find that this class of cosmological modifications is insufficient on its own to explain the galaxy excess. Our study underscores the importance of holistic testing for any beyond-$Λ$CDM proposal; apparent success in one observational regime does not guarantee overall viability. By demonstrating the limitations of a purely cosmological solution, our results strengthen the case that evolving astrophysical properties are a necessary ingredient for solving the challenge of early galaxy formation.

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arXiv2025

ELAIS-N1 deep field uGMRT Band-2: constraints on diffuse Galactic synchrotron emission power spectrum

We present high sensitivity, low radio frequency continuum observations of the ELAIS-N1 field with 32 hours of observations of the uGMRT Band-2 ($120-250$ MHz) covering $5.86\,\text{deg}^2$ area, achieving a central off-source RMS noise of $237\,μ\mathrm{Jy}/\mathrm{beam}$ with a resolution of $11.45''$ at the central frequency of 183 MHz. A radio source catalogue of 1027 sources statistically matches with similar observations at different frequencies within the sensitivity range of the uGMRT. The calibrated data is further used to characterise the dominant foreground, the Diffuse Galactic Synchrotron Emission (DGSE), in angular scale and frequency regime. We derived the angular power spectrum (APS) of DGSE in two ways: image-based estimator (i-APS) and visibility-based Tapered Gridded Estimator (TGE; hereafter as t-APS). We assess the characteristics of DGSE with a power-law form of $C_{\ell} = A({1000}/{\ell})^β$. Combining data from Band-2 and earlier Band-3 observations, we derived a spectral variation of $C_{\ell}$ in the form of $C_{\ell} = A{ν^{-2α}}{\ell^{-β}}$. Our result indicates a spectral break at $ν= 230\,{\pm}\,5$ MHz, corresponding to a synchrotron age of $t_\text{syn} = 106\,{\pm}\,1$ Myr for the cosmic-ray electrons (CRe). This break result suggests a low-energy cutoff in the CRe population, leading to spectral curvature at low frequencies. Using both of the techniques, i-APS and t-APS, we find that the mean spectral index $α$ and power-law index $β$ are consistent within the frequency range $120-500$ MHz.

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arXiv2025

Probing Reionization-Era Galaxies with JWST UV Luminosity Functions and Large-Scale Clustering

JWST has transformed our understanding of early galaxy formation, providing an unprecedented view of the first billion years of cosmic history. In this work, we build upon our previously developed semi-analytical framework that self-consistently models the evolving UVLF of galaxies and the global reionization history while incorporating the effects of radiative feedback. Comparing our predictions with JWST and HST data, we identify a fundamental tension: models that match the UVLF fail to reproduce the observed evolution of galaxy clustering (bias) with redshift, and vice versa. To resolve this, we introduce a redshift- and mass-dependent duty cycle linked to the duration of star formation. This duty cycle increases towards higher redshifts, requiring either an enhanced production of UV radiation or increased star formation efficiency at z>10 to match the JWST UVLFs, but declines at lower redshifts (5<z<=9) and towards low-mass halos to remain consistent with the bias and HST UVLF measurements. Reconciling theory with observations requires the characteristic star formation timescale to be longer in massive halos, and to decrease with redshift at fixed halo mass, evolving from ~ 85 Myr at z=6 to ~ 45 Myr at z=14 for $10^{10} M_\odot$ halos. Finally, our extended model, assuming a halo mass-dependent escaping ionizing efficiency ($\varepsilon_{\rm esc} \equiv f_{\rm esc} \times ξ_{\rm ion}$), whose population-averaged value gradually increases with redshift and corresponds to $\langle f_{\rm esc} \rangle \sim$ 15% at z=5 for a fixed value of $ξ_{\rm ion} = 10^{25.23}$ erg$^{-1}$ Hz across all galaxies, produces a reionization history consistent with current constraints. These findings underscore the importance of jointly constraining high-redshift galaxy models using both UVLF and bias statistics to accurately interpret JWST data and refine our understanding of early cosmic evolution.

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arXiv2025

Capturing Small-Scale Reionization Physics: A Sub-Grid Model for Photon Sinks with SCRIPT

The epoch of reionization represents a major phase transition in cosmic history, during which the first luminous sources ionized the intergalactic medium (IGM). However, the small-scale physics governing ionizing photon sinks - particularly the interplay between recombinations, photon propagation, and self-shielded regions - remains poorly understood. Accurately modeling these processes requires a framework that self-consistently links ionizing emissivity, the clumping factor, mean free path, and photoionization rate. In this work, we extend the photon-conserving semi-numerical framework, SCRIPT, by introducing a self-consistent sub-grid model that dynamically connects these quantities to the underlying density field, enabling a more realistic treatment of inhomogeneous recombinations and photon sinks. We validate our model against a comprehensive set of observational constraints, including the UV luminosity function from HST and JWST, CMB optical depth from Planck, and Lyman-$α$ forest measurements of the IGM temperature, photoionization rate, and mean free path. Our fiducial model also successfully reproduces Lyman-$α$ opacity fluctuations, reinforcing its ability to capture large-scale inhomogeneities in the reionization process. Notably, we demonstrate that traditionally independent parameters, such as the clumping factor and mean free path, are strongly correlated, with implications for the timing, morphology, and thermal evolution of reionization. Looking ahead, we will extend this framework to include machine learning-based parameter inference. With upcoming 21cm experiments poised to provide unprecedented insights, SCRIPT offers a powerful computational tool for interpreting high-redshift observations and refining our understanding of the last major phase transition in the universe.

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arXiv2025

Accelerating HI density predictions during the Epoch of Reionization using a GPR-based emulator on N-body simulations

Building fast and accurate ways to model the distribution of neutral hydrogen during the Epoch of Reionization (EoR) is essential for interpreting upcoming 21 cm observations. A key component of semi-numerical models of reionization is the collapse fraction field $f_{\text{coll}}(\mathbf{x})$, which represents the fraction of mass within dark matter halos at each location. Using high-dynamic range N-body simulations to obtain this is computationally prohibitive and semi-analytical approaches, while being fast, end up compromising on accuracy. In this work, we bridge the gap by developing a machine learning model that can generate $f_{\text{coll}}$ maps by sampling from the full distribution of $f_{\text{coll}}$ conditioned on the dark matter density contrast $δ$. The conditional distribution functions and the input density field to the model are taken from low-dynamic range N-body simulations that are more efficient to run. We evaluate the performance of our ML model by comparing its predictions to a high-dynamic range N-body simulation. Using these $f_{\text{coll}}$ maps, we compute the HI and HII maps through a semi-numerical code for reionization. We are able to recover the large-scale HI density field power spectra $(k \lesssim 1\ h\,{\rm Mpc}^{-1})$ at the $\lesssim10\%$ level, while the HII density field is reproduced with errors well below 10% across all scales. Compared to existing semi-analytical prescriptions, our approach offers significantly improved accuracy in generating the collapse fraction field, providing a robust and efficient alternative for modeling reionization.

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arXiv2025

A White Paper on The Multi-Messenger Science Landscape in India

The multi-messenger science using different observational windows to the Universe such as Gravitational Waves (GWs), Electromagnetic Waves (EMs), Cosmic Rays (CRs), and Neutrinos offer an opportunity to study from the scale of a neutron star to cosmological scales over a large cosmic time. At the smallest scales, we can explore the structure of the neutron star and the different energetics involved in the transition of a pre-merger neutron star to a post-merger neutron star. This will open up a window to study the properties of matter in extreme conditions and a guaranteed discovery space. On the other hand, at the largest cosmological scales, multi-messenger observations allow us to study the long-standing problems in physical cosmology related to the Hubble constant, dark matter, and dark energy by mapping the expansion history of the Universe using GW sources. Moreover, the multi-messenger studies of astrophysical systems such as white dwarfs, neutron stars, and black holes of different masses, all the way up to a high redshift Universe, will bring insightful understanding into the physical processes associated with them that are inaccessible otherwise. This white paper discusses the key cases in the domain of multi-messenger astronomy and the role of observatories in India which can explore uncharted territories and open discovery spaces in different branches of physics ranging from nuclear physics to astrophysics.

astro-ph.HE↗

arXiv2025

Reionization and its sources

Reionization represents an important phase in the history of our Universe when ultraviolet radiation from the first luminous sources, primarily stars and accreting black holes, ionized the neutral hydrogen atoms in the intergalactic medium (IGM). This process follows the ``Dark Ages'', a period with no luminous sources, and is initiated by the formation of the first sources, marking the ``Cosmic Dawn''. Reionization proceeds through multiple stages: initially, ionized bubbles form around galaxies, then expand and overlap across the IGM, culminating in a fully ionized state, with neutral hydrogen remaining only in dense regions. Understanding reionization involves a diverse range of physical concepts, from large-scale structure formation and star formation to radiation propagation through the IGM. Observationally, reionization can be explored using the cosmic microwave background (CMB), Lyman-$α$ absorption, high-redshift galaxy surveys, and emerging 21~cm experiments, which together offer invaluable insights into this transformative epoch.

astro-ph.CO↗

arXiv2024

Efficient hybrid technique for generating sub-grid haloes in reionization simulations

Simulating the distribution of cosmological neutral hydrogen (HI) during the epoch of reionization requires a high dynamic range and is hence computationally expensive. The size of the simulation is dictated by the largest scales one aims to probe, while the resolution is determined by the smallest dark matter haloes capable of hosting the first stars. We present a hybrid approach where the density and tidal fields of a large-volume, low-resolution simulation are combined with small haloes from a small-volume, high-resolution box. By merging these two boxes of relatively lower dynamic range, we achieve an effective high-dynamic range simulation using only 13% of the computational resources required for a full high-dynamic range simulation. Our method accurately reproduces the one- and two-point statistics of the halo field, its cross-correlation with the dark matter density field, and the two-point statistics of the HI field computed using a semi-numerical code, all within 10% accuracy at large scales and across different redshifts. Our technique, combined with semi-numerical models of reionization, provides a resource-efficient tool for modeling the HI distribution at high redshifts.

astro-ph.CO↗

arXiv2024

Modelling the star-formation activity and ionizing properties of high-redshift galaxies

Early results from the JWST observations have reported a surprisingly high number of UV-bright galaxies at $z \geq 10$, which appears to challenge the theoretical predictions from standard galaxy formation models in the $Λ$CDM framework at these redshifts. To alleviate this tension, several cosmological and astrophysical interpretations have been advanced. However, all of these proposed scenarios carry noteworthy consequences for other large-scale processes in the early Universe, particularly cosmic reionization, since high-redshift galaxies are believed to be the primary ionizing sources during the Epoch of Reionization (EoR). To investigate this, we introduce a semi-analytical model of galaxy formation and evolution that explains the evolving galaxy UV luminosity function (UVLF) over $6 \lesssim z \lesssim 15$, and also jointly tracks the time evolution of the globally averaged neutral hydrogen fraction in the intergalactic medium. The model self-consistently accounts for the suppression of star formation in low-mass galaxies due to reionization feedback and is constrained by comparing the model predictions with various observational probes like the UVLF data from HST and JWST, recent measurements of the neutral hydrogen fraction, and the CMB scattering optical depth. Our analysis confirms that a rapid enhancement in the star-formation rate efficiency and/or UV luminosity per stellar mass formed is necessary for consistency with the JWST UVLF estimates at $z \geq 10$. We further find that it is possible to jointly satisfy the current reionization constraints when the escape fraction is assumed to be halo-mass dependent, requiring higher Lyman-continuum leakage from low-mass galaxies. We also examine the relative contribution of galaxies with different UV luminosities towards the ionizing photon budget for the EoR and investigate the large-scale bias of high-$z$ galaxies.

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arXiv2024

Simulating the epoch of Helium Reionization in photon-conserving semi-numerical code SCRIPT

The reionization of the second electron of helium (HeII) leaves important imprints on the thermal and ionization state of the intergalactic medium (IGM). Observational evidence suggests that HeII reionization ended at $z \simeq 3$ due to ionizing photons emitted predominantly by quasars. We present efficient semi-numerical simulations of helium reionization in a $230 \ \mathrm{h^{-1}~Mpc}$ box, that takes into account the spatial patchiness of reionization coupled with photoheating of the IGM. Dark matter haloes are assigned quasars using empirical measurements of the quasar luminosity function, assuming a universal quasar lifetime consistent with duty cycle values inferred from measurements of the quasar clustering. The ionizing photon field from quasars is then included in the semi-numerical Code for ReionIzation with PhoTon conservation (SCRIPT), which was originally developed for modeling hydrogen reionization. In this work, we make appropriate modifications to SCRIPT for modeling inhomogenous HeII reionization and the corresponding thermal history of the IGM is modelled via a subgrid prescription. Our model has three main free parameters i.e. the global clumping factor $\mathcal{C}_{HeIII}$, the temperature increase due to photoheating $T^{re}_{He}$ and the quasar spectral energy distribution (SED) index, $α_{UV}$. Our fiducial model with $\mathcal{C}_{HeIII}=15.6$ and $T^{re}_{He} \sim 6000 \ K$ gives reasonable values for the empirical measurements of the temperature density equation of state at these redshifts, assuming that quasars brighter than $\mathrm{M_{1450}}<-21$ and having $α_{UV}=1.7$ contribute to HeII reionization. The efficiency of our code shows promising prospects for performing parameter estimation in future, for models of HeII reionization using observations of the Ly$α$ forest.

astro-ph.CO↗

arXiv2024

A modified lognormal approximation of the Lyman-$α$ forest: comparison with full hydrodynamic simulations at $2\leq z\leq 2.7$

Observations of the Lyman-$α$ forest in distant quasar spectra with upcoming surveys are expected to provide significantly larger and higher-quality datasets. To interpret these datasets, it is imperative to develop efficient simulations. One such approach is based on the assumption that baryonic densities in the intergalactic medium (IGM) follow a lognormal distribution. We extend our earlier work to assess the robustness of the lognormal model of the Lyman-$α$ forest in recovering the parameters characterizing IGM state, namely, the mean-density IGM temperature ($T_0$), the slope of the temperature-density relation ($γ$), and the hydrogen photoionization rate ($Γ_{12}$), by comparing with high-resolution Sherwood SPH simulations across the redshift range $2 \leq z \leq 2.7$. These parameters are estimated through a Markov Chain Monte Carlo technique, using the mean and power spectrum of the transmitted flux. We find that the usual lognormal distribution of IGM densities cannot recover the parameters of the SPH simulations. This limitation arises from the fact that the SPH baryonic density distribution cannot be described by a simple lognormal form. To address this, we extend the model by scaling the linear density contrast by a parameter $ν$. While the resulting baryonic density is still lognormal, the additional parameter gives us extra freedom in setting the variance of density fluctuations. With this extension, values of $T_0$ and $γ$ implied in the SPH simulations are recovered at $\sim 1-σ$ ($\lesssim$ 10%) of the median (best-fit) values for most redshifts bins. However, this extended lognormal model cannot recover $Γ_{12}$ reliably, with the best-fit value discrepant by $\gtrsim 3-σ$ for $z > 2.2$. Despite this limitation in the recovery of $Γ_{12}$, we argue that the model remains useful for constraining cosmological parameters.

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