Joining bits and pieces of reionization history

TL;DR

This paper combines CMB and astrophysical data to reconstruct the reionization history, achieving precise measurements of optical depth and duration, and finds no evidence for complex reionization scenarios beyond monotonic models.

Contribution

It introduces a novel, free-form reconstruction method of reionization history using combined CMB and astrophysical data, improving constraints on optical depth and duration.

Findings

Optical depth τ = 0.051 with tight error bounds.

Reionization duration between redshifts 6 and 10 is well constrained.

No evidence supports multi-phase reionization scenarios.

Abstract

Cosmic Microwave Background temperature and polarization anisotropies from Planck have estimated a lower value of the optical depth to reionization ($\tau$) compared to WMAP. A significant period in the reionization history would then fall within $6<z< 10$, where detection of galaxies with Hubble Frontier Fields program and independent estimation of neutral hydrogen in the inter galactic medium by Lyman-$\alpha$ observations are also available. This overlap allows an analysis of cosmic reionization which utilizes a direct combination of CMB and these astrophysical measurements and potentially breaks degeneracies in parameters describing the physics of reionization. For the first time we reconstruct reionization histories by assuming photo-ionization and recombination rates to be free-form and by allowing underlying cosmological parameters to vary with CMB (temperature and polarization anisotropies and lensing) data from Planck 2018 release and a compilation of astrophysical data. We find an excellent agreement between the low-$\ell$ Planck 2018 HFI polarization likelihood and astrophysical data in determining the integrated optical depth. By combining both data, we report for a minimal reconstruction $\tau=0.051^{+0.001+0.002}_{-0.0012-0.002}$ at 68\% and 95\% CL, which, for the errors in the current astrophysical measurements quoted in the literature, is nearly twice better than the projected cosmic variance limited CMB measurements. For the duration of reionization, redshift interval between 10\% and complete ionization, we get $2.9^{+0.12+0.29}_{-0.16-0.26}$ at 68\% and 95\% CL, which improves significantly on the corresponding result obtained by using Planck 2015 data. By a Bayesian analysis of the combined results we do not find evidence beyond monotonic reionization histories, therefore multi-phase reionization scenario is disfavored compared to minimal alternatives.

Tables

Table 1. Table 1:   Best fit χ eff 2 = − 2 ​ ln ⁡ ℒ subscript superscript 𝜒 2 eff 2 ℒ \chi^{2}_{\mathrm{eff}}=-2\ln{\cal L} from MCMC and the bounds on the optical depth τ 𝜏 \tau (68.3% C.L.) and duration of reionization Δ z reion subscript superscript Δ reion 𝑧 \Delta^{\mathrm{reion}}_{z} (both 68.3% and 95%C.L. for skewed posterior) obtained in the reconstructions for different data combinations. The Bayes factors ( ln ⁡ B 𝐵 \ln B ) w.r.t. the minimal model B0 are also provided.

Model/data		P18	P18+UV17	P18+UV17+QHII
B0 Minimal	${\chi }_{\mathrm{eff}}^2$ $\ln B$ $\tau$ ${\Delta }_z^{\mathrm{reion}}$	2779.9 0 ${0.051}_{-0.009}^{+0.006}$ $3_{-1.2-1.8}^{+0.79+2.0}$	2783.4 0 $0.05\pm 0.001$ ${2.8}_{-0.15-0.25}^{+0.11+0.27}$	2792 0 $0.051\pm 0.001$ ${2.9}_{-0.16-0.26}^{+0.12+0.29}$
B1 1 node	${\chi }_{\mathrm{eff}}^2$ $\ln B$ $\tau$ ${\Delta }_z^{\mathrm{reion}}$	2780.5 -0.4 ${0.052}_{-0.009}^{+0.006}$ ${3.08}_{-1.3-2.0}^{+0.77+2.1}$	2782 -0.1 $0.05\pm 0.001$ ${2.8}_{-0.15-0.24}^{+0.11+0.31}$	2790.3 0 $0.051\pm 0.001$ ${2.9}_{-0.16-0.26}^{+0.12+0.29}$
B2 2 nodes	${\chi }_{\mathrm{eff}}^2$ $\ln B$ $\tau$ ${\Delta }_z^{\mathrm{reion}}$	2778.8 -2.2 $0.05\pm 0.008$ ${3.3}_{-2.7-3}^{+0.03+7}$	2782 -3.5 ${0.049}_{-0.006}^{+0.007}$ ${2.7}_{-0.32-0.8}^{+0.2+1.3}$	2789 -3.2 ${0.052}_{-0.002}^{+0.0008}$ ${3.05}_{-0.53-0.7}^{+0.08+1.2}$
B3 3 nodes	${\chi }_{\mathrm{eff}}^2$ $\ln B$ $\tau$ ${\Delta }_z^{\mathrm{reion}}$	- - - -	2781.8 -6.6 $0.05\pm 0.005$ ${2.9}_{-0.82-2.1}^{+0.086+4.3}$	2786.5 -8.2 ${0.052}_{-0.003}^{+0.0006}$ ${2.86}_{-0.6-0.86}^{+0.07+1.5}$

Equations

$$\frac{d{Q}_{\mathrm{HII}}}{dt}=\frac{{\dot{n}}_{\mathrm{ion}}}{⟨n_{\mathrm{H}}⟩}-\frac{Q_{\mathrm{HII}}}{t_{\mathrm{rec}}},$$

Full text

I Introduction

The two cosmic transitions between ionized and neutral state for the hydrogen atom are imprinted in key astrophysical and cosmological observations. The first transition from ionized plasma to neutral state for atoms, cosmological recombination occurred around 13.8 billion of years ago (or equivalently at a redshift $z\sim 1100$). After half a billion years, the hydrogen became ionized again during cosmic reionization, which followed the so called dark ages. Evidence for cosmic reionization comes from astrophysical measurements, such as the Gunn-Peterson test in high redshift quasars or the declining visibility of Lyman-$\alpha$ high redshift galaxies, and from cosmological observations as the large angular scale polarization pattern of CMB anisotropies. While astrophysical measurements mostly encode the central stage and the completion of cosmic reionization, the CMB anisotropy pattern is mostly sensitive to its duration through the integrated optical depth ($\tau$), and marginally to its early stage.

Recent determinations of $\tau$ from Planck assuming a nearly instantaneous transition for the ionization fraction Aghanim et al. (2018a, 2019); Adam et al. (2016); Aghanim et al. (2016a); Lattanzi et al. (2017) have revealed preferences for lower values compared to WMAP, owing to the understanding of the Galactic dust contamination to microwave polarization at large angular scales. Recent works demonstrate that star forming galaxies detected till $z\simeq 10$ as a source of reionization offer a consistent scenario with this optical depth Ishigaki et al. (2018). Observation of galaxies at high redshifts ($z\sim 6-10$), mainly with recent six cluster observations (Abell 2744, MACSJ0416, MACSJ0717, MACSJ1149, AbellS1063, Abell370) by Hubble Frontier Fields program Coe et al. (2015); Lotz et al. (2017); Livermore et al. (2017) upto a limiting AB magnitude of 29, provides shape of UV luminosity densities that determine the ionizing photon emission history. On the other hand, Gunn-Peterson optical depth Fan et al. (2006a, b); ionized near-zone around high redshift quasars Mortlock et al. (2011); Bolton et al. (2011); dark gaps in quasar spectra McGreer et al. (2015); damping wings of Gamma-Ray Burst 050904 Totani et al. (2006); McQuinn et al. (2008) and quasars Schroeder et al. (2013); Greig et al. (2017); Davies et al. (2018); Lyman-$\alpha$ emitters McQuinn et al. (2007); Ouchi et al. (2010), Lyman-$\alpha$ emission from galaxies Ono et al. (2012); Caruana et al. (2014); Schenker et al. (2014); Tilvi et al. (2014); Pentericci et al. (2014); Sobacchi and Mesinger (2015); Mason et al. (2018, 2019) provide measurement of remaining neutral hydrogen in the inter-galactic medium (IGM) between redshift $5-8$. Redshift overlap of HFF and Lyman-$\alpha$ observations with the reionization as measured by Planck calls for a joint analysis in a model independent framework. Since physics describing cosmic reionization is partially degenerate with cosmological parameters Zaldarriaga et al. (1997); Hazra et al. (2018a), it is important to perform this analysis by allowing the underlying cosmological model to vary as well (see Robertson et al. (2010); Mitra et al. (2012, 2015); Bouwens et al. (2015a); Robertson et al. (2015); Ishigaki et al. (2015); Price et al. (2016); Gorce et al. (2018); Ishigaki et al. (2018); Mitra et al. (2018) for previous works in which all but the reionization parameters are kept fixed).

In this Letter we perform for the first time a joint analysis using updated CMB anisotropy and a combination of astrophysical data to reconstruct reionization histories, where solutions to ionization equation of hydrogen with free-form ionization and recombination rates are used instead of conventional free-electron fraction parametrization Hu and Holder (2003); Mortonson and Hu (2008); Douspis et al. (2015); Hazra and Smoot (2017); Hazra et al. (2018a, b); Obied et al. (2018); Heinrich and Hu (2018); Villanueva-Domingo et al. (2018); Millea and Bouchet (2018). Our analysis removes parametric model dependence with this generic construct. Use of the ionization equation allows us to include all three types of data (CMB, UV luminosities and neutral fraction) in a single framework. At the same time use of complete CMB data and freedom in the cosmological parameters exploits the degeneracies and provide conservative constraints.

II Reconstruction of reionization history: the framework

We directly solve the reionization equation Wyithe and Loeb (2003) for the volume filling factor of ionized regions:

[TABLE]

where the source term $\dot{n}_{\rm ion}$ is the ionizing photon production rate and is defined by the product of the UV luminosity density ($\rho_{\rm UV}$), the photon production efficiency ($\xi_{\rm ion}$) and the escape fraction ($f_{\rm esc}$). We keep the magnitude averaged product $\log_{10}\langle f_{\rm esc}\xi_{\rm ion}\rangle=24.85$ from Ishigaki et al. (2015), also consistent with other analyses Price et al. (2016); Ishigaki et al. (2018); Madau (2017). The recombination time is defined as $t_{\rm rec}=1/\left[C_{\rm HII}\alpha_{\rm B}(T)(1+Y_{p}/(4X_{p}))\langle n_{\rm H}\rangle(1+z)^{3}\right]$ using the clumping factor ($C_{\rm HII}$), recombination coefficient ($\alpha_{\rm B}(T)$), density of hydrogen atom ($\langle n_{\rm H}\rangle$) and hydrogen ($X_{p}$) and helium abundances ($Y_{p}$). In this work, instead of using analytical forms, we define $\rho_{\rm UV}$ and $t_{\rm rec}$ to be free parameters in different nodes which are allowed to vary between a conservative redshift range for the reionization process, i.e. $z=5.5-30$ (see, Hazra and Smoot (2017)). Different nodes are connected using Piecewise Cubic Hermite Interpolating Polynomial. Fixed nodes are located at $z=0,5.5$ and 30 and values of source and recombination terms are fixed to be consistent to best fit logarithmic double power law (see Eq. (39) of Ishigaki et al. (2015)) and also consistent with Becker and Bolton (2013) when interpolated at smaller redshifts. However as we allow the intermediate source and recombination terms to be free, values at fixed nodes do not limit our general construct.

We allow upto three nodes in this moving-bin reconstruction denoted as $B1,~{}B2$ and $B3$ respectively and each node comes with three parameters, namely, the intermediate redshift ($z_{\rm int}$) and $\rho_{\rm UV}$ and recombination timescale defined in that redshift. Since reionization progresses with the competition between ionization and recombination, in our analysis we have used the ratios $(1/t_{\rm rec})/(\dot{n}_{\rm ion}/\langle n_{\rm H}\rangle)$ as free parameter instead of $t_{\rm rec}$. From the reconstruction, $t_{\rm rec}$ can be obtained as a function of redshift and assuming certain IGM temperature, the clumping factor can also be derived. For a minimal construct we also consider $B0$ where we impose at $z=z_{\rm int}$, $\dot{n}_{\rm ion}t_{\rm rec}=\langle n_{\rm H}\rangle$ (a mathematical limit of B1 that generalizes the ionization balance). The optical depth is a derived parameter in our approach and is given by the integral from onset of reionization ($z_{\rm begin}$) till today: $\tau=\int_{0}^{z_{\rm begin}}\frac{c(1+z)^{2}}{H(z)}Q_{\rm HII}(z)\sigma_{\rm Thomson}\langle n_{\rm H}\rangle(1+\frac{Y_{p}}{4X_{p}})$, where $\sigma_{\rm Thomson}$ is the Thomson scattering cross-section. We fuse our integrator with CAMB Lewis et al. (2000) maintaining the standard treatment for Helium reionization.

III Datasets and priors

Three different datasets have been mostly used in this work. For CMB we use the latest publicly available likelihoods in temperature, polarization and lensing from the Planck 2018 release (hereafter P18) Aghanim et al. (2019); Aghanim et al. (2018b) and the Planck 2015 release (hereafter P15)  Aghanim et al. (2016b); Ade et al. (2016). We use the full angular power spectrum data in order to fully account for non-negligible correlations between reionization history and other cosmological parameters Hazra et al. (2018a). For UV luminosity density, we use Bouwens et al. (2015b); Ishigaki et al. (2018) data spanning $z\sim 6-11$ derived from Hubble Frontier Fields HFF ; Lotz et al. (2017) observations. The density is obtained by integrating the UV luminosity function by fitting Schechter function till a truncation magnitude of $-17$ (hereafter UV17) [We use the recent data compiled by Ishigaki et. al (2018) exploiting the full six-cluster HFF data]. We also use direct $Q_{\rm HII}$ constraints (hereafter QHII) from McGreer et al. (2015); Totani et al. (2006); McQuinn et al. (2008); Schroeder et al. (2013); Greig et al. (2017); Davies et al. (2018); McQuinn et al. (2007); Ouchi et al. (2010); Schenker et al. (2014); Tilvi et al. (2014); Mason et al. (2018, 2019). These data cover a redshift range of $5.6-8$ and thereby overlap with the UV density. For B0 and B1 the intermediate redshift is allowed to vary between the entire range $z=5.5-30$. For B2, $z_{\rm int}^{1}$ can vary between $z=5.5-12$ and $z_{\rm int}^{2}$ between $z=12-30$; for B3, $z_{\rm int}^{1}$, $z_{\rm int}^{2}$ and $z_{\rm int}^{3}$ move within $5.5-8$, $8-12$ and $12-30$ respectively. The redshift ranges for nodes in B2 and B3 are chosen in a way that the UV data can constrain the source term variation till $z=12$ and the CMB constrains the last node by constraining the integrated optical depth. We allow $\Omega_{\rm b}h^{2}$, $\Omega_{\rm CDM}h^{2}$, $\theta$, $A_{\rm s}$, $n_{\rm s}$, foregrounds and calibration nuisance parameters in Planck likelihood to vary. We use publicly available CosmoMC Lewis and Bridle (2002) for parameter estimation. We also consider the stability of our results allowing $\log_{10}\langle f_{\rm esc}\xi_{\rm ion}\rangle$ to vary and using UV luminosity data with truncation magnitude of $-15$.

IV Constraints and concordances

We first show the consistency of the low-$\ell$ Planck 2018 polarisation likelihood lowE and astrophysical data UV17+QHII in determining the integrated optical depth $\tau$. By combining Planck 2018 TT and astrophysical data we obtain for B0, $\tau=0.051\pm 0.001$ at 68 %CL. This determination of $\tau$ is in excellent agreement, but much more precise, than the 68 % CL estimate obtained by Planck 2018 TT + lowE, $\tau=0.052\pm 0.008$ The consistency of Planck lowE and astrophysical data in determining $\tau$ is robust to the addition of Planck high-$\ell$ polarisation and/or lensing and to the addition of nodes in the rates.

We now proceed with the joint constraints using P15, P18, P18+UV17 and P18+UV17+QHII in minimal single node (B0), single node (B1), 2 nodes (B2) cases and P15+UV17, P18+UV17 and P18+UV17+QHII in three nodes (B3) case. For B3 we do not explore CMB-only constraints owing to its inability to provide reasonable constraints in such an extended parameter space.

In Table 1 we provide the constraints on $\tau$, $\Delta^{\mathrm{reion}}_{z}$, i.e. the redshift interval between 10% and complete ionization, the fit to the data ($\chi^{2}_{\mathrm{eff}}$) and the Bayesian evidences ($\ln B$) calculated from the chains using MCEvidence Heavens et al. (2017a, b). We find that $\ln B$ for B1 is close to 0 w.r.t B0. While B2 and B3 improve the fit to all data combinations, the addition of extra parameters is penalized by the Bayes factor and become disfavored compared to B1. It can be readily identified for B0 and B1, that allow monotonic histories, $\tau$ can be constrained with much better precision when our compilation of astrophysical data from UV17 and UV17+QHII are combined compared to P18 alone. In all the cases, mean values of $\tau$ remains similar and the low-$\ell$ polarization likelihood from P18 using HFI, plays an important role making the histories consistent with the astrophysical data. With more general histories allowed in B2, addition of UV17 and UV17+QHII only improves the constraint by 50% compared to P18. Since in B2 and B3, the last node is only constrained by P18, upper bound on optical depth gets worse. In Gorce et al. (2018), keeping fixed the underlying cosmology and using five (four star formation history parameters and clumping factor) and six (allowing $f_{\rm esc}$ to vary alongside) parameters, the authors report the standard deviations of $\tau$ to be $0.0019$ and $0.002$ respectively, when all datasets are used. In B2 and B3, that allows 6 and 9 parameters to describe the reionization, we obtain standard deviation$\sim 0.003-0.0035$, 50% wider compared to Gorce et al. (2018). The constraints become more conservative as our reconstructions allow more flexibilities compared to fixed form parametrization. While our framework allows a wide range of $\Delta^{\mathrm{reion}}_{z}$, the constraints demonstrate that in a P18+UV17+QHII data combination, even the most flexible model (B3) must have $\Delta^{\mathrm{reion}}_{z}>2$ at 95% C.L.

In Fig. 1 we plot the constraints from the MCMC analyses. We plot 95% bounds on $Q_{\rm HII}$ (top row) for all the data combinations as a function of $z$. Constraints from P15 for B0-B2 and P15+UV17 for B3 are provided in the background stripes. The improvement in P18 constraints using HFI polarization compared to P15 data is significant, as anticipated in Aghanim et al. (2016a); Adam et al. (2016). While B0 produces monotonic power law reionization histories, B1 allows extended and step like histories. B2 and B3 with extra nodes provide the scopes for non-monotonic and complex histories. As we know, P18 mainly constrains the integrated optical depth, therefore the ionization histories are not well constrained in all three cases (B0, B1 and B2). UV luminosity density data allow only a small subset of histories from P18 and the derived bounds on $\tau$ improve significantly in P18+UV17. In middle row, we plot 68% and 95% constraints on corresponding source term, $\rho_{\rm UV}$ and on top we display the Bouwens et. al. (2014) and Ishigaki et. al. (2018) and data points used. Since luminosity densities at higher redshifts for B2 and B3 are not constrained well and therefore we plot samples only till $z=12$. In bottom row we plot the marginalized constraints on $\tau$: for all the cases the improvement due to our compilation of astrophysical data w.r.t CMB alone is evident. The optical depth from P18+UV17 and P18+UV17+QHII agree well in all cases. The agreement is better in P18 compared to P15 as the later inclines towards higher mean values of $\tau$, although with larger uncertainties.

In Fig. 2 we plot reconstructed samples of recombination timescale and its lower limits obtained from P18+UV17+QHII. Within $z=6-8$, the lower limit on the timescale is found to be about $1$ Gyr. The evolution of ionization is mapped with colored samples.

In Fig. LABEL:fig:QHII, we plot the 68.3% and 95% C.L. on the reionization histories for all the reconstructions using P18+UV17+QHII. On top of the bounds, we plot $Q_{\rm HII}$ data points from different observations that we have used. While B0 and B1 reconstruct similar histories, constraints on histories in B2 and B3 are wider at higher redshifts. Significant improvement compared to P15 is evident in B3.

Note that, in this framework we can also reconstruct the clumping factor at different redshifts. Since our sample provide free form reconstruction of $t_{\mathrm{rec}}$, we can obtain the clumping factor for any assumed value of IGM temperature. For $T_{\mathrm{IGM}}=2\times 10^{4}K$, we find $C_{\rm HII}\lesssim 3$ within $6<z<8$ and monotonically increasing with decrease in redshift. This result is completely consistent with parametric $C_{\rm HII}=2.9\left[\frac{1+z}{6}\right]^{-1.1}$ fit to simulation Shull et al. (2012). This bound is expected to be degenerate with $\langle f_{\rm esc}\xi_{\rm ion}\rangle$ if allowed to be free (for more discussion on $f_{\rm esc}$ see, Dayal and Ferrara (2018)).

Our analysis finds correlations between other background cosmological parameters with the reconstructed reionization histories and in Fig. LABEL:fig:nstau we present the correlation between derived parameter $\tau$ and $n_{\rm s}$ and between $\sigma_{8}$ and $\Delta^{\mathrm{reion}}_{z}$ for B1. HFI polarization data on large angular scales and astrophysical data helps in breaking the degeneracies and provide tighter constraint on the reionization histories and therefore on $\tau$. Tighter constraints on $n_{\rm s}$ and $\sigma_{8}$ are also important to compare CMB with large scale structure data.

V Conclusion

In this paper we have reconstructed the history of reionization using CMB and a compilation of astrophysical data. This free form reconstruction allows more conservative variation in ionizing UV flux (the source term $\dot{n}_{\rm ion}$) and recombination times/rates ($t_{\rm rec}$) at flexible redshift nodes. We are also able to combine data from UV luminosities in addition to those for CMB and neutral hydrogen. This framework allows sharp to highly extended reionization histories that also involves non-monotonic changes in the ionization fraction.

Below we summarize the main results of our analysis:

We find an excellent consistency between the low-$\ell$ Planck 2018 HFI polarization likelihood and our compilation of astrophysical data in determining the integrated optical depth $\tau$. When considering jointly P18+UV17+QHII data, we obtain $\tau=0.051^{+0.006+0.013}_{-0.009-0.014}$. We note that with the nominal errors of our compilation of astrophysical data, we obtain a joint constraint tighter by a nearly a factor of 2 w.r.t. the projected constraint from cosmic variance limited proposed CMB space missions Di Valentino et al. (2018); Hazumi et al. (2019); CMB (2018); Sutin et al. (2018). 2. 2.

A joint analysis that includes Planck 2018 data, UV luminosity density integrated upto -17 magnitude and Lyman-$\alpha$ observations, does not allow sharp reionization histories (with Tanh model defined as sharp, $\Delta^{\rm reion}_{z}\sim 1.7$ between 10% to 99% ionization). We report at 95% C.L. $2.6<\Delta^{\rm reion}_{z}<3.2$ (in the single node reconstruction), and $2<\Delta^{\rm reion}_{z}<4.4$ (three nodes reconstruction allowing conservative constraints). 3. 3.

There are no evidences for non-monotonic or multi-step reionization histories. Bayesian evidence disfavors complex reionization histories with more than one intermediate node between the beginning and completion of reionization. Use of HFI large scale E-mode polarization in P18 results in substantially tighter constraints at the high redshift tail of ionization histories compared to P15. 4. 4.

Samples of recombination timescales from P18+UV17+QHII reveals that clumping factor $C_{\mathrm{HII}}\lesssim 3$ within redshift $6-8$, assuming a IGM temperature of 20000K (correspondingly, we find 95% lower bound of $t_{\mathrm{rec}}\sim 1$ Gyr). 5. 5.

When $\langle f_{\rm esc}\xi_{\rm ion}\rangle$ is allowed to vary, the combined datasets constrains $\tau=0.052\pm 0.002$ at 95% in the single node case.

Allowing $\langle f_{\rm esc}\xi_{\rm ion}\rangle$ as free parameter use of different UV luminosity data with truncation magnitude of -15 (UV15) provides $\tau=0.054\pm 0.003$ at 95% in single node case. Contribution towards ionization from dimmer sources is reflected in the higher value of optical depth. Higher uncertainty in the UV15 leads to relaxed bounds w.r.t UV17 case.

Discussions on such extensions and constraints on $f_{\rm esc}$, $\xi_{\rm ion}$ in different cases and data combinations will be provided in a detailed paper (Paoletti et al., in preparation). 6. 6.

We find that for simple monotonic models that can be described by a single intermediate node, degeneracies between reionization history and other cosmological parameters can be lifted completely with current astrophysical data.

Our analysis opens up to conservative constraints on reionization allowing the combination of astrophysical measurements with CMB in this newly introduced free-form reconstruction. It will be interesting to see the performance of such method for constraining physical models of reionization in the perspective of future cosmological measurements.

Acknowledgements.

The authors would like to acknowledge the use of Nandadevi cluster in the Institute of Mathematical Science’s High Performance Computing (hpc.imsc.res.in), APC cluster (https://www.apc.univ-paris7.fr/FACeWiki/pmwiki.php?n=Apc-cluster.Apc-cluster) and INAF OAS Bologna cluster. The authors thank Masami Ouchi and Masafumi Ishigaki for providing their UV luminosity density data compilation. DKH would like to thank Sourav Mitra for important discussions. The authors would like to thank Andrea Ferrara and Tirthankar Roy Choudhury for their valuable comments on the manuscript. DKH has received fundings from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 664931. DP and FF acknowledge financial support by ASI Grant 2016-24-H.0 and partial financial support by the ASI/INAF Agreement I/072/09/0 for the Planck LFI Activity of Phase E2. GFS acknowledge Laboratoire APC-PCCP, Université Paris Diderot and Sorbonne Paris Cité (DXCACHEXGS) and also the financial support of the UnivEarthS Labex program at Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02).