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Yin-Zhe Ma

Publications and source records attributed to Yin-Zhe Ma.

At least 19 recordsLinked to original sources

Constraining primordial non-Gaussianity and energy injection with the thermal Sunyaev-Zeldovich effect and integrated Sachs-Wolfe effect cross-correlation

Constraining primordial non-Gaussianity (PNG) provides key insights into the physics of cosmic inflation and the initial conditions of the Universe, which remain central topics in cosmology. In this study, we use the cross-correlation between the integrated Sachs-Wolfe (ISW) effect and the thermal Sunyaev-Zeldovich (tSZ) effect derived from Ibitoye et al. (2024) to jointly constrain PNG and early-Universe energy injection, including the standard intergalactic medium contribution. For scale-independent PNG we obtain $f_{\rm NL} = -358^{+140}_{-114}$ ($68\%$~C.L.). For a scale-dependent model ($f_{\rm NL}=f_{\rm NL}^{0}(\ell/\ell_{0})^{n_{\rm NL}}$, with $\ell_{0}=200$), we find $f^{0}_{\rm NL} = -296^{+173}_{-157}$ and $n_{\rm NL} = 0.62^{+1.02}_{-0.64}$, both consistent with Gaussian initial conditions. We also constrain the early-Universe energy injection amplitude to be $α_{\rm inj} = -3.93^{+1.34}_{-0.99}$, with uncertainty reduced by a factor of $\sim\!2.6$ if Planck 2018 $f_{\rm NL}$ constraint is applied as a prior. Future surveys such as Simons Observatory and Euclid will tighten these constraints further. Complementary to conventional probes, this work provides the first ISW-tSZ constraint on exotic energy injection and enables precision tests of early-Universe physics while probing late-time gravitational potential and thermal energy perturbations.

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Exploring the High-Redshift 21-cm Signal via Self-Consistent Simulations using Artificial Neural Network Emulation

We present a novel, self-consistent, semi-numeric Cosmic Dawn (CD) simulation in which small-scale star formation (SF) is calibrated to the \emph{AEOS} and \emph{Renaissance} hydrodynamic simulations. SF proceeds within dark matter (DM) halos via neural network emulation while considering large-scale fluctuations in density and feedback. We translate the resulting 3D distribution of galaxies into predictions for the 21-cm brightness temperature, \Tb, and power spectrum, \PS. We simulate several unique realizations to study the impact of varying astrophysics on \Tb, finding that more efficient Population II (PopII) SF largely yields stronger Lyman-$α$ coupling, resulting in a shallower and wider absorption trough. However, we find that PopII SF dominates \PS\ at $z \lesssim 20$ and on smaller scales at intermediate redshifts ($k \gtrsim 0.2\ \mathrm{Mpc^{-1}}$ at $z \simeq 34-20$) while Population III (PopIII) SF dominates \PS\ at $z\gtrsim34$ and on larger scales at intermediate redshifts. Compared with previous works, we find that the combination of hydrodynamic SF calibration, a critical halo mass for SF considering \Htwo\ self-shielding, and stochastic DM halo merger histories results in both earlier SF and higher SF rates across CD. Further, we find that the delay period separating PopIII and PopII SF (\tdelay) significantly impacts \Tb, and that one must include DM halo merger histories to properly account for this transition. Finally, we find our fiducial \Tb\ to be detectable at $z\lesssim25$ with 1080 hours of HERA observations under moderate foreground assumptions, and the lack of such a detection at $z \gtrsim 20$ would suggest \tdelay\ $\gtrsim$ 30 Myr.

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Sensitivity of the Hongmeng 21cm experiment to scattering dark matter

Scattering between dark matter and baryons can cool the intergalactic medium temperature and deepen the 21cm signal. Such interactions have been proposed to explain the unusually deep 21cm absorption signal reported by EDGES in 2018. We explore the potential to detect dark matter - baryon scattering with the Hongmeng project, an upcoming Moon-orbiting satellite experiment dedicated to measuring the global 21cm signal between redshifts $11-46$. We self-consistently forward-model the simulated sky-temperature data, jointly varying the astrophysical and foreground models. We show that even with a very conservative observational strategy in which the experiment only takes data when the Earth and the Sun are both shielded by the Moon, Hongmeng can tighten the current constraints on the cross section of dark matter - baryon scattering $σ_0$ by a factor of 39 after the full mission, which lasts for five years. The prospective upper limit on $σ_0$ can reach $5.4 \times 10^{-43} {\rm cm^2}$ for dark matter masses between 0.1 MeV and 0.3 GeV. Even after only one month of operation, an improvement by a factor of 4 relative to current $σ_0$ limits can be expected.

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Probing the Baryon Distribution with Fast Radio Bursts

Baryonic feedback redistributes matter on small to mid cosmological scales, ultimately limiting inferences from Stage IV galaxy surveys. Direct baryon tracers are crucial for recovering cosmological signals masked by astrophysical effects, and vice versa: galaxy formation and other astrophysical processes must be interpreted cosmologically. Fast radio bursts (FRBs) serve as such tracers: their dispersion measure (DM) records the line-of-sight integrated ionised electron density. The Square Kilometre Array (SKA) will be the only radio telescope capable of detecting many FRBs in the southern hemisphere, significantly enhancing synergy with surveys such as Rubin Observatory. This chapter completes the FRB trilogy by forecasting the SKA's potential to constrain the baryon distribution from cosmological to galactic scales and across cosmic time. We tackle this question by investigating the DM scatter as a function of redshift. We also study the statistical properties of the DM field and its cross-correlation with Stage IV galaxy surveys. Our focus is on cosmic shear and galaxy clustering. This shows that the SKA can play a crucial role in pinpointing baryonic feedback models, thereby greatly enhancing the cosmological constraining power of Stage IV galaxy surveys. Furthermore, we show that the SKA will be able to measure the properties of the circumgalactic medium using the scattering timescale of FRBs. Lastly, the large redshift range of FRB detections with the SKA can improve our understanding of the epoch of reionisation. It may also clarify the mechanism behind FRBs.

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Cosmology with Intensity Mapping via Statistics Beyond the Power Spectrum in the SKAO Era

The cosmological distribution of neutral hydrogen (HI) during the post-reionization era is highly non-Gaussian due to the underlying non-linear structure formation, complex galaxy biasing, and potential primordial non-Gaussianity. One needs higher-order (beyond two-point) statistics to maximally extract the non-Gaussian information out of the 21-cm intensity maps. This chapter summarizes the potential of several higher-order statistics, including voxel intensity distribution, emission line stacking, probability density functions, $\ell_1$-norm, bispectrum, and various marked statistics. Additionally, image-based morphological descriptors, such as the Largest Cluster Statistic, local dimensions, and Minkowski functionals, etc., can potentially characterize the morphology and geometry of the cosmic web encoded in the 21-cm intensity maps. This chapter presents forecasts of the detectability of these higher-order statistics in the context of the future SKAO observations. These forecasts incorporate instrumental noise, observational effects, and, in some cases, foreground removal in their analyses. With its unprecedented sensitivity, the future SKAO 21-cm observations will enable us to measure these higher-order statistics more precisely, possibly helping to break degeneracies between astrophysical and cosmological parameters, and maximizing the science outcome from these surveys.

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Fast Radio Bursts as Cosmological Probes

Fast radio bursts (FRBs) are brief, coherent radio pulses of extragalactic origin. They typically last from microseconds to milliseconds and have energies large enough to be visible over cosmological distances. Since FRBs interact with free electrons along their paths, the original burst is dispersed (Dispersion Measure, DM) and broadened (scattering). Furthermore, the burst's polarization is altered by Faraday rotation. Consequently, FRBs are excellent probes of the cosmological distribution of baryons, the expansion of the Universe, magnetic fields, and minuscule effects of fundamental physics that accumulate over vast distances. This chapter is the second of a trilogy of FRB chapters and discusses FRBs as a standalone probe. We first introduce the foundation of FRB observables related to those questions. Next, we lay the groundwork for forecasting SKA's potential by describing the method to simulate the expected FRB population observable with the SKA. These synthetic FRB catalogues are then used to investigate the SKA's potential to probe the Universe's expansion rate and fundamental physics, such as the equivalence principle and the existence of massive photons. Furthermore, we investigate the possibility of tracing cosmic magnetic fields and investigating different dark matter candidates.

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The MeerKAT Thousand-Pulsar Polarisation Array II: Searches for Ultralight Axion-Like Dark Matter

We construct Pulsar Polarisation Arrays (PPA), using regular pulsars monitored in MeerKAT's Thousand Pulsar Array (TPA) Programme, to search for Axion-like Dark Matter (ALDM) within Milky Way. Specifically, from a catalogue of 1237 regular pulsars, we select the 50 ones with the highest signal-to-noise ratio and set upper limits on the ALDM Chern-Simons coupling. We find no signals with statistical significance over the mass range of $[10^{-23},10^{-20}]\,{\rm eV}$ in the six-year MeerKAT's data. By combining the high-quality TPA pulsars and the accurate ionospheric subtraction of spinifex, we establish the most sensitive upper limits to the date on the ALDM Chern-Simons coupling, namely $\lesssim 10^{-14} - 3\times 10^{-13}\,{\rm GeV}^{-1}$, for the mass range of $[10^{-23},10^{-21}]\,{\rm eV}$ except at $m_a \sim 1.3 \times 10^{-22}\,$eV. This study underscores the great potential of constructing regular-pulsar PPAs for scientific tasks.

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The MeerKAT Thousand-Pulsar Polarization Array I: Properties of the Polarization and Rotation Measure Time Series Data

The polarimetry of recent pulsar observations has provided a wealth of observational data with which to test physical theories of emission mechanisms, radiative transfer and even theories that extend beyond the Standard Model. In this work, we have outlined the data analysis of the polarisation time series data of a population of 513 pulsars from the Thousand Pulsar Array observing programme, laying the foundation for building the MeerKAT Thousand-Pulsar Polarization Array as a probe for ultralight Axion-Like Dark Matter (ALDM). From this large dataset we have focused on the temporal trends in the observed polarisation angle (PA) through a measure we call the PA offset, and characterised the trends due to the effects of Faraday Rotation within the interstellar medium and the Earth's ionosphere, as well as generic white and red noise models that are estimated within a Bayesian MCMC analysis. Then, motivated by potential extra contributions to the rotation of the PA that may not be Faraday-like, arising from the proposed ALDM field, we have investigated a derived time dependence for the rotation measure (RM) required to explain the observed PA offset. Comparison of these estimates to RM values that are measured in typical pulsar studies, through a technique known as RM Synthesis, provides a probe of any wavelength-independent contribution to the rotation of the PA. Although we find no evidence for oscillatory behaviour within our dataset's observation timespan, we do find cases of deviation from the usual RM values in several `pulsars of interest', as well as long-term linear trends in the time evolution of Faraday rotation that have been presented in the literature before.

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Primordial black holes and the velocity acoustic oscillations features in 21 cm signals from the cosmic Dark Ages

Astrophysical luminous objects such as the first stars have not yet formed in the Dark Ages. However, primordial black holes (PBHs) always exist throughout cosmic history since the inflation epoch. During the Dark Ages, PBHs may accrete the ambient gas and release radiation like astrophysical luminous objects, change the cosmic radiation field, the thermal status of the intergalactic medium (IGM), and the hydrogen spin temperature. The accretion rate is modulated by the relic supersonic relative streaming velocities between dark matter (DM) and baryons, imprinting Velocity Acoustic Oscillations (VAOs) features in the 21 cm power spectrum. Such VAOs features could be a promising probe for detecting the PBHs in Dark Ages. We find that even if PBHs comprise only a small fraction of DM, they can generate VAOs wiggles with a relative amplitude up to about 30% in Dark Ages. For example, for PBHs with a mass at recombination of 200 solar masses and mass fraction in the total DM f_PBH,rec around 1e-13 at the recombination era, VAOs features appear at redshift around 20; if f_PBH,rec is around 3e-10, then VAOs features could appear as early as redshift around 40. Moreover, the redshift evolution of the VAOs features exhibits clearly separated stages dominated by inhomogeneous Ly-alpha scattering, and inhomogeneous X-ray heating, respectively. It reflects the characteristics of PBHs (mass and fraction in total DM) and their interactions with the IGM. We also estimate that, the VAOs wiggles at redshift around 20 are detectable for the upcoming SKA-low AA*, while wiggles at redshift around 40 are detectable for an hypothetic lunar surface-based interferometer array in the future.

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Cosmological Prediction from the joint observation of MeerKAT and CSST at $z$ = 0.4 $\sim$ 1.2

Cross-correlating neutral hydrogen (HI) 21cm intensity mapping with galaxy surveys provides an effective probe of astrophysical and cosmological information. This work presents a cross-correlation analysis between MeerKAT single-dish HI intensity mapping and Chinese Space Station Survey Telescope (CSST) spectroscopic galaxy surveys in $z=0.4\sim1.2$, which will share a survey area of several thousand square degrees. Utilizing Jiutian-1G cosmological simulation, we simulate the observational data of MeerKAT and CSST with survey areas from $\sim1600$ to $600$ deg$^2$ at $z=0.5$, 0.7, and 1. The effects of beam pattern, polarization leakage, and different foregrounds in the MeerKAT HI intensity mapping are considered in the simulation. After employing foreground removal with the principal component analysis (PCA) method and performing signal compensation, we derive the cross-power spectra of MeerKAT and CSST. We perform the joint constraint using the CSST galaxy auto-power spectra and MeerKAT-CSST cross-power spectra with the least-squares fitting method. The constraint results show that, in the simulated survey area, the relative accuracy can achieve $6\%\sim 8\%$ for the parameter products $Ω_{\rm HI}b_{\rm HI}b_{g}r_{\mathrm{HI},g}$ and $Ω_{\rm HI}b_{\rm HI}r_{\mathrm{HI},g}$ at the three redshifts, which is $3\sim4$ times smaller than the current result. These findings indicate that the full MeerKAT-CSST joint observation with thousands of square degrees overlapping survey area can be a powerful probe of large-scale structure, and has the ability to provide information of cosmic evolution of HI and galaxies in a wide redshift range.

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Cross Correlation between the Thermal Sunyaev-Zel'dovich Effect and Projected Galaxy Density Field

We present a joint analysis of the power spectra of the Planck Compton $y$-parameter map and the projected galaxy density field using the Wide Field Infrared Survey Explorer (WISE) all-sky survey. We detect the statistical correlation between WISE and Planck data (g$y$) with a significance of $21.8\,σ$. We also measure the auto-correlation spectrum for the tSZ ($yy$) and the galaxy density field maps (gg) with a significance of $150\,σ$ and $88\,σ$, respectively. We then construct a halo model and use the measured correlations $C^{\rm gg}_{\ell}$, $C^{yy}_{\ell}$ and $C^{{\rm g}y}_{\ell}$ to constrain the tSZ mass bias $B\equiv M_{500}/M^{\rm tSZ}_{500}$. We also fit for the galaxy bias, which is included with explicit redshift and multipole dependencies as $b_{\rm g}(z,\ell)=b_{\rm g}^0(1+z)^α(\ell/\ell_0)^β$, with $\ell_0=117$. We obtain the constraints to be $B =1.50{\pm 0.07}\,(\textrm{stat}) \pm{0.34}\,(\textrm{sys})$, i.e. $1-b_{\rm H}=0.67\pm 0.03\,({\rm stat})\pm 0.16\,({\rm sys})$ (68\% confidence level) for the hydrostatic mass bias, and $b_{\rm g}^0=1.28^{+0.03}_{-0.04}\,(\textrm{stat}) \pm{0.11}\,(\textrm{sys})$, with $α=0.20^{+0.11}_{-0.07}\,(\textrm{stat}) \pm{0.10}\,(\textrm{sys})$ and $β=0.45{\pm 0.01}\,(\textrm{stat}) \pm{0.02}\,(\textrm{sys})$ for the galaxy bias. Incoming data sets from future CMB and galaxy surveys (e.g. Rubin Observatory) will allow probing the large-scale gas distribution in more detail.

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Squared-field cross-correlation between kinetic Sunyaev-Zel'dovich effect and 21-cm intensity mapping

Neutral hydrogen (HI) 21-cm intensity mapping is an effective method to track the distribution of baryonic matter, and extract astrophysical and cosmological information. The 21-cm intensity field has a nonvanishing cross-correlation with the kinetic Sunyaev-Zel'dovich (kSZ) effect that traces the velocity and density perturbations of free electrons. By using the linear perturbation theory, in this paper we calculate analytically, for the first time, the cross-correlation between the squared kSZ field and the projection of the squared HI intensity mapping field with the flat-sky approximation. This statistic remains nonvanishing even after the long-wavelength line-of-sight modes ($k_{\parallel}$) are removed due to foreground contamination. We further forecast for the prospects of detection with the SKA-MID 21-cm intensity mapping experiments (redshifts in range of $0.3 < z < 1$), and the kSZ maps measured by the Atacama Cosmology Telescope (ACT) and Simons Observatory (SO). The predicted cumulative signal-to-noise ratio is $1.92$ for SKA-ACT and $3.99$ for SKA-SO. These results show a possible on-the-edge detection on the cross-correlation signal at low redshifts, which in turn could serve as a validation step toward using it for the Epoch of Reionization studies.

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Signatures of inhomogeneous dark matter annihilation on 21-cm

The energy released from dark matter (DM) annihilation leads to additional ionization and heating of the intergalactic gas, impacting the hydrogen 21-cm signal during the cosmic dawn. The dark matter annihilation rate scales with its density squared and becomes inhomogeneously boosted with structure formation. This paper examines the inhomogeneity in DM annihilation rate induced by the growth of DM halo structures, and we show that this effect can significantly amplify the spatial fluctuations in temperature and ionization fraction of the gas. Consequently, the fluctuations in the 21-cm brightness temperature may also be enhanced. We showcase these effects for a DM mass of 100 MeV annihilating into $\rm{e}^-\rm{e}^+$ at a rate of $\left<σv\right>/m_χ\sim 10^{-27} {\rm cm^3 s^{-1} GeV^{-1}}$, which is consistent with current constraints set by the cosmic microwave background. We find that, compared to the homogeneous calculations, inhomogeneous annihilation can enhance the 21-cm power spectrum by up to a factor of 130 over the scales of $k \in [0.05, 3]\ {\rm{Mpc^{-1}}}$ at redshifts $11-16$. Such signatures could potentially be detected by upcoming radio observatories such as the Square Kilometer Array telescope.

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First Results from HERA Phase II

We report the first upper limits on the power spectrum of 21-cm fluctuations during the Epoch of Reionization and Cosmic Dawn from Phase II of the Hydrogen Epoch of Reionization Array (HERA) experiment. HERA Phase II constitutes several significant improvements in the signal chain compared to Phase I, most notably resulting in expanded frequency bandwidth, from 50-250 MHz. In these first upper limits, we investigate a small two-week subset of the available Phase II observations, with a focus on identifying new systematic characteristics of the instrument, and establishing an analysis pipeline to account for them. We report 2$σ$ upper limits in eight spectral bands, spanning $5.6 \leq z \leq 24.4$ that are consistent with thermal noise at the $2σ$ level for $k \gtrsim 0.6-0.9 h{\rm Mpc}^{-1}$ (band dependent). Our tightest limit during Cosmic Dawn ($z>12$) is $1.13\times 10^6 {\rm mK}^2$ at ($k=0.55 h{\rm Mpc}^{-1}, z=16.78$), and during the EoR ($5.5<z<12$) it is $1.78\times 10^3 {\rm mK}^2$ at ($k=0.70 h{\rm Mpc}^{-1}, z=7.05$). We find that mutual coupling has become our dominant systematic, leaking foreground power that strongly contaminates the low-$k$ modes, resulting in the loss of modes from $k=0.35-0.55$ compared to Phase I data.

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In search of an interaction in the dark sector through Gaussian Process and ANN approaches

Whether the current observational data indicate any evidence of interaction between the dark sector is a matter of supreme interest at the present moment. This article searched for an interaction in the dark sector between a pressure-less dark matter and a dark energy fluid with constant equation of state, $w_{\rm DE}$. For this purpose, two non-parametric approaches, namely, the Gaussian Process (GP) and the Artificial Neural Networks (ANN) have been employed and using the Hubble data from Cosmic Chronometers (CC), Pantheon+ from Supernovae Type Ia and their combination we have reconstructed the interaction function. We find that for $w_{\rm DE} =-1$, the interaction in the dark sector is not prominent while for $w_{\rm DE} \neq -1$, evidence of interaction is found depending on the value of $w_{\rm DE}$. In particularly, we find that if we start deviating from $w_{\rm DE} = -1$ either in the quintessence ($w_{\rm DE} > -1$) or phantom ($w_{\rm DE} < -1$) direction, an emergence of dark interaction is observed from both GP and ANN reconstructions. We further note that ANN which is applied for the first time in this context seems to play a very efficient role compared to GP.

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A demonstration of the effect of fringe-rate filtering in the Hydrogen Epoch of Reionization Array delay power spectrum pipeline

Radio interferometers targeting the 21cm brightness temperature fluctuations at high redshift are subject to systematic effects that operate over a range of different timescales. These can be isolated by designing appropriate Fourier filters that operate in fringe-rate (FR) space, the Fourier pair of local sidereal time (LST). Applications of FR filtering include separating effects that are correlated with the rotating sky vs. those relative to the ground, down-weighting emission in the primary beam sidelobes, and suppressing noise. FR filtering causes the noise contributions to the visibility data to become correlated in time however, making interpretation of subsequent averaging and error estimation steps more subtle. In this paper, we describe fringe rate filters that are implemented using discrete prolate spheroidal sequences, and designed for two different purposes -- beam sidelobe/horizon suppression (the `mainlobe' filter), and ground-locked systematics removal (the `notch' filter). We apply these to simulated data, and study how their properties affect visibilities and power spectra generated from the simulations. Included is an introduction to fringe-rate filtering and a demonstration of fringe-rate filters applied to simple situations to aid understanding.

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MeerKAT HI observations of Low Surface Brightness/Ultradiffuse Galaxy Candidates Projected around Two Southern Loose Groups

A large catalogue of low surface brightness galaxies (LSBGs) from the Dark Energy Survey showed significant clustering around nearby galaxy groups and clusters. Using the HIPASS survey, we tried to determine the redshift of a sub-sample of these LSBGs and determine whether they were members of the groups they were projected near, but this was hampered by HIPASS's high spectral rms. This letter reports on MeerKAT H I observations to determine the redshifts of 52 LSBG candidates projected in the vicinity of two groups from our previous HIPASS study. The main goal is to investigate and ascertain whether these LSBGs are genuine group members. H I was detected with MeerKAT and redshifts were determined for only five of the 52 candidates within a velocity range of $\pm$ 2500 km/s of their respective group velocities. All five H I detections were blue LSBGs and two of them were confirmed to be ultradiffuse galaxies (UDGs). Both these UDGs were group members, while the other three detections were either foreground or background galaxies. In this letter we explore scenarios that can explain the 90% non-detection. MeerKAT's excellent sensitivity allows us to conclude that the majority of the non-detected candidates, particularly the blue galaxies, are not group members but lie at higher redshifts. However, this still leaves the open question as why Tanoglidis LSBG candidates, in particular the red ones, appear to be clustered in projection around nearby groups.

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An independent estimate of H(z) at z = 0.5 from the stellar ages of brightest cluster galaxies

Several cosmological observations (e.g., Cosmic Microwave Background (CMB), Supernovae Type Ia, and local distance ladder measurements such as Cepheids) have been used to measure the global expansion rate of the Universe, i.e., the Hubble constant, $H_{0}$. However, these precision measurements have revealed tensions between different probes that are proving difficult to solve. Independent, robust techniques must be exploited to validate results or mitigate systematic effects. We use the Cosmic Chronometer (CC) method, which leverages the differential age evolution of passive galaxies, to measure $H(z)$, without any assumption of the underlying cosmology. Unlike previous CC studies, we used only brightest cluster galaxies (BCGs), the oldest and most massive galaxies in the Universe, to construct a pure and homogeneous sample. In this work we used a sample of 53 BCGs in massive, Sunyaev-Zel'dovich selected galaxy clusters (0.3 $< z <$ 0.7) with Southern African Large Telescope (SALT) spectroscopic observations. We used optical spectra to measure D4000$_{\rm n}$ of the BCGs to obtain a new direct measurement of $H(z) = 72.1 \pm 33.9(\rm stat) \pm 7.3$(syst) km s$^{-1}$ Mpc$^{-1}$ at $z=0.5$. By using BCGs, we significantly reduced the systematic errors to 10% by minimising the stellar mass and metallicity dependence of the method. The dominant uncertainty, and limitation for our study, is statistical, and we need larger, homogeneous samples of the oldest, most massive galaxies. By using the $Planck$+BAO prior of $Ω_{m}$ and $Ω_Λ$, the projected Hubble constant is $H_{0}$ = $54.6 \pm 25.7(\rm stat) \pm 5.5$(syst) km s$^{-1}$ Mpc$^{-1}$, consistent with both CMB and Cepheid measurements.

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