Leo W. H. Fung
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The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
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Leo W. H. Fung has a total of 100 co-authors. Here are the top co-authors by number of shared publications:
- Lingfeng Li
International Center of Theoretical Physics-Asia Pacific
3 shared publications - Tao Liu
The Hong Kong University of Science and Technology
3 shared publications - Yu-Cheng Qiu
Shanghai Jiao Tong University
3 shared publications - Andrew Robertson
Carnegie Observatories , 813 Santa Barbara Street, Pasadena, CA 91101
2 shared publications - Anton M. Koekemoer
Space Telescope Science Institute
2 shared publications - Hoang Nhan Luu
Donostia International Physics Center
2 shared publications
Latest Publications
- Open Access
- 28 October 2026
- Published by American Astronomical Society in The Astrophysical Journal
- Vol. 1011 (1) , 102
Lensing in the Blue. III. Weak-lensing Shape Catalogs of 30 Merging Galaxy Clusters
We present the weak gravitational lensing dataset from the Super-pressure Balloon-Borne Imaging Telescope (SuperBIT), which imaged 30 galaxy clusters during its 45 night flight in 2023 April–May. SuperBIT is a first-of-its-kind balloon-borne imaging telescope that achieved near-diffraction-limited observations in near-space conditions above 98% of the Earth’s atmosphere. We use the metacalibration algorithm to obtain calibrated galaxy shapes for our target clusters and several calibration fields, enabling unbiased reconstruction of the weak-lensing signal. We employ several diagnostics throughout the pipeline, including assessments of point-spread function (PSF) modeling residuals and their impact on weak-lensing measurements, as well as tests for correlations between galaxy shapes and measured galaxy and PSF properties. To assess the multiplicative shear bias of the pipeline, we analyze a parallel set of simulated images that incorporate the real observing conditions from the flight, including measured SuperBIT PSFs, observed sky backgrounds, and detector noise, yielding a bias of (1.54% ± 6.00%).
- 28 October 2026
- Published by American Astronomical Society in The Astrophysical Journal
- Vol. 1011 (1) , 102
Open AccessLensing in the Blue. III. Weak-lensing Shape Catalogs of 30 Merging Galaxy Clusters
We present the weak gravitational lensing dataset from the Super-pressure Balloon-Borne Imaging Telescope (SuperBIT), which imaged 30 galaxy clusters during its 45 night flight in 2023 April–May. SuperBIT is a first-of-its-kind balloon-borne imaging telescope that achieved near-diffraction-limited observations in near-space conditions above 98% of the Earth’s atmosphere. We use the metacalibration algorithm to obtain calibrated galaxy shapes for our target clusters and several calibration fields, enabling unbiased reconstruction of the weak-lensing signal. We employ several diagnostics throughout the pipeline, including assessments of point-spread function (PSF) modeling residuals and their impact on weak-lensing measurements, as well as tests for correlations between galaxy shapes and measured galaxy and PSF properties. To assess the multiplicative shear bias of the pipeline, we analyze a parallel set of simulated images that incorporate the real observing conditions from the flight, including measured SuperBIT PSFs, observed sky backgrounds, and detector noise, yielding a bias of (1.54% ± 6.00%).
- Open Access
- 10 December 2025
- Published by Oxford University Press (OUP) in Monthly Notices of the Royal Astronomical Society
- Vol. 545 (4)
Probing dark matter substructures with free-form modelling: a case study of the ‘Jackpot’ strong lens
Characterizing the population and internal structure of sub-galactic haloes is critical for constraining the nature of dark matter. These haloes can be detected near galaxies that act as strong gravitational lenses with extended arcs, as they perturb the shapes of the arcs. However, this method is subject to false-positive detections and systematic uncertainties, particularly degeneracies between an individual halo and larger scale asymmetries in the distribution of lens mass. We present a new free-form lens modelling code, developed within the framework of the open-source software pyautolens, to address these challenges. Our method models mass perturbations that cannot be captured by parametric models as pixelized potential corrections and suppresses unphysical solutions via a Matérn regularization scheme that is inspired by Gaussian process regression. This approach enables the recovery of diverse mass perturbations, including subhaloes, line-of-sight haloes, external shear, and multipole components that represent the complex angular mass distribution of the lens galaxy, such as boxiness/disciness. Additionally, our fully Bayesian framework objectively infers hyperparameters associated with the regularization of pixelized sources and potential corrections, eliminating the need for manual fine-tuning. By applying our code to the well-known ‘Jackpot’ lens system, SLACS0946+1006, we robustly detect a highly concentrated subhalo that challenges the standard cold dark matter model. This study represents the first attempt to independently reveal the mass distribution of a subhalo using a fully free-form approach.
- 10 December 2025
- Published by Oxford University Press (OUP) in Monthly Notices of the Royal Astronomical Society
- Vol. 545 (4)
Open AccessProbing dark matter substructures with free-form modelling: a case study of the ‘Jackpot’ strong lens
Characterizing the population and internal structure of sub-galactic haloes is critical for constraining the nature of dark matter. These haloes can be detected near galaxies that act as strong gravitational lenses with extended arcs, as they perturb the shapes of the arcs. However, this method is subject to false-positive detections and systematic uncertainties, particularly degeneracies between an individual halo and larger scale asymmetries in the distribution of lens mass. We present a new free-form lens modelling code, developed within the framework of the open-source software pyautolens, to address these challenges. Our method models mass perturbations that cannot be captured by parametric models as pixelized potential corrections and suppresses unphysical solutions via a Matérn regularization scheme that is inspired by Gaussian process regression. This approach enables the recovery of diverse mass perturbations, including subhaloes, line-of-sight haloes, external shear, and multipole components that represent the complex angular mass distribution of the lens galaxy, such as boxiness/disciness. Additionally, our fully Bayesian framework objectively infers hyperparameters associated with the regularization of pixelized sources and potential corrections, eliminating the need for manual fine-tuning. By applying our code to the well-known ‘Jackpot’ lens system, SLACS0946+1006, we robustly detect a highly concentrated subhalo that challenges the standard cold dark matter model. This study represents the first attempt to independently reveal the mass distribution of a subhalo using a fully free-form approach.
- Open Access
- 29 July 2025
- Published by American Astronomical Society in The Astrophysical Journal
- Vol. 988 (2) , 264
Explaining JWST Counts with Galaxy Formation Models
A distinct power-law break is apparent at m AB ∼ 21 in the deep near-infrared PEARLS-JWST galaxy counts. The break becomes more pronounced at longer wavelengths, with the slope flattening smoothly with apparent magnitude in the shortest band used at 0.9 μm, and trending toward an increasingly broken slope by the longest wavelength passband of JWST’s Near Infrared Camera, 4.4 μm. This behaviour is remarkably well predicted by the GALFORM semi-analytical model of galaxy formation. We use the model to diagnose the origin of this behaviour. The features that are responsible for the break are (1) the inherent break in the luminosity function (LF); (2) the change in the volume element with redshift; (3) the redshift-dependent nature of the k-correction (with 1 contributing to the existence of the break and 2–3 contributing to its shape). We study the contribution to these effects by their morphology using the bulge-to-total stellar mass ratio. The way in which bulge-dominated galaxies populate the bright end of the LF while disk-dominated galaxies dominate the faint end is preserved in the galaxy number counts, with a characteristic stellar mass at a break of ∼1010 M ⊙. The shape of the number counts is mainly driven by galaxies with relatively low redshift (z ≲ 2) for a limit of m AB ≲ 28. We give a comprehensive description of why the galaxy number counts in the near-infrared PEARLS-JWST observation look the way they do and which population of galaxies is dominant at each apparent magnitude.
- 29 July 2025
- Published by American Astronomical Society in The Astrophysical Journal
- Vol. 988 (2) , 264
Open AccessExplaining JWST Counts with Galaxy Formation Models
A distinct power-law break is apparent at m AB ∼ 21 in the deep near-infrared PEARLS-JWST galaxy counts. The break becomes more pronounced at longer wavelengths, with the slope flattening smoothly with apparent magnitude in the shortest band used at 0.9 μm, and trending toward an increasingly broken slope by the longest wavelength passband of JWST’s Near Infrared Camera, 4.4 μm. This behaviour is remarkably well predicted by the GALFORM semi-analytical model of galaxy formation. We use the model to diagnose the origin of this behaviour. The features that are responsible for the break are (1) the inherent break in the luminosity function (LF); (2) the change in the volume element with redshift; (3) the redshift-dependent nature of the k-correction (with 1 contributing to the existence of the break and 2–3 contributing to its shape). We study the contribution to these effects by their morphology using the bulge-to-total stellar mass ratio. The way in which bulge-dominated galaxies populate the bright end of the LF while disk-dominated galaxies dominate the faint end is preserved in the galaxy number counts, with a characteristic stellar mass at a break of ∼1010 M ⊙. The shape of the number counts is mainly driven by galaxies with relatively low redshift (z ≲ 2) for a limit of m AB ≲ 28. We give a comprehensive description of why the galaxy number counts in the near-infrared PEARLS-JWST observation look the way they do and which population of galaxies is dominant at each apparent magnitude.
- Open Access
- 23 July 2025
- Published by American Astronomical Society in The Astrophysical Journal
- Vol. 988 (2) , 178
Constraining the z ∼ 1 Initial Mass Function with HST and JWST Lensed Stars in MACS J0416.1−2403
Our understanding of galaxy properties and evolution is contingent on knowing the initial mass function (IMF), and yet to date the IMF is constrained only to local galaxies. Individual stars are now becoming routinely detected at cosmological distances, where luminous stars such as supergiants in background galaxies strongly lensed by galaxy clusters are temporarily further magnified by huge factors (up to 104) by intracluster stars, thus being detected as transients. The detection rate of these events depends on the abundance of luminous stars in the background galaxy and is thus sensitive to the IMF and the star formation history (SFH), especially for the blue supergiants detected as transients in the rest-frame ultraviolet/optical filters. As a proof of concept, we use simple SFH and IMF models constrained by spectral energy distributions (SEDs) to see how well we can predict the Hubble Space Telescope and James Webb Space Telescope transient detection rate in a lensed arc dubbed “Spock” (z = 1.0054). We find that demanding a simultaneous fit of the SED and the transient detection rate places constraints on the IMF, independent of the assumed simple SFH model. We conclude that our likelihood analysis indicates that the data definitively prefers the “Spock” galaxy to have a Salpeter IMF (α = 2.35) rather than a top-heavy IMF (α = 1)—which is thought to be the case in the early universe—with no clear excess of supergiants above the standard IMF.
- 23 July 2025
- Published by American Astronomical Society in The Astrophysical Journal
- Vol. 988 (2) , 178
Open AccessConstraining the z ∼ 1 Initial Mass Function with HST and JWST Lensed Stars in MACS J0416.1−2403
Our understanding of galaxy properties and evolution is contingent on knowing the initial mass function (IMF), and yet to date the IMF is constrained only to local galaxies. Individual stars are now becoming routinely detected at cosmological distances, where luminous stars such as supergiants in background galaxies strongly lensed by galaxy clusters are temporarily further magnified by huge factors (up to 104) by intracluster stars, thus being detected as transients. The detection rate of these events depends on the abundance of luminous stars in the background galaxy and is thus sensitive to the IMF and the star formation history (SFH), especially for the blue supergiants detected as transients in the rest-frame ultraviolet/optical filters. As a proof of concept, we use simple SFH and IMF models constrained by spectral energy distributions (SEDs) to see how well we can predict the Hubble Space Telescope and James Webb Space Telescope transient detection rate in a lensed arc dubbed “Spock” (z = 1.0054). We find that demanding a simultaneous fit of the SED and the transient detection rate places constraints on the IMF, independent of the assumed simple SFH model. We conclude that our likelihood analysis indicates that the data definitively prefers the “Spock” galaxy to have a Salpeter IMF (α = 2.35) rather than a top-heavy IMF (α = 1)—which is thought to be the case in the early universe—with no clear excess of supergiants above the standard IMF.
- 1 April 2025
- Published by SPIE-Intl Soc Optical Eng in Journal of Astronomical Telescopes, Instruments, and Systems
- Vol. 11 (02) , 028007-17
False alarm rate-based statistical detection limit for astronomical photon detectors
In ultra-fast astronomical observations featuring fast transients on sub-μs time scales, the conventional signal-to-noise ratio (SNR) threshold, often fixed at 5σ, becomes inadequate as observational window timescales shorten, leading to unsustainably high false alarm rates (FAR). We provide a basic statistical framework that captures the essential noise generation processes relevant to the analysis of time series data from photon-counting detectors. In particular, we establish a protocol of defining detection limits in astronomical photon-counting experiments, such that a FAR-based criterion is preferred over the traditional SNR-based threshold scheme. We developed statistical models that account for noise sources such as dark counts, sky background, and crosstalk and established a probabilistic detection criterion applicable to high-speed detectors. The model is tested against the on-site data obtained in the Single-Photon Imager for Nanosecond Astrophysics (SPINA) experiment, and consistency is confirmed. We compare the performance of several detector technologies, including photon-counting CMOS/CCDs, SPADs, SiPMs, and PMTs, in detecting faint astronomical signals. These findings offer insights into optimizing detector choice for future ultra-fast astronomical instruments and suggest pathways for improving detection fidelity under rapid observational conditions.
- 1 April 2025
- Published by SPIE-Intl Soc Optical Eng in Journal of Astronomical Telescopes, Instruments, and Systems
- Vol. 11 (02) , 028007-17
False alarm rate-based statistical detection limit for astronomical photon detectors
In ultra-fast astronomical observations featuring fast transients on sub-μs time scales, the conventional signal-to-noise ratio (SNR) threshold, often fixed at 5σ, becomes inadequate as observational window timescales shorten, leading to unsustainably high false alarm rates (FAR). We provide a basic statistical framework that captures the essential noise generation processes relevant to the analysis of time series data from photon-counting detectors. In particular, we establish a protocol of defining detection limits in astronomical photon-counting experiments, such that a FAR-based criterion is preferred over the traditional SNR-based threshold scheme. We developed statistical models that account for noise sources such as dark counts, sky background, and crosstalk and established a probabilistic detection criterion applicable to high-speed detectors. The model is tested against the on-site data obtained in the Single-Photon Imager for Nanosecond Astrophysics (SPINA) experiment, and consistency is confirmed. We compare the performance of several detector technologies, including photon-counting CMOS/CCDs, SPADs, SiPMs, and PMTs, in detecting faint astronomical signals. These findings offer insights into optimizing detector choice for future ultra-fast astronomical instruments and suggest pathways for improving detection fidelity under rapid observational conditions.
- Open Access
- 21 February 2025
- Published by Oxford University Press (OUP) in Monthly Notices of the Royal Astronomical Society
- Vol. 538 (2) , 711-725
Pure gravitational wave estimation of Hubble’s constant using neutron star–black hole mergers
It has been predicted before the first detection of such events by LIGO (Laser Interferometer Gravitational-Wave Observatory) that the gravitational wave (GW) emission from compact binary coalescence can be used to constrain the expansion rate of the Universe. Here we show how $H_0$ can be derived purely from the GW of neutron star–black hole (NSBH) mergers using a Bayesian hierarchical inference framework. This method provides an estimate of $H_0$ spanning the redshift range $z<0.25$ with current GW sensitivity and without the need for any afterglow detection. We utilize the inherent distribution of neutron star masses together with the NSBH waveform amplitude and frequency to estimate distance and redshift, respectively, thereby measuring $H_0$ for the NSBH events up to systematics and observational uncertainties. With the most updated observations, our first estimate is $H_0 = 96^{+44}_{-50}$ km s−1 Mpc−1 for the likely NSBH events GW190426 and GW200115 using the proof-of-concept analysis outlined in this work. Taking into account of realistic detectors response and sensitivity, we forecast that soon, with 10 more such NSBH events, we can reach competitive precision of $\delta H_0/H_0 \lesssim 20 {{\, \rm per \, cent}}$. It is valuable to have a completely independent, both in terms of observations and relevant physics, method of determining $H_0$ as is emphasized by the current controversy on $H_0$ and attempts to explain it by invoking new physics.- 21 February 2025
- Published by Oxford University Press (OUP) in Monthly Notices of the Royal Astronomical Society
- Vol. 538 (2) , 711-725
Open AccessPure gravitational wave estimation of Hubble’s constant using neutron star–black hole mergers
It has been predicted before the first detection of such events by LIGO (Laser Interferometer Gravitational-Wave Observatory) that the gravitational wave (GW) emission from compact binary coalescence can be used to constrain the expansion rate of the Universe. Here we show how $H_0$ can be derived purely from the GW of neutron star–black hole (NSBH) mergers using a Bayesian hierarchical inference framework. This method provides an estimate of $H_0$ spanning the redshift range $z<0.25$ with current GW sensitivity and without the need for any afterglow detection. We utilize the inherent distribution of neutron star masses together with the NSBH waveform amplitude and frequency to estimate distance and redshift, respectively, thereby measuring $H_0$ for the NSBH events up to systematics and observational uncertainties. With the most updated observations, our first estimate is $H_0 = 96^{+44}_{-50}$ km s−1 Mpc−1 for the likely NSBH events GW190426 and GW200115 using the proof-of-concept analysis outlined in this work. Taking into account of realistic detectors response and sensitivity, we forecast that soon, with 10 more such NSBH events, we can reach competitive precision of $\delta H_0/H_0 \lesssim 20 {{\, \rm per \, cent}}$. It is valuable to have a completely independent, both in terms of observations and relevant physics, method of determining $H_0$ as is emphasized by the current controversy on $H_0$ and attempts to explain it by invoking new physics. - Open Access
The Hubble Constant in the Axi-Higgs Universe
The $\Lambda$CDM model provides an excellent fit to the CMB data. However, a statistically significant tension emerges when its determination of the Hubble constant $H_0$ is compared to the local distance-redshift measurements. The axi-Higgs model, which couples an ultralight axion to the Higgs field, offers a specific variation of the $\Lambda$CDM model. It relaxes the $H_0$ tension as well as explains the $^7$Li puzzle in Big-Bang nucleosynthesis, the clustering $S_8$ tension with the weak-lensing data, and the observed isotropic cosmic birefringence in CMB. In this paper, we demonstrate how the $H_0$ and $S_8$ tensions can be relaxed simultaneously, by correlating the axion impacts on the early and late universe. In a benchmark scenario ($m=2 \times 10^{-30}$ eV) selected for experimental tests soon, the analysis combining the CMB+BAO+WL+SN data yields $H_0 = 69.9 \pm 1.5$ km/s/Mpc and $S_8 = 0.8045 \pm 0.0096$. Combining this (excluding the SN (supernovae) part) with the local distance-redshift measurements yields $H_0 = 72.42 \pm 0.76$ km/s/Mpc, while $S_8$ is slightly more suppressed.Open AccessThe Hubble Constant in the Axi-Higgs Universe
The $\Lambda$CDM model provides an excellent fit to the CMB data. However, a statistically significant tension emerges when its determination of the Hubble constant $H_0$ is compared to the local distance-redshift measurements. The axi-Higgs model, which couples an ultralight axion to the Higgs field, offers a specific variation of the $\Lambda$CDM model. It relaxes the $H_0$ tension as well as explains the $^7$Li puzzle in Big-Bang nucleosynthesis, the clustering $S_8$ tension with the weak-lensing data, and the observed isotropic cosmic birefringence in CMB. In this paper, we demonstrate how the $H_0$ and $S_8$ tensions can be relaxed simultaneously, by correlating the axion impacts on the early and late universe. In a benchmark scenario ($m=2 \times 10^{-30}$ eV) selected for experimental tests soon, the analysis combining the CMB+BAO+WL+SN data yields $H_0 = 69.9 \pm 1.5$ km/s/Mpc and $S_8 = 0.8045 \pm 0.0096$. Combining this (excluding the SN (supernovae) part) with the local distance-redshift measurements yields $H_0 = 72.42 \pm 0.76$ km/s/Mpc, while $S_8$ is slightly more suppressed. - Open Access
- 22 June 2023
- Published by American Physical Society (APS) in Physical Review Research
- Vol. 5 (2) , L022059
Hubble constant in the axi-Higgs universe
The -cold-dark-matter model provides an excellent fit to the cosmic microwave background (CMB) data. However, a statistically significant tension emerges when its determination of the Hubble constant is compared to the local distance-redshift measurements. The axi-Higgs model, which couples an ultralight axion to the Higgs field, offers a specific variation of the model. It relaxes the tension as well as explains the puzzle in big bang nucleosynthesis, the clustering tension with the weak-lensing (WL) data, and the observed isotropic cosmic birefringence in CMB. In this Letter, we demonstrate how the and tensions can be relaxed simultaneously, by correlating the axion impacts on the early and late universe. In a benchmark scenario selected for upcoming experimental tests, the analysis combining the (baryon acoustic oscillation) data yields km/s/Mpc and . Combining this [excluding the SN (supernovae) part] with the local distance-redshift measurements yields km/s/Mpc, while is slightly more suppressed.- 22 June 2023
- Published by American Physical Society (APS) in Physical Review Research
- Vol. 5 (2) , L022059
Open AccessHubble constant in the axi-Higgs universe
The -cold-dark-matter model provides an excellent fit to the cosmic microwave background (CMB) data. However, a statistically significant tension emerges when its determination of the Hubble constant is compared to the local distance-redshift measurements. The axi-Higgs model, which couples an ultralight axion to the Higgs field, offers a specific variation of the model. It relaxes the tension as well as explains the puzzle in big bang nucleosynthesis, the clustering tension with the weak-lensing (WL) data, and the observed isotropic cosmic birefringence in CMB. In this Letter, we demonstrate how the and tensions can be relaxed simultaneously, by correlating the axion impacts on the early and late universe. In a benchmark scenario selected for upcoming experimental tests, the analysis combining the (baryon acoustic oscillation) data yields km/s/Mpc and . Combining this [excluding the SN (supernovae) part] with the local distance-redshift measurements yields km/s/Mpc, while is slightly more suppressed. - Open Access
Axi-Higgs Cosmology
If the electroweak Higgs vacuum expectation value $v$ in early universe is $\sim 1 \%$ higher than its present value $v_0=246$ GeV, the $^7$Li puzzle in BBN and the CMB/$\Lambda$CDM tension with late-universe measurements on Hubble parameter are mitigated. We propose a model of an axion coupled to the Higgs field, named ``axi-Higgs'', with its mass $m_a \sim 10^{-30} - 10^{-29}\,{\rm eV}$ and decay constant $f_a \sim 10^{17} - 10^{18}\,{\rm GeV}$, to achieve this goal. The axion initial value $a_{\rm ini}$ yields an initial $\Delta v_{\rm ini}/v_0 \sim 0.01$ throughout the BBN-recombination epoch and a percent level contribution to the total matter density today. Because of its very large de Broglie wavelength, this axion matter density $\omega_a$ suppresses the matter power spectrum, alleviating the CMB/$\Lambda$CDM $S_8/\sigma_8$ tension with the weak-lensing data. It also explains the recently reported isotropic cosmic birefringence by its coupling with photons. Adding the axion ($m \sim 10^{-22}\,$eV) in the fuzzy dark matter model to the axi-Higgs model allows bigger $\Delta v_{\rm rec}$ and $\omega_a$ to address the Hubble and $S_8/\sigma_8$ tensions simultaneously. The model predicts that $\Delta v$ may be detected by the spectral measurements of quasars, while its oscillation may be observed in the atomic clock measurements.Open AccessAxi-Higgs Cosmology
If the electroweak Higgs vacuum expectation value $v$ in early universe is $\sim 1 \%$ higher than its present value $v_0=246$ GeV, the $^7$Li puzzle in BBN and the CMB/$\Lambda$CDM tension with late-universe measurements on Hubble parameter are mitigated. We propose a model of an axion coupled to the Higgs field, named ``axi-Higgs'', with its mass $m_a \sim 10^{-30} - 10^{-29}\,{\rm eV}$ and decay constant $f_a \sim 10^{17} - 10^{18}\,{\rm GeV}$, to achieve this goal. The axion initial value $a_{\rm ini}$ yields an initial $\Delta v_{\rm ini}/v_0 \sim 0.01$ throughout the BBN-recombination epoch and a percent level contribution to the total matter density today. Because of its very large de Broglie wavelength, this axion matter density $\omega_a$ suppresses the matter power spectrum, alleviating the CMB/$\Lambda$CDM $S_8/\sigma_8$ tension with the weak-lensing data. It also explains the recently reported isotropic cosmic birefringence by its coupling with photons. Adding the axion ($m \sim 10^{-22}\,$eV) in the fuzzy dark matter model to the axi-Higgs model allows bigger $\Delta v_{\rm rec}$ and $\omega_a$ to address the Hubble and $S_8/\sigma_8$ tensions simultaneously. The model predicts that $\Delta v$ may be detected by the spectral measurements of quasars, while its oscillation may be observed in the atomic clock measurements.