Revisiting the Hubble Constant, Spatial Curvature, and Cosmography with Strongly Lensed Quasar and Hubble Parameter Observations

Although the Hubble constant H0 and spatial curvature ΩK have been measured with very high precision, they still suffer from some tensions. In this paper, we propose an improved method to combine the observations of ultracompact structure in radio quasars and strong gravitational lensing with quasars acting as background sources to determine H0 and ΩK simultaneously. By applying the distance sum rule to the time-delay measurements of seven strong lensing systems and 120 intermediate-luminosity quasars calibrated as standard rulers, we obtain stringent constraints on the Hubble constant (H0 = 78.3 ± 2.9 km s−1 Mpc−1) and the cosmic curvature (ΩK = 0.49 ± 0.24). On the one hand, in the framework of a flat universe, the measured Hubble constant ($H_0=73.6^{+1.8}_{-1.6} \mathrm{\,km\,s^{-1}\,Mpc^{-1}}$) is strongly consistent with that derived from the local distance ladder, with a precision of 2 per cent. On the other hand, if we use the local H0 measurement as a prior, our results are marginally compatible with zero spatial curvature ($\Omega _K=0.23^{+0.15}_{-0.17}$) and there is no significant deviation from a flat universe. Finally, we also evaluate whether strongly lensed quasars would produce robust constraints on H0 and ΩK in the non-flat and flat Λ cold dark matter model, if the compact radio structure measurements are available from very long baseline interferometry observations.

In this paper we use a newly compiled sample of ultra-compact structure in radio quasars and strong gravitational lensing systems with quasars acting as background sources to constrain six spatially flat and non-flat cosmological models (ΛCDM, PEDE, and DGP). These two sets of quasar data (time-delay measurements of six strong lensing systems and 120 intermediate-luminosity quasars calibrated as standard rulers) could break the degeneracy between the cosmological parameters (H0, Ωm, and Ωk), and therefore provide more stringent cosmological constraints for the six cosmological models we study. A joint analysis of the quasar sample provides model-independent measurements of the Hubble constant H0, which are strongly consistent with that derived from the local distance ladder by the SH0ES collaboration in the ΛCDM and PEDE model. However, in the framework of the DGP cosmology (especially for a flat universe), the measured Hubble constant is in good agreement with that derived from the recent Planck 2018 results. In addition, our results show that zero spatial curvature is supported by the current lensed and unlensed quasar observations and that there is no significant deviation from a flat universe. For most of the cosmological models we study (flat ΛCDM, non-flat ΛCDM, flat PEDE, and non-flat PEDE), the derived matter density parameter is completely consistent with Ωm ∼ 0.30 in all the data sets, as expected based on the latest cosmological observations. Finally, according to the statistical deviance information criterion (DIC), the joint constraints provide substantial observational support to the flat PEDE model; however, they do not rule out dark energy being a cosmological constant and non-flat spatial hypersurfaces.

In recent years, model-independent approaches have gained increasing attention as powerful tools to investigate persistent tensions between cosmological observations and the predictions of the standard ΛCDM model. Notably, recent data from the DESY5 Type Ia Supernovae (SNIa) sample and the latest Baryon Acoustic Oscillation (BAO) measurements from the DESI collaboration challenge the validity of the cosmological constant, and under the assumption of standard pre-recombination physics, they still remain in tension with the SH0ES local distance ladder measurements. Building on our previous work, Mon. Not. Roy. Astron. Soc. 523 (2023) 3, 3406-3422, we present a follow-up analysis of the model-independent calibration of both the local and inverse distance ladders using cosmic chronometers (CCH) data and the Gaussian Processes technique. We constrain the SNIa absolute magnitude, M, and the comoving sound horizon at the baryon-drag epoch, rd, while simultaneously deriving a measurement of the spatial curvature parameter, Ωk, using CCH with DESY5 and DESI DR1 and DR2 data releases. Our results show that this data combination is compatible with a flat universe at ∼1.7σ, with Ωk = −0.143 ± 0.085, indicating a weaker compatibility than that observed with SNIa from Pantheon+, while the ladders calibrators read $M=-19.324_{-0.095}^{+0.092}$ and $r_d = (144.00^{+5.38}_{-4.88}$) Mpc. Although current uncertainties limit the precision of our constraints and prevent us from arbitrating the Hubble tension, it is nevertheless instructive to explore the constraining power of our methodology with future SNIa, CCH, and BAO observations from surveys such as Vera C. Rubin Observatory - LSST, Euclid, and DESI. Thus, for the first time, we present a forecast analysis for the triad (M, Ωk, rd). Our results indicate that, in an optimistic scenario, upcoming data will improve agnostic constraints on the ladder calibrators - M by ∼54 %, rd by ∼66 % - which enable us to constrain the Hubble parameter, H0, at a 2 % level. Precision on Ωk will increase by $\sim 50~{{\ \rm per\ cent}}$. Our analysis outlines which improvements in future data – whether in quality, quantity, or redshift coverage – are likely to have the greatest impact on tightening these constraints.

Model-independent measurements for the cosmic spatial curvature, which is related to the nature of cosmic spacetime geometry, play an important role in cosmology. On the basis of the distance sum rule in the Friedmann–Lemaître–Robertson–Walker metric, (distance ratio) measurements of strong gravitational lensing (SGL) systems, together with distances from SNe Ia observations, have been proposed to directly estimate the spatial curvature without any assumptions for the theories of gravity and contents of the universe. However, previous studies indicated that a spatially closed universe was strongly preferred. In this paper, we re-estimate the cosmic curvature with the latest SGL data, which includes 163 well-measured systems. In addition, possible factors, e.g., a combination of SGL data from different surveys and stellar masses of the lens galaxy, which might affect estimations for the spatial curvature, are considered in our analysis. We find that, except for the case where only SGL systems from the Sloan Lens ACS Survey are considered, a spatially flat universe is consistently favored at very high confidence levels by the latest observations. It has been suggested that an increasing number of well-measured strong lensing events might significantly reduce the bias of estimation for the cosmic curvature.

Strong gravitational lensing along with the distance sum rule method can constrain both cosmological parameters as well as density profiles of galaxies without assuming any fiducial cosmological model. To constrain galaxy parameters and cosmic curvature $(\Omega_{k0})$, we use the distance ratio data from a recently compiled database of $161$ galactic scale strong lensing systems. We use databases of supernovae type-Ia (Pantheon) and Gamma Ray Bursts (GRBs) for calculating the luminosity distance. To study the model of the lens galaxy, we consider a general lens model namely, the Extended Power-Law model. Further, we take into account two different parametrisations of the mass density power-law index $(\gamma)$ to study the dependence of $\gamma$ on redshift. The best value of $\Omega_{k0}$ suggests a closed universe, though a flat universe is accommodated at $68\%$ confidence level. We find that parametrisations of $\gamma$ have a negligible impact on the best fit value of the cosmic curvature parameter. Furthermore, measurement of time delay can be a promising cosmographic probe via "time delay distance" that includes the ratio of distances between the observer, the lens and the source. We again use the distance sum rule method with time-delay distance dataset of H0LiCOW to put constraints on the Cosmic Distance Duality Relation (CDDR) and the cosmic curvature parameter $(\Omega_{k0})$. For this we consider two different redshift-dependent parametrisations of the distance duality parameter $(\eta)$. The best fit value of $\Omega_{k0}$ clearly indicates an open universe. However, a flat universe can be accommodated at $95\%$ confidence level. Further, at $95\%$ confidence level, no violation of CDDR is observed. We believe that a larger sample of strong gravitational lensing systems is needed in order to improve the constraints on the cosmic curvature and distance duality parameter.

In this work, we use a model-independent approach to determine the Hubble constant and the spatial curvature simultaneously. We use the cosmic chronometers in combination with the time delay measurements in this method, and the cosmography method is used to reconstruct distances from cosmic chronometer observations instead of the polynomial expansion. This approach avoids the problems of the physical meaning of polynomial coefficients and the calibration of "nuisance parameter". Our results show that the measured Hubble constant $H_0=72.24^{+2.73}_{-2.52} km/s/Mpc$ is in good agreement with that derived from the local distance ladder measurement. In addition, our results show that a zero spatial curvature $\Omega_k=0.062^{+0.117}_{-0.078}$ and an accelerating expansion of the Universe $q_0=-0.645^{+0.126}_{-0.124}$ are supported by the current time delay of strong lensing system and cosmic chronometer observations. If we assume a flat universe in advance, we can infer that the $H_0=70.47^{+1.14}_{-1.15} km/s/Mpc$, falls between the SH0ES and Plank CMB observation results. Moreover, the convergence and goodness of the fourth order fitting is worse than the third order fitting case. Finally, the cosmic expansion history is reconstructed within a wide range of redshifts.

Combining the ‘time-delay distance’ (DΔt) measurements from galaxy lenses and other distance indicators provides model-independent determinations of the Hubble constant (H0) and spatial curvature (ΩK, 0), only based on the validity of the Friedmann–Lemaître–Robertson–Walker (FLRW) metric and geometrical optics. To take the full merit of combining DΔt measurements in constraining H0, we use gamma-ray burst (GRB) distances to extend the redshift coverage of lensing systems much higher than that of Type Ia Supernovae (SNe Ia) and even higher than quasars, whilst the general cosmography with a curvature component is implemented for the GRB distance parametrizations. Combining Lensing + GRB yields $H_0=71.5^{+4.4}_{-3.0}$ km s−1 Mpc−1 and $\Omega _{K,0} = -0.07^{+0.13}_{-0.06}$ (1σ). A flat-universe prior gives slightly an improved $H_0 = 70.9^{+4.2}_{-2.9}$ km s−1Mpc−1. When combining Lensing+GRB + SN Ia, the error bar ΔH0 falls by 25 per cent, whereas ΩK, 0 is not improved due to the degeneracy between SN Ia absolute magnitude, MB, and H0 along with the mismatch between the SN Ia and GRB Hubble diagrams at z ≳ 1.4. Future increment of GRB observations can help to moderately eliminate the MB–H0 degeneracy in SN Ia distances and ameliorate the restrictions on cosmographic parameters along with ΩK, 0 when combining Lensing+SN Ia + GRB. We conclude that there is no evidence of significant deviation from a (an) flat (accelerating) universe and H0 is currently determined at 3 per cent precision. The measurements show great potential to arbitrate the H0 tension between the local distance ladder and cosmic microwave background measurements and provide a relevant consistency test of the FLRW metric.

Multiple measurements on the cosmic curvature using Gaussian process regression without calibration and a cosmological model

Applying the distance sum rule in strong gravitational lensing (SGL) and SN Ia observations, one can provide an interesting cosmological model-independent method to determine the cosmic curvature parameter Ω k . In this paper, with the newly compiled data sets including 161 galactic-scale SGL systems and 1048 SN Ia data, we place constraints on Ω k within the framework of three types of lens models extensively used in SGL studies. Moreover, to investigate the effect of different mass lens samples on the results, we divide the SGL sample into three subsamples based on the center velocity dispersion of intervening galaxies. In the singular isothermal sphere (SIS) and extended power-law lens models, a flat universe is supported with an uncertainty of about 0.2, while a closed universe is preferred in the power-law lens model. We find that the choice of lens models and the classification of SGL data actually can influence the constraints on Ω k significantly.

Although the cosmic curvature has been tightly constrained in the standard cosmological model using observations of cosmic microwave background anisotropies, it is still of great importance to independently measure this key parameter using only late-Universe observations in a cosmological model-independent way. The distance sum rule in strong gravitational lensing (SGL) provides such a way, provided that the three distances in the sum rule can be calibrated by other observations. In this paper, we propose that gravitational waves (GWs) can be used to provide the distance calibration in the SGL method, which can avoid the dependence on distance ladder and cover a wider redshift range. Using the simulated GW standard siren observation by the Einstein Telescope as an example, we show that this scheme is feasible and advantageous. We find that ΔΩk ≃ 0.17 with the current SGL data, which is slightly more precise than the case of using SN to calibrate. Furthermore, we consider the forthcoming LSST survey that is expected to observe many SGL systems, and we find that about 104 SGL data could provide the precise measurement of ΔΩk ≃ 10−2 with the help of GWs. In addition, our results confirm that this method of constraining Ωk is strongly dependent on lens models. However, obtaining a more accurate phenomenological model for lens galaxies is highly predictable as future massive surveys observe more and more SGL samples, which will significantly improve the constraint of cosmic curvature.

The Hubble constant is a key cosmological parameter that sets the present-day expansion rate as well as the age, size, and critical density of the Universe. Intriguingly, there is currently a tension in the measurements of its value in the standard flat ΛCDM model – observations of the Cosmic Microwave Background with the Planck satellite lead to a value of the Hubble constant that is lower than the measurements from the local Cepheids-supernovae distance ladder and strong gravitational lensing. Precise and accurate Hubble constant measurements from independent probes, including water masers, are necessary to assess the significance of this tension and the possible need of new physics beyond the current standard cosmological model. We present the progress toward an accurate Hubble constant determination.

Although the spatial curvature has been measured with very high precision, it still suffers from the well-known cosmic curvature tension. In this paper, we use an improved method to determine the cosmic curvature, by using the simulated data of binary neutron star mergers observed by the second generation space-based DECi-hertz Interferometer Gravitational-wave Observatory (DECIGO). By applying the Hubble parameter observations of cosmic chronometers to the DECIGO standard sirens, we explore different possibilities of making measurements of the cosmic curvature referring to a distant past: one is to reconstruct the Hubble parameters through the Gaussian process without the influence of hypothetical models, and the other is deriving constraints on Ω K in the framework of the non-flat Λ cold dark matter model. It is shown that in the improved method DECIGO could provide a reliable and stringent constraint on the cosmic curvature (Ω K = −0.007 ± 0.016), while we could only expect the zero cosmic curvature to be established at the precision of ΔΩ K = 0.11 in the second model-dependent method. Therefore, our results indicate that in the framework of methodology proposed in this paper, the increasing number of well-measured standard sirens in DECIGO could significantly reduce the bias of estimations for cosmic curvature. Such a constraint is also comparable to the precision of Planck 2018 results with the newest cosmic microwave background (CMB) observations (ΔΩ K ≈ 0.018), based on the concordance ΛCDM model.

Although the spatial curvature has been measured with very high precision, it still suffers from the well known cosmic curvature tension. In this paper, we propose an improved method to determine the cosmic curvature, by using the simulated data of binary neutron star mergers observed by the second generation space-based DECi-hertz Interferometer Gravitational-wave Observatory (DECIGO). By applying the Hubble parameter observations of cosmic chronometers to the DECIGO standard sirens, we explore different possibilities of making measurements of the cosmic curvature referring to a distant past: one is to reconstruct the Hubble parameters through the Gaussian process without the influence of hypothetical models, and the other is deriving constraints on $\Omega_K$ in the framework of non-flat $\Lambda$ cold dark matter model. It is shown that in the improved method DECIGO could provide a reliable and stringent constraint on the cosmic curvature ($\Omega_{K} = -0.007\pm0.016$), while we could only expect the zero cosmic curvature to be established at the precision of $\Delta \Omega_K=0.12$ in the second model-dependent method. Therefore, our results indicate that in the framework of methodology proposed in this paper, the increasing number of well-measured standard sirens in DECIGO could significantly reduce the bias of estimations for cosmic curvature. Such constraint is also comparable to the precision of Planck 2018 results with the newest cosmic microwave background (CMB) observations ($\Delta \Omega_{K} \approx 0.018$), based on the concordance $\Lambda$CDM model.

Inflation predicts that the Universe is spatially flat. The Planck 2018 measurements of the cosmic microwave background anisotropy favour a spatially closed universe at more than 2σ confidence level. We use model-independent methods to study the issue of cosmic curvature. The method reconstructs the Hubble parameter H(z) from cosmic chronometers data with the Gaussian process method. The distance modulus is then calculated with the reconstructed function H(z) and fitted by Type Ia supernovae data. Combining the cosmic chronometers and Type Ia supernovae data, we obtain Ωk0h2 = 0.102 ± 0.066 that is consistent with a spatially flat universe at the 2σ confidence level. By adding the redshift-space distortions data to the Type Ia supernovae data with a proposed novel model-independent method, we obtain $\Omega _{k0}h^2=0.117^{+0.058}_{-0.045}$ and no deviation from Λ cold dark matter (ΛCDM) model is found.

The different spatial curvatures of the universe affect the measurement of cosmological distances, which may also contribute to explaining the observed dimming of type Ia supernovae. This phenomenon may be caused by the opacity of the universe. Similarly, the opacity of the universe can lead to a bias in our measurements of curvature. Thus, it is necessary to measure cosmic curvature and opacity simultaneously. In this paper, we propose a new model-independent method to simultaneously measure the cosmic curvature and opacity by using the latest observations of HII galaxies acting as standard candles and the latest Hubble parameter observations. The machine learning method-Artificial Neural Network is adopted to reconstruct observed Hubble parameter H(z) observations. Our results support a slightly opaque and flat universe at 1sigma confidence level by using previous 156 HII regions sample. However, the negative curvature is obtained by using the latest 181 HII regions sample in the redshift range zsim 2.5. More importantly, we obtain the simultaneous measurements with precision on the cosmic opacity mathrm Delta tau sim 10^{-2} and curvature mathrm Delta Omega _Ksim 10^{-1}. A strong degeneracy between the cosmic opacity and curvature parameters is also revealed in this analysis.