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Register a new userMachine Learning the Cosmic Curvature in a Model-independent Way
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In this work, we achieve the determination of the cosmic curvature $Omega_K$ in a cosmological model-independent way, by using the Hubble parameter measurements $H(z)$ and type Ia supernovae (SNe Ia). In our analysis, two nonlinear interpolating tools are used to reconstruct the Hubble parameter, one is the Artificial Neural Network (ANN) method, and the other is the Gaussian process (GP) method. We find that $Omega_K$ based on the GP method can be greatly influenced by the prior of $H_0$, while the ANN method can overcome this. Therefore, the ANN method may have more advantages than GP in the measurement of the cosmic curvature. Based on the ANN method, we find a spatially open universe is preferred by the current $H(z)$ and SNe Ia data, and the difference between our result and the value inferred from Planck CMB is $1.6sigma$. In order to test the reliability of the ANN method, and the potentiality of the future gravitational waves (GW) standard sirens in the measurement of the cosmic curvature, we constrain $Omega_K$ using the simulated Hubble parameter and GW standard sirens in a model-independent way. We find that the ANN method is reliable and unbiased, and the error of $Omega_K$ is $sim0.186$ when 100 GW events with electromagnetic counterparts are detected, which is $sim56%$ smaller than that constrained from the Pantheon SNe Ia. Therefore, the data-driven method based on ANN has potential in the measurement of the cosmic curvature.
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We use current measurements of the expansion rate $H(z)$ and cosmic background radiation bounds on the spatial curvature of the Universe to impose cosmological model-independent constraints on cosmic opacity. To perform our analyses, we compare opacity-free distance modulus from $H(z)$ data with those from two supernovae Ia compilations: the Union2.1 plus the most distant spectroscopically confirmed SNe Ia (SNe Ia SCP-0401 $z=1.713$) and two Sloan Digital Sky Survey (SDSS) subsamples. The influence of different SNe Ia light-curve fitters (SALT2 and MLCS2K2) on the results is also verified. We find that a completely transparent universe is in agreement with the largest sample in our analysis (Union 2.1 plus SNe Ia SCP-0401). For SDSS sample a such universe it is compatible at $< 1.5sigma$ level regardless the SNe Ia light-curve fitting used.
Given observations of the standard candles and the cosmic chronometers, we apply Pad{e} parameterization to the comoving distance and the Hubble paramter to find how stringent the constraint is set to the curvature parameter by the data. A weak informative prior is introduced in the modeling process to keep the inference away from the singularities. Bayesian evidence for different order of Pad{e} parameterizations is evaluated during the inference to select the most suitable parameterization in light of the data. The data we used prefer a parameterization form of comoving distance as $D_{01}(z)=frac{a_0 z}{1+b_1 z}$ as well as a competitive form $D_{02}(z)=frac{a_0 z}{1+b_1 z + b_2 z^2}$. Similar constraints on the spatial curvature parameter are established by those models and given the Hubble constant as a byproduct: $Omega_k = 0.25^{+0.14}_{-0.13}$ (68% confidence level [C.L.]), $H_0 = 67.7 pm 2.0$ km/s/Mpc (68% C.L.) for $D_{01}$, and $Omega_k = -0.01 pm 0.13$ (68% C.L.), $H_0 = 68.8 pm 2.0$ km/s/Mpc (68% C.L.) for $D_{02}$. The evidence of different models demonstrates the qualitative analysis of the Pad{e} parameterizations for the comoving distance.
Applying the distance sum rule in strong gravitational lensing (SGL) and type Ia supernova (SN Ia) observations, one can provide an interesting cosmological model-independent method to determine the cosmic curvature parameter $Omega_k$. In this paper, with the newly compiled data sets including 161 galactic-scale SGL systems and 1048 SN Ia data, we place constraints on $Omega_k$ 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 sub-samples 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 the uncertainty 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 $Omega_k$ significantly.
The question of whether Cosmic Microwave Background (CMB) temperature and polarization data from Planck favor a spatially closed Universe with curvature parameter $Omega_K<0$ has been the subject of recent intense discussions. Attempts to break the geometrical degeneracy combining Planck data with external datasets such as Baryon Acoustic Oscillation (BAO) measurements all point towards a spatially flat Universe, at the cost of significant tensions with Planck, which make the resulting dataset combination problematic. Settling this issue requires identifying a dataset which can break the geometrical degeneracy while not incurring in these tensions. We argue that cosmic chronometers (CC), measurements of the expansion rate $H(z)$ from the relative ages of massive early-type passively evolving galaxies, are the dataset we are after. Furthermore, CC come with the additional advantage of being virtually free of cosmological model assumptions. Combining Planck 2018 CMB temperature and polarization data with the latest CC measurements, we break the geometrical degeneracy and find $Omega_K=-0.0054 pm 0.0055$, consistent with a spatially flat Universe and competitive with the Planck+BAO constraint. Our results are stable against minimal parameter space extensions and CC systematics, and we find no substantial tension between Planck and CC data within a non-flat Universe, making the resulting combination reliable. Our results allow us to assert with confidence that the Universe is spatially flat to the ${cal O}(10^{-2})$ level, a finding which might possibly settle the ongoing spatial curvature debate, and lends even more support to the already very successful inflationary paradigm.
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 2$sigma$ 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 $Omega_{k0}h^2=0.102pm 0.066$ which is consistent with a spatially flat universe at the 2$sigma$ 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 $Lambda$CDM model is found.
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