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Register a new userEppur `e piatto? The cosmic chronometer take on spatial curvature and cosmic concordance
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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.
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The concordance of the $Lambda$CDM cosmological model in light of current observations has been the subject of an intense debate in recent months. The 2018 Planck Cosmic Microwave Background (CMB) temperature anisotropy power spectrum measurements appear at face value to favour a spatially closed Universe with curvature parameter $Omega_K<0$. This preference disappears if Baryon Acoustic Oscillation (BAO) measurements are combined with Planck data to break the geometrical degeneracy, although the reliability of this combination has been questioned due to the strong tension present between the two datasets when assuming a curved Universe. Here, we approach this issue from yet another point of view, using measurements of the full-shape (FS) galaxy power spectrum, $P(k)$, from the Baryon Oscillation Spectroscopic Survey DR12 CMASS sample. By combining Planck data with FS measurements, we break the geometrical degeneracy and find $Omega_K=0.0023 pm 0.0028$. This constrains the Universe to be spatially flat to sub-percent precision, in excellent agreement with results obtained using BAO measurements. However, as with BAO, the overall increase in the best-fit $chi^2$ suggests a similar level of tension between Planck and $P(k)$ under the assumption of a curved Universe. While the debate on spatial curvature and the concordance between cosmological datasets remains open, our results provide new perspectives on the issue, highlighting the crucial role of FS measurements in the era of precision cosmology.
An approach to estimate the spatial curvature $Omega_k$ from data independently of dynamical models is suggested, through kinematic parameterizations of the comoving distance ($D_{C}(z)$) with third degree polynomial, of the Hubble parameter ($H(z)$) with a second degree polynomial and of the deceleration parameter ($q(z)$) with first order polynomial. All these parameterizations were done as function of redshift $z$. We used SNe Ia dataset from Pantheon compilation with 1048 distance moduli estimated in the range $0.01<z<2.3$ with systematic and statistical errors and a compilation of 31 $H(z)$ data estimated from cosmic chronometers. The spatial curvature found for $D_C(z)$ parametrization was $Omega_{k}=-0.03^{+0.24+0.56}_{-0.30-0.53}$. The parametrization for deceleration parameter $q(z)$ resulted in $Omega_{k}=-0.08^{+0.21+0.54}_{-0.27-0.45}$. The $H(z)$ parametrization has shown incompatibilities between $H(z)$ and SNe Ia data constraints, so these analyses were not combined. The $D_C(z)$ and $q(z)$ parametrizations are compatible with the spatially flat Universe as predicted by many inflation models and data from CMB. This type of analysis is very appealing as it avoids any bias because it does not depend on assumptions about the matter content of the Universe for estimating $Omega_k$.
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.
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.
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.
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