Testing backreaction effects with observational Hubble parameter data
Abstract
The spatially averaged inhomogeneous Universe includes a kinematical backreaction term \({\mathcal {Q}}_{\mathcal {D}}\) that is relate to the averaged spatial Ricci scalar \({\langle \mathcal {R}} \rangle _{\mathcal {D}}\) in the framework of general relativity. Under the assumption that \({\mathcal {Q}}_{\mathcal {D}}\) and \({\langle \mathcal {R}} \rangle _{\mathcal {D}}\) obey the scaling laws of the volume scale factor \(a_{\mathcal {D}}\), a direct coupling between them with a scaling index n is remarkable. In order to explore the generic properties of a backreaction model for explaining the accelerated expansion of the Universe, we exploit two metrics to describe the late time Universe. Since the standard FLRW metric cannot precisely describe the late time Universe on small scales, the template metric with an evolving curvature parameter \(\kappa _{\mathcal {D}}(t)\) is employed. However, we doubt the validity of the prescription for \(\kappa _{\mathcal {D}}\), which motivates us apply observational Hubble parameter data (OHD) to constrain parameters in dust cosmology. First, for FLRW metric, by getting best-fit constraints of \(\varOmega ^{{\mathcal {D}}_0}_m = 0.25^{+0.03}_{-0.03}\), \(n = 0.02^{+0.69}_{-0.66}\), and \(H_{\mathcal {D}_0} = 70.54^{+4.24}_{-3.97}\ \mathrm{km \ s^{-1} \ Mpc^{-1}}\), the evolutions of parameters are explored. Second, in template metric context, by marginalizing over \(H_{\mathcal {D}_0}\) as a prior of uniform distribution, we obtain the best-fit values of \(n=-1.22^{+0.68}_{-0.41}\) and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.12^{+0.04}_{-0.02}\). Moreover, we utilize three different Gaussian priors of \(H_{\mathcal {D}_0}\), which result in different best-fits of n, but almost the same best-fit value of \({{\varOmega }_{m}^{\mathcal {D}_{0}}}\sim 0.12\). Also, the absolute constraints without marginalization of parameter are obtained: \(n=-1.1^{+0.58}_{-0.50}\) and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.13\pm 0.03\). With these constraints, the evolutions of the effective deceleration parameter \(q^{\mathcal {D}}\) indicate that the backreaction can account for the accelerated expansion of the Universe without involving extra dark energy component in the scaling solution context. Nevertheless, the results also verify that the prescription of \(\kappa _{\mathcal {D}}\) is insufficient and should be improved.
1 Introduction
The universe is homogeneous and isotropic on very large scales. According to Einstein’s general relativity, one can obtain a homogeneous and isotropic solution of Einstein’s field equations, which is called Friedmann–Lemaitre–Robertson–Walker (FLRW) metric. Since on smaller scales, the universe appears to be strongly inhomogeneous and anisotropic, Larena et al. [1] doubt that the FLRW cosmology describes the averaged inhomogeneous universe at all times. They assume that FLRW metric may not hold at late times especially when there are large matter inhomogeneities existed, even though it may be suitable at early times. Therefore they introduce a template metric that is compatible with homogeneity and isotropy on large scales of FLRW cosmology, and also contains structure on small scales. In other words, this metric is built upon weak, instead of strong, cosmological principle.
The observations of type Ia supernovae (SNe Ia) [2, 3] suggest that the universe is in a state of accelerated expansion, which implies that there exists a latent component so called dark energy (DE) with negative pressure that causes the accelerated expansion of the universe. There are many scenarios proposed to account for the observations. The simplest one is the positive cosmological constant in Einstein’s equations, which in common assumption is equivalent to the quantum vacuum. Since the measured cosmological constant is much smaller than the particle physics predicted, some other scenarios are proposed, such as the phenomenological models which explained DE as a late time slow rolling scalar field [4] or the Chaplygin gas [5], and the modified gravity models. Recently, a new scenario [6, 7] is raised to consider DE as a backreaction effect of inhomogeneities on the average expansion of the universe. Here we specifically focus on the backreaction model without involving perturbation theory.
According to Buchert [8], the averaged equations of the averaged spatial Ricci scalar \({\langle \mathcal {R}} \rangle _{\mathcal {D}}\) and the ‘backreaction’ term \({\mathcal {Q}}_{\mathcal {D}}\) can be solved to obtain the exact scaling solutions in which a direct coupling between \({\langle \mathcal {R}} \rangle _{\mathcal {D}}\) and \({\mathcal {Q}}_{\mathcal {D}}\) with a scaling index n is significant. With that solution, a domain-dependent Hubble function (effective volume Hubble parameter) \(H_{\mathcal {D}}\) can be expressed with the scaling index n and the present effective matter density parameter \({{\varOmega }_{m}^{\mathcal {D}_{0}}}\). Also, as mentioned in [1], the pure scaling ansatz is not what we expected in a realistic evolution of backreaction.
So as to explore the generic properties of a backreaction model for explaining the observations of the Universe, we, in this paper, exploit two metrics to describe the late time Universe. Although the template metric proposed by Larena et al. [1] is reasonable, the prescription of the so-called “geometrical instantaneous spatially-constant curvature” \(\kappa _{\mathcal {D}}\) is skeptical, based on the discrepancies between our results and theirs. Comparing the FLRW metric with the smoothed template metric, we use observational Hubble parameter data (OHD) to constrain the scaling index n (corresponding to constant equation of state for morphon field \(w^{\mathcal {D}}_{\Phi }\) [9]) and the present effective matter density parameter \({{\varOmega }_{m}^{\mathcal {D}_{0}}}\) without involving perturbation theory. In the latter case, according to [10], we choose to marginalize over both the top-hat prior of \(H_{{\mathcal {D}}_0}\) with a uniform distribution in the interval [50, 90] and three different Gaussian priors of \(H_{{\mathcal {D}}_0}\), where we also obtain the absolute constraint results without the marginalization of the parameters. Combining both the FLRW geometry and the template metric with the backreaction model, we obtain the fine relation between effective Hubble parameter \(H_{\mathcal {D}}\) and effective scale factor \(a_{\mathcal {D}}\) by utilizing Runge–Kutta method to solve the differential equations of the latter, in order to acquire the link between \(a_{\mathcal {D}}\) and effective redshift \(z_{\mathcal {D}}\). At last, a conflict, as expected, arises. Our results show that it needs a higher instead of lower amount of backreaction to interpret the effective geometry, even though accelerated expansion of \(a_{\mathcal {D}}\) still remains. The power law prescription of \(\kappa _{\mathcal {D}}\) certainly need to be improved, since it only evolves from 0 to \(-1\), which is insufficient. Of course, we should point out that the power law ansatz is not the realistic case and the results are expected to be inaccurate. For simplicity, we only deliberate the situation under the assumption of power-law ansatz here.
The paper is organized as follows. The backreaction context is demonstrated in Sect. 2. In Sect. 3, we introduce the template metric and computation of observables along with the effective Hubble parameter \(H_{\mathcal {D}}\), and demonstrate how to relate effective redshift \({z_{\mathcal {D}}}\) to effective scale factor \(a_{\mathcal {D}}\). We also refer to overall cosmic equation of state \(w^{\mathcal {D}}_\mathrm{eff}\) [9] and how it differs from constant equation of state w. In Sect. 4, according to the effective Hubble parameter, we apply OHD with both the FLRW metric and the template metric, and make use of Metropolis–Hastings algorithm of the Markov-Chain-Monte-Carlo (MCMC) method and mesh-grid method, respectively, to obtain the constraints of the parameters. In former case, we employ the best-fits to illustrate the evolutions of \({q}^{\mathcal {D}}\), \(w^{\mathcal {D}}_\mathrm{eff}\), \(\kappa _{\mathcal {D}}\) and density parameters. In latter case, we test the effective deceleration parameter with the best-fit values. After analysis of the results in Sect. 4, we summarize our conclusion and discussion in Sect. 5.
We use the natural units \(c = 1\) throughout the paper, and assign that Greek indices such as \(\alpha \), \(\mu \) run through \(0\cdots 3\), while Latin indices such as i, j run through \(1\cdots 3\).
2 The backreaction model
3 Effective geometry
3.1 The template metric
3.2 Computation of observables
4 Constraints with OHD
4.1 The flat FLRW model
The current available OHD dataset. The method I is the differential ages method, and II represents the radial Baryon acoustic oscillation (BAO) method. H(z) is in unit of \(\mathrm{km \ s^{-1} \ Mpc^{-1}}\) here
z | H(z) | Method | References |
---|---|---|---|
0.0708 | \(69.0\pm 19.68\) | I | Zhang et al. [14] |
0.09 | \(69.0\pm 12.0\) | I | Jimenez et al. [15] |
0.12 | \(68.6\pm 26.2\) | I | Zhang et al. [14] |
0.17 | \(83.0\pm 8.0\) | I | Simon et al. [16] |
0.179 | \(75.0\pm 4.0\) | I | Moresco et al. [17] |
0.199 | \(75.0\pm 5.0\) | I | Moresco et al. [17] |
0.20 | \(72.9\pm 29.6\) | I | Zhang et al. [14] |
0.240 | \(79.69\pm 2.65\) | II | Gaztañaga et al. [18] |
0.27 | \(77.0\pm 14.0\) | I | Simon et al. [16] |
0.28 | \(88.8\pm 36.6\) | I | Zhang et al. [14] |
0.35 | \(84.4\pm 7.0\) | II | Xu et al. [19] |
0.352 | \(83.0\pm 14.0\) | I | Moresco et al. [17] |
0.3802 | \(83.0\pm 13.5\) | I | Moresco et al. [20] |
0.4 | \(95\pm 17.0\) | I | Simon et al. [16] |
0.4004 | \(77.0\pm 10.2\) | I | Moresco et al. [20] |
0.4247 | \(87.1\pm 11.2\) | I | Moresco et al. [20] |
0.43 | \(86.45\pm 3.68\) | II | Gaztanaga et al. [18] |
0.44 | \(82.6\pm 7.8\) | II | Blake et al. [21] |
0.4497 | \(92.8\pm 12.9\) | I | Moresco et al. [20] |
0.4783 | \(80.9\pm 9.0\) | I | Moresco et al. [20] |
0.48 | \(97.0\pm 62.0\) | I | Stern et al. [22] |
0.57 | \(92.4\pm 4.5\) | II | Samushia et al. [23] |
0.593 | \(104.0\pm 13.0\) | I | Moresco et al. [17] |
0.6 | \(87.9\pm 6.1\) | II | Blake et al. [21] |
0.68 | \(92.0\pm 8.0\) | I | Moresco et al. [17] |
0.73 | \(97.3\pm 7.0\) | II | Blake et al. [21] |
0.781 | \(105.0\pm 12.0\) | I | Moresco et al. [17] |
0.875 | \(125.0\pm 17.0\) | I | Moresco et al. [17] |
0.88 | \(90.0\pm 40.0\) | I | Stern et al. [22] |
0.9 | \(117.0\pm 23.0\) | I | Simon et al. [16] |
1.037 | \(154.0\pm 20.0\) | I | Moresco et al. [17] |
1.3 | \(168.0\pm 17.0\) | I | Simon et al. [16] |
1.363 | \(160.0\pm 33.6\) | I | Moresco [24] |
1.43 | \(177.0\pm 18.0\) | I | Simon et al. [16] |
1.53 | \(140.0\pm 14.0\) | I | Simon et al. [16] |
1.75 | \(202.0\pm 40.0\) | I | Simon et al. [16] |
1.965 | \(186.5\pm 50.4\) | I | Moresco [24] |
2.34 | \(222.0\pm 7.0\) | II | Delubac et al. [25] |
The confidence regions are shown in Fig. 2, where the best fits are \(\varOmega ^{{\mathcal {D}}_0}_m = 0.25^{+0.03}_{-0.03}\), \(n = 0.02^{+0.69}_{-0.66}\), and \(H_{\mathcal {D}_0} = 70.54^{+4.24}_{-3.97}\ \mathrm{km \ s^{-1} \ Mpc^{-1}}\). As opposed to Fig. 2 of [1], the degeneracy direction [26] of the contour (\(n, \varOmega ^{{\mathcal {D}}_0}_m\)) is different in our Fig. 2. In this paper, the best-fit values of \(\varOmega ^{{\mathcal {D}}_0}_m=0.25\) and \(n=0.03\), while in [1], \(\varOmega ^{{\mathcal {D}}_0}_m = 0.26\) and \(n = 0.24\) were given for the flat FLRW model. The best-fits of \(\varOmega ^{{\mathcal {D}}_0}_m\) are almost the same, however, the values of n are variant, the reason of which may be the lack of precision caused by the insufficient amount of OHD. Nevertheless, as for best-fit of \(\varOmega ^{{\mathcal {D}}_0}_m\), in comparison with [27], \(\varOmega _m = 0.263^{+0.042}_{-0.042}(1\sigma \ \mathrm stat)^{+0.032}_{-0.032}(\mathrm sys)\) for a flat \(\Lambda \)CDM model, and with [2], \(\varOmega ^{flat}_m = 0.28^{+0.09}_{-0.08}(1\sigma \ \mathrm stat)^{+0.05}_{-0.04}(\mathrm sys)\) for a flat cosmology, the proportions of matter density are not much of differences.
4.2 The template metric model
Our results show that it demands lower values of \(\varOmega ^{\mathcal {D}_0}_m\) for the models to be compatible with data, and on the contrary with Larena’s conclusion, a larger amount of backreaction is required to account for effective geometry. As mentioned in [1], a DE model in FLRW context with \(n=-1\) is compatible with the data at 1\(\sigma \) for \(\varOmega ^{\mathcal {D}_0}_m \sim 0.1\), and as calculated in [33, 34], the leading perturbative model (\(n=-1\)) is marginally at 1\(\sigma \) for \(\varOmega ^{\mathcal {D}_0}_m \sim 0.3\). As expected, purely perturbative estimate of backreaction could not provide sufficient geometrical effect to account for observations. What is not expected is that the values of \(\varOmega ^{\mathcal {D}_0}_m\) is higher or lower compared to the standard DE models with a FLRW geometry. The following subsection will explore the effective deceleration parameter \(q^{\mathcal {D}}\) evolves over \(a_{\mathcal {D}}\) in many cases, in order to pinpoint the hinge of the issue.
4.3 Testing the effective deceleration parameter
5 Conclusions and discussions
In this paper, we delve the backreaction model of dust cosmology with both FLRW metric and smoothed template metric to constrain parameters with observational Hubble parameter data (OHD), the purpose of which is to explore the generic properties of a backreaction model for explaining the observations of the Universe. Unlike the work [35], first, in the FLRW model, we constrain two of three parameters with MCMC method by marginalizing the likelihood function over the rest one parameter, and obtain the best-fits: \(\varOmega ^{{\mathcal {D}}_0}_m = 0.25^{+0.03}_{-0.03}\), \(n = 0.02^{+0.69}_{-0.66}\), and \(H_{\mathcal {D}_0} = 70.54^{+4.24}_{-3.97}\ \mathrm km \ s^{-1} \ Mpc^{-1}\). We employ these best-fit values of n and \(\varOmega ^{{\mathcal {D}}_0}_m\) to study the evolutions of \(q^{\mathcal {D}}\), \(w^{\mathcal {D}}_\mathrm{eff}\), \(\kappa _{\mathcal {D}}\), and effective density parameters. The results compared with other models are slightly biased, which is natural as for the inconsistency between FLRW geometry and averaged model. Second, with template metric and the specific method for computing the observables along null geodesic, we choose a top-hat prior, i.e., uniform distribution of \(H_{{\mathcal {D}}_0}\) to be marginalized, in order to attain the posterior PDF of parameters. By making use of classical mesh-grid method, we plot the likelihood contour in the subspace of \((n,\varOmega _m^{\mathcal {D}_0})\), and obtain the best-fit values, which are \(n=-1.22^{+0.68}_{-0.41}\) and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.12^{+0.04}_{-0.02}\). The value of \({{\varOmega }_{m}^{\mathcal {D}_{0}}}\) is considerably small in comparison with the one of Larena et al. [1].
Our results show that it demands lower values of \(\varOmega ^{\mathcal {D}_0}_m\) for the models to be compatible with data, which means that on the contrary with Larena’s conclusion, a larger amount of backreaction is required to account for effective geometry. The reasons for this discrepancy may be the wrong prescription of \(\kappa _{\mathcal {D}}\), the chosen prior of \(H_{{\mathcal {D}}_0}\), or the lack of amount for OHD. To test the probability of the second reason, we select three different Gaussian prior distributions of \(H_{{\mathcal {D}}_0}\), where the three sets of best-fits are: \(n=-\,0.88^{+0.26}_{-0.23}\), and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.12^{+0.03}_{-0.03}\), for \(H_{{\mathcal {D}}_0}=69.32\pm 0.80\ \mathrm km \ s^{-1} \ Mpc^{-1}\) [30]; \(n=-1.04^{+0.27}_{-0.31}\), and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.13^{+0.02}_{-0.03}\) with \(H_{{\mathcal {D}}_0}=67.3\pm 1.2\ \mathrm km \ s^{-1} \ Mpc^{-1}\) [31]; \(n=-\,0.58^{+0.36}_{-0.34}\), and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.12^{+0.02}_{-0.03}\), related to \(H_{{\mathcal {D}}_0}=73.24\pm 1.74\ \mathrm km \ s^{-1} \ Mpc^{-1}\) [32]. As a result, we find out that although three Gaussian priors lead to different best-fit values of n, the best-fits of \({{\varOmega }_{m}^{\mathcal {D}_{0}}}\) are still compatible with the result of top-hat prior. In addition, we also constrain the parameters without marginalization of any parameter, and obtain the best-fit values: (\(n=-1.2^{+0.61}_{-0.58}, {{H}_{\mathcal {D}_{0}}}=66^{+5.3}_{-4.2}\ \mathrm km\ s^{-1}\ Mpc^{-1}\)), (\({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.13\pm 0.03, {{H}_{\mathcal {D}_{0}}}=67^{+5.1}_{-4.4}\ \mathrm km\ s^{-1}\ Mpc^{-1}\)), and (\(n=-1.1^{+0.58}_{-0.50},{{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.13\pm 0.03\)). The best-fits results, \(n=-1.1\) and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.13\), are consistent with both the ones with the Gaussian prior of \(H_{{\mathcal {D}}_0}=67.3\pm 1.2\ \mathrm km \ s^{-1} \ Mpc^{-1}\) and the ones with top-hat prior of \(H_{{\mathcal {D}}_0}\), and also in contrary with the absolute constraint results of [1], i.e., \(n=0.12\) and \({{\varOmega }_{m}^{\mathcal {D}_{0}}}=0.38\). Therefore, the prior issue can be excluded. Since the same set of data shared by FLRW case result in a reasonable conclusion, the lack of amount for OHD can also be neglected.
Finally, we believe that the prescription of \(\kappa _{\mathcal {D}}\) should be modified, or some other scenarios should be considered. On the one hand, we can modify the prescription into the forms of Eqs. (38) and (39), which are not different from each other in this context but are distinct beyond the scaling solutions. On the other hand, we can add an appropriate positive constant to the expression of \(\kappa _{\mathcal {D}}\) for complimenting the problem of not including positive possibility.
In order to further pinpoint the hinge of the issue, we explore the evolutions of \(q^{\mathcal {D}}\) over effective scale factor \(a_{\mathcal {D}}\) with best-fit values of both us and Larena et al. It turns out that both results are similar in the tendency of the evolutions. The only differences lie in the different turning points for the slow evolutions and the present values of \(q^{\mathcal {D}_0}\). The larger n becomes, the earlier for the evolutionary curves change from fast to slow. In other words, despite of the constraints of the effective parameters, there are not much differences, which leaves both the constraints for mutual contradiction. It just proves our point that we must remain skeptical on the prescription of \(\kappa _{\mathcal {D}}\) and consider other options as mentioned above.
Notes
Acknowledgements
This work was supported by the National Science Foundation of China (Grants nos. 11573006, 11528306), National Key R&D Program of China (2017YFA0402600), the Fundamental Research Funds for the Central Universities and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase).
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