Testing backreaction effects with type Ia supernova data and observational Hubble parameter data

A large number of observations riess1998 ; perlmutter1999measurements ; spergel2003first ; spergel2007three ; dunkley2009five ; larson2011seven ; hinshaw2013nine , from low redshift to high redshift, suggest that the Universe is in a state of accelerated expansion, which implies that there exists a sector named as dark energy with negative pressure accounting for the invisible fuel that accelerates the expansion rate of the current universe. There are many scenarios have been proposed to account for these observations, the simplest one is the cosmological constant scenario, in which cosmological constant or, equivalently, vacuum energy plays the role of dark energy. However, because of the huge discrepancy between the theoretical expected vacuum energy density and the observed one, other alternative scenarios have been proposed, including scalar field models such as quintessenceCaldwell1998Cosmological , phantomCaldwell1999A , dilatonicPiazza2004Dilatonic , tachyonPadmanabhan2002Accelerated and quintomBo2006Oscillating etc. and modified gravity models such as braneworlds maartens2010brane , scalar-tensor gravity esposito2001scalar , higher-order gravitational theories capozziello2005reconciling ; das2006curvature . Recently, a third alternative has been considered to explain the dark energy phenomenon as backreaction effect(a Large scale effect caused by small scale inhomogeneities of the universe)rasanen2004dark ; kolb2006cosmic .

In order to consider backreaction effect, it is necessary to answer a longstanding question that how to average a general inhomogeneous spacetime. To date the macroscopic gravity(MG) approach zalaletdinov1992averaging ; zalaletdinov1993towards ; mars1997space is probably the most well known attempt at averaging in space-time. Although it provides a prescription for the correlation functions which emerge in an averaging of the Einstein’s field equations, so far it required a number of assumptions about the correlation functions which make the theory less convictive. Therefore, many researchers adopt another averaged approach put forward by Buchert buchert2000average ; buchert2001average , in despite of its foliation dependent nature, such approach is quite simple and hence becomes the most well studied averaged approach. Since the averaged field equations in such approach do not form a closed set, one needs to make some assumptions about the backreaction term appeared in the averaged equations. In buchert2006correspondence , by taking the assumption that the backreaction term ${\cal Q}_{\cal D}$ and the averaged spatial Ricci scalar $\left\langle{\cal R}\right\rangle_{\cal D}$ obey the scaling laws of the volume scale factor $a_{\cal D}$, Buchert proposed a simple backreaction model(we will refer it as scaling backreaction model in the follow). In order to test the ability of such backreaction model to correctly describe observations of the large scale properties of the Universe, Larena et. al. in larena2009testing introduced a template metric with an evolving curvature parameter and confronted the scaling backreaction model with such template metric against supernova data and the position of the Cosmic Microwave Backround (CMB) peaks. Because of doubting the validity of the prescription for such evolving curvature parameter, Cao et. al.cao2018testing fitted the scaling backreaction model using OHD with the FLRW metric and the template metric, and found out that, for the template metric case, their fitting results is in contrary with the constraint results of Larena et al.larena2009testing , and for the FLRW metric case, two constraint results are basically consistent, which, to a certain extent, supporting their guess that the prescription for the geometrical instantaneous spatially-constant curvature $\kappa_{{\cal D}}$ should be modified. However, as was pointed out by Larena et. al. in larena2009testing , the pure scaling ansatz for ${\cal Q}_{\cal D}$ and $\left\langle{\cal R}\right\rangle_{\cal D}$ is not what we expected in a realistic evolution of backreaction, therefore the conclusion that the prescription for $\kappa_{{\cal D}}$ needs to be modified was drawn too early. In order to prove Cao et. al.’s conjecture in a more persuasive way, we describe the backreaction effects with an effective perfect fluid, whose EoS was assumed as $w_{eff}^{{\cal D}}=w_{0}^{{\cal D}}+w_{a}^{{\cal D}}(1-a_{\cal D})$ which having one more parameter than that of the scaling backreaction model. Then we fit such CPL backreaction model using SN Ia data and OHD with the FLRW metric and the template metric, and the constraint results agree with Cao et. al.’s conjecture.

The paper is organized as follows. In Section II, the spatial averaged approach of Buchert is demonstrated with presentation of the averaged equations for the volume scale factor $a_{\cal D}$, and under this theoretical framework, the CPL backreaction model is introduced. In Section III, we introduce the template metric proposed by Larena et. al., which is a necessary tool to test the theoretical predictions with observations, although we doubt the validity of the prescription for $\kappa_{{\cal D}}$. We also show the way to compute observables with such template metric in this section. In Section IV, we apply a likelihood analysis of the CPL backreaction model by confronting it using latest SN Ia data and OHD with the FLRW metric and the template metric. After analysis of the results in Section IV, we summarize our results in the last section.

In buchert2000average , Buchert considers a universe filled with irrotational dust with energy density $\varrho$. By foliating space-time with the use of Arnowitt-Deser-Misner(ADM) procedure and defining an averaging operator that acts on any spatial scalar $\Psi$ function as

$$\left\langle\Psi\right\rangle_{\cal D}:=\frac{1}{V_{{\cal D}}}\int_{{\cal D}}\Psi Jd^{3}X,$$

(1)

where $V_{{\cal D}}:=\int_{{\cal D}}Jd^{3}X$ is the domain’s volume, Buchert obtains two averaged equations here we need, the averaged Raychaudhuri equation

$$3\frac{{\ddot{a}}_{\cal D}}{a_{\cal D}}+4\pi G\left\langle\varrho\right\rangle_{\cal D}={{\cal Q}}_{\cal D},$$

(2)

and the averaged Hamiltonian constraint

$$3\left(\frac{{\dot{a}}_{\cal D}}{a_{\cal D}}\right)^{2}-8\pi G\left\langle\varrho\right\rangle_{\cal D}=-\frac{\left\langle{\cal R}\right\rangle_{\cal D}+{{\cal Q}}_{\cal D}}{2}.$$

(3)

In these two equations, $a_{\cal D}(t)=\left(\frac{V_{\cal D}(t)}{V_{{{\cal D}_{\rm\bf 0}}}}\right)^{1/3}$ is the volume scale factor, where $V_{{{\cal D}_{\rm\bf 0}}}=|{{{\cal D}_{\rm\bf 0}}}|$ denotes the present value of the volume, and ${\cal Q}_{\cal D}$, $\left\langle{\cal R}\right\rangle_{\cal D}$ represent the backreaction term and the averaged spatial Ricci scalar respectively, which are related by the following integrability condition

$$\frac{1}{a_{\cal D}^{6}}\partial_{t}\left(\,{{\cal Q}}_{\cal D}\,a_{\cal D}^{6}\,\right)\;+\;\frac{1}{a_{\cal D}^{2}}\;\partial_{t}\left(\,\left\langle{\cal R}\right\rangle_{\cal D}a_{\cal D}^{2}\,\right)\,=0\;.$$

(4)

Now we can define the effective prefect fluid by

	$\displaystyle\varrho_{\rm eff}^{{\cal D}}:$	$\displaystyle=$	$\displaystyle-\frac{1}{16\pi G}({{\cal Q}}_{\cal D}+\left\langle{\cal R}\right\rangle_{\cal D}),$		(5)
	$\displaystyle p_{\rm eff}^{{\cal D}}:$	$\displaystyle=$	$\displaystyle-\frac{1}{16\pi G}({{\cal Q}}_{\cal D}-\frac{\left\langle{\cal R}\right\rangle_{\cal D}}{3}),$		(6)

the averaged Raychaudhuri equation and the averaged Hamiltonian constraint then can formally be recast into standard Friedmann equations for a total perfect fluid energy momentum tensor

$$3\frac{{\ddot{a}}_{\cal D}}{a_{\cal D}}+4\pi G(\left\langle\varrho\right\rangle_{\cal D}+\varrho_{\rm eff}^{{\cal D}}+3p_{\rm eff}^{{\cal D}})=0,$$

(7)

$$3\left(\frac{{\dot{a}}_{\cal D}}{a_{\cal D}}\right)^{2}=8\pi G(\left\langle\varrho\right\rangle_{\cal D}+\varrho_{\rm eff}^{{\cal D}}),$$

(8)

given the effective energy density and pressure, the effective EoS reads

$$w_{\rm eff}^{{\cal D}}:=\frac{p_{\rm eff}^{{\cal D}}}{\varrho_{\rm eff}^{{\cal D}}}=\frac{{{\cal Q}}_{\cal D}-\frac{\left\langle{\cal R}\right\rangle_{\cal D}}{3}}{{{\cal Q}}_{\cal D}+\left\langle{\cal R}\right\rangle_{\cal D}}.$$

(9)

One can then obtain a specific backreaction model with an extra ansatz about the form of $w_{\rm eff}^{{\cal D}}$, for example, for the scaling backreaction model, we have $w_{\rm eff}^{{\cal D}}=-\frac{n+3}{3}$, in which $n$ is the scaling index. In this paper, in order to model the realistic evolution of backreaction, we assume that the effective EoS follows the CPL form, i.e.

$$w_{\rm eff}^{{\cal D}}=w_{0}^{{\cal D}}+w_{a}^{{\cal D}}(1-a_{\cal D}),$$

(10)

so the Eq. (8) can be rewrite as

$$H_{{\cal D}}^{2}=H_{{{\cal D}_{\rm\bf 0}}}^{2}[\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}a_{{\cal D}}^{-3}+\Omega_{X}^{{{\cal D}_{\rm\bf 0}}}a_{{\cal D}}^{-3(1+w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}e^{-3w_{a}^{{\cal D}}(1-a_{{\cal D}})}],$$

(11)

here $H_{\cal D}:={\dot{a}}_{\cal D}/a_{\cal D}$ denotes the volume Hubble parameter and $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}:=\frac{8\pi G}{3H_{{{\cal D}_{\rm\bf 0}}}^{2}}\left\langle\varrho\right\rangle_{{\cal D}_{\rm\bf 0}}$, $\Omega_{X}^{{{\cal D}_{\rm\bf 0}}}:=\frac{8\pi G}{3H_{{{\cal D}_{\rm\bf 0}}}^{2}}\varrho_{\rm eff}^{{{\cal D}_{\rm\bf 0}}}=1-\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$. Combine the above equations, we have the following formulas for the backreaction term and the averaged spatial Ricci scalar

$${\cal Q}_{\cal D}=-\frac{9}{2}(1-\Omega_{m}^{{{\cal D}_{\rm\bf 0}}})H_{{{\cal D}_{\rm\bf 0}}}^{2}e^{-3w_{a}^{{\cal D}}(1-a_{{\cal D}})}[(\frac{1}{3}+w_{0}^{{\cal D}}+w_{a}^{{\cal D}})a_{{\cal D}}^{-3(1+w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}-w_{a}^{{\cal D}}a_{{\cal D}}^{-2-3(w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}],$$

(12)

$$\left\langle{\cal R}\right\rangle_{\cal D}=-\frac{9}{2}(1-\Omega_{m}^{{{\cal D}_{\rm\bf 0}}})H_{{{\cal D}_{\rm\bf 0}}}^{2}e^{-3w_{a}^{{\cal D}}(1-a_{{\cal D}})}[(1-w_{0}^{{\cal D}}-w_{a}^{{\cal D}})a_{{\cal D}}^{-3(1+w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}+w_{a}^{{\cal D}}a_{{\cal D}}^{-2-3(w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}].$$

(13)

In this section, we perform a likelihood analysis on the parameters of the CPL backreaction model with both the FLRW metric and the template metric using SN Ia data and OHD respectively.

1.SN Ia data

We use the Pantheon samplescolnic2018complete with 1048 data points to fit the model, and its $\chi^{2}$ function is

$$\tilde{\chi}_{SN}^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})=\sum_{i=1}^{1048}\frac{\left(\mu_{obs_{i}}-5\log_{10}\left[H_{{{\cal D}_{\rm\bf 0}}}d_{L}({z_{{\cal D}}}_{i})\right]-\mu_{0}\right)^{2}}{\sigma_{\mu_{i}}^{2}},$$

(23)

where $\mu_{obs_{i}}$ represents the observed distance modulus, $\sigma_{\mu_{i}}$ denotes its statistical uncertainty and $\mu_{0}$ is a free parameter.

We can rewrite the $\chi^{2}$ as

$$\tilde{\chi}_{SN}^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})=R-2\mu_{0}S+\mu_{0}^{2}T,$$

(24)

in which $R$, $S$ and $T$ is defined as

	$\displaystyle R$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{1048}\frac{\left(\mu_{obs_{i}}-5\log_{10}\left[H_{{{\cal D}_{\rm\bf 0}}}d_{L}({z_{{\cal D}}}_{i})\right]\right)^{2}}{\sigma_{\mu_{i}}^{2}},$		(25)
	$\displaystyle S$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{1048}\frac{\left(\mu_{obs_{i}}-5\log_{10}\left[H_{{{\cal D}_{\rm\bf 0}}}d_{L}({z_{{\cal D}}}_{i})\right]\right)}{\sigma_{\mu_{i}}^{2}},$		(26)
	$\displaystyle T$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{1048}\frac{1}{\sigma_{\mu_{i}}^{2}}.$		(27)

Since $\mu_{0}$ is an irrelevant parameter, we can get rid of it by marginalizing it, or we can find the extreme point of the $\chi^{2}$ with respect to the independent variable $\mu_{0}$ and substitute it in Eq. (24). Here we choose the second method, in fact both methods give similar results. Since the extreme point of the $\chi^{2}$ with respect to the independent variable $\mu_{0}$ is $\mu_{0}=S/T$, we have the final formula for the $\chi^{2}$ as follows,

$$\chi_{SN}^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})=R-\frac{S^{2}}{T}.$$

(28)

2.OHD

For the observed Hubble parameter dataset in Table 1, the best-fit values of the parameters $(H_{{{\cal D}_{\rm\bf 0}}},\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})$ can be determined by a likelihood analysis based on the calculation of

$$\chi^{2}_{H}=\sum_{i=1}^{38}\frac{(H_{{{\cal D}_{\rm\bf 0}}}E(z_{{\cal D}_{i}};\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})-H_{i})^{2}}{\sigma_{i}^{2}}.$$

(29)

Here we assume that each Hubble parameter data point is independent. However, the covariance matrix of data is not necessarily diagonal, as discussed in yu2013nonparametric , and if it is not, the problem will become complicated and should be treated by means of the method mentioned by yu2013nonparametric .

Since $\chi_{SN}^{2}$ does not contain $H_{{{\cal D}_{\rm\bf 0}}}$, in order to directly compare the two sets of fitting results, we can marginalize the $H_{{{\cal D}_{\rm\bf 0}}}$ in the $\chi^{2}_{H}$ by using the following formula:

$$\begin{split}P(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}}|\{H_{i}\})&=\int P(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}},H_{{{\cal D}_{\rm\bf 0}}}|\{H_{i}\})dH_{{{\cal D}_{\rm\bf 0}}}\\ &=\int\mathcal{L}(\{H_{i}\}|\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}},H_{{{\cal D}_{\rm\bf 0}}})P(H_{{{\cal D}_{\rm\bf 0}}})dH_{{{\cal D}_{\rm\bf 0}}},\end{split}$$

(30)

where $\mathcal{L}(\{H_{i}\}|\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}},H_{{{\cal D}_{\rm\bf 0}}})=(\prod_{i=1}^{38}\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}})\exp(-\frac{\chi^{2}_{H}}{2})$ is the likelihood function before $H_{{{\cal D}_{\rm\bf 0}}}$ is marginalized, $P(H_{{{\cal D}_{\rm\bf 0}}})=\frac{1}{\sqrt{2\pi\sigma_{H}^{2}}}\exp[-\frac{(H_{{{\cal D}_{\rm\bf 0}}}-\mu_{H})^{2}}{\sigma_{H}^{2}}]$ is the prior of $H_{{{\cal D}_{\rm\bf 0}}}$, where $\mu_{H}=73.8$ and $\sigma_{H}=1.1$ wong2020h0licow . After marginalization, we can get the probability distribution of the three parameters $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$, $w_{0}^{{\cal D}}$ and $w_{a}^{{\cal D}}$ as

$$P(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}}|\{H_{i}\})=\frac{1}{\sqrt{A}}[{\rm erf}(\frac{B}{\sqrt{A}})+1]e^{\frac{B^{2}}{A}},$$

(31)

where

$$A=\frac{1}{2\sigma_{H}^{2}}+\sum_{i=1}^{38}\frac{E^{2}(z_{{{\cal D}}_{i}};\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})}{2\sigma_{i}^{2}},B=\frac{\mu_{H}}{2\sigma_{H}^{2}}+\sum_{i=1}^{38}\frac{E(z_{{{\cal D}}_{i}};\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})H_{i}}{2\sigma_{i}^{2}}.$$

(32)

By which we can obtain the $\chi^{2}$ of OHD after $H_{{{\cal D}_{\rm\bf 0}}}$ is marginalized.

$$\chi_{OHD}^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}})=-2\ln(P(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}},w_{a}^{{\cal D}}|\{H_{i}\}))$$

(33)

Table 3 show the constriant results of the CPL backreaction model with both the FLRW metric and template metric by using SN Ia data and OHD respectively, and Table 3 show the prior of the parameters in the CPL backreaction model with both the FLRW metric and template metric. From the given fitting results one can infer that, for the template metric case, the tensions in parameters $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$, $w_{0}^{{\cal D}}$ and $w_{a}^{{\cal D}}$ between two different data sets are 2.91$\sigma$, 0.05$\sigma$ and 0.49$\sigma$ respectively, and for the FLRW metric case, the corresponding tensions are 1.7$\sigma$, 0.17$\sigma$ and 0.07$\sigma$ respectively. It is obvious that the CPL backreaction model with the FLRW metric has smaller parameter tensions in generally, and these tensions can actually be attributed to the lack of precision caused by the insufficient amount of OHD. The difference in the degree of parameter tensions in two different cases can also be inferred from Figure 1 and Figure 2. In cao2018testing , Cao et. al. show with the scaling backreaction model that, for the template metric case, their fitting results, which is produced by using OHD, is in contrary with the constraint results of Larena et al.larena2009testing , which is produced by using SN Ia data and the position of CMB peaks, and for the FLRW metric case, two constraint results are basically consistent. What our work further show is that even for a more realistic backreaction model like the CPL backreaction model, similar results reproduce. Therefore, we draw the same conclusion that Cao et. al make that the prescription of $\kappa_{{\cal D}}$ should be modified, or some other scenarios should be introduced.

In this paper, in order to further prove the conclusion drawn by Cao et. al. in cao2018testing that the prescription of $\kappa_{{\cal D}}$ needs to be modified, we equivalently regard the backreaction effects as the direct results of an effective perfect fluid whose EoS is parameterized as CPL form by us to model the realistic evolution of backreaction as accurately as possible. As a comparison, we employ two metrics to describe the late time Universe, one of them is the FLRW metric, and the other is the template metric that contain the geometrical instantaneous spatially-constant curvature $\kappa_{{\cal D}}$ whose prescription needs to be tested. We then fit such CPL backreaction model using both SN Ia data and OHD with these two metrics, and from the the constraint results we find that, for the template metric case, the tensions in parameters $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$, $w_{0}^{{\cal D}}$ and $w_{a}^{{\cal D}}$ between two different data sets are 2.91$\sigma$, 0.05$\sigma$ and 0.49$\sigma$ respectively, and for the FLRW metric case, the corresponding tensions are 1.7$\sigma$, 0.17$\sigma$ and 0.07$\sigma$ respectively. Obviously, the CPL backreaction model with the FLRW metric has smaller parameter tensions in generally, and these tensions can actually be attributed to the insufficient amount of OHD, however, for the other case, tensions are too large to be attributed to the insufficient amount of OHD. Therefore, we believe that the prescription of $\kappa_{{\cal D}}$ needs to be modified.

The paper is partially supported by the Natural Science Foundation of China.