Testing backreaction effects with observational Hubble parameter data

Testing backreaction effects with observational Hubble parameter data

Shu-Lei Cao\thanksrefaddr1 Department of Astronomy, Beijing Normal University, Beijing, 100875, China Tsung-Dao Lee Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China School of Information Management, School of Physics and Electric Information, Shandong Provincial Key Laboratory of Biophysics, Dezhou University, Dezhou 253023,China    Huan-Yu Teng\thanksrefaddr1 Department of Astronomy, Beijing Normal University, Beijing, 100875, China Tsung-Dao Lee Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China School of Information Management, School of Physics and Electric Information, Shandong Provincial Key Laboratory of Biophysics, Dezhou University, Dezhou 253023,China    Hao-Ran Yu\thanksrefaddr2 Department of Astronomy, Beijing Normal University, Beijing, 100875, China Tsung-Dao Lee Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China School of Information Management, School of Physics and Electric Information, Shandong Provincial Key Laboratory of Biophysics, Dezhou University, Dezhou 253023,China    Hao-Yi Wan\thanksrefaddr1,addr3 Department of Astronomy, Beijing Normal University, Beijing, 100875, China Tsung-Dao Lee Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China School of Information Management, School of Physics and Electric Information, Shandong Provincial Key Laboratory of Biophysics, Dezhou University, Dezhou 253023,China    Tong-Jie Zhang\thanksrefe1,addr1,addr4 Department of Astronomy, Beijing Normal University, Beijing, 100875, China Tsung-Dao Lee Institute, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China School of Information Management, School of Physics and Electric Information, Shandong Provincial Key Laboratory of Biophysics, Dezhou University, Dezhou 253023,China
Received: date / Accepted: date
Abstract

The spatially averaged inhomogeneous Universe includes a kinematical backreaction term that is relate to the averaged spatial Ricci scalar in the framework of general relativity. Under the assumption that and obey the scaling laws of the volume scale factor , a direct coupling between them with a scaling index is remarkable. In order to explore the generic properties of a backreaction model to explain the observations 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 is employed. However, we doubt that the evolving curvature parameter also obeys the scaling law, thus we make use of observational Hubble parameter data (OHD) to constrain parameters in dust cosmology to testify it. First, in FLRW model, after getting best-fit constraints of , , and , evolutions of parameters are studied. Second, in template metric context, by marginalizing over as a prior of uniform distribution, we obtain the best-fit values as and . Moreover, we utilize three different Gaussian priors of , which result in different best-fits of , but almost the same best-fit value of . With these constraints, evolutions of the effective deceleration parameter indicate that the backreaction can account for the accelerated expansion of the Universe without involving extra dark energy component in the scaling solution context. However, the results also imply that the prescription of the geometrical instantaneous spatially-constant curvature of the template metric is insufficient and should be improved.

journal: Eur. Phys. J. C

\thankstext

e1e-mail: tjzhang@bnu.edu.cn

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. Larena2009 () 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 structuring 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) Perlmutter1999 (); Riess1998 () suggest that the universe is in a state of accelerated expansion, which implies that there exists a latent component so called Dark Energy with quality of negative pressure that operates 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 Dark Energy as a late time slow rolling scalar field Copeland2006 () or the Chaplygin gas Gorini2009 (), and the modified gravity models. Recently, a new scenario Rasanen2004 (); Kolb2006 () is raised to consider dark energy 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 Buchert2000 (), averaged equations of the averaged spatial Ricci scalar and the ‘backreaction’ term can be solved to obtain the exact scaling solutions in which a direct coupling between and with a scaling index is significant. With that solution, a domain-dependent Hubble function (effective volume Hubble parameter) can be expressed with the scaling index and the present effective matter density parameter . Also, as mentioned in Larena2009 (), the pure scaling ansatz is not what we expected in a realistic evolution of backreaction.

In order to explore the generic properties of a backreaction model to explain 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. Larena2009 () is reasonable, the prescription of the so-called “geometrical instantaneous spatially-constant curvature" 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 (corresponding to constant equation of state for morphon field Buchert2006 ()) and the present effective matter density parameter without involving perturbation theory. In the latter case, according to Ma2011 (), we choose to marginalize over both the top-hat prior of with a uniform distribution in the interval [50, 90] and three different Gaussian priors of . Combining both the FLRW geometry and the template metric with the backreaction model, we obtain the fine relation between effective Hubble parameter and effective scale factor , and utilize Runge-Kutta method to solve the differential equations of latter, in order to acquire the link between and effective redshift . 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 still remains. The power law prescription of 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.

The paper is organized as follows. The backreaction context is demonstrated in Section 2. In Section 3, we introduce the template metric and computation of observables along with the effective Hubble parameter , and demonstrate how to relate effective redshift to effective scale factor . We also refer to overall cosmic equation of state Buchert2006 () and how it differs from constant equation of state . In Section 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 , , and density parameters. In latter case, we test the effective deceleration parameter with the best-fit values. After analysis of the results in Section 4, we summarize our conclusion and discussion in Section 5.

We apply the natural units everywhere, and assign that Greek indices such as , run through 0…3, while Latin indices such as run through 1…3.

2 The bacreaction model

Buchert Buchert2000 () introduced a model of dust cosmologies, which leads to two averaged equations that we need, the averaged Raychaudhuri equation

(1)

and the averaged Hamiltonian constraint

(2)

where , , and represent averaged matter density in the domain , gravitational constant, and cosmological constant, respectively. The over-dot represents partial derivative as to proper time here after. A effective scale factor is introduced via volume (normalized by the volume of the initial domain ),

(3)

The averaged spatial Ricci scalar and the ‘backreaction’ are domain-dependent constants, which are time-dependent functions. The ‘backreaction’ term is expressed as

(4)

with two scalar invariants

(5)

and

(6)

where is the expansion tensor, with the trace-free symmetric shear tensor , the rate of shear and the expansion rate . Substituting Eq. 1 into Eq. 2, one can get

(7)

where the dot over the parentheses represents partial derivative over time , and the solutions are

(8)

where and are real numbers. There are two types of solutions to be considered as mentioned in Buchert2000 (). The first type is that when and , the solution is corresponding to a quasi-FLRW universe at late time, which means backreaction is negligible. The second type is a direct coupling between and , which states that when , the exact relations between them are

(9)

and

(10)

A domain-dependent Hubble function is defined to be , and dimensionless (‘effective’) averaged cosmological parameters are also given by

(11)
(12)
(13)

and

(14)

Thus, according to Eq. 2, one can have

(15)

The components that are not included in Friedmann equation read

(16)

If , then is considered to be the Dark Energy contribution, and usually dubbed X-matter. By considering Eq. 9 and Eq. 10, one can get

(17)

Furthermore, one can easily obtain

(18)

where denotes the domain at present time, and from now on.

In comparison with the deceleration parameter in standard cosmology, an effective volume deceleration parameter is interpreted as

(19)

3 Effective geometry

3.1 The template metric

Larena et al. Larena2009 () proposes the template metric (space-time metric) as follows,

(20)

where is introduced as the size of the horizon at present time, so that the coordinate distance is dimensionless, and the domain-dependent effective 3-metric reads

(21)

with solid angle element . Under their assumption, the template 3-metric is identical to the spatial part of a FLRW space-time at any given time, except for the time-dependent scalar curvature. According to their discussion, must be related to , then in analogy with a FLRW metric, the correlation is given by

(22)

Notice that this template metric does not need to be a dust solution of Einstein’s equations, since Einstein’s field equations are satisfied locally for any space-time metric. Nevertheless, this prescription is insufficient, as cannot be positive in this case, which is assertive and skeptical.

3.2 Computation of Observables

The computation of effective distances along the approximate smoothed light cone associated with the travel of light is very different from general that of distances Bonvin2006 (). Firstly, an effective volume redshift is defined as

(23)

where the O and the S represent the evaluation of the quantities at the observer and source, respectively, is the template effective metric (29), is the 4-velocity of the matter content () with respect to comoving reference, and is the wave vector of a light ray that travels from the source S to the observer O (). Normalizing the wave vector to get and defining the scaled vector , one can obtain the following relation

(24)

where obeys the null geodesics equation , which leads to

(25)

As light travels along null geodesic, we have

(26)

which is slightly different from Eq. (30) of Larena2009 (), since what they have chosen is actually . By solving Eq. 26 one can get the coordinate distance , and then substitutes it into Eq. 25 to find the relation between and . Combining equations in Section 3 and Section 2, one can obtain

(27)
(28)
Figure 1: The evolutions of the coordinate distance (panel (a)), the redshift (panel (b)) and the ratio between the FLRW redshift to the effective redshift, , in the averaged model as function of the effective scale factor with , , (dashed line). The flat FLRW model with same parameters are the solid lines shown in panels (a) and (b), respectively. Panel (d) represents for Larena’s best-fit averaged model with , , .

When Larena et al. Larena2009 () substituted their Eq. (39) into Eq. (30), they missed one element and the negative sign, even though they used the right one as our Eq. 28. To check that out, we repeat the process of Larena’s work, and find out a deviation, as shown in Fig. 1. Panel (a) and panel (d) of our Fig. 1 are identical with Larena’s, and although their panels (b) and (c) have same evolutionary trends, our initial point of is around , which is obviously different from theirs around . Since panel (c) and panel (d) are obtained with the same method and just different equations, we remain skeptical about their results.

In dark energy context, a constant equation of state is correlated with the component as However, as for the time-dependent curvature of the backreaction model, due to Buchert2006 (), the overall ‘cosmic equation of state’ is given by

(29)

where , the constant equation of state for the morphon field, represents the effect of the averaged geometrical degrees of freedom.

4 Constraints with OHD

4.1 The flat FLRW model

In flat FLRW model, one can have

(30)

where , and are assumed to be identical to , and , respectively. As a result, Eq. 18 becomes

(31)

where , and the unknown parameters are , , and . We consider the prior distributions of these parameters to be all uniform distributions, and with a range of , , from -3 to 3, 0.0 to 0.7, and 50.0 to 90.0, respectively. The constraints on () can be obtained by minimizing ,

(32)

Here we assume that each measurement in is independent. However, we note that the covariance matrix of data is not necessarily diagonal, as discussed in Yu2013 (), and if not, the case will become complicated and should be treated by means of the method mentioned by Yu2013 (). Despite that, if interested in the constraint of and , one could marginalize to obtain the probability distribution function of and , i.e., the likelihood function is

(33)

where is the prior distribution function for the present effective volume Hubble constant. Table 1 shows all 38 available OHD and reference therein, which includes data obtained by both the differential galactic ages method and the radial Baryon Acoustic Oscillation (BAO) method.

Method Ref.
I Zhang et al. (2014)-Zhang2014 ()
I Jimenez et al. (2003)-Jimenez2003 ()
I Zhang et al. (2014)-Zhang2014 ()
I Simon et al. (2005)-Simon2005 ()
I Moresco et al. (2012)-Moresco2012 ()
I Moresco et al. (2012)-Moresco2012 ()
I Zhang et al. (2014)-Zhang2014 ()
II Gaztaaga et al. (2009)-Gaztanaga2009 ()
I Simon et al. (2005)-Simon2005 ()
I Zhang et al. (2014)-Zhang2014 ()
II Xu et al. (2013)-Xu2013 ()
I Moresco et al. (2012)-Moresco2012 ()
I Moresco et al. (2016)-Moresco2016 ()
I Simon et al. (2005)-Simon2005 ()
I Moresco et al. (2016)-Moresco2016 ()
I Moresco et al. (2016)-Moresco2016 ()
II Gaztanaga et al. (2009)-Gaztanaga2009 ()
II Blake et al. (2012)-Blake2012 ()
I Moresco et al. (2016)-Moresco2016 ()
I Moresco et al. (2016)-Moresco2016 ()
I Stern et al. (2010)-Stern2010 ()
II Samushia et al. (2013)-Samushia2013 ()
I Moresco et al. (2012)-Moresco2012 ()
II Blake et al. (2012)-Blake2012 ()
I Moresco et al. (2012)-Moresco2012 ()
II Blake et al. (2012)-Blake2012 ()
I Moresco et al. (2012)-Moresco2012 ()
I Moresco et al. (2012)-Moresco2012 ()
I Stern et al. (2010)-Stern2010 ()
I Simon et al. (2005)-Simon2005 ()
I Moresco et al. (2012)-Moresco2012 ()
I Simon et al. (2005)-Simon2005 ()
I Moresco (2015)-Moresco2015 ()
I Simon et al. (2005)-Simon2005 ()
I Simon et al. (2005)-Simon2005 ()
I Simon et al. (2005)-Simon2005 ()
I Moresco (2015)-Moresco2015 ()
II Delubac et al. (2015)-Delubac2015 ()
Table 1: The current available OHD dataset. The method I is the differential galactic ages method, and II represents the radial Baryon Acoustic Oscillation (BAO) method. is in units of here.
Figure 2: The 1, 2 and 3 confidence regions of the effective parameters , , and for the backreaction model, along with their own probability density fuction.

The confidence regions are demonstrated in Fig. 2. As shown in the figures, the best fits are , , and . As opposed to the Fig. 2 of Larena2009 (), the direction of the contour () is different in our Fig. 2. In this paper, the best-fit values of and , while in Larena2009 (), they were given by and for the flat FLRW model. The best-fits of are alike, however, the values of have nothing in common. The reason of which could be the lack of precision caused by the insufficient amount of OHD. Nevertheless, as for best-fit of , in comparison with Astier2006 () with for a flat CDM model, and Perlmutter1999 () with for a flat cosmology, the backreaction model along with FLRW metric seems reasonable in this sense.

Figure 3: (a)The evolution of with best-fit values of = 0.03, and = 0.25; (b)The evolution of with the same best-fit values as in (a); (c)The relation between and with the same best-fit values as in (a); (d) The evolution of (solid line) and (dashed line) with the same best fits as in (a).

Evolution of and the dimensionless averaged cosmological parameters are illustrated in Fig. 3 (c) and (d), respectively, with the best fits of = 0.03, and = 0.25. It obviously evolves from 0 to -1, which is biased. Furthermore, we substitute the best-fit values into the effective volume deceleration parameter

(34)

Consequently, we plot this relation into the - figure. Fig. 3 (a) illustrates how the volume deceleration parameter evolves with redshift with best-fit values of and . The transition redshift from a deceleration to accelerated phase is approximately 0.815, and the present value of the volume deceleration parameter is about -0.636. While the transition value in Riess2004 () is constrained to be , and the transition value in Santos2016 () is corresponding to the present value of the deceleration parameter with flat prior on , we find our transition redshift and are both higher than them. Of course, since the flat FLRW relation does not fit averaged model, the result is foreseeable. The overall cosmic equation of state is given by

(35)

With the best-fit values of and , we can also plot the - figure as shown in Fig. 3 (b). The present value is approximately -0.758.

4.2 The template metric model

In the template metric context, as mentioned above, one cannot directly constrain parameters with the expression of Eq. 18. However, by using Runge-Kutta method to solve Eq. 28, Eq. 25 and subsequently Eq. 24 with certain values of and , one can form a one-to-one correspondence of and , and then the constraint on parameters of Eq. 18 with OHD could be performed. Thus, one can rewrite Eq. 18 as follows

(36)

After marginalizing the likelihood function over , one can obtain parameter constraints in (, ) subspace. As Ma et. al. Ma2011 () stated, with top-hat prior of over the interval , the posterior probability density function (PDF) of parameters given the dataset {} by Bayes’ theorem reads

(37)

where

and

where , , and erf represents the error function. In the process, we utilize so called mesh-grid method to scan spots of () subspace with the range of , and to be [-2, 4] and [0.0, 0.7], respectively. Eventually, as shown in Fig. 4, we attain the constraints with , and . Apparently, our results are quite different from Larena’s results with and for averaged model. Since these are all based on the power law ansatz of , it leaves the issue to be the wrong prescription of (power law ansatz), the chosen prior of , or the lack of amount for OHD.

Figure 4: The 1, 2 and 3 confidence regions of the effective parameters , , marginalizing over the interval [50, 90]. Where the best-fits are , and .

To test the probability of the second reason, we select three different Gaussian prior distributions of to evaluate the results. At this circumstance, as described in Ma2011 (), the posterior PDF of parameters becomes

(38)

where

where and denote prior expectation and deviation of , respectively. First, we make use of Bennett2013 () to obtain the constraints. As a result, Fig. 5 illustrates the confidence regions with 1 constraints , and . Second, as shown in Fig. 6, we acquire the constraints and with Planck2014 (). Last, we attain the constraints with Gaussian prior of Riess2016 (), as depicted in Fig. 7, and . As a result, we find out that although three Gaussian priors lead to different best-fit values of , the best-fits of are compatible with the result of top-hat prior. 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. Therefore, the power law ansatz should not be applied on , and some other scenarios should be introduced.

Figure 5: The 1, 2 and 3 confidence regions of the effective parameters , with Gaussian prior of . Here the best-fits are , and .
Figure 6: Same as Fig. 5, but with Gaussian prior of . Here the best-fits are , and .
Figure 7: Same as Fig. 5, but with Gaussian prior of . Here the best-fits are , and .

Our results indicate that it demands lower values of 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. As mentioned in Larena2009 (), a dark energy model in FLRW context with is compatible with the data at 1 for , and as calculated in Li2007 () and Li2008 (), the leading perturbative model () is marginally at 1 for . 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 is higher or lower compared to the standard dark energy models with a FLRW geometry. The following subsection explores the effective deceleration parameter evolves against in many cases, in order to pinpoint the hinge of the issue.

4.3 Testing the effective deceleration parameter

Fig. 8 shows the evolutions of against effective scale factor with our best-fit values. One can gain the information that in each case tends to 0.5 as to be smaller, and they all have sufficient backreaction to meet the observations, at leat in this perspective. This indicate that the observational data do not disfavour the constraints. However, we also illustrate the same evolutions by using the absolute and the marginalized best-fits of Larena et. al., as shown in Fig. 9, and surprisingly find out the similar conclusion. This further favours our doubt about the power law prescription of .

Figure 8: The evolutions of against effective scale factor with best-fit values of (a) = -1.22, = 0.12, (b) = -0.88, = 0.12, (c) = -1.04, = 0.13, and (d) = -0.58, = 0.12, respectively.
Figure 9: Evolutions of against effective scale factor with best-fit values of (a) = 0.12, = 0.38, (b) = 0.5, = 0.397.

5 Conclusions and discussions

In this paper, we combine 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 to explain the observations of the Universe. Unlike Chiesa2014 (), 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 as , , and . At this best-fit model, we employ the best-fit values of and to study the evolutions of , , , and effective density parameters. The results compared with other models are slightly biased, which is normal as for the inconsistency between FLRW geometry and averaged model.

Second, with template metric and the specific method of how to compute the observables along null geodesic, we choose a top-hat prior, i.e., uniform distribution of to be marginalized, in order to attain the posterior PDF of parameters. By making use of classical mesh-grid method, we programme to scan point-by-point in the subspace of to obtain the probability of every point, and subsequently plot the contours. By means of that, we obtain the best-fit values, which are and . The value of is considerably small in comparison with Larena’s. Our results indicate that it demands lower values of 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. Since these are all based on the power law ansatz of , it leaves the issue to be the wrong prescription of (power law ansatz), the chosen prior of , or the lack of amount for OHD. Since the same set of data shared by FLRW case result in a reasonable conclusion, the lack of amount for OHD can be neglected in a way. To test the probability of the second reason, we select three different Gaussian prior distributions of to evaluate the results. Subsequently, we obtain three sets of best-fits: , and , corresponding to Bennett2013 (); , and with Planck2014 (); , and , related to Riess2016 (). As a result, we find out that although three Gaussian priors lead to different best-fit values of , the best-fits of are still compatible with the result of top-hat prior. 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. Eventually, we believe that the power law ansatz should not be applied on , and some other scenarios should be introduced.

In order to further pinpoint the hinge of the issue, we explore the evolutions of against effective scale factor 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. In other words, despite of the constraints of the effective parameters, there are not much differences, which leaves both the constraints less meaningful. That just prove our point that one must remain skeptical on the power law ansatz of and consider other options.

Acknowledgements.
We are grateful to Yu Liu for useful discussion. This work was supported by the National Science Foundation of China (Grants No. 11573006, 11528306), the Ministry of Science and Technology National Basic Science program (project 973) under grant No. 2012CB821804.

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