Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Age problem in Lemaitre-Tolman-Bondi void models

Xiao-Peng Yana, De-Zi Liub, Hao Wei3'*

a School of Physics, Beijing Institute of Technology, Beijing 100081, China b Department of Astronomy, Peking University, Beijing 100871, China

CrossMark

A R T I C L E I N F 0

Article history:

Received 23 November 2014 Received in revised form 21 January 2015 Accepted 21 January 2015 Available online 23 January 2015 Editor: M. Trodden

A B S T R A C T

As is well known, one can explain the current cosmic acceleration by considering an inhomogeneous and/or anisotropic universe (which violates the cosmological principle), without invoking dark energy or modified gravity. The well-known one of this kind of models is the so-called Lemaitre-Tolman-Bondi (LTB) void model, in which the universe is spherically symmetric and radially inhomogeneous, and we are living in a locally underdense void centered nearby our location. In the present work, we test various LTB void models with some old high redshift objects (OHROs). Obviously, the universe cannot be younger than its constituents. We find that an unusually large r0 (characterizing the size of the void) is required to accommodate these OHROs in LTB void models. There is a serious tension between this unusually large r0 and the much smaller r0 inferred from other observations (e.g. SNIa, CMB and so on). However, if we instead consider the lowest limit 1.7 Gyr for the quasar APM 08279+5255 at redshift z = 3.91, this tension could be greatly alleviated.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

Since the discovery of the current accelerated expansion of the universe [1-6], various models have been proposed to explain this mysterious phenomenon. As is well known, the modern cosmology is based on general relativity and the cosmological principle. The well-known Einstein field equations read

G pv = 8n GTiJ,V,

where G^v and are the Einstein tensor and the stress-energy tensor respectively, and we set the speed of light c = 1 throughout this work. According to the pillars of modern cosmology, these theoretical models can be categorized into the following three major types.

The first one is to modify the right hand side of Einstein field equations. That is, one can introduce an exotic energy component, namely dark energy with negative pressure [7-9], while general relativity still holds. The simplest candidate of dark energy is a tiny cosmological constant [10,11] introduced by Einstein himself in 1917. As is well known, it seriously suffers from the fine-turning problem and the cosmological coincidence problem [11-14]. To alleviate these problems, various dynamical models of dark energy were proposed, such as quintessence [15-17], phantom [18,19],

* Corresponding author.

E-mail addresses: 764644314@qq.com (X.-P. Yan), haowei@bit.edu.cn (H. Wei).

k-essence [20-22], quintom [23], Chaplygin gas [24,25], vector-like dark energy [26-28], holographic dark energy [29], (new) age-graphic dark energy [30-32], hessence [33,34], spinor dark energy [35-37], and so on.

The second one is to modify the left hand side of Einstein field equations, namely to modify general relativity on cosmolog-ical scale. Einstein's general relativity is checked to hold in the range from large scales like the solar system to small scales in the order of millimeter. However, there is no a priori reason to believe that general relativity cannot be modified on cosmological scales. In the literature, various modified gravity theories were proposed to account for the cosmic acceleration, for instance, f (R) theory [38-40], scalar-tensor theory [40,41], Dvali-Gabadadze-Porrati (DGP) model [42-44], Galileon gravity [45-47], Gauss-Bonnet gravity [48,49], f (T) theory [50,51], massive gravity [52-54].

The third one is to give up the cosmological principle, and consider an inhomogeneous and/or anisotropic universe, without invoking dark energy or modified gravity. As a tenet, the cosmo-logical principle is known to be partly satisfied on large scales. However, it has not been proven on cosmic scales > 1 Gpc [55]. Obviously, our local universe is inhomogeneous and anisotropic on small scales. On the other hand, the nearby sample has been examined for evidence of a local "Hubble Bubble" [56]. It is reasonable to imagine that we are living in a locally underdense void. If the cosmological principle is relaxed, it is possible to explain the apparent cosmic acceleration in terms of a peculiar distribution of matter centered upon our location [57-64]. In the

http://dx.doi.org/10.1016/j.physletb.2015.01.029

0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

literature, the cosmological principle has been tested by using e.g. type la supernovae (SNIa) [57,65,135], cosmic microwave background (CMB) [55,66-71,130], time drift of cosmological redshifts [72,129], baryon acoustic oscillations (BAO) [73-75], integrated Sachs-Wolfe effect [76], galaxy surveys [77], kinetic Sunyaev Zel'-dovich effect [78-80,131,134], observational H(z) data [81,82], gamma-ray bursts [83], growth of large-scale structure [132], and so on. lt is found that the violation of cosmological principle can be consistent with most of these observations (in fact few observations slightly favor the violation of cosmological principle). Therefore, it is reasonable to consider an inhomogeneous and/or anisotropic universe. ln the literature, the well-known models violating cosmological principle are the so-called Lemaitre-Tolman-Bondi (LTB) void models [84-86]. In LTB void models, the universe is spherically symmetric and radially inhomogeneous, and we are living in a locally underdense void centered nearby our location. The Hubble diagram inferred from lines-of-sight originating at the center of the void might be misinterpreted to indicate cosmic acceleration. In fact, LTB void models can be consistent with (even slightly favored by) the observations mentioned above.

In the present work, we try to test LTB void models with the age of the universe. Obviously, the universe cannot be younger than its constituents. ln history, the age problem played an important role in cosmology for many times. However, we should clarify the two meanings of age problem. The first meaning is that the total age of the universe (namely the age measured at present day, or, redshift z = 0) cannot be smaller than the age of the oldest known objects (e.g. globular clusters, galaxies, quasars) in our universe. Historically, the matter-dominated FriedmannRobertson-Walker (FRW) model without cosmological constant can be ruled out [87] because its total age is smaller than the ages inferred from old globular clusters, unless the Hubble constant is extremely low or the universe is extremely open. ln the literature, one might consider a variant of this type of age problem. For instance, the authors of [88,89] reconstructed LTB model from ^CDM model by requiring they share the same expansion history (luminosity distance, light-cone mass density, angular diameter distance dA(z), Hubble parameter H(z)), and found that the total age of the universe inferred from LTB model is much smaller than the one inferred from ^CDM model (t^CDM — fLTB ~ 2.4 Gyr). However, strictly speaking, this variant of age problem is not the real age problem, since LTB model is the reconstructed one, and the total age of the universe is not compared with the real age of old objects (e.g. globular clusters, galaxies, quasars). So, we do not consider this kind of age problem in the present work.

lnstead, here we consider the second meaning of age problem, namely the age of the universe at any high redshift z > 0 (rather than the total age at present day, z = 0) cannot be younger than its constituents at the same redshift. Obviously, in this case the age problem becomes more serious than the first one. There are some old high redshift objects (OHROs) considered extensively in the literature, for instance, the 3.5 Gyr old galaxy LBDS 53W091 at redshift z = 1.55 [90,91], the 4.0 Gyr old galaxy LBDS 53W069 at redshift z = 1.43 [92]. In addition, the old quasar APM 08279+5255 at redshift z = 3.91 [93,94] is also used extensively. Its age is estimated to be 2.0-3.0 Gyr [93,94]. ln [95], by using a different method, its age is reevaluated to be 2.1 Gyr. To assure the robustness of our analysis, we use the most conservative lower age estimate 2.0 Gyr for the old quasar APM 08279+5255 at redshift z = 3.91 throughout the present work. In the literature, these three OHROs have been extensively used to test various dark energy models (see e.g. [87,95-102]) and modified gravity models (see e.g. [103-107]). In the present work, we will use them to test various LTB void models.

The rest of this paper is organized as follows. In Section 2, we briefly review the main points of LTB model. In Section 3, we test various LTB void models with OHROs. ln Section 4, we discuss the possibility to alleviate the age problem. In Section 5, we give the brief conclusion and discussion.

2. The LTB model

In the LTB void model, the universe is spherically symmetric and radially inhomogeneous, and we are living in a locally under-dense void centered nearby our location. The dynamic of a spherically symmetric dust universe is described by the LTB solution to Einstein field equations. It was firstly proposed by Lemaitre [84], then was further discussed by Tolman [85] and Bondi [86]. The LTB metric, in comoving coordinates (r, 6, 0) and synchronous time t, is given by [84-86] (see also e.g. [81,82,110,133])

ds2 = —dt2 +

A 2(r, t )

dr2 + A (r, t) dQ2,

1 — k(r)

where dQ2 = d62 + sin2 6 d02; a prime denotes a derivative with respect to r, and k(r) is an arbitrary function of r, playing the role of spatial curvature. Note that it reduces to the well-known FRW metric if A(r, t) = a(t)r and k(r) = kr2. The stress-energy tensor of the mass source is given by

Tv =-pM (r, t)5ov5° ,

where pM is the energy density of dust matter. The Einstein field equations read [82,108-110,115]

H 2 + 2H x H, +

k(r) k'(r)

+ AA7 = 8n G Pm ,

A2 + 2 AA + k(r) = 0,

where a dot denotes a derivative with respect to t, and

A (r, t)

Hx(r, t) =

H,,(r, t) =

A(r, t)' A'(r, t)

A'(r, t)'

are the expansion rates at the transverse and longitudinal directions, respectively. Integrating Eq. (4), we obtain [108-110, 113-115]

A2 (r, t) = ^ — k(r), (7)

( , ) A(r, t) ( ), y '

where M(r) is an arbitrary function (the factor 2 is introduced just for convenience; one should be aware of the different symbol conventions in the relevant references). If M(r) and k(r) are given, one can obtain A(r, t) by directly solving Eq. (7). For convenience, we instead try to find the parametric solutions for it. Following e.g. [111-114], we recast Eq. (7) as

A2 (r, t) 2M(r)

( , ) =-k+ .s(/,s,, (8)

\k(r)\

A(r, t)\k(r)\

to normalize k = k(r)/\k(r)\ = +1, —1, 0 for k(r) > 0, k(r) < 0, k(r) = 0, respectively. The solutions of Eq. (8) can be written implicitly in terms of an auxiliary variable n as [111]

A(r, t) =

M (r) ds(n)

with t — tB (r) =

M (r)s(n)

\k(r)\ dn ' ........"v/ \k(r)\3/2

where tB (r) is actually a "constant" of integration. Therefore, Eq. (8) becomes an ordinary differential equation of the function s(n),

"d2 s(n)' 2 —k 'ds(n)'

dn2 dn

ds(n) dn

whose solutions are given by [111]

s(n) =

n — sin n for k = +1, sinh n — n for fc = —1, n3/6 for fc = 0.

Substituting Eq. (11) into Eq. (9), the parametric solutions of Eq. (7) read (see e.g. [82,110,112-114])

A(r, t) = T(-L(1 - cosh n),

t — tB (r) =

—k(r)]3/2

(sinh n — n) for k(r) < 0,

A(r, t) = (1 — cos n),

t — tB (r) =

A(r, t) =

[k(r)]3/2 9 M (r)'

(n — sin n) for k(r) > 0,

[t — tB (r)]2/3 for k(r) = 0,

where tB (r) is an arbitrary function of r, usually interpreted as the "bang time" due to singularity behavior at t = tB. Substituting Eq. (7) into Eq. (3), we have [108-110,113-115]

2M'(r) A' A2

= 8n G pm .

Considering Eq. (7) at the present day (t = t0), it can be recast as 2M(r) k(r)

H 2w(r) A0(r)3 H 2w(r) A0(r)

= Vm (r) + Vk (r),

where the subscript "0" indicates the present value of corresponding quantity, i.e., A0(r) = A(r, t = t0), Hx0(r) = Hx(r, t = t0). Therefore, we can parameterize the functions M(r) and k(r) as [108-110]

2M(r) = H2±0(r)^M (r) A30(r), (17)

-k(r) = H2w(r)QK (r) A2(r), (18)

where (r) = 1 — QM(r). Noting Eq. (15), it is easy to see that QM and QK defined in Eqs. (17) and (18) can reduce to the present fractional densities of FRW cosmology if A(r, t) = a(t)r and k(r) = kr2 while Hx0 and QM are spatially homogeneous. So, the above parameterizations are justified. Substituting Eqs. (17), (18) into Eqs. (12)-(14), we obtain the total cosmic age as a function of r [82], namely

t0 — tB (r) =

f (vm ) H±0(r),

in which the function F(x) is defined by

F (x) =

— VX—1+X sin— 1 y

(x—1)3/2

V1—x—x sinh— 1

(1—x)3/2

for x> 1, for x = 1,

for x < 1.

Furthermore, to compare our theoretical models with observations, we need to associate the coordinates with redshift z. For an observer located at the center r = 0, by symmetry, incoming light travels along radial null geodesics, ds2 = dQ2 = 0, and hence we have [110]

A'(r, t)

where the minus sign is due to dt/dr < 0, namely time decreases when going away. Together with the redshift equation [108-110, 115]

d ln(1 + z) A'(r, t)

dr V1 — fc(r) we can write a parametric set of differential equations [110]

d ln(1 + z) dr

A'(r, t) A'(r, t), V1 — k(r)

d ln(1 + z) A'(r, t)

Once the functions QM (r) and Hx0(r) characterizing LTB model are given, substituting Eqs. (17) and (18) into Eq. (7), the scale function A(r, t) can be found by solving the resulting differential equation. Then, one can obtain t(z) and r(z) as functions of red-shift z from Eqs. (23) and (24) with the initial conditions r(z = 0) = 0 and t(z = 0) = t0. Note that in solving Eq. (7), the parametric solutions given in Eqs. (12)-(14) are useful. One can do this numerically using a modified version of the code easyLTB [110] (see e.g. [82] for a brief technical illustration; however, one should be careful of the typos in [82], and the different symbol conventions in the relevant references, e.g. [82,108-110,112-115], as well as the difference between the relevant references and the code easyLTB [110]). It is worth noting that the present scale function A0(r) = A(r, t = t0) of LTB model can be chosen to be any smooth and invertible positive function. Following [82,108-110], we choose the conventional gauge A0(r) = A(r, t = t0) = r, which actually corresponds to set the present scale factor a0 = a(t = t0) = 1 in FRW cosmology.

3. Testing various LTB void models with OHROs

In the LTB void models, we are living at a special space point, which is close to the center of a large local underdense region of the universe [84-86,115-117]. At very large distances from the observer, the inhomogeneous LTB region goes to an external FRW space. Obviously, it violates the Copernican principle that states we do not occupy any special place in the universe. In the literature, it is found that the LTB void models can be consistent with (even slightly favored by) various observations mentioned in Section 1. Here, we try to test various LTB void models with three OHROs mentioned in Section 1, namely the 3.5 Gyr old galaxy LBDS 53W091 at redshift z = 1.55 [90,91], the 4.0 Gyr old galaxy LBDS 53W069 at redshift z = 1.43 [92], and the 2.0 Gyr old quasar APM 08279+5255 at redshift z = 3.91 [93,94].

3.1. The Gaussian model

The gradient in the bang time tB (r) corresponds to a currently non-vanishing decaying mode [118,119], which might imply an in-homogeneous early universe that violates inflation, and lead to inhomogeneities in the galaxy formation time. To be simple, one might assume that the big bang is spatially homogeneous, namely tB is a constant. Following e.g. [82,110], we can set tB = 0 for convenience. In this case, Eq. (19) becomes

H±0(r) = H0F (Vm ),

where the function F is given in Eq. (20), and H0 = 1/t0.

So, in this case, one only needs to specify QM (r), and then Hx0(r) can be found from Eq. (25).

0.7 0.6

0.8 0.6

0.7 0.4

Fig. 1. The 3D plot of the allowed parameter space of the Gaussian model for (a) OHRO at z = 1.43, (b) OHRO at z = 1.55, (c) OHRO at z = 3.91, respectively. The blue contours indicate the model parameters making the theoretical cosmic age equal to the age of OHRO at the same redshift. The allowed parameter spaces are the upper regions of these contours. Note that r0 is in units of Gpc. See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

At first, we consider the simplest Gaussian LTB void model [65, 82], in which the matter density function QM(r) has a Gaussian profile, namely

Qm (r) = 1 + (Qin — 1) exp( — —2

where Qin is the matter density at the center of the void, and r0 describes the size of the void. ln this work, we only consider the case of Qin < 1. From Eq. (25), it is easy to obtain

H w(r) = H 0-

-JQk (r) — Qm (r) sinh

Qk (r) Qm (r)

[Qk (r)]3/2

where QK (r) = 1 — QM (r), and H0 actually plays the role of Hubble constant. So, there are three free model parameters, namely Qin, r0, and h (which is the Hubble constant H0 in units of 100 km/s/Mpc).

To test the Gaussian model with the three OHROs at red-shift z = 1.43, 1.55, 3.91, we scan a fairly wide parameter space 0.01 < Qin < 0.99, 1.0 Gpc < r0 < 501.0 Gpc, and 0.4 < h < 1.0. At every point, we numerically calculate the theoretical cosmic age at redshift z = 1.43, 1.55, 3.91 for the Gaussian model with the corresponding parameters Qin, r0, and h. Then, we obtain three

contours which indicate the model parameters making the theoretical cosmic age equal to the age of OHRO at the same redshift z = 1.43, 1.55, 3.91. We present them in Fig. 1. Only the parameters corresponding to a theoretical cosmic age larger than (or equal to) the age of OHRO at the same redshift are allowed. In fact, the allowed parameter spaces are the upper regions of the contours shown in Fig. 1. From Fig. 1, it is easy to see that a large r0 is required to accommodate the three OHROs. ln particular, from the panel (c) of Fig. 1, we find that r0 > 10 Gpc is required to accommodate OHRO at redshift z = 3.91. To see this clearer, in Fig. 2 we show the 2D slices of the allowed parameter space with fixed h = 0.738, 0.673, 0.623. Note that h = 0.738 is the best-fit value of the Hubble constant from SHOES SNla project [120]; h = 0.673 is the one from Planck CMB data [121]. On the other hand, Sandage et al. advocated a lower Hubble constant from HST SNla, and the best-fit value of their final result is h = 0.623 [122]. The allowed parameter spaces are the upper regions of the contour lines in Fig. 2. Note that the absence of green-solid contour line in Fig. 2 means that the entire plotted parameter space of the Gaussian model with a fixed h = 0.623 is allowed for OHRO at z = 1.43. This fact can be seen clearly from Fig. 3, in which we scan the parameter space 0.01 < Qin < 0.99, 0.1 Gpc < r0 < 25 Gpc with fixed h = 0.738, 0.673, 0.623, and plot cosmic age as function of

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

z = 3.91. The unusually large r0 brings a serious crisis to the Gaussian LTB void model. As is well known, the Hubble radius (Hubble horizon) H~ 3.0h—1 Gpc [123] characterizes the size of our observable universe. The size of the void should be much larger than the size of our observable universe to accommodate OHROs (see the discussion in Section 5 however). On the other hand, there is a serious tension between this unusually large r0 and the much lower r0 of order 1.0 Gpc inferred from other observations (e.g. SNla, CMB and so on) mentioned in Section 1. If the Gaussian LTB void model can be consistent with other observations (e.g. SNla, CMB and so on), it cannot accommodate OHROs. Of course, it is known that the lower Hubble constant, the larger cosmic age is. However, as shown in the panel (c) of Fig. 1, r0 > 10 Gpc is still required even for a very low h = 0.4. So, this serious crisis cannot be alleviated with a lower Hubble constant.

3.2. The CGBH model

Fig. 2. The 2D plot of the allowed parameter space of the Gaussian model with fixed h = 0.738 (black contour lines), 0.673 (red contour lines), 0.623 (green contour lines) for OHROs at z= 1.43 (solid contour lines), z= 1.55 (dashed contour lines), z = 3.91 (dash-dotted contour lines), respectively. The contour lines indicate the model parameters making the theoretical cosmic age equal to the age of OHRO at the same redshift. The allowed parameter spaces are the upper regions of these contour lines. Note that r0 is in units of Gpc. See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

/i=0.738

^m /1=0.673 ^m /1=0.623

r0: 0.1-25.0 a„: 0.01-0.99

0 1 2 3 4 5

Redshift

Fig. 3. Cosmic age as function of redshift z for the parameter space 0.01 < Qin < 0.99, 0.1 Gpc < ro < 25 Gpc of the Gaussian model with fixed h = 0.738 (cyan), 0.673 (red), 0.623 (green). The three OHROs at redshift z = 1.43, 1.55, 3.91 are also indicated by black stars. See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

redshift z. It is clear that OHRO at z = 1.43 is below the lower boundary of the green region, and hence all the parameter space is allowed in this case. Similarly, the absence of red-dashed and green-dashed contour lines in Fig. 2 means that the entire plotted parameter space of the Gaussian model with fixed h = 0.673, 0.623 is allowed for OHRO at z = 1.55, and it can also be seen clearly from Fig. 3, since OHRO at z = 1.55 is below the lower boundaries of both the red and green regions. On the other hand, from Fig. 3, one can also see that at least r0 > 25 Gpc is required to accommodate OHRO at z = 3.91, since it is above all the upper boundaries of the cyan, red, and green regions. ln fact, from the small panel in Fig. 2, at least r0 > 30 Gpc is required to accommodate OHRO at

Next, we consider a simplified version of the so-called Garcia-Bellido-Haugb0lle (GBH) model [110], namely the constrained GBH (CGBH) model [110] (see also e.g. [82]). In CGBH model, one also assumes that the big bang is spatially homogeneous, namely tB is a constant which can be set to zero. So, Eq. (25) is valid in the CGBH model. The matter density function QM(r) is given by [110] (see also e.g. [82])

Qm (r) = 1 + (Qin — 1)

1 — tanh [(r — r0)/2Ar]l 1 + tanh(r0/2Ar) y

where Qin is the matter density at the center of the void; r0 describes the size of the void; Ar characterizes the transition to uniformity. ln this work, we only consider the case of Qin < 1. From Eq. (25), we get

H±0(r) = H0

VQk (r) — Qm (r) sinh 1J [Qk (r)]3/2

Qk (r)

Qm (r)

where QK(r) = 1 — QM(r), and H0 actually plays the role of Hubble constant. So, there are four free model parameters, namely Qin, r0, h (which is the Hubble constant H0 in units of 100 km/s/Mpc), and Sr = Ar/r0 (which is equivalent to Ar in fact).

Similar to the previous subsection, we firstly scan the full parameter space to test this model with OHROs. However, since there are four free parameters in the CGBH model, it is difficult to plot a 4D parameter space. lnstead, we consider the 3D plot of the allowed parameter space of the CGBH model with a fixed h = 0.673 coming from Planck CMB data [121], and we present it in Fig. 4. Note that the wide parameter ranges we scanned are 0.01 < Qin < 0.99, 1.0 Gpc < r0 < 501.0 Gpc, and 0.1 < Sr < 0.9. The absence of plot for OHRO at redshift z = 1.55 in Fig. 4 means that the entire plotted parameter space of the CGBH model with a fixed h = 0.673 is allowed for OHRO at z = 1.55. This can be seen clearly from Fig. 5 in which OHRO at z = 1.55 is below the lower boundary of the red region, and hence the entire parameter space is allowed in this case. Note that from the panel (b) in Fig. 4, a large r0 > 10 Gpc is required to accommodate OHRO at z = 3.91. Also, in Fig. 6 we show the 2D slices of the allowed parameter space with fixed h = 0.623, 0.673, 0.738, and Sr = 0.40, 0.64, 0.80. Note that Sr = 0.64 is the best-fit value from SNla, CMB and BAO [110], and Sr = 0.40 and 0.80 are close to the edges of its 2a confidence region. The absence of the contour lines for OHROs at z = 1.43 and 1.55 in the left panel of Fig. 6 means that the entire plotted parameter space of the CGBH model with a fixed h = 0.623 is allowed for these two OHROs. And the absence of the contour lines for OHRO at z = 1.55 in the middle panel of Fig. 6 means that the entire plotted parameter space of the CGBH

Fig.4. The same as in Fig. 1, except for the CGBH model with a fixed h = 0.673, and (a) OHRO at z = 1.43, (b) OHRO at z = 3.91. See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. The same as in Fig. 3, except for the CGBH model with an additional parameter 0.2 < Sr < 0.9. See the text for details. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

model with a fixed h = 0.673 is allowed for OHRO at z = 1.55. This can be seen clearly from Fig. 5 in which OHRO at z = 1.43 is below the lower boundary of the green region, and OHRO at z = 1.55 is below the lower boundaries of both the red and green regions. From the three small panels of Fig. 6 and the panel (b) of Fig. 4, we see that at least r0 > 10 Gpc is required to accommodate OHRO at z = 3.91. Therefore, the same crisis also exists in the CGBH model. Again, the size of the void should be much larger than the size of our observable universe (characterized by the Hubble radius/horizon H-1 ~ 3.0h-1 Gpc [123]) to accommodate OHROs (see the discussion in Section 5 however). On the other hand, there is a serious tension between this unusually large r0 > 10 Gpc and the much lower r0 of order 1.0 Gpc inferred from other observations (e.g. SNIa, CMB and so on) mentioned in Section 1. If the CGBH LTB void model can be consistent with other observations (e.g. SNIa, CMB and so on), it cannot accommodate OHROs.

3.3. The GBH model

Finally, we consider the original version of GBH model [110], in which one does not assume that the big bang is spatially homogeneous. Therefore, Eqs. (25) and (26) are invalid, and hence i2M(r) and H±0(r) should be specified independently. In GBH model, they are given by [110]

o ^ o xro o J1 - tanh [(r - ro)/2Ar] 1

Gm (r) = Gout + (Gin - Gout)-1 , , , , ,- , (31)

[ 1 + tanh(ro/2Ar) J

u ^ u ^ru u J1 - tanh [(r - rp)/2Ar] 1 H±o(r) = Hout + (Hin - Hout) —1 , , , , ,— , (32)

[ 1 + tanh(ro/2Ar) J

where Qout is the asymptotic value of the matter density; Qin is the matter density at the center of the void; Hout and Hin describe the Hubble expansion rate outside and inside the void, respectively; r0 describes the size of the void; Ar characterizes the transition to uniformity. Following [110], we fix Qout = 1. So, there are five free model parameters, namely Qin, r0, Sr = Ar/r0 (which is equivalent to Ar in fact), hin and hout (which are Hin and Hout in units of 100 km/s/Mpc).

Similar to the previous subsections, we try to scan the full parameter space to test this model with OHROs. However, since there are five free parameters in the GBH model, it is very difficult to plot a 5D parameter space. Instead, in Fig. 7 we show the 2D slices of the allowed parameter space with fixed hin = 0.50, 0.58, 0.70, and hout = 0.60, 0.49, 0.40, as well as Sr = 0.40, 0.62, 0.80. Note that hin = 0.58, hout = 0.49, Sr = 0.62 are the best-fit values from SNIa, CMB and BAO [110], and we appropriately vary these parameters to see their effect on the allowed parameter space. From Fig. 7, it is easy to see that the parameters Sr and hout have fairly minor effects on the allowed parameter space. On the other hand, comparing the three columns of Fig. 7, we find that the parameter hin plays a considerable role. The smaller hin, the wider parameter space can be allowed. From the middle and right columns of Fig. 7 (especially from the small panels), it is easy to see that for hin > 0.58, a large r0 > 10 Gpc is required to accommodate OHRO at z = 3.91. In this case, as in the previous two LTB void models, the serious crisis also exists in the GBH LTB void model. That is, the size of the void should be much larger than the size of

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Fig. 6. The same as in Fig. 2, except for the CGBH model with fixed h = 0.623 (left panel), 0.673 (middle panel), 0.738 (right panel), and Sr = 0.40 (green contour lines), 0.64 (red contour lines), 0.80 (black contour lines), for OHROs at z = 1.43 (solid contour lines), z = 1.55 (dashed contour lines), z = 3.91 (dash-dotted contour lines). See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

our observable universe to accommodate OHROs (see the discussion in Section 5 however); there exists a serious tension between this unusually large r0 > 10 Gpc and the much lower r0 of order 1.0 Gpc inferred from other observations (e.g. SNIa, CMB and so on) mentioned in Section 1. However, for a very low hin = 0.50, the required r0 can be in a lower range ~ 4-6 Gpc to accommodate OHROs, as shown in the left column of Fig. 7. Note that the Hubble radius/horizon H-1 ~ 3.0h-1 Gpc ~ 6 Gpc for a very low h ~ 0.50. So, in this case, it is possible to accommodate OHROs while the size of the void is smaller than the size of our observable universe. However, if we further consider the constraints from other observations (e.g. SNIa, CMB and so on), the situation becomes subtle. In [110], the best-fit parameters with 2a uncertainties from SNIa, CMB and BAO are given by hin = 0.58 ± 0.03, hout = 0.49 ± 0.2, Qin = 0.13 ± 0.06, r0 = 2.3 ± 0.9 Gpc, Sr = 0.62 ± (> 0.20). There exists still a remarkable tension far beyond 2a between OHROs and other observations, because hin = 0.50 and r0 ~ 4-6 Gpc can be excluded by other observations (e.g. SNIa, CMB and BAO) far beyond 2a regions. Even worse, the effect of hin is in contrast to the one of r0 actually. If we increase hin, the required r0 to accommodate OHRO at z = 3.91 will increase correspondingly, as shown in Fig. 7. Therefore, it is very difficult to conciliate both hin and r0 with the higher hin = 0.58 ± 0.03 and the smaller r0 = 2.3 ± 0.9 inferred from SNIa, CMB and BAO [110] at the same time. We are in a dilemma. The tension between the constraints from OHROs and other observations (e.g. SNIa, CMB and so on) is fairly serious. So, the age problem cannot be completely alleviated in the GBH LTB void model, although it is in a situation slightly better than the Gaussian model and the CGBH model (but at the price of having more model parameters).

4. Alleviating the age problem

In the previous section, it is easy to find that the age problem in three LTB void models is mainly due to the OHRO at z = 3.91. In fact, this OHRO has ruled out (at least brought trouble to) most

cosmological models (see e.g. [87,95-107]), including the well-known concordance ^CDM model. Naturally, one might doubt on the validity of this quasar APM 08279+5255 at redshift z = 3.91 (we thank the referee for pointing out this issue). In fact, its age was estimated model-dependently. In [94], the age 2.0-3.0 Gyr was obtained by using the giant elliptical model (M4a) and the extreme model (M6a), which are two of 12 chemical evolution models considered in [128]. In [98], its age was re-evaluated following [128]. They found the age of APM 08279+5255 since the initial star formation and stellar evolution in the galaxy: (1) the best estimated value is 2.1 Gyr; (2) 1a lower limit is 1.8 Gyr; (3) the lowest limit is 1.5 Gyr (although this is highly improbable as noted in [98]). However, this is not the age since the beginning of the universe, because the initial star formation started about 0.2-0.3 Gyr after the big bang. Therefore, in [98] they concluded the age of APM 08279+5255 since the beginning of the universe: (1) the best estimated value is 2.3 Gyr; (2) 1a lower limit is 2.0 Gyr; (3) the lowest limit is 1.7 Gyr (although this is highly improbable as noted in [98]).

In the previous section, we used the age 2.0 Gyr for APM 08279+5255 at z = 3.91, which is just the 1a lower limit given in [98]. Here, we would like to consider the lowest limit 1.7 Gyr [98], and see whether the age problem can be alleviated. Because the age problem is most serious in the Gaussian LTB void model as mentioned above, and not to break the length limit, we only consider the Gaussian model here. In Fig. 8, we show the 3D plot and the 2D slices of the allowed parameter space of the Gaussian model only for OHRO at z = 3.91 whose age has been changed to 1.7 Gyr. Although r0 > 20 Gpc is still required for h > 0.623 (see the right panel of Fig. 8), we find from the left panel of Fig. 8 that r0 can reach ~ 5 Gpc for a lower h < 0.55. In this case, r0 ~ 5 Gpc to accommodate OHROs can be slightly smaller than the Hubble radius/horizon H-1 ~ 3.0h-1 Gpc [123]. However, r0 ~ 5 Gpc is still larger than the one of order 1.0 Gpc inferred from other observations (e.g. SNIa, CMB and so on) mentioned in Section 1, the tension still exists. Nevertheless, comparing with the previous sec-

- (z,t)=(1.43,4.00)

- ■ (z,t)=(1.55,3.50) ■ ■ (z,t)=(3.91,2.00)

<5r=0.40 5r=0.62 <5r=0.80

hir =0.70 K.„=0.60

0.01 0.03 0.05 0.07

ftin=0.70 ft,„„=0.49

• i ' _l_

0.01 0.03 0.05 0.07

fcin=0.70 h„Lt=0A0

p.01 0.03,0.05 0.07

0.0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Fig. 7. The same as in Fig. 2, except for the GBH model with fixed hin = 0.50 (left column), 0.58 (middle column), 0.70 (right column), hout = 0.60 (top row), 0.49 (middle row), 0.40 (bottom row), Sr = 0.40 (black contour lines), 0.62 (red contour lines), 0.80 (green contour lines), for OHROs at z = 1.43 (solid contour lines), z = 1.55 (dashed contour lines), z = 3.91 (dash-dotted contour lines). See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

tion, it is fair to say that the tension has been greatly soften and the age problem is alleviated in some sense.

Since the age 2.0 Gyr for APM 08279+5255 at z = 3.91 has also ruled out most cosmological models including ^CDM model, it is of interest to see whether the age problem can also be alleviated in ^CDM model, if the age of APM 08279+5255 is changed to the lowest limit 1.7 Gyr [98]. For the flat ^CDM model, its age at red-shift z is given by [87,96,99,123]

T(z) =( d! , , () J (1 + z) H (z)'

where H(z) = H„VGm0(1 + z)3 + (1 - Gm0). (33)

There are two free parameters, namely Qm0 and h (the Hubble constant H0 in units of 100 km/s/Mpc). In Fig. 9, we scan the

parameter space (a) 0.5 < h < 0.8, 0.25 < Qm0 < 0.45 and (b) 0.661 < h < 0.685, 0.298 < Qm0 < 0.332, and plot cosmic age as function of redshift z. Note that the latter (b) is in fact the 1a region from Planck CMB data [121], namely h = 0.673 ± 0.012, Qm0 = 0.315 ± 0.017. From Fig. 9, we see that all the three OHROs can be well accommodated in ^CDM model for the wide parameter space (a), since the cosmic age can be larger than the ones of all OHROs. Even for the narrow parameter space (b), OHRO at z = 3.91 is just on the edge. Clearly, the age problem can also be alleviated in ^CDM model if the age of APM 08279+5255 is changed to the lowest limit 1.7 Gyr.

5. Conclusion and discussion

As is well known, one can explain the current cosmic acceleration by considering an inhomogeneous and/or anisotropic universe

h=0.623 fc=0.673 fc=0.738

0.05 0.10 0.15 0.20

Fig. 8. The same as in Fig. 1 (left panel) and Fig. 2 (right panel), except only for OHRO at z = 3.91 whose age has been changed to 1.7 Gyr. See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

It is worth noting that in addition to the three OHROs used in this work, there are other OHROs in the literature, for instance, the 4.0 Gyr old radio galaxy 3C 65 at z = 1.175 [124], and the high redshift quasar B1422+231 at z = 3.62 whose best-fit age is 1.5 Gyr with a lower bound of 1.3 Gyr [125]. However, they cannot be used to constrain the models as restrictive as the three OHROs used in this work. So, we do not consider them here. On the other hand, 9 extremely old globular clusters in M31 galaxy [126,127] were considered in [102]. Note that their ages are estimated to be in the range 14-16 Gyr [126,127], which is much larger than the total age of the universe ~ 13.8 Gyr inferred from the CMB observations (e.g. WMAP [4] and Planck [121]). Of course, this does not mean that they cannot be used in the relevant works. However, since as mentioned in Section 1 we only use OHROs in the present work, and hence we also have not considered these 9 extremely old globular clusters in M31 galaxy.

In Section 3, we find that an unusually large r0 > 10 Gpc (or even larger) is required to accommodate OHROs, which means that the size of the void should be much larger than the size of our observable universe (characterized by the Hubble radius/horizon H-1 ~ 3.0h-1 Gpc [123]). However, this does not make the LTB void models invalid (we thank the referee for pointing out this issue). Instead, it just means that the whole void is unobservable, or likewise that the void is a super-horizon mode perturbation. But since the variation of density inside the horizon is not negligible, such a model is physically distinct from FRW model, and hence is meaningful in principle (we thank the referee for pointing out this issue). In the present work, the age problem manifests itself mainly in the serious tension between the constraints from OHROs and other observations (e.g. SNIa, CMB and so on).

Redshift

Fig. 9. Cosmic age as function of redshift z for the parameter space (a) 0.5 < h < 0.8, 0.25 < Qm0 < 0.45 (green) and (b) 0.661 < h < 0.685, 0.298 < Qm0 < 0.332 (red) of the flat ACDM model. The three OHROs at redshift z = 1.43, 1.55, 3.91 are also indicated by black stars. Note that the age of OHRO at z = 3.91 has been changed to 1.7 Gyr. See the text for details. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(which violates the cosmological principle), without invoking dark energy or modified gravity. The well-known one of this kind of models is the so-called Lemaitre-Tolman-Bondi (LTB) void model, in which the universe is spherically symmetric and radially inho-mogeneous, and we are living in a locally underdense void centered nearby our location. In the present work, we test various LTB void models with some old high redshift objects (OHROs). Obviously, the universe cannot be younger than its constituents. We find that an unusually large r0 (characterizing the size of the void) is required to accommodate these OHROs in LTB void models. There is a serious tension between this unusually large r0 and the much smaller r0 inferred from other observations (e.g. SNIa, CMB and so on). However, if we instead consider the lowest limit 1.7 Gyr for the quasar APM 08279+5255 at redshift z = 3.91, this tension could be greatly alleviated.

Acknowledgements

We thank the anonymous referee for quite useful comments and suggestions, which helped us to improve this work. We are grateful to Profs. Rong-Gen Cai, Shuang Nan Zhang and Tong-Jie Zhang for helpful discussions. We also thank Hao Wang, Si-Qi Liu, as well as Zu-Cheng Chen, Jing Liu, Ya-Nan Zhou, Xiao-Bo Zou and Hong-Yu Li for kind help and discussions. This work was supported in part by NSFC under Grants No. 11175016 and No. 10905005, as well as NCET under Grant No. NCET-11-0790.

References

[1] A.G. Riess, et al., Astron. J. 116 (1998) 1009, arXiv:astro-ph/9805201.

[2] S. Perlmutter, et al., Astrophys. J. 517 (1999) 565, arXiv:astro-ph/9812133.

[3] D.N. Spergel, et al., Astrophys. J. Suppl. 148 (2003) 175, arXiv:astro-ph/ 0302209.

[4] G. Hinshaw, et al., Astrophys. J. Suppl. 208 (2013) 19, arXiv:1212.5226.

[5] M. Tegmark, et al., Phys. Rev. D 69 (2004) 103501, arXiv:astro-ph/0310723.

[6] M. Tegmark, et al., Phys. Rev. D 74 (2006) 123507, arXiv:astro-ph/0608632.

[7] E.J. Copeland, M. Sami, S. Tsujikawa, Int. J. Mod. Phys. D 15 (2006) 1753, arXiv:hep-th/0603057.

[8] M. Kamionkowski, arXiv:0706.2986 [astro-ph].

[9] T. Padmanabhan, Curr. Sci. 88 (2005) 1057, arXiv:astro-ph/0411044.

[10] S. Nobbenhuis, Found. Phys. 36 (2006) 613, arXiv:gr-qc/0411093.

[11] V. Sahni, A.A. Starobinsky, Int. J. Mod. Phys. D 9 ( 2000 ) 373, arXiv:astro-ph/ 9904398.

[12] P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75 (2003) 559, arXiv:astro-ph/0207347.

[13] A. Albrecht, et al., arXiv:astro-ph/0609591.

[14] S.M. Carroll, AIP Conf. Proc. 743 (2005) 16, arXiv:astro-ph/0310342.

[15] R.R. Caldwell, R. Dave, P.J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582, arXiv: astro-ph/9708069.

[16] P.J. Steinhardt, L.M. Wang, I. Zlatev, Phys. Rev. D 59 (1999) 123504, arXiv: astro-ph/9812313.

[17] I. Zlatev, P.J. Steinhardt, Phys. Lett. B 459 (1999) 570, arXiv:astro-ph/9906481.

[18] R.R. Caldwell, Phys. Lett. B 545 (2002) 23, arXiv:astro-ph/9908168.

[19] R.R. Caldwell, et al., Phys. Rev. Lett. 91 (2003) 071301, arXiv:astro-ph/0302506.

[20] C. Armendariz-Picon, et al., Phys. Rev. Lett. 85 (2000) 4438, arXiv:astro-ph/ 0004134.

[21] C. Armendariz-Picon, et al., Phys. Rev. D 63 (2001) 103510, arXiv:astro-ph/ 0006373.

[22] T. Chiba, T. Okabe, M. Yamaguchi, Phys. Rev. D 62 (2000) 023511, arXiv: astro-ph/9912463.

[23] B. Feng, X.L. Wang, X.M. Zhang, Phys. Lett. B 607 (2005) 35, arXiv:astro-ph/ 0404224.

[24] A.Y. Kamenshchik, U. Moschella, V. Pasquier, Phys. Lett. B 511 (2001) 265, arXiv:gr-qc/0103004.

[25] M.C. Bento, O. Bertolami, A.A. Sen, Phys. Rev. D 66 (2002) 043507, arXiv:gr-qc/ 0202064.

[26] C. Armendariz-Picon, J. Cosmol. Astropart. Phys. 0407 (2004) 007, arXiv: astro-ph/0405267.

[27] H. Wei, R.G. Cai, Phys. Rev. D 73 (2006) 083002, arXiv:astro-ph/0603052.

[28] H. Wei, R.G. Cai, J. Cosmol. Astropart. Phys. 0709 (2007) 015, arXiv:astro-ph/ 0607064.

[29] M. Li, Phys. Lett. B 603 (2004) 1, arXiv:hep-th/0403127.

[30] R.G. Cai, Phys. Lett. B 657 (2007) 228, arXiv:0707.4049.

[31] H. Wei, R.G. Cai, Phys. Lett. B 660 (2008) 113, arXiv:0708.0884.

[32] H. Wei, R.G. Cai, Phys. Lett. B 663 (2008) 1, arXiv:0708.1894.

[33] H. Wei, R.G. Cai, D.F. Zeng, Class. Quantum Gravity 22 (2005) 3189, arXiv: hep-th/0501160.

[34] H. Wei, R.G. Cai, Phys. Rev. D 72 (2005) 123507, arXiv:astro-ph/0509328.

[35] C.G. Boehmer, J. Burnett, Mod. Phys. Lett. A 25 (2010) 101, arXiv:0906.1351.

[36] C.G. Boehmer, J. Burnett, D.F. Mota, D.J. Shaw, J. High Energy Phys. 1007 (2010) 053, arXiv:1003.3858.

[37] H. Wei, Phys. Lett. B 695 (2011) 307, arXiv:1002.4230.

[38] A. De Felice, S. Tsujikawa, Living Rev. Relativ. 13 (2010) 3, arXiv:1002.4928.

[39] T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. 82 (2010) 451, arXiv:0805.1726.

[40] T. Clifton, P.G. Ferreira, A. Padilla, C. Skordis, Phys. Rep. 513 (2012) 1, arXiv: 1106.2476.

[41] D. Saez-Gomez, arXiv:0812.1980 [hep-th].

[42] G.R. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485 (2000) 208, arXiv: hep-th/0005016.

[43] C. Deffayet, Phys. Lett. B 502 (2001) 199, arXiv:hep-th/0010186.

[44] C. Deffayet, G.R. Dvali, G. Gabadadze, Phys. Rev. D 65 (2002) 044023, arXiv: astro-ph/0105068.

[45] A. Nicolis, R. Rattazzi, E. Trincherini, Phys. Rev. D 79 (2009) 064036, arXiv: 0811.2197.

[46] A. De Felice, S. Tsujikawa, Phys. Rev. D 84 (2011) 124029, arXiv:1008.4236.

[47] A. De Felice, S. Tsujikawa, J. Cosmol. Astropart. Phys. 1202 (2012) 007, arXiv: 1110.3878.

[48] T. Koivisto, D.F. Mota, Phys. Rev. D 75 (2007) 023518, arXiv:hep-th/0609155.

[49] S. Nojiri, S.D. Odintsov, M. Sasaki, Phys. Rev. D 71 (2005) 123509, arXiv: hep-th/0504052.

[50] G.R. Bengochea, R. Ferraro, Phys. Rev. D 79 (2009) 124019, arXiv:0812.1205.

[51] E.V. Linder, Phys. Rev. D 81 (2010) 127301, arXiv:1005.3039; E.V. Linder, Phys. Rev. D 82 (2010) 109902 (Erratum).

[52] M. Fierz, W. Pauli, Proc. R. Soc. Lond. Ser. A 173 (1939) 211.

[53] C. de Rham, G. Gabadadze, A.J. Tolley, Phys. Rev. Lett. 106 (2011) 231101, arXiv:1011.1232.

[54] S.F. Hassan, R.A. Rosen, Phys. Rev. Lett. 108 (2012) 041101, arXiv:1106.3344.

[55] R.R. Caldwell, A. Stebbins, Phys. Rev. Lett. 100 (2008) 191302, arXiv:0711.3459.

[56] I. Zehavi, A.G. Riess, R.P. Kirshner, A. Dekel, Astrophys. J. 503 (1998) 483, arXiv:astro-ph/9802252.

[57] M.N. Celerier, Astron. Astrophys. 353 (2000 ) 63, arXiv:astro-ph/9907206.

[58] M.N. Celerier, arXiv:astro-ph/0005594.

[59] M.N. Celerier, arXiv:astro-ph/0006273.

[60] M.N. Celerier, arXiv:astro-ph/0702416.

[61] R.K. Barrett, C.A. Clarkson, Class. Quantum Gravity 17 (2000) 5047, arXiv: astro-ph/9911235.

[62] K. Tomita, Mon. Not. R. Astron. Soc. 326 (2001) 287, arXiv:astro-ph/0011484.

[63] K. Tomita, Prog. Theor. Phys. 106 (2001) 929, arXiv:astro-ph/0104141.

[64] H. Iguchi, T. Nakamura, K.i. Nakao, Prog. Theor. Phys. 108 (2002) 809, arXiv: astro-ph/0112419.

[65] T. Clifton, P.G. Ferreira, K. Land, Phys. Rev. Lett. 101 (2008) 131302, arXiv: 0807.1443.

[66] F.K. Hansen, et al., Mon. Not. R. Astron. Soc. 354 (2004) 641, arXiv:astro-ph/ 0404206.

[67] A.K. Singal, arXiv:1305.4134 [astro-ph.CO].

[68] J.W. Moffat, J. Cosmol. Astropart. Phys. 0510 (2005) 012, arXiv:astro-ph/ 0502110.

[69] H. Alnes, M. Amarzguioui, Phys. Rev. D 74 (2006) 103520, arXiv:astro-ph/ 0607334.

[70] T. Clifton, P.G. Ferreira, J. Zuntz, J. Cosmol. Astropart. Phys. 0907 (2009) 029, arXiv:0902.1313.

[71] C. Clarkson, M. Regis, J. Cosmol. Astropart. Phys. 1102 (2011) 013, arXiv: 1007.3443.

[72] J.P. Uzan, C. Clarkson, G.F.R. Ellis, Phys. Rev. Lett. 100 (2008) 191303, arXiv: 0801.0068.

[73] K. Bolejko, J.S.B. Wyithe, J. Cosmol. Astropart. Phys. 0902 (2009) 020, arXiv: 0807.2891.

[74] S. February, C. Clarkson, R. Maartens, J. Cosmol. Astropart. Phys. 1303 (2013) 023, arXiv:1206.1602.

[75] J.P. Zibin, A. Moss, D. Scott, Phys. Rev. Lett. 101 (2008) 251303, arXiv: 0809.3761.

[76] K. Tomita, K.T. Inoue, Phys. Rev. D 79 (2009) 103505, arXiv:0903.1541.

[77] F.S. Labini, Y.V. Baryshev, J. Cosmol. Astropart. Phys. 1006 (2010) 021, arXiv: 1006.0801.

[78] P. Zhang, A. Stebbins, Phys. Rev. Lett. 107 (2011) 041301, arXiv:1009.3967.

[79] W. Valkenburg, et al., Mon. Not. R. Astron. Soc. 438 (2014) L6, arXiv: 1209.4078.

[80] P. Bull, T. Clifton, P.G. Ferreira, Phys. Rev. D 85 (2012) 024002, arXiv:1108. 2222.

[81] T.J. Zhang, H. Wang, C. Ma, arXiv:1210.1775 [astro-ph.CO].

[82] H. Wang, T.J. Zhang, Astrophys. J. 748 (2012) 111, arXiv: 1111.2400.

[83] A. Meszaros, L.G. Balazs, Z. Bagoly, P. Veres, arXiv:0906.4034 [astro-ph.CO].

[84] G. Lemaitre, Ann. Soc. Sci. Brux. Ser. A 53 (1933 ) 51;

see G. Lemaitre, Gen. Relativ. Gravit. 29 (1997) 641, for English translation.

[85] R.C. Tolman, Proc. Natl. Acad. Sci. 20 (1934) 169;

see R.C. Tolman, Gen. Relativ. Gravit. 29 (1997) 935, for English translation.

[86] H. Bondi, Mon. Not. R. Astron. Soc. 107 (1947) 410.

[87] J.S. Alcaniz, J.A.S. Lima, Astrophys. J. 521 (1999) L87, arXiv:astro-ph/9902298.

[88] M.X. Lan, M. Li, X.D. Li, S. Wang, Phys. Rev. D 82 (2010) 023516, arXiv:1002. 0978.

[89] S. Liu, T.J. Zhang, Phys. Lett. B 733 (2014) 69, arXiv:1311.1600.

[90] J. Dunlop, et al., Nature 381 (1996) 581.

[91] H. Spinrad, et al., Astrophys. J. 484 (1997) 581, arXiv:astro-ph/9702233.

[92] J. Dunlop, in: H. Rottgering, et al. (Eds.), The Most Distant Radio Galaxies, Dordrecht, 1999, p. 71.

[93] G. Hasinger, N. Schartel, S. Komossa, Astrophys. J. 573 (2002) L77, arXiv: astro-ph/0207005.

[94] S. Komossa, G. Hasinger, arXiv:astro-ph/0207321.

[95] A. Friaca, et al., Mon. Not. R. Astron. Soc. 362 (2005) 1295, arXiv:astro-ph/ 0504031.

[96] J.S. Alcaniz, et al., Mon. Not. R. Astron. Soc. 340 (2003) L39, arXiv:astro-ph/ 0301226;

J.A.S. Lima, J.S. Alcaniz, Mon. Not. R. Astron. Soc. 317 (2000) 893, arXiv: astro-ph/0005441;

D. Jain, A. Dev, Phys. Lett. B 633 (2006) 436, arXiv:astro-ph/0509212;

J.S. Alcaniz, D. Jain, A. Dev, Phys. Rev. D 67 (2003) 043514, arXiv:astro-ph/

0210476.

[97] J.V. Cunha, R.C. Santos, Int. J. Mod. Phys. D 13 (2004) 1321, arXiv:astro-ph/ 0402169;

J.F. Jesus, Gen. Relativ. Gravit. 40 (2008) 791, arXiv:astro-ph/0603142;

M.A. Dantas, et al., Astron. Astrophys. 467 (2007) 421, arXiv:astro-ph/

0607060;

S. Rahvar, M.S. Movahed, Phys. Rev. D 75 (2007) 023512, arXiv:astro-ph/ 0604206.

[98] R.J. Yang, S.N. Zhang, Mon. Not. R. Astron. Soc. 407 (2010) 1835, arXiv:0905. 2683.

[99] H. Wei, S.N. Zhang, Phys. Rev. D 76 (2007) 063003, arXiv:0707.2129.

[100] Y. Zhang, H. Li, X. Wu, H. Wei, R.G. Cai, arXiv:0708.1214 [astro-ph].

[101] H. Wei, J. Cosmol. Astropart. Phys. 1104 (2011) 022, arXiv:1012.0883.

[102] S. Wang, X.D. Li, M. Li, Phys. Rev. D 82 (2010) 103006, arXiv:1005.4345.

[103] S. Capozziello, P.K.S. Dunsby, E. Piedipalumbo, C. Rubano, arXiv:0706.2615 [astro-ph].

[104] M.S. Movahed, S. Baghram, S. Rahvar, Phys. Rev. D 76 (2007) 044008, arXiv: 0705.0889.

[105] M.S. Movahed, M. Farhang, S. Rahvar, Int. J. Theor. Phys. 48 (2009) 1203, arXiv:astro-ph/0701339.

[106] M.S. Movahed, S. Ghassemi, Phys. Rev. D 76 (2007) 084037, arXiv:0705. 3894.

[107] N. Pires, Z.H. Zhu, J.S. Alcaniz, Phys. Rev. D 73 (2006) 123530, arXiv:astro-ph/ 0606689.

[108] K. Enqvist, T. Mattsson, J. Cosmol. Astropart. Phys. 0702 (2007) 019, arXiv: astro-ph/0609120.

[109] K. Enqvist, Gen. Relativ. Gravit. 40 (2008) 451, arXiv:0709.2044.

[110] J. Garcia-Bellido, T. Haugboelle, J. Cosmol. Astropart. Phys. 0804 (2008) 003, arXiv:0802.1523.

[111] S.W. Goode, J. Wainwright, Phys. Rev. D 26 (1982) 3315.

[112] A. Krasinski, Inhomogeneous Cosmological Models, Cambridge University Press, 1997.

[113] M.N. Celerier, Astron. Astrophys. 543 (2012) A71, arXiv:1108.1373.

[114] M.N. Celerier, J. Phys. Conf. Ser. 484 (2014) 012005, arXiv:1203.2814.

[115] H. Alnes, M. Amarzguioui, O. Gron, Phys. Rev. D 73 ( 2006 ) 083519, arXiv: astro-ph/0512006.

[116] M.N. Celerier, et al., Astron. Astrophys. 518 (2010) A21, arXiv:0906.0905.

[117] R.A. Vanderveld, et al., Phys. Rev. D 74 (2006) 023506, arXiv:astro-ph/ 0602476.

[118] J.P. Zibin, Phys. Rev. D 78 (2008) 043504, arXiv:0804.1787.

[119] J. Silk, Astron. Astrophys. 59 (1977) 53.

[120] A.G. Riess, et al., Astrophys. J. 730 (2011) 119, arXiv:1103.2976; A.G. Riess, et al., Astrophys. J. 732 (2011) 129 (Erratum).

[121] P.A.R. Ade, et al., Astron. Astrophys. 571 (2014) A16, arXiv:1303.5076.

[122] A. Sandage, et al., Astrophys. J. 653 (2006) 843, arXiv:astro-ph/0603647.

[123] E.W. Kolb, M.S. Turner, The Early Universe, Addison-Wesley Publishing Company, USA, 1990.

[124] A. Stockton, M. Kellogg, S.E. Ridgway, Astrophys. J. 443 (1995) L69.

[125] Y. Yoshii, T. Tsujimoto, K. Kawara, Astrophys. J. 507 (1998) L113, arXiv: astro-ph/9809047.

[126] J. Ma, et al., Astron. J. 137 (2009) 4884, arXiv:0904.0553.

[127] S. Wang, et al., Astron. J. 139 (2010) 1438, arXiv:1001.3939.

[128] F. Hamann, G. Ferland, Astrophys. J. 418 (1993) 11.

[129] M. Quartin, L. Amendola, Phys. Rev. D 81 (2010) 043522, arXiv:0909.4954.

[130] A. Moss, J.P. Zibin, D. Scott, Phys. Rev. D 83 (2011) 103515, arXiv:1007.3725.

[131] J.P. Zibin, A. Moss, Class. Quantum Gravity 28 (2011) 164005, arXiv:1105.0909.

[132] M. Ishak, A. Peel, M.A. Troxel, Phys. Rev. Lett. 111 (2013) 251302, arXiv:1307. 0723.

[133] R. Lapiedra, J.A. Morales-Lladosa, Phys. Rev. D 89 (2014) 064033, arXiv:1303. 1123.

[134] V. Marra, A. Notari, Class. Quantum Gravity 28 (2011) 164004, arXiv:1102. 1015.

[135] V. Marra, M. Paakkonen, J. Cosmol. Astropart. Phys. 1012 (2010) 021, arXiv: 1009.4193.