Can we remove the systematic error due to isotropic inhomogeneities?

Can we remove the systematic error due to isotropic inhomogeneities?

Hiroyuki Negishi111Electronic address:negishi@sci.osaka-cu.ac.jp and Ken-ichi Nakao222Electronic address:knakao@sci.osaka-cu.ac.jp Department of Mathematics and Physics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan
Abstract

Usually, we assume that there is no inhomogeneity isotropic in terms of our location in our universe. This assumption has not been observationally confirmed yet in sufficient accuracy, and we need to consider the possibility that there are non-negligible large-scale isotropic inhomogeneities in our universe. The existence of large-scale isotropic inhomogeneities affects the determination of the cosmological parameters. In particular, from only the distance-redshift relation, we can not distinguish the inhomogeneous isotropic universe model from the homogeneous isotropic one, because of the ambiguity in the cosmological parameters. In this paper, in order to avoid such ambiguity, we consider three observables, the distance-redshift relation, the fluctuation spectrum of the cosmic microwave background radiation(CMBR) and the scale of the baryon acoustic oscillation(BAO), and compare these observables in two universe models; One is the inhomogeneous isotropic universe model with the cosmological constant and the other is the homogeneous isotropic universe model with the dark energy other than the cosmological constant. We show that these two universe models can not predict the same observational data of all three observables but the same ones of only two of three, as long as the perturbations are adiabatic. In principle, we can distinguish the inhomogeneous isotropic universe from the homogeneous isotropic one through appropriate three observables, if the perturbations are adiabatic.

OCU-PHYS-450 AP-GR-131


I Introduction

Usually, the modern physical cosmology adopts the Copernican principle which states that we do not live in the privileged domain in the universe. The Copernican principle and the observed high isotropy of the CMBR provide the high homogeneity and isotropy of our universe in the globally averaged sense. By contrast, if we remove the Copernican principle, the isotropy around us does not necessarily imply the homogeneity of our universe. The Copernican principle can not be directly confirmed by observations since in order to do so we have to move to the other clusters of galaxies from our galaxy. Hence there is the possibility that there are large-scale isotropic inhomogeneities in our universe. The existence of isotropic inhomogeneities around us affects the determination of the cosmological parameters.

The universe model with large-scale isotropic inhomogeneities has been studied in the context of the scenario to explain the observed distance-redshift relation without introducing dark energy components within the framework of general relativity. There are several severe observational constraints on the scenario without dark energyTomita:2000jj (); Tomita:2001gh (); Celerier:1999hp (); Iguchi:2001sq (); Yoo:2008su (); Bull:2012zx (); Clifton:2008hv (); Vanderveld:2006rb (); Yoo:2010qn (), in particular, these universe models are constrained by observations of the kinetic Sunyaev-Zeldovich effectZhang (); Zibin:2010a (); Zibin:2011ma (); The scenario with adiabatic isotropic inhomogeneities has already been ruled out, even though both growing and decaying modes are assumed to exist. On the other hand, the scenario with non-adiabatic isotropic inhomogeneities has not been ruled out yet, although there is an argument on whether the initial condition is contrived.

Even if there are dark energy components, not so large isotropic inhomogeneities may exist and significantly affect observational resultsRomano:2010nc (); Romano:2011mx (); Marra:2010pg (); Sinclair:2010sb (); Valkenburg:2013qwa (); Valkenburg:2012td (); deLavallaz:2011tj (); Valkenburg:2011ty (); Marra:2012pj (); Tokutake:2016hod (); Negishi:2015oga (). Since our observation is confined on a past light cone, a spatially more distant event we observe occurred a longer time ago. If the universe is homogeneous and isotropic, the temporal evolution of the universe is revealed by observing events of various distances. By contrast, if the universe is inhomogeneous and isotropic, observational data contain the information about not only the temporal evolution of the universe but also its spatial inhomogeneities. An intrinsic degeneracy between temporal evolution and isotropic inhomogeneities around us may cause systematic errors. Denoting the total energy density and the total pressure of dark energy components by and , respectively, its equation of state is given by

(1)

with , is a function less than . The case of corresponds to the cosmological constant. We can not distinguish the inhomogeneous isotropic universe model with the cosmological constant from the homogeneous isotropic universe model with dark energy of , if we have the observational date of the distance-redshift relation only; isotropic inhomogeneities may cause systematic errors on the amount of dark energy and its equation of state. Some authors studied systematic errors due to isotropic inhomogeneities and have evaluated the magnitude of themMarra:2010pg (); Valkenburg:2012td (); Romano:2010nc (); Romano:2011mx (); Sinclair:2010sb (); Valkenburg:2013qwa (); deLavallaz:2011tj (); Valkenburg:2011ty (); Marra:2012pj (); Tokutake:2016hod (); Negishi:2015oga (). However, nobody has not shown how to remove the systematic errors caused by isotropic inhomogeneities. It is just the main purpose of the present paper.

The homogenous isotropic universe model is often called the Friedmann-Lemaître-Robertson-Walker (FLRW) universe model, and hereafter we call so.

In this paper, we study whether we can distinguish the inhomogeneous isotropic universe model with the cosmological constant from the FLRW universe model with dark energy of and remove the systematic error due to isotropic inhomogeneities, if we use multiple observables: the distance-redshift relation, the fluctuation spectrum of the CMBR and the scale of the BAO in the distribution of galaxies. In other words, we investigate whether there is an inhomogeneous isotropic universe model whose distance-redshift relation, fluctuation spectrum of the CMBR and BAO scale are identical with those of the FLRW universe model. We assume that the inhomogeneous isotropic universe model is filled with non-relativistic matter which is cold dark matter(CDM) and baryonic matter and a positive cosmological constant. Furthermore, we restrict ourselves to the case that the amplitude of isotropic inhomogeneities is so small that they can be treated by the linear perturbation approximation of the flat FLRW universe model, and the scale of isotropic inhomogeneities are larger than the BAO scale but should be smaller than the present horizon scale. If we can find such an inhomogeneous isotropic universe model, it is impossible to distinguish these two universe models from each other, otherwise we can.

The organization of this paper is as follows. In Sec. II, we derive the basic equations for the inhomogeneous isotropic universe model. In Sec. III, we derive the null geodesic equations which are used to construct the inhomogeneous isotropic universe model from observables given from the FLRW universe model. In Sec. IV, we show expressions of observables in the inhomogeneous isotropic universe model. In Sec. V, we give the observables in the FLRW model and derive conditions to determine the inhomogeneous isotropic universe model. We explain the numerical procedure and the numerical result in Sec. VI. Finally, Sec. VII is devoted to the summary and discussion.

In this paper, we adopt the sign conventions of the metric and Riemann tensor of Ref.wald () and the geometrized unit in which the speed of light and Newton’s gravitational constant are one.

Ii Inhomogeneous isotropic universe model

As mentioned in Sec.I, the inhomogeneous isotropic universe model is described by the flat FLRW universe model with isotropic linear perturbations. By adopting the synchronous comoving gauge, the infinitesimal world interval is written in the form,

(3)
(4)

where is the scale factor scaled so as to be unity at the present time and is the line element of the unit 2-sphere.

We assume that this universe model is filled with non-relativistic matter and the cosmological constant . The stress-energy tensor of the non-relativistic matter is given by

(5)

where , and are the energy density of the background, the density contrast and the 4-velocity, respectively. The coordinate system is chosen so that the components of the 4-velocity is given by .

The Einstein equations lead to the Friedmann equation for the background;

(6)

where is the background energy density at . Denoting the present value of by , Eq. (6) is rewritten in the form

(7)

where

(8)

The Einstein equations lead to the equations for the linear perturbations;

(9)
(10)
(11)
(12)

where a dot denotes a partial differentiation with respect to .

The general solution of Eq. (12) is represented by the linear superposition of the growing factor and the decaying factor , which are defined as

(13)

Hereafter, we assume that the decaying mode does not exisit, since this assumption is consistent with the inflationary universe scenario. Thus, we have

(14)

where is an arbitrary function of the radial coordinate .

From Eqs. (9) – (11), and are expressed by

(15)
(16)

where and are arbitrary functions of the radial coordinate . A gauge freedom remains in Eq. (4), so we fix the gauge so that . Through Eq. (11), we have

(17)

Iii Null geodesics in inhomogeneous isotropic universe model

In order to calculate some observable, we consider a past-directed radial null geodesic which emanates from the observer. We assume that the observer in the inhomogeneous isotropic universe model stays at the symmetry center , so that the observer recognizes the universe to be isotropic. By virtue of the isotropy in terms of the observer, both and should vanish. One of the non-trivial components of the geodesic equations is given by

(18)

where is the affine parameter. Equation (18) determines , whereas the null condition determines in the manner

(19)

Then we have equations for and as

(20)
(21)

The redshift for the observer is given by

(22)

where subscripts s and o mean the quantities evaluated at the source and the observer, respectively, and we have chosen the affine parameter so that is unity.

We rewrite the equations for the radial null geodesic in the forms appropriate for later analyses. Equation (18) is rewritten in the form,

(23)

From Eqs. (22) and (23), we have

(24)

From the null condition (19), we have

(25)

We express the radial null geodesic as a function of ;

(26)
(27)

where the quantities with a horizontal bar represent the background solution. From Eqs. (24) and (25), we see that the background solutions and satisfy

(28)
(29)

where

Equation (24) leads to the equation for the linear perturbations as

(30)

whereas Eq. (25) leads to

(31)

where we have used Eqs. (28) and (29).

The boundary conditions for Eqs. (28)–(31) are given by

(32)

If the inhomogeneous isotropic universe model is completely fixed, Eqs. (30) and (31) determine and .

Iv Observables

In this Section, we show observables of our interest in the inhomogeneous isotropic universe model.

iv.1 Angular diameter distance-redshift relation

By observing many supernovae, we have the correlation between the luminosity distance and redshift . In this paper, we treat the angular diameter distance instead of ; is obtained from through

(33)

In the inhomogeneous isotropic universe model, the angular diameter distance from some light source to the observer is equal to the areal radius at which the light is emitted;

(35)
(36)

Equation (36) implies that the angular diameter distance-redshift relation depends on two parameters , and one arbitrary function , since and are determined by the geodesic equations (28)–(31), if we give , and

iv.2 Cosmic microwave background radiation

Anisotropies of the CMBR are important observables. They come from the anisotropies of the last scattering surface and the Integrated Sachs-Wolfe(ISW) effect. In this paper, we are interested in the high multipoles of the anisotropies of the CMBR for which the ISW effect is not important, since there is a slight information in low multipoles because of their large cosmic variance. Hereafter, we ignore the ISW effect, and follow Clarkson-Regis () to calculate the anisotropies of the CMBR in the inhomogeneous isotropic universe model.

Hereafter, we adopt the following assumptions;

  • The inhomogeneous isotropic universe model well agrees with the background universe model in the vicinity of the last scattering surface(LSS);

  • The background universe model well agrees with the Einstein-de Sitter(EdS) universe model at the LSS;

  • The non-relativistic matter is composed of baryonic matter and CDM whose background energy densities are represented by and respectively;

  • The primordial density fluctuations is adiabatic and its power spectrum is characterized by an amplitude and a spectral index ;

    (37)

    where .

If these assumptions are valid, the CMBR angular power spectrum is given by

(38)

where is the CMBR angular power spectrum of the EdS universe model filled with baryonic matter and CDM whose energy densities at the LSS are the same as and , respectively, where superscripts LSS mean quantities at the LSS, and the primordial power spectrum is the same as that given by Eq.(37), and

(39)

where is the redshift at the LSS, and is the angular diameter distance at in the EdS universe model. We choose the present temperature of the CMBR to be and the spectral index to be . Equation (38) means that the CMBR angular power spectrum of the inhomogeneous isotropic universe model depends on , and which characterize and . Usually, depends on the function in the inhomogeneous isotropic universe model, but Eq. (38) depends on a finite parameters, since we ignore the ISW effect.

iv.3 Baryon Acoustic Oscillation

Large scale redshift surveys of galaxies tell us the BAO scaleEisenstein:2005su (); Percival et al.(2010) (); Blake et al.(2011) (). Observables related to the BAO scale are which is the angular diameter of the BAO scale and which is the BAO scale in the redshift space. Most papers on the BAO observations quote

(40)

In order to calculate , first of all, we need the BAO scale at the decoupling time, which is denoted by . In the inhomogeneous isotropic universe model, the isotropic density perturbation has been assumed to be composed of the only growing mode which is so small that the liner perturbation approximation is valid until the present time. We assume that the ratio of baryonic matter and CDM is everywhere constant, so that we regard that the inhomogeneous and isotropic universe model is almost homogeneous and isotropic at the decoupling time and is everywhere constant. This assumption is the same as that in ref. Zumalacarregui-Bellido-Lapuente (). The values of the BAO scale in transverse and radial directions, and , at an event on the past light cone are respectively given by

(42)
(44)
(45)
(47)
(49)
(50)

where is the decoupling time, is scale factor of decoupling time, and in the second equality in the above equations we have used the fact that the BAO scale is comoving in the gauge adopted here and the scale of isotropic inhomogeneities are larger than the BAO scale. From Eq. (45), we have

(52)
(53)

is obtained from integrating geodesic equations (24) – (27)

(55)
(57)
(60)

where in the second equality in the above equations we have used the fact that the BAO scale is comoving and the scale of isotropic inhomogeneities are larger than the BAO scale. Thus, we obtain

(63)

Equation (63) means that in the inhomogeneous isotropic universe model depends on , , , and . is determined by and , where

(64)
(65)

V How to construct the inhomogeneous isotropic universe model

As mentioned in Sec I, we study whether we can distinguish the inhomogeneous isotropic universe model with the cosmological constant from the FLRW universe model with dark energy other than the cosmological constant, if we consider multiple observables.

We choose the FLRW universe model as follows; The FLRW universe model is filled with non-relativistic matter which is composed of baryonic matter and CDM and dark energy whose in the equation of state (1) is written in the form

(66)

where is the scale factor of the FLRW universe model, and is constant and ; We assume that for ; Furthermore, for simplicity, we assume that the FLRW universe model has flat space, . The equation of state of this form was studied in the cosmological context by M. Chevallier and D. PolarskiChevallier:2000qy (). This FLRW universe model is characterized by five parameters, Hubble constant , , , the present value of energy density of baryonic matter and of dark energy .

In accordance with the standard scenario of the structure formation, we assume the adiabatic primordial density fluctuations in the FLRW universe model, whose power spectrum is given by

(67)

where and are an amplitude and a spectral index, respectively.

We investigate whether we can construct the inhomogeneous isotropic universe model which satisfies the following conditions; on the angular diameter distance-redshift relation

(68)

on the angular power spectrum of the CMBR

(69)

on the averaged angular scale of the BAO

(70)

where characters with a hat denote quantities of the FLRW universe model. Note that the condition on the angular diameter distance-redshift relation is restricted in the domain . This is because we do not have any observational data of the distance-redshift relation in the domain of .

For convenience, we define a new variable

(71)

The condition (68) together with Eq. (36) gives us the relation between , , and ,

(72)

where

(73)

By substituting Eq. (72) into Eq. (31), we eliminate from Eq. (31) and obtain the differential equation for . By eliminating from Eq. (30), we obtain the differential equation for . As a result, we obtain the following system of differential equations to determine the inhomogeneous isotropic universe model which satisfies the condition (68);

(75)
(77)
(78)

where

(80)
(82)
(83)

The boundary conditions for Eqs. (75)–(78) are given as follows. The boundary condition on is given by Eq. (32). Imposing regularity for at , i.e., , we obtain

(84)

The value of can be made zero by a rescaling of the coordinate. Hence we impose

(85)

In the next section, we explain how to solve Eqs. (75)–(78) under the boundary conditions (32), (84) and (85). We numerically solve these equations and investigate whether the obtained inhomogeneous isotropic universe model satisfies the conditions (69) and (70).

Vi numerical procedure and result

Before performing numerical integral, we choose the parameters in the FLRW universe model consistent with Planck resultsAde:2015xua (), , , , , . We give the parameters and in the domain and . Then, we numerically integrate Eqs. (75)–(78) under the boundary conditions (32), (84) and (85) and check that the obtained inhomogeneous isotropic universe model can satisfy the conditions (69) and (70) as follows.

To integrate Eqs. (75)–(78), we need to give the parameters, and , of the inhomogeneous isotropic universe model. We fix these parameters as follows. Equation (77) has a regular singular point at which is a root of . Since is a background quantity, we can obtain without the knowledge about inhomogeneities, and have found that is larger than unity for the cases of our interest. The function should satisfy

(86)

so that the solutions of Eqs. (75)–(78) and their derivatives with respect to are continuous at . The condition (86) leads to a relation between , , and . We assume and then Eq. (86) gives a relation between the remaining three quantities, , and . We choose the two of three quantities and so that the solutions of three differential equations (75)–(78) are everywhere continuous. We solve Eqs. (75)–(78) from to by imposing the boundary conditions (32), (84) and (85) and, at the same time, from to , by making a guess at , and and then fixing so that Eq. (86) is satisfied; If we fail to get continuous solutions of Eqs. (75)–(78), we select different values of , and in accordance with the Newton method in the three-dimensional parameter space and then again integrate Eqs. (75)–(78) from to and, at the same time, from to ; We iterate this procedure until the discrepancies between the values at obtained by the integrals from to and from to are sufficiently small; Next, we integrate Eqs. (75)–(78) outward from with the values of , , and which guarantee the continuity of the solutions. Note that Eqs. (75)–(78) implies the smoothness of solutions. As a result, if we give , we obtain and which characterize the inhomogeneous isotropic universe model in the domain and and which characterize null geodesic in the domain from Eqs. (75)–(78).

In order to fix remaining freedoms of the inhomogeneous isotropic universe model, we use the condition (69). If , and are the same as , the energy density of non-relativistic matter and baryonic matter at the LSS in the FLRW universe model, respectively, the condition (69) is satisfied up to the overall factor. We fix to fit the height of the first peak of the CMBR angular power spectrum with that of the FLRW universe model. As a result, , , and are uniquely determined.

We check that the inhomogeneous isotropic universe model can satisfy the condition (70). Equation (63) determines for arbitrary . The R.H.S of Eq. (63) is composed of , , , , , and the background quantities , , , and . The background Hubble constant is determined through Eqs. (75)–(78) by fixing . The growing factor and the background quantities , and are completely determined by fixing . Equation (64) leads to

(87)

Since is determined by the condition (69), Eq. (87) implies that is also determined by fixing . The perturbed angular diameter distance is determined by the condition (68). is equal to the BAO scale at LSS , since we assume that is everywhere constant, where the BAO scale at LSS is determined by the condition (69). The remaining perturbative variables , , are determined by Eqs. (75)–(78) once is fixed. Here note that there is no condition to determine : is still a free parameter. Hence we may rewrite in the form

(88)

Here, by using Newton method, we find a root of the one of two conditions in Eq. (70),

(89)

Note that, at this stage, there is no free parameter in the inhomogeneous isotropic universe model. Then, we have checked whether the root of Eq. (89) satisfies another one of two conditions in Eq. (70), i.e., . In order to evaluate the difference of the BAO scale between the inhomogeneous isotropic universe model and the FLRW universe model, we use defined as

(90)

In Fig. 1, we depict as a heat map on the - plane. In Fig. 2, we depict as a function of with various to see more details about -dependence of . It can be seen from these figures that vanishes along a curve on - plane. We denote this curve by . In Fig. 3, we depict as a function of in the domain , for several paris of and on the curve . It can be seen from Fig. 3 that the equation is satisfied only if or . Thus, if we impose the conditions (68)–(70), the inhomogeneous isotropic universe model can not satisfy at neither nor . This fact implies that we can, in principle, distinguish the inhomogeneous isotropic universe model from the FLRW universe model. Accordingly, we can remove the systematic error, if we use distance redshift relation, the CMBR angular power spectrum and the BAO scale at three distinct redshifts as observables.

Figure 1: We depict as a heat map on the - plane. The difference in the form of a point represent the difference of the density contrast at the observer . Circle means a point that absolute value of the density contrast at the observer is less than 0.1, whereas triangles means a point that at the observer is more than 0.1. The lower left corner of the blank is the domain in which the liner perturbation approximation is not vaild.
Figure 2: We depict as a function of the with various .
Figure 3: We depict as a function of in the domain , for several paris of and on the curve in - plane along which vanishes.

We compare the difference of the BAO scale between the inhomogeneous isotropic universe model and the FLRW universe model with the error in the observational data. The WiggleZ Dark Energy Survey has revealed Blake et al.(2011) (), and the ratio of the median and the error is 0.0328. Hence we find from Figs. 1 and 2 that it is impossible to distinguish the inhomogeneous isotropic universe model from the FLRW universe model by using the present observational data of the BAO scale, as long as and , since the maximum value of is less than We again consider the case that and on the curve . In this case, we need to compare the difference of the BAO scale between the inhomogeneous isotropic universe model and the FLRW universe model with the error in the observational data at other than . For example, we compare them at . The WiggleZ Dark Energy Survey has revealed Blake et al.(2011) (), and the ratio of the median and the error is 0.0468. It is larger than in Fig. 3, so that it is impossible to distinguish two universe models by using the present observational data.

In Fig. 4, we depict as a function of the redshift , if the inhomogeneous isotropic universe model satisfies the conditions in the domain , Eq. (69) and Eq. (70) only at of the FLRW universe model with various of the dark energy with . It can be seen from this figure that the larger is, the larger becomes, if is larger that 0.2. Thus, it is important to get observational data of the distance-redshift relation and the BAO scale at larger than unity.

Figure 4: We depict as a function of the redshift z. The FLRW universe model is filled with the non-relativistic matter and the dark energy of various and .

Vii Summary and discussion

We studied whether we can distinguish the inhomogeneous isotropic universe model with the cosmological constant from the FLRW universe model with dark energy other than the cosmological constant and remove the systematic error due to isotropic inhomogeneities, by considering multiple observables: the distance-redshift relation, the fluctuation spectrum of the CMBR and the BAO scale. We found that we can do so; There is no inhomogeneous isotropic universe model whose distance-redshift relation, fluctuation spectrum of the CMBR and the BAO scale are identical with those of the FLRW universe model, as long as the density perturbations are adiabatic. It is nontrivial that we can distinguish the inhomogeneous isotropic universe model from the FLRW universe model by using a finite number of observables, since the inhomogeneous isotropic universe model has a functional degree of freedom.

Here it should be noted that we have used not only the information about the universe on the past light cone but also that inside the past light cone, since we assumed that the BAO scale at the decoupling time is everywhere constant. However, there is a possibility that the BAO scale at the decoupling time is inhomogeneous, if the ratio between the energy densities of baryonic matter and CDM has been inhomogeneous at the decoupling time. If the ratio between the energy densities of baryonic matter and CDM is inhomogeneous and isotropic, we can not distinguish the inhomogeneous isotropic universe model from the FLRW universe model by only the distance-redshift relation, the fluctuation spectrum of the CMBR and the BAO scale. In the case of and , the inhomogeneous isotropic universe model can also explain observations, if the fluctuation of the ratio between the energy densities of baryonic matter and CDM at is (see Appendix A). It is very important to observe the ratio between the energy densities of baryonic matter and CDM in the domain .

Acknowledgments

We are grateful to Chul-Moon Yoo, Ryusuke Nishikawa, Hideki Ishihara and colleagues in the group of elementary particle physics and gravity at Osaka City University for useful discussions and helpful comments. K. N. was supported in part by JSPS KAKENHI Grant No. 25400265.

Appendix A The ratio of baryonic matter and CDM

Here, we discuss whether the inhomogeneous isotropic universe model can satisfy the all of the conditions (68)–(70), if the ratio between the energy densities of baryonic matter and CDM depends on . We assume that the comoving length scale of fluctuation of ratio between the energy densities of baryonic matter and CDM is the order of that of the present horizon and the energy density of radiation does not depend on .

The energy densities of non-relativistic matter and baryonic matter are write in the form,

(91)
(92)

where is the density contrast of baryonic matter. For simplicity, we assume that

(93)

where is arbitrary constant.

The condition (68) does not impose any constraints on the ratio between the energy densities of baryonic matter and CDM. By the assumption, the ratio between the energy densities of baryonic matter and CDM is almost spatially constant in the vicinity of the LSS, so that the condition (69) fixes and in the same way as in the case that the ratio between the energy densities of baryonic matter and CDM does not fluctuate. By contrast, the fluctuation of the ratio between the energy densities of baryonic matter and CDM affects the sound velocity of baryonic matter, hence may depend on . Since we consider the inhomogeneous isotropic universe model whose comoving length scale of inhomogeneities is comparable to that of the present horizon, can be obtained by regarding our inhomogeneous isotropic universe model as a homogenous isotropic universe model in each domain of BAO scale at decoupling time. Hence at each is approximately determined by the fitting formulae given in Ref. Eisenstein:1997ik (), since we have assumed that at the decoupling time is equal to the value at the LSS.

Here note that is a free parameter. We may choose so that the inhomogeneous isotropic universe model satisfies the conditions (68)–(70). For example, in the case of the FLRW universe model with and , the inhomogeneous isotropic universe model satisfies the conditions (68)–(70), if we assume . The fluctuation of the ratio between the energy densities of baryonic matter and CDM

(94)

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