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Distance-Redshift Relations in an Anisotropic Cosmological Model

R. S. Menezes Jr.1, C. Pigozzo2 and S. Carneiro2,31Instituto Federal da Bahia, Salvador, BA, Brazil

2Instituto de Fsica, Universidade Federal da Bahia, Salvador, Bahia, Brazil3International Centre for Theoretical Physiscs, Trieste, Italy

(Dated: October 11, 2012)

In this paper we study an anisotropic model generated from a Bianchi III metric, generalization of

Godels metric, which is an exact solution of Einsteins field equations. In particular, we analysesupernova Ia data, namely the SDSS sample calibrated with the MLCS2k2 fitter, verifying in whichlimits of distances and redshifts the anisotropy of the model could be observed, and in which limitsthe model approaches the CDM model. We found that redshifts above z 2 has a particularimportance in setting the point at which the anisotropy would be noticed, as well as the point atwhich the present model begins to diverge from the CDM. We conclude that data above this limit,as well as an increasing accuracy in measuring these distances, might indicate the existence of suchan anisotropy in the universe.

PACS numbers:

I. INTRODUCTION

The search for evidence of possible anisotropies in the universe has been studied by several authors [1, 2, 3, 4, 5].The importance of these studies lies in the fact that the discovery of a possible anisotropy could lead to a betterunderstanding of the mechanism of the Big Bang and of the small temperature fluctuations detected in the cosmicmicrowave background radiation (CMB) [6]. In addition, such a discovery would bring down the belief in the validityof the cosmological principle that is the basis of many interpretations of observations.

In order to detect possible signatures of anisotropy, in this paper we consider an anisotropic model constructedfrom an anisotropic (but homogeneous) metric with expansion, which is an exact solution of Einsteins equations.This metric, whose first particular case was proposed by Godel in 1949 [7], is classified as Bianchi III and was studiedin a more general form by M. Reboucas and J. Tiomno [8], and by V. Korotkii and Y. Obukhov [9] (from now on wewill call it the RTKO metric). In general it has non-zero rotation (but zero shear), as well as a conformal expansion.Here we will take the particular case where the rotation is made zero [10, 11]. In this case, it becomes1

ds2 = a2()

d2 (dx2 + e2xdy2 + dz2)

, (1)

where is the conformal time.As is well known, a perfect fluid does not lead to anisotropies. Therefore, in order to account for them, a scalar field

is introduced, which will be responsible for creating a pressure difference in the preferred direction, which we define asbeing the z direction. In addition, we will introduce a cosmological constant responsible for the present accelerationin the expansion of the universe. Although the metric is anisotropic, it provides an strictly isotropic CMB (at thebackground level) due to the existence of a conformal Killing vector parallel to the fluid 4-velocity [9, 12]. In whatconcerns the distance-redshift relations, we will consider here 288 supernovae Ia with redshifts between 0.0218 and1.551, collected by the Sloan Digital Sky Survey II (SSDS-II) [ 13] and calibrated with the Multicolor Light CurveShapes (MLCS2k2) fitter [14].

The work is organized as follows. In section 2 we present the main properties of the RTKO metric. In Section 3 webuild the correspondent cosmological model. In Section 4 we show the method of determining the model parameters.In section 5 we obtain the results for the anisotropic model and, for comparison, for the CDM model. Finally, insection 6 we present our conclusions.

rsmjr@ifba.edu.br cpigozzo@ufba.br saulo.carneiro.ufba@gmail.com Associate Member1 We are using the natural sistem of units, with 8G = c = 1.

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II. THE RTKO METRIC

Before proceeding to the construction of the anisotropic model, let us discuss some properties of the RTKO metric.If we apply the coordinate transformations

ex = cosh r + cos sinh r, (2)

yex = sin sinh r, (3)

to metric (1), we put it in the cylindrical form

ds2 = a2()(d2 dr2 sinh2 rd2 dz2), (4)which is invariant under rotations around the z-axis. Now, making use of the transformations

r = sin , (5)

z = cos , (6)

the above line element can be expressed in spherical coordinates as

ds2 = a2()[d2 d2 2d2 sinh2( sin )d2]. (7)The anisotropy is clear in the last term. It will lead to an angular dependence in the angular-diameter distance,

as we shall see. Interestingly enough, the metric expressed in the later form reduces to the spatially flat Friedmann-Lematre-Robertson-Walker (FLRW) space-time in the limit of small distances. Indeed, taking the expansion of thehyperbolic sine function up to first-order terms, we obtain

ds2 a2()[d2 d2 2(d2 sinh2 d2)], (8)i.e., the flat FLRW metric expressed in spherical coordinates. As a consequence, the anisotropic model to be builtwill reduce to the standard model in the limit of small distances.

The RTKO metric has Killing vectors

(1) = x yy, (2) = y, (3) = z, (9)which classify it as a spatially homogeneous Bianchi type III metric. It also has a conformal Killing vector

conf = 0 . (10)

The 4-velocity of a comoving fluid in this space-time is given by

u =dx

ds=

0a()

=confa()

. (11)

Let us consider now the Einstein field equations

R1

2R = T

. (12)

When we use metric (1), we obtain for the energy-momentum tensor the diagonal components

T00 a4 = 3a2 a2, (13)T11 a

4 = T22 a4 = 2aa a2, (14)

T33 a2 = T11 a

2 1, (15)while all the non-diagonal ones are zero. Here, the prime denotes derivative with respect to the conformal time. Inthe next section we discuss the cosmological models arising from the above metric.

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III. THE ANISOTROPIC MODEL

Let us define

T00 = , Tii = pi (16)

as the total energy density and pressures of the universe content, where the index i takes the values 1, 2 and 3,representing the pressures in the directions x, y e z, respectively. Equations (13)-(15) then become

a4 = 3a2 a2, (17)p1a

4 = p2a4 = a2 2aa, (18)

p3a2 = p1a

2 + 1. (19)

This set of equations shows clearly the anisotropy of the metric, since the pressure along direction z differs fromthe others two by an additional term 1/a2. The usual components of the universe (radiation, matter and cosmologicalconstant) can not generate such an anisotropy. Let us show that it can be generated, in a self-consistent way, by amassless scalar field minimally coupled to gravity. Such a scalar field satisfies the Klein-Gordon equation

; =1g (

g,g), = 0, (20)

and has an energy-momentum tensor given by

T = ,,g 1

2,,g

. (21)

Here g is the determinant of the metric tensor g, and is the delta of Kronecker. The commas and semicolons

refer, respectively, to ordinary and covariant derivatives.Our scalar field will be given by

(z) = Cz , (22)

where C is a constant. It is easy to verify that it is solution of equation (20). On the other hand, the non-zerocomponents of its energy-momentum tensor (the diagonal ones) are

T00 = T11 = T22 = T33 = C2

2a2, (23)

which lead, through (16), to

(s) =C2

2a2, (24)

p(s)1 = p

(s)2 = p(s)3 =

C2

2a2, (25)

where the upper index (s) refers to the contribution of the scalar field. We note that the pressures are anisotropic,and the energy density falls with the square of the scale factor. Using this fact we can work out the anisotropy presentin Einsteins equations.

Let us define

+ (s), pi pi + p(s)i , (26)where the bar refers to the isotropic content (radiation, matter and cosmological constant). Equation (15) takes theform

p3a2 = p1a

2 C2 + 1. (27)We then see that the anisotropy of the model is absorbed by the scalar field if (and only if) C2 = 1. In this case wehave p1 = p2 = p3 = p, and the scalar field is simply given by

(z) = z. (28)

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The energy density and pressures are now given by

(s) =1

2a2, (29)

p(s)1 = p

(s)2 = p(s)3 =

1

2a2, (30)

and the Einstein equations can be rewritten as

a4

= 3a2

3

2 a2

, (31)

pa4 = a2 2aa + a2

2. (32)

Notably, these equations are precisely those of an open FLRW model with curvature k = 1/2. In fact, from (31)we obtain the Friedmann equation

H2 =

3+

1

2a2, (33)

where we have defined the Hubble parameter H = a/a2.On the other hand, from the continuity equation

a+ 3H( + p) = 0 (34)

we can write the energy densities of radiation, non-relativistic matter and the cosmological constant respectively asr =

Aa4 , m =

Ba3 , and = , with A, B and constants. The total energy density of the isotropic content is then

= r + m + =A

a4+

B

a3+ . (35)

The Friedmann equation (33) assumes now the form

H2 =A

3a4+

B

3a3+

3+

1

2a2. (36)

It can also be written as

(s) = 1r m , (37)where, for matter, radiation and the cosmological constant, we have defined j = j/3H

2 (j = r, m or ), and

(s) 12H2a2

. (38)

Owing to its particular dependence on the scale factor, we will call this last term the curvature of the anisotropicmodel. However, let us point out that it contains, besides the space-time curvature itself, the contribution of theenergy density of the scalar field responsible for the anisotropy.

By using the relation

1 + z =a0a

, (39)

we obtain, from (36) and (37),

H(z)

H0 E(z) =

r0(1 + z)4 + m0(1 + z)3 + (1r0 m0 )(1 + z)2 + , (40)

where the index 0 refers to the present time. On the other hand, the deceleration parameter is defined by

q 1 aa

a2, (41)

or, as a function of the redshift, by

q(z) = 1 + (1 + z)H(z)

dH

dz, (42)

while the age parameter is given by

H0t0 =

0

1

1 + z

H0H(z)

dz. (43)

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IV. PARAMETERS ESTIMATION

In order to establish a comparison between the anisotropic model and the standard one (here denominated k-CDMto include the spatially curved cases), we have performed an analysis of 288 supernovas Ia compiled by the SDSSSupernova Survey with the fitter MLCS2k2. The best-fit values of the free parameters of both models were determinedby means of the usual 2 statistics.

A. The k-CDM case

In this case we use the modulus distance

(z) =

5log

1 + z|k0 | sin

|k0|

z0

dz

E(z)

5log(h) + 42.38 (k = +1)

5log

(1 + z)

z0

dz

E(z)

5log(h) + 42.38 (k = 0)

5log

1 + z

|k0 |sinh

|k0|

z0

dz

E(z)

5log(h) + 42.38 (k = 1),

(44)

where h H0/100 km/s-Mpc, and

E(z) =

m0(1 + z)3 + (1 r0 m0 )(1 + z)2 + . (45)After marginalising h and fixing r0 with the CMB constraints, the free parameters are m0 and .

B. The anisotropic case

The FLRW metric is isotropic. This means that, given two galaxies with the same redshift, they are at the sameluminosity distance from us. In the case of an anisotropic metric that is not true. For a given redshift we can obtaindifferent distances, i.e., the brightness depends on the distance and on the angle of observation. We need therefore toobtain the correct luminosity distance for our metric.

With this goal we will define the angular-diameter distance by [4, 12]

dA2 =

dApd

, (46)

where dAp is the proper-area element formed by the solid angle d = sin dd. The reason for this choice is the factthat, in an anisotropic metric, the usual definition

dA =dl

d(47)

(where d is the angle associated to the distance dl between two observed points) would lead to different distances dlfor the same dA as the observation angle changes.

The area element in the anisotropic case can be obtained by taking our spatial metric and fixing , which leads tothe line element

dl2 = a2()[2d2 + sinh2( sin )d2], (48)

from which we can see that the proper-area element is given by

dAp() = a2() sinh( sin )dd, (49)

which, after dividing by d, leads to

dA() = a()

sinh( sin )

sin

1/2. (50)

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In order to perform an observational analysis, we can rewrite equation (50) in terms of the redshift,

dA(z) =a0

1 + z

sinh( sin )

sin

1/2, (0, ). (51)

On the other hand, the coordinate can be obtained by taking a null geodesic,

d = d

=

1

a0H0

Z(z), (52)

where we have defined

Z(z) z0

dz

E(z). (53)

Therefore, the luminosity distance will be given by

dL(z) = (1 + z)2dA(z) =

(1 + z)Z(z)

H0

sinh(

|k0|Z(z)sin )|k0|Z(z)sin

1/2, (0, ). (54)

We can note that, in the limit 0 or 1, the luminosity distance reduces to that of the spatially flat FLRWmodel. Let us also remind that

|k0

|= 2

(s)0 (see (38)), which permits to rewrite the above equation in terms of

(s)0 .

In order to fix the free parameters of the present model,

m0, (s)0

(after marginalizing h), we use the angular

average of the luminosity distance in the interval [0, ], for each value of z. The average modulus distance will begiven by

(z) = 5 log[H0dL(z)] 5log(h) + 42.38, (55)where dL(z) is the average of (54) in the interval [0, ].

V. RESULTS AND DISCUSSION

With the procedure described above, we have obtained the best fit for the k-CDM model with parameters m0 =0.39, = 0.58 and k0 = 0.03, with

2r = 0.84. Therefore, in the absence of other priors, the data favors a slightly

negative curvature. Within 2 confidence level, we obtain 0.16 < m0 < 0.58 and 0.12 < < 0.95. For thecurvature we have, within the same confidence level, 0.51 < k0 < 0.71.

For the anisotropic case, the best fit corresponds to m0 = 0.38, = 0.58 and (s) = 0.04, with 2r = 0.84. The

2 intervals are 0 < m0 < 0.48, 0.04 < < 0.67 and 0 < (s) < 0.95. The confidence regions for both models

are presented in Fig. 1. In Table I we compare the parameters best values for both models, as well as the presentdeceleration parameters q0 and the age parameters H0t0, obtained from equations (42) and (43), respectively.

In order to set the level of agreement between the anisotropic model and the observational data, in Fig. 2a we showa comparison between the observed distance moduli and the theoretical predictions. The solid lines correspond tothe maximum ( = /2), average and minimum ( = 0) values of the distance modulus. In this comparison we haveused (s) = 0.52, which is the maximum curvature allowed (within the 2 level) when we fix the matter densityparameter in m0 = 0.25. We can see that, for the range of redshifts available in SDSS, the theoretical predictionsare within the observed limits.

As shown above, the best fit requires a small curvature (a spatially open universe) in the k-CDM model. There

is then space for confusing a possible anisotropy with an actual space curvature. In order to verify in which range ofredshifts the two models may be confused, we show in Fig. 2b the average distance modulus as a function of z in both

Model m0 k0|(s)0 r

2 q0 H0t0

k-CDM 0.39 0.03 0.58 0.840 -0.386 0.888

Anisotropic 0.38 0.04 0.58 0.840 -0.384 0.890

TABLE I: Parameters best values for the k-CDM and anisotropic models fitted with the sample SDSS (MLCS2k2).

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0.0 0.2 0.4 0.6 0.8

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

m0

0.0 0.2 0.4 0.6 0.8 1.0

0.5

0.0

0.5

1.0

m0

k0

0

s

FIG. 1: Confidence levels 1, 2 and 3 for the k-CDM (white) and anisotropic (blue and white) models with SDSS(MLCS2k2).

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FIG. 2: (a) Distance modulus versus redshift for the anisotropic model (minimum, medium and maximum) and (b) for themodels with best fit parameters for z 2.

models, for the corresponding best-fit values of their parameters. We can note that both agree with the observationaldata, and that the curves superpose even for redshifts above z 2 (see Fig. 3a). However, when we use the maximumcurvature allowed (within 2 level) for m0 = 0.25 (which corresponds to

(s) = 0.52 and k0 = 0.64), the modelspredictions start to diverge (see Fig. 3b), indicating that, above the upper limit of the used data, the test is moresensitive to a possible anisotropy, which could, in principle, be detected.

VI. CONCLUSION

We have studied the observational viability of an anisotropic cosmology based on a Bianchi III metric whichgeneralizes the Godel solution, in the particular case where the cosmic rotation is made zero. As shown elsewhere,such a metric leads to an exactly isotropic CMB (at the background level) and, in addition, it is an exact solution ofthe Einstein equations in the presence of an anisotropic, minimally coupled scalar field.

This metric induces an anisotropy in the luminosity distance of galaxies with same redshifts but observed in differentdirections. This effect can, in principle, be used to detect a preferential axis in the sky through the observation ofsupernovas Ia at high redshifts. In the present work we have made use of the SDSS supernova sample calibrated withthe MLCS2k2 fitter, since it is less dependent on a CDM fiducial model as compared to SALT or SALT II fitters.For comparison, we have also analysed the CDM model, with any possible curvature.

Our analysis have shown that, in both cases, a small negative curvature (or anisotropy) is allowed by observation.

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FIG. 3: (a) Distance modulus versus redshift for the models with best fit parameters for z 10 and (b) for the models withmaximum curvature parameters. The CDM is represented by the blue line and the anisotropic model by the red line.

On the other hand, the present data do not distinguish between the two models, even when we consider the maximum

permitted curvature/anisotropy. A more robust conclusion about the existence of an anisotropy at cosmological scalesmay, however, be reached when supernovas data with higher redshifts are available.

Acknowledgements

The authors acknowledge CNPq - Brazil for the grants under which this work was carried out.

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http://arxiv.org/abs/astro-ph/0008106http://arxiv.org/abs/astro-ph/0008106