Cosmological model with local symmetry of very special relativity and constraints on it from supernovae

Abstract

Based on Cohen & Glashow's very special relativity (Cohen and Glashow in Phys. Rev. Lett. 97:021601, 2006 ), we propose an anisotropic modification to the Friedmann-Robertson-Walker (FRW) line element. An arbitrarily oriented 1-form is introduced and the FRW spacetime becomes of the Randers-Finsler type. The 1-form picks out a privileged axis in the universe. Thus, the cosmological redshift as well as the Hubble diagram of the type Ia supernovae (SNe Ia) becomes anisotropic. By directly analyzing the Union2 compilation, we obtain the privileged axis pointing to (l,b)=(304^sup ^±43^sup ^,-27^sup ^±13^sup ^) (68 % C.L.). This privileged axis is close to those obtained by comparing the best-fit Hubble diagrams in pairs of hemispheres. It should be noticed that the result is consistent with isotropy at the 1σ level since the anisotropic magnitude is D=0.03±0.03.

Eur. Phys. J. C (2013) 73:2459DOI 10.1140/epjc/s10052-013-2459-x

Regular Article - Theoretical Physics

Cosmological model with local symmetry of very special relativityand constraints on it from supernovae

Zhe Chang1,2,a, Ming-Hua Li1,b, Xin Li1,2,c, Sai Wang1,d

1Institute of High Energy Physics, Chinese Academy of Sciences, 100049 Beijing, China

2Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, 100049 Beijing, China Received: 26 February 2013 / Revised: 21 May 2013 / Published online: 4 June 2013 Springer-Verlag Berlin Heidelberg and Societ Italiana di Fisica 2013

Abstract Based on Cohen & Glashows very special relativity (Cohen and Glashow in Phys. Rev. Lett. 97:021601, 2006), we propose an anisotropic modication to the Fried-mannRobertsonWalker (FRW) line element. An arbitrarily oriented 1-form is introduced and the FRW spacetime becomes of the RandersFinsler type. The 1-form picks out a privileged axis in the universe. Thus, the cosmological red-shift as well as the Hubble diagram of the type Ia super-novae (SNe Ia) becomes anisotropic. By directly analyzing the Union2 compilation, we obtain the privileged axis pointing to (l, b) = (304 43, 27 13) (68 % C.L.). This

privileged axis is close to those obtained by comparing the best-t Hubble diagrams in pairs of hemispheres. It should be noticed that the result is consistent with isotropy at the 1 level since the anisotropic magnitude is D = 0.03 0.03.

1 Introduction

The cosmological principle is one of foundations of the standard cosmological model, i.e., the CDM model [1]. It says that the universe is statistically homogeneous and isotropic at large scale. The CDM model is well consistent with present cosmological observations, such as the Wilkinson Microwave Anisotropy Probe (WMAP) [2] and the Sloan Digital Sky Survey (SDSS) [3], etc. However, there exist challenges for the standard cosmological model (see review in Ref. [4]), such as the large-scale cosmic ows [5], the alignment of low multipoles in the CMB spectra [68], the large-scale alignment of the quasar polarization vectors [9].

a e-mail: mailto:[email protected]

Web End [email protected]

b e-mail: mailto:[email protected]

Web End [email protected]

c e-mail: mailto:[email protected]

Web End [email protected]

d e-mail: mailto:[email protected]

Web End [email protected]

One of their resolutions refers to a privileged axis at large scale in the universe [10].

The type Ia supernovae (SNe Ia) have been used to search for the possible anisotropy of the universe [1122], since they were employed to discover the cosmic acceleration [23, 24]. Especially, Antoniou & Perivolaropoulos [10] used the hemisphere comparison method to analyze the Union2 data [25] and found a direction (l, b) =

(309+233 , 18+11

10 ) for the maximum accelerating expan

sion of the universe. Similarly, Cai & Tuo [26] found a preferred direction (l, b) = (314+20

33 ) for the

cosmological deceleration parameter. Most recently, Kalus et al. [27] also found that the highest expansion rate of the universe towards the direction (l, b) (35, 19)

(95 % C.L.).

A privileged axis, as is mentioned above, may account for the anisotropic phenomenologies of the observational astrophysics and cosmology. Actually, the issue of privileged axis has been studied extensively in the very special relativity (VSR) [28]. The Finslerian spacetime structure d = ( dx dx)

1b

13 , 28+11

2 (n dx )b was proved to be invariant under the DISIMb(2) group [29]. There is a preferred axis n in the above line element. It could characterize the anisotropy of the at spacetime, which leads to the Lorentz invariance violation (LIV). Locally, the Randers metric was proved to possess the symmetry of the group TE(2) [30, 31]. The group TE(2) is a semi-product of T (4) and E(2) [32]. The group E(2) denotes a proper subgroup of the Lorentz group with three generators [28]. Therefore, Randers space-time could possess local symmetry of the generic VSR.

In this paper, we propose a Randers line element (structure) [33] with local symmetry of the generic VSR to estimate the possible anisotropy of the Hubble diagram of SNe Ia. At rst order, we show the related Friedmann equation which is the same as that in the standard model. A modied cosmological redshift formula is presented. The red-

Page 2 of 6 Eur. Phys. J. C (2013) 73:2459

shift is direction dependent and refers a privileged axis. This may imply that the universe undergoes an anisotropic expansion. Next, we show a modied luminosity-distance vs. red-shift relation for the SNe Ia. It is certainly anisotropic. The Union2 compilation [25] is used to constrain the direction of the privileged axis. The rest of the paper is arranged as follows. In Sect. 2, the Friedmann equation is presented in the Randers spacetime. The anisotropic Hubble diagram is showed at rst order in Sect. 3. In Sect. 4, we investigate the Union2 dataset of the supernovae to constrain the level of the anisotropy of the universe. Conclusions and discussions are listed in Sect. 5.

2 The Friedmann equation in Randers spacetime

In the CDM model, the cosmic spacetime is described by the spatially at FriedmannRobertsonWalker (FRW) line element [1]

d = [radicalBig]dt2 a2(t)[parenleftbig]dx2 + dy2 + dz2[parenrightbig], (1) where a(t) denotes the scale factor of the universe at the time t. The cosmological redshift z is given by

1 + z(t) =

a(t0)

a(t)

[parenleftbig][parenleftbig]a2 2a a[parenrightbig] b23 a a[parenleftbig] b23[parenrightbig][parenrightbig] = 8G, (5)

where the dots denote the temporal derivative d

dx0 and is the energy density of cosmic inventory. We notice that the second term on the left hand side involves only the second-order effects, which are proportional to b23 and its temporal

derivative. This term could be dropped at the rst-order approximation. Thus, the Friedmann equation (5) reduces to the conventional Friedmann equation as

[parenleftbigg] a

a

1 a4

1a(t), (2)

where we set a(t0) 1 for today. The redshift describes

the expansion rate of the universe between t and t0. The overline denotes physical objects in the FRWRiemannian spacetime.

The FRW line element should be modied when there exists a privileged axis in the universe. We postulate that an extra 1-form is added into the FRW structure. The 1-form singles out a privileged axis in the universe. The spacetime structure becomes

d d + b(x) dx. (3) Actually, the spacetime structure (3) belongs to Randers type [33]. This is a class of Finsler spacetime [34]. The 1-form could be viewed as an arbitrarily oriented electromagnetic 4-potential in the universe. It may be the relic of a primordial magnetic eld at large scales [3538].

In the following sections, we will show that the anisotropy of Hubble diagram stems from the spatial components (3-vector) of the preferred axis b. In addition, we can

choose the coordinate system such that the spatial 3-vector is the third spatial axis. In this way, the 1-form becomes

b dx = b3(t) dz. Here b3 is set to evolve with only t.

Without loss of generality, thus, the Randers structure (3) could be simplied as

d = [radicalBig]dt2 a2(t)[parenleftbig]dx2 + dy2 + dz2[parenrightbig] + b3(t) dz. (4)

Following Stavrinos et al.s approach [39], the above Randers structure could be approximated as one osculating Riemann metric via the osculating Riemannian method [40]. The Randers spacetime evolves following along the conventional Einsteins gravitational eld equations in this approach [41].

In the osculating Riemannian method, we could view the velocity coordinates as functions of the position, i.e. y = y(x), by restricting the velocity y dxd to an individ

ual tangent space for the given position x. In this way, the Finsler geometric quantities are approximated by the corresponding osculating Riemannian quantities. The Einsteins gravitational eld equations could be obtained by computing the connection and the curvature of the osculating Riemannian metric g(x) g(x, y(x)), see details in the refer

ences [3942]. After a calculation of the time-time component of Einsteins eld equations, one obtains the Friedmann equation in the Randers spacetime. The Friedmann equation is given by

3

[parenleftbigg] a

a

2

+

=

2 8G

3 . (6)

This result reveals that the anisotropic effect is secondary for the dynamical evolution of the Randers spacetime (4). However, the anisotropic effect could have signicant inuence on the kinematical behaviors of the Randers spacetime. We will discuss this issue in the following section.

3 The anisotropic Hubble diagram

In the FRWRanders spacetime (3), the cosmological red-shift z has been derived via resolving the Finslerian geodesic equations, which are given by [43]

d2x0

d2 + ij aa

dxi d

dxjd +

dx0d f [parenleftbigg]x,

dx d

[parenrightbigg] = 0, (7)

d2xi

d2 + 2ij a

a

[parenrightbigg] = 0, (8)

where f (x, dxd ) b| dxd dxd /F . Here b| denotes the co

variant derivative of b(x) with respect to the FRW met

ric [34]. The above geodesic equations have solutions as

dx0 d

dxjd +

dxid f [parenleftbigg]x,

dx d

Eur. Phys. J. C (2013) 73:2459 Page 3 of 6

a dx0d J1 and a2 dxid J1, where J1 = 1 b[hatwide]

p [43].

Thus, the redshift z could be written as

1 + z(t, [hatwide]

p) =

1 b[hatwide]p[parenrightbig], (9)

where the unit 4-vector

p denotes a light-like direction towards each SN Ia. It is anisotropic since there is a privileged axis b(x) in the above formula.

When the privileged axis is a spatial 3-vector, the modied redshift z could be rewritten as

1 + z(t,[hatwide]

p) =

1 a(t)

dt a(t )

, (15)

which is the same as the conventional one. It could be rewritten as

dL = (1 + z) [integraldisplay]

1

(1+z)

da

ma3 + , (16)

where m (8G/3H20)m0 and /3H20, m0 is the

critical mass density, is the cosmological constant, and H0 is the Hubble constant. Here we assume that the energy density of cosmic inventory is critical today, i.e., m + = 1.

In the derivation of (16), we have used the conventional Friedmann equation and the denition H a/a as a rst-

order approximation.

By substituting the redshift (11) into Eq. (16), we obtain the anisotropic distanceredshift relation as

H0dL = (1 + z) [integraldisplay]

z

1 a(t)

1 D([hatwide]n [hatwide] p)

, (10)

where

n is the unit 3-vector for the spatial components of

b, and D denotes the magnitude which is smaller than one. To simplify the following discussions, we choose the spatial 3-vector

n as the third spatial axis of the coordinate system. Thus, the redshift z becomes

1 + z(t, cos ) =

1 a2H0

1a [1 D cos ], (11) where denotes the angle between

n and p. This formula shows clearly that the cosmological redshift is direction dependent and anisotropic. Thus, the light undergoes an anisotropic propagation of the spatial dipole form in the universe.

The Hubble diagram would be anisotropic and have a corresponding privileged axis which exists in the Randers spacetime. We will discuss this proposition in the following paragraphs. For the null geodesic, the spacetime line element vanishes, i.e., d = 0. In the FRWRanders spacetime,

the Randers structure (3) should vanish. Thus, we obtain

dt2 a2 dr2 = (D cos dr)2, (12) where we use the polar coordinates and

n is set again as the third spatial axis. Here the angular coordinates are discarded since they are irrelative to the distanceredshift relation. Finally, the above equation could be simplied as

dt = [radicalbig]a2 + D2 cos2 dr. (13) The privileged axis emerges again, which affects the propagation of the cosmic light. This result is different from that of dt = a(t) dr in the CDM model. The modica

tion is quadrupolar in Eq. (13). Obviously, the cosmic light would propagate with different speeds in different directions in the space. However, the quadrupolar effect is a second-order effect. It just affects the propagation of the cosmic light slightly. Thus, we could disregard it in the following discussion and Eq. (13) becomes the conventional one, dt = a(t) dr.

In the observational universe, the luminosity-distance dL of a SN Ia is given by [44]

dL (1 + z)r, (14)

where r denotes the comoving distance between us and the SN Ia. We could obtain the comoving distance r by integrating Eq. (13) at the rst order of D cos . Therefore, the luminosity-distance vs. redshift relation can be obtained as

dL = (1 + z) [integraldisplay]

t0

t(z)

A 1 dz

0 m( A1+z )3 + , (17)

where A 1 D cos . At rst order, the above equation

becomes

H0dL = (1 + z) [integraldisplay]

z

B(z , cos ) dz

0 m(1 + z )3 + , (18)

where B denotes the dipolar effect,

B(z, cos ) 1 +

1/2 + /m(1 + z)3

1 + /m(1 + z)3

D cos . (19)

These relations refer to the anisotropic Hubble diagram of SNe Ia. The direction dependence is obvious in the above equations since different modications are introduced to the distanceredshift relation of SNe Ia towards different spatial directions. Once again, we see that the 1-form in the Randers structure leads to the privileged axis in the universe. When D vanishes, both A and B reduce back to 1. Thus, the

FRWRanders spacetime returns back to the CDM model.

4 Numerical results from supernovae

We dene a Cartesian coordinate system for the unit 3-vector

p corresponding to each SN Ia with the equatorial coordinates (, ), in which

p is given as

p cos() cos()i + cos() sin()j + sin()k, (20) where i, j and k are basis 3-vectors. Suppose

n is given by

n cos(0) cos(0)i + cos(0) sin(0)j + sin(0)k, (21) then the cosine of the angle between these two vectors is

Page 4 of 6 Eur. Phys. J. C (2013) 73:2459

cos [hatwide]

n [hatwide]

p. (22)

Substituting the relation (22) into (19), the theoretical distance modulus could be calculated from (18). It is given as

th(z, , ; 0, 0, D) = 5 log10[bracketleftbig]dL(Mpc)[bracketrightbig] + 25. (23) Here th has the direction dependence and the privileged axis points towards the direction (0, 0) in the equatorial coordinate system.

We base our numerical study on the Union2 dataset [25]. The Union2 compilation consists of 557 SNe Ia. Thus, we perform the least-2 t to the data of these SNe Ia to determine the privileged axis (0, 0) and the geometrical param-

Fig. 1 The distance modulus vs. redshift relation of the SNe Ia in the FRWRanders spacetime. The (black) dots and errorbars denote experimental data which comes from Union2 compilation [25]. The (red) triangles denote theoretical predictions for the SNe Ia in our model (Color gure online)

eter D (viewed as a constant for simplicity). The 2 statistic in our t is

2

557 [th(zi, i, i; 0, 0, D) obs(zi)]2

(zi)2

, (24)

where th(zi, i, i; 0, 0, D) is the theoretical distance

modulus. obs(zi) and (zi), respectively, denote the observational values of the distance modulus and measurement errors, which are obtained from the Union2 compilation. In the FRWRanders spacetime, D is at the level of 0.01,

which leads to only rst-order term to Eq. (18).

We rst carry the least-2 t of the -CDM model to the Union2 compilation. We obtain the best-t parameters,i.e., the matter component m = 0.27 0.02, and the Hub

ble parameter H0 = 70.0 0.4 (the unit is km s1 Mpc1

throughout the paper). Note that these tted parameters are consistent with those obtained by the WMAP [2]. In the following, the parameters m and H0 are xed to the above values as the center values. Then we employ the least-2 method (24) to obtain the anisotropic magnitude D and the direction of privileged axis (0, 0). The results are given as

D = 0.03 0.03, and (0, 0) = (263 43, 91 13)

in the equatorial coordinate system or (l, b) = (304

43, 27 13) in the galactic coordinate system, for a

minimum value of 2min 0.97. All the results are pre

sented in a sense of 68 % C.L.. Correspondingly, the distance modulus vs. redshift relation of the SNe Ia is shown in Fig. 1. The direction (point A) of privileged axis obtained by this direct analysis is close to those obtained by comparing the best-t Hubble diagrams in pairs of hemispheres, see Fig. 2. For example, Antoniou & Perivolaropoulos [10] showed (l, b) = (309+23

3 , 18+11

10 ), Cai & Tuo [26] got

(l, b) = (314+20

33 ), and Kalus et al. [27] obtained

(l, b) (35, 19) (95 % C.L.).

13 , 28+11

Fig. 2 The direction of privileged axis as well as its possible variation in the galactic coordinate. The point A denotes the direction of privileged axis obtained in this paper (l, b) = (304 43, 27

13) (68 % C.L.) with the xed parameters m = 0.27 and H0 = 70.0

as the center values. For the 1 -error variation of m (H0 = 70.0), the

privileged axis points to the direction (l, b) = (294, 31) (point B)

with m = 0.29, and to (l, b) = (317, 26) (point C) with m =

0.25. For the 1 -error variation of H0 (m = 0.27), it points to the

direction (l, b) = (283, 34) (point D) with H0 = 70.4, and to

(l, b) = (326, 22) (point E) with H0 = 69.6. For contrast, we also

show other possible directions of the privileged axis obtained via the hemisphere comparison method [10, 26, 27]

Eur. Phys. J. C (2013) 73:2459 Page 5 of 6

We have xed the parameters m and H0 in the above t to nd the privileged axis. Related to these center values, the direction of privileged axis is denoted by the point A in Fig. 2 as is mentioned above. To see the possible variation of the direction of privileged axis, we vary, respectively, the m and H0 by 1 error with respect to their center values,i.e., m = 0.27 and H0 = 70.0. For the 1 -error variation of

m (H0 = 70.0), the privileged axis points towards to the di

rection (l, b) = (294, 31) (point B) with m = 0.29, and

to (l, b) = (317, 26) (point C) with m = 0.25. For the

1 -error variation of H0 (m = 0.27), it points to the direc

tion (l, b) = (283, 34) (point D) with H0 = 70.4, and to

(l, b) = (326, 22) (point E) with H0 = 69.6. The above

obtained directions are also depicted in Fig. 2. We could see that all these directions are close to each other. Thus, the above results reveal that the direction of privileged axis remains almost same, when the cosmological parameters m and H0 are allowed to vary by 1 error, respectively.

5 Conclusions and remarks

In this paper, we proposed an anisotropic modication to the FRW line element. The modied line element refers to the Randers spacetime, which possesses the local symmetry of Cohen & Glashows VSR. The local symmetry involves the group TE(2). The Euclidean group E(2) contains three generators T1 Kx + Jy, T2 Ky Jx, and Jz, where Ki

and Ji (i = x, y, z) denote the generators of boosts and ro

tations, respectively. The former two generators form a two-parameter group of translations in the xy plane. Thus, the local FRWRanders spacetime is cylindrically symmetric.However, the parity violates in the z-direction. Otherwise, the E(2) would be enlarged to the Lorentz group [28]. Actually, the 1-form b dx in the Randers structure changes

its sign under the direction reversal dx/d dx/d.

This reveals that the Randers structure is asymmetric. On the other hand, the indicatrix of the FRWRanders spacetime is quadratic, while the center of the indicatrix is displaced in the z-direction [33]. The above two accounts imply a privileged axis in the FRWRanders spacetime.

The existence of the privileged axis contradicts with the cosmological principle and implies a statistically anisotropic universe. We extracted the direction (l, b) = (304 43, 27 13) (68 % C.L.) for the privileged axis bi, based on

the Union2 compilation of the SNe Ia. This direction is close to those obtained by comparing the best-t Hubble diagrams in pairs of hemispheres. It is noteworthy that the preferred direction coincides within its error bars with the direction

d1 of partial mirror antisymmetry of CMB temperature uctuations [45]. In addition, one should note that our result is consistent with isotropy at the 1 level. The reason is that the anisotropic magnitude was obtained as D = 0.03 0.03

in our best-t.

It is noteworthy that the Randers spacetime belongs to Finsler geometry [34]. Actually, Finsler geometry gets rid of the quadratic restriction on the spacetime structure (line element) [46]. It is a natural generalization of Riemann geometry and includes Riemann geometry as a special case. Finsler spacetime could depend on certain preferred directions of the spacetime background [29, 4752]. This could also be revealed via the isometric transformation [30, 53, 54]. There are no more than (d(d1)2 + 1) Killing vectors in the d-

dimensional Finsler spacetime [53]. Otherwise, the Finsler spacetime becomes Riemannian. Thus, it might be a reasonable candidate to account for the privileged axis and the anisotropic properties of the universe.

Acknowledgements We are thankful for useful discussions with Jian-Ping Dai, Yunguo Jiang, Danning Li, and Hai-Nan Lin. We are very grateful to Prof. Shuang-Nan Zhang who provides us the galactic coordinates of SNe Ia in the Union2 compilation. This work is supported by the National Natural Science Fund of China under Grant No. 11075166.

References

1. S. Weinberg, Cosmology (Oxford University Press, New York,

2008)

2. E. Komatsu, et al. (WMAP collaboration), Astrophys. J. Suppl. Ser. 192, 18 (2011)

3. B.A. Reid et al., Mon. Not. R. Astron. Soc. 404, 60 (2010)4. L. Perivolaropoulos. arXiv:1104.05395. R. Watkins, H.A. Feldman, M.J. Hudson, Mon. Not. R. Astron. Soc. 392, 743 (2009)

6. C.L. Bennett et al., Astrophys. J. Suppl. Ser. 192, 17 (2011)7. K. Land, J. Magueijo, Phys. Rev. Lett. 95, 071301 (2005)8. M. Tegmark, A. de Oliveira-Costa, A. Hamilton, Phys. Rev. D 68, 123523 (2003)

9. D. Hutsemekers, R. Cabanac, H. Lamy, D. Sluse, Astron. Astrophys. 441, 915 (2005)

10. I. Antoniou, L. Perivolaropoulos, J. Cosmol. Astropart. Phys. 1012, 012 (2010)

11. T.S. Kolatt, O. Lahav, Mon. Not. R. Astron. Soc. 323, 859 (2001)12. C. Bonvin, R. Durrer, M. Kunz, Phys. Rev. Lett. 96, 191302 (2006)

13. C. Gordon, K. Land, A. Slosar, Phys. Rev. Lett. 99, 081301 (2007)14. D.J. Schwarz, B. Weinhorst, Astron. Astrophys. 474, 717 (2007)15. S. Gupta, T.D. Saini, T. Laskar, Mon. Not. R. Astron. Soc. 388,

242 (2008)

16. T.S. Koivisto, D.F. Mota, J. Cosmol. Astropart. Phys. 0806, 018 (2008)

17. M. Blomqvist, J. Enander, E. Mortsell, J. Cosmol. Astropart. Phys. 10, 018 (2010)

18. A. Cooray, D.E. Holz, R. Caldwell, J. Cosmol. Astropart. Phys. 11, 015 (2010)

19. S. Gupta, T.D. Saini, Mon. Not. R. Astron. Soc. 407, 651 (2010)20. T.S. Koivisto, D.F. Mota, M. Quartin, T.G. Zlosnik, Phys. Rev. D 83, 023509 (2011)

21. J. Colin, R. Mohayaee, S. Sarkar, A. Shaeloo, Mon. Not. R. Astron. Soc. 414, 264 (2011)

22. L. Campanelli, P. Cea, G.L. Fogli, A. Marrone, Phys. Rev. D 83, 103503 (2011)

23. A.G. Riess et al., Astron. J. 116, 1009 (1998)

Page 6 of 6 Eur. Phys. J. C (2013) 73:2459

24. S. Perlmutter et al., Astron. J. 517, 565 (1999)25. R. Amanullah et al., Astrophys. J. 716, 712 (2010)26. R.-G. Cai, Z.-L. Tuo, J. Cosmol. Astropart. Phys. 1202, 004 (2012)

27. B. Kalus, D.J. Schwarz, M. Seikel, A. Wiegand, arXiv:1212.369128. A.G. Cohen, S.L. Glashow, Phys. Rev. Lett. 97, 021601 (2006)29. G.W. Gibbons, J. Gomis, C.N. Pope, Phys. Rev. D 76, 081701 (2007)

30. X. Li, Z. Chang, Differ. Geom. Appl. 30, 737 (2012)31. L. Zhang, X. Xue, arXiv:1205.113432. L. Zhang, X. Xue, arXiv:1204.642533. G. Randers, Phys. Rev. 59, 195 (1941)34. D. Bao, S.S. Chern, Z. Shen, An Introduction to RiemannFinsler Geometry. Graduate Texts in Mathematics, vol. 200 (Springer, New York, 2000)

35. T. Kahniashvili, G. Lavrelashvili, B. Ratra, Phys. Rev. D 78, 063012 (2008)

36. J.D. Barrow, P.G. Ferreira, J. Silk, Phys. Rev. Lett. 78, 3610 (1997)

37. L. Campanelli, Phys. Rev. D 80, 063006 (2009)38. J. Kim, P. Naselsky, J. Cosmol. Astropart. Phys. 07, 041 (2009)

39. P.C. Stavrinos, A.P. Kouretsis, M. Stathakopoulos, Gen. Relativ. Gravit. 40, 1403 (2008)

40. H. Rund, The Differential Geometry of Finsler Spaces (Springer, Berlin, 1959)

41. G.S. Asanov, Finsler Geometry, Relativity and Gauge Theories (Reidel, Dordrecht, 1995)

42. Z. Chang, S. Wang, arXiv:1303.605843. Z. Chang, S. Wang, X. Li, Eur. Phys. J. C 72, 1838 (2012)44. S. Dodelson, Modern Cosmology (Elsevier, Singapore, 2008)45. F. Finelli, A. Gruppuso, F. Paci, A.A. Starobinsky, J. Cosmol. Astropart. Phys. 07, 049 (2012)46. S.S. Chern, Not. AMS 959 (1996)47. G.Yu. Bogoslovsky, Il Nuovo Cimento B 40, 99 (1977)48. G.Yu. Bogoslovsky, Il Nuovo Cimento B 40, 116 (1977)49. G.Yu. Bogoslovsky, Il Nuovo Cimento B 43, 377 (1978)50. Z. Chang, S. Wang, Eur. Phys. J. C 72, 2165 (2012)51. Z. Chang, S. Wang, Eur. Phys. J. C 73, 2337 (2013)52. V. Balan, G.Yu. Bogoslovsky, S.S. Kokarev, D.G. Pavlov, S.V. Siparov, N. Voicu, J. Mod. Phys. 3, 1314 (2012)

53. H.C. Wang, J. Lond. Math. Soc. s1-22(1), 5 (1947)54. S.F. Rutz, Contemp. Math. 196, 289 (1996)