Constraints on dark energy from H II starburst galaxy apparent magnitude versus redshift data

Physics Letters B 715 (2012) 9–14

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Physics Letters B www.elsevier.com/locate/physletb

Constraints on dark energy from H II starburst galaxy apparent magnitude versus redshift data Data Mania a,b,∗ , Bharat Ratra a a b

Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA Center for Elementary Particle Physics, Ilia State University, 3-5 Cholokashvili Ave., Tbilisi 0179, Georgia

a r t i c l e

i n f o

Article history: Received 12 January 2012 Received in revised form 30 June 2012 Accepted 6 July 2012 Available online 20 July 2012 Editor: S. Dodelson

a b s t r a c t In this Letter we use H II starburst galaxy apparent magnitude versus redshift data from Siegel et al. (2005) [74] to constrain dark energy cosmological model parameters. These constraints are generally consistent with those derived using other data sets, but are not as restrictive as the tightest currently available constraints. © 2012 Elsevier B.V. All rights reserved.

1. Introduction There is significant observational evidence supporting the idea that we live in a spatially-flat Universe that is currently undergoing accelerated cosmological expansion. Most cosmologists believe that this accelerated expansion is powered by dark energy which dominates the current cosmological energy budget (for reviews of dark energy see [7,36,71,79], and references therein).1 The energy budget of the “standard” model of cosmology – the spatially-flat ΛCDM model [46] – is currently dominated by far by dark energy: Einstein’s cosmological constant Λ contributes more than 70% of the budget. Nonrelativistic cold dark matter (CDM) is the next largest contributor (more than 20%), followed by nonrelativistic baryons (around 5%). For a review of the standard model see [62] and references therein. It has been know for some time that the ΛCDM model is reasonably consistent with most observational constraints (see, e.g., [1,18,30,84], for early indications).2 However, the ΛCDM model appears to leave some conceptual questions unanswered. One of these is that the naive energy density scale we expect from quantum field theory considerations is many orders of magnitude higher than the measured cosmological constant energy density scale. Another is what is known as the coincidence puzzle: The energy density of a cosmological constant

*

Corresponding author at: Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, KS 66506, USA. E-mail addresses: [email protected] (D. Mania), [email protected] (B. Ratra). 1 Some, however, feel that this accelerated expansion is better viewed as an indication that the general relativistic description of gravitation needs to be improved on (see [20,29,79], and references therein). In this Letter we assume that general relativity provides an adequate description of gravitation on cosmological scales. 2 The ΛCDM model assumes the “standard” CDM structure formation picture, which might be in some observational difficulty (see, e.g., [48,50]). 0370-2693/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physletb.2012.07.011

does not change over time but the matter density decreases with the cosmological expansion, so it is puzzle why we observers live at this seemingly special time, when the dark energy and nonrelativistic matter densities happen to be of comparable size. These puzzles, and perhaps others, could be alleviated if we assume that the dark energy density was higher in the past and slowly decreased in time, thus remaining comparable to the nonrelativistic matter density for a longer time [61]. Many such timevarying dark energy models have been proposed.3 In this Letter, for illustrative purposes, we consider two dark energy models and one dark energy parametrization. In the ΛCDM model, time-independent dark energy – the cosmological constant – may be modeled as a spatially homogeneous fluid with equation of state parameter w Λ = p Λ /ρΛ = −1 (where p Λ and ρΛ are the fluid pressure and energy density). In an attempt to describe slowly decreasing dark energy, the dark energy fluid equation of state is extended to the XCDM X -fluid parametrization. Here, dark energy is again modeled as a spatially homogeneous ( X ) fluid, but now with an equation of state parameter w X = p X /ρ X , where w X (< −1/3) is an arbitrary constant and p X and ρ X are the pressure and energy density of the X -fluid. When w X = −1 the XCDM parametrization reduces to the complete and consistent ΛCDM model. For any other value of w X (< −1/3), the XCDM parametrization is incomplete as it cannot describe spatial inhomogeneities (see, e.g., [59,60]). For computational simplicity, here we study the XCDM parametrization in only the spatially-flat cosmological case. A slowly rolling scalar field φ , with decreasing (in φ ) potential energy density V (φ), can be used to consistently model a gradually decreasing, time-evolving, dark energy density. An inverse

3

For recent discussions see, e.g., [16,24,27,32,37,44,52], and references therein.

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D. Mania, B. Ratra / Physics Letters B 715 (2012) 9–14

power-law potential energy density V (φ) ∝ φ −α , where α is a non-negative constant, provides a simple realization of this φ CDM scenario [47,61]. When α = 0 the φ CDM model reduces to the corresponding ΛCDM case. For computational simplicity, here we study the φ CDM model in only the spatially-flat cosmological case. Current observational data convincingly indicate that the cosmological expansion is accelerating. The main support for accelerated expansion comes from three types of data: supernova Type Ia (SNIa) apparent magnitude versus redshift measurements (see, e.g., [2,26,28,80]); cosmic microwave background (CMB) anisotropy data (see, e.g., [33,34,38,58]) combined with low estimates of the cosmological matter density (see, e.g., [13]); and, baryon acoustic oscillation (BAO) peak length scale estimates (see, e.g., [6,49,67,82]). However, the error bars associated with these three kinds of data are still too large to allow for a significant observational discrimination between the ΛCDM model and the two simple time-varying dark energy models discussed above. There are two major reasons to consider additional data sets. First, it is of great interest to check the above results by comparing to results derived from other data. If the new constraints differ significantly from the old ones, this could mean that the model being tested is observationally inconsistent, or it could mean that one of the data sets had an undetected systematic error. Either of these is a significant result. On the other hand, if the constraints from the new and the old data are consistent, then a joint analysis of all the data could result in significantly tighter constraints, and so possibly result in significantly discriminating between constant and time-varying dark energy. Other data that have recently been used to constrain dark energy models include strong gravitational lensing measurements (e.g., [5,11,35,86]), angular size as a function of redshift observations (e.g., [8,12,25]), Hubble parameter as a function of redshift measurements (e.g., [31,45,63,65,73]), galaxy cluster gas mass fraction data (e.g., [1,23,66,78]), and large-scale structure observations (e.g., [4,9,10,19,43]). While constraints from these data are less restrictive than those derived from the SNeIa, CMB and BAO data, both types of data result in largely compatible parameter restrictions that generally support a currently accelerating cosmological expansion. This gives us confidence that the broad outlines of the “standard” cosmological model are now in place. However, there is still ambiguity. Current observations are unable to differentiate between different dark energy models. For instance, while current data favor a timeindependent cosmological constant, they are unable to rule out time-varying dark energy. More and higher-quality data is needed to accomplish this important task. It is anticipated that future space missions will result in significantly more and better SNeIa, BAO, and CMB anisotropy data (see, e.g., [3,57,69,83]). A complementary approach is to develop cosmological tests that make use of different sets of objects. Recent examples include the lookback time test (e.g., [17,53]) and the gamma-ray burst luminosity versus redshift test (see, e.g., [72,81,85]). Gamma-ray bursts, in particular, are very luminous and can be seen to much higher redshift than the SNeIa and so probe an earlier cosmological epoch. H II starburst galaxies are also very luminous and can be seen to redshifts exceeding 3. Recent work has indicated that H II starburst galaxies might be standardizable candles [40,42,77], because of the correlation between their velocity dispersion, H β luminosity, and metallicity [39,41,76]. In this Letter we use H II galaxy data from [74] (the data itself is derived from [21,51]) to constrain parameters of the three dark energy models mentioned above. Plionis et al. [54–56] have used the Siegel et al. [74] data to constrain the XCDM parametrization. Here we also constrain parameters of the consistent and physically

motivated ΛCDM and φ CDM cosmological models. We also derive constraints on the parameters of these models and the XCDM parametrization from a joint analysis of the Siegel et al. [74] H II galaxy data and the Percival et al. [49] BAO peak length scale measurements. In the next section we list the basic equations for the dark energy models we consider. In Section 3 we discuss the analysis method used for the H II galaxy data and the resulting constraints and how they compare with those from other cosmological tests. Section 4 presents constraints from a joint analysis of H II, SNeIa and BAO data. Section 5 summarizes our results. 2. Dark energy model equations The Friedmann equation for the ΛCDM model with spatial curvature is



H ( z, H 0 , p) = H 0 Ωm (1 + z)3 + ΩΛ + Ω R (1 + z)2 .

(1)

Here H 0 is the Hubble constant and H ( z) is the Hubble parameter at redshift z, Ωm , ΩΛ , and Ω R = 1 − Ωm − ΩΛ are the nonrelativistic (baryonic and cold dark) matter, cosmological constant, and space curvature energy density parameters, respectively, and the model parameter set p = (Ωm , ΩΛ ). For the spatially-flat XCDM parametrization, the Hubble parameter is



H ( z, H 0 , p) = H 0 Ωm (1 + z)3 + (1 − Ωm )(1 + z)3(1+ w X ) .

(2)

Here w X < −1/3 and the model parameter set p = (Ωm , w X ). For the spatially-flat φ CDM case we have a coupled set of equations governing the scalar field and Hubble parameter evolution

κα

φ¨ + 3H φ˙ −

φ −(α +1) = 0,  H ( z, H 0 , p) = H 0 Ωm (1 + z)3 + Ωφ ( z).

(3)

2G

(4)

Here we consider the inverse power-law scalar field (φ) potential energy density [47]. In the early universe φ field accepts solution that we can take as an initial condition:



φ∝

2 3

3

α (α + 2)a α+2 .

(5)

Density parameter is

Ωφ ( z ) =

1

  κ ˙ 2 + φ −α , φ 2

(6)

G

12H 0

where G is the Newtonian gravitational constant and α > 0 is a free parameter (that determines κ ). Equation of state parameter is given by

wφ =

φ˙ 2 − kφ −α /G . φ˙ 2 + kφ −α /G

(7)

In this case the model parameter set is p = (Ωm , α ). It should be noted that φ CDM model cannot be approximated with XCDM parametrization (see, e.g., [59]). The distance modulus as a function of redshift is





μ(z, H 0 , p) = 25 + 5 log 3000(1 + z) y (z) − 5 log h,

(8)

where h is the dimensionless Hubble constant defined from H 0 = 100h km s−1 Mpc−1 . In the spatially open case y ( z) is

y ( z, p) = √

1

ΩR

sinh



z ΩR

dz E ( z )

 ,

(9)

0

where

z  E (z) = H (z)/ H 0 . When Ω R = 0 this reduces to y (z) = dz / E ( z ). The angular diameter distance d A = c y ( z)/[ H 0 (1 + z)]. 0

D. Mania, B. Ratra / Physics Letters B 715 (2012) 9–14

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Table 1 [74] H II starburst galaxy distance moduli and uncertainties. Galaxy

z

μobs ± σ

Q0201-B13

2.17

2.10 47.49+ −3.43

Q1623-BX432

2.18

Q1623-MD107

2.54

Q1700-BX717

2.44

CDFa C1

3.11

Q0347-383 C5

3.23

B2 0902+343 C12

3.39

Q1422+231 D81

3.10

SSA22a-MD46

3.09

SSA22a-D3

3.07

DSF2237+116a C2

3.32

B2 0902+343 C6

3.09

MS1512-CB58

2.73

1.97 45.45+ −3.07 0.31 44.82+ −1.58

0.31 46.64+ −1.58 0.31 45.77+ −1.58 0.44 47.12+ −0.32

0.71 46.96+ −0.81 0.38 48.81+ −0.40 0.56 46.76+ −0.51 0.43 49.71+ −0.41

Fig. 1. H II galaxy data 1, 2, and 3σ confidence level contours in the (Ωm , ΩΛ ) plane for the ΛCDM model. The dotted line corresponds to the spatially-flat ΛCDM case and the shaded area in the upper left hand corner is the part of parameter space 2 without a big bang. The best-fit point with χmin = 53.3 is indicated by the solid black circle at Ωm = 0.19 and ΩΛ = 0.98.

0.25 47.73+ −0.25

1.38 45.22+ −1.76

1.22 47.49+ −1.57

3. Constraints from H II galaxy apparent magnitude data We use the 13 μobs ( zi ) measurements of [74], listed in Table 1, to constrain cosmological parameters by minimizing

χH2 II ( H 0 , p) =

13  [μobs ( zi ) − μpred ( zi , H 0 , p)]2

σi2

i =1

.

(10)

Here zi is the redshift at which μobs ( zi ) is measured, μpred ( zi , H 0 , p) is the predicted distance modulus at the same redshift for the model under consideration, and σi is the average of the upper and lower error bars listed in Table 1. The Siegel et al. [74] measurements listed in Table 1 are computed from



μobs = 2.5 log

σ

5

F Hβ





− 2.5 log

O H



− A Hβ + Z0

(11)

where F H β and A H β are the H β flux and extinction and O / H is a metallicity. Following [55], for the zero point magnitude we use Z 0 = −26.60, use Hubble constant value H 0 = 73 km s−1 Mpc−1 (and do not account for the associated uncertainty), and also exclude two H II galaxies (Q1700-MD103 and SSA22a-MD41) that show signs of a considerable rotational velocity component [22]. The best fit parameter set p∗ is where χH2 II is at a mini2 mum, χmin . The 1σ , 2σ , and 3σ confidence intervals, in the twodimensional parameter space, are enclosed by the contours where 2 χ 2 = χmin + χ 2 with χ 2 = 2.30, χ 2 = 6.17, and χ 2 = 11.8, respectively. The H II galaxy data constraints on cosmological parameters of the three models are shown in Figs. 1–3. Our results generally agree with [55] for the XCDM parametrization (cf. our Fig. 2 and their Fig. 10). The small differences are due to the fact that in our analysis we do not use specially weighted sigmas, but rather average distance moduli uncertainties, and also ignore gravitational lensing effects. Both analyses ignore small H 0 uncertainties; these are not important for our illustrative purposes here, but should be accounted for in an analysis of improved near-future H II galaxy data. The H II galaxy data constraints in Figs. 1–3 are not as restrictive as those that follow from SNeIa, BAO, or CMB anisotropy data. They are, however, comparable to those from Hubble parameter observations (see [15], and references therein) or lookback time observations (see [64], and references therein), and somewhat more restrictive than angular diameter distance constraints

Fig. 2. H II galaxy data 1, 2, and 3σ confidence level contours in the (Ωm , w X ) plane for the spatially-flat XCDM parametrization. The dotted line corresponds to 2 the spatially-flat ΛCDM case. The best-fit point with χmin = 53.3 is indicated by the solid black circle at Ωm = 0.17 and w X = −0.86.

Fig. 3. H II galaxy data 1, 2, and 3σ confidence level contours in the (Ωm , α ) plane for the spatially-flat φ CDM model. α = 0 corresponds to the spatially-flat ΛCDM 2 case. The best-fit point with χmin = 53.3 is indicated by the solid black circle at Ωm = 0.17 and α = 0.39.

(see [14], and references therein) and gamma-ray burst luminosity distance ones (see [68], and references therein). We again note that we have not accounted for the uncertainty in H 0 in our analysis, so the H II galaxy constraints derived here are a little more restrictive than they really are. However, our analysis clearly illustrates the potential constraining power of near-future H II galaxy luminosity distance data.

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Fig. 4. Joint H II galaxy and SNeIa data (solid lines) and SNeIa data only (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , ΩΛ ) plane for the ΛCDM model. Conventions and notation are as in Fig. 1. The best-fit point for the first case with −2 log(Lmax ) = 610.4 is indicated by the solid black circle at Ωm = 0.23 and ΩΛ = 0.84 and for the second −2 log(Lmax ) = 555.9 is indicated as a diamond at Ωm = 0.36 and ΩΛ = 1.03.

Fig. 5. Joint H II galaxy and SNeIa data (solid lines) and SNeIa data only (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , w X ) plane for the spatiallyflat XCDM parametrization. Conventions and notation are as in Fig. 2. The best-fit point for the first case with −2 log(Lmax ) = 608.7 is indicated by the solid black circle at Ωm = 0.30 and w x = −1.34 and for the second −2 log(Lmax ) = 553.1 is indicated as a diamond at Ωm = 0.37 and w x = −1.65.

Fig. 6. Joint H II galaxy and SNeIa data (solid lines) and SNeIa data only (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , α ) plane for the spatiallyflat φ CDM model. Conventions and notation are as in Fig. 3. The best-fit point for the first case with −2 log(Lmax ) = 610.7 and for the second −2 log(Lmax ) = 557.4 is indicated by the solid black circle at Ωm = 0.21 and α = 0.

Fig. 7. Joint H II galaxy and BAO data (solid lines) and BAO data only (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , ΩΛ ) plane for the ΛCDM model. Conventions and notation are as in Fig. 1. The best-fit point with −2 log(Lmax ) = 55.2 is indicated by the solid black circle at Ωm = 0.25 and ΩΛ = 0.95.

4. Joint constraints from SNeIa, BAO and H II galaxy data For the SNeIa analysis we use the SCP Union2.1 compilation of redshift versus distance modulus relation (580 points) [75]. We minimize the function 2 χSN = (μ)T C −1 μ

where μ is a vector consisting of differences μi = μobs ( zi ) − μpred (zi , H 0 , p), the Hubble constant value used is H 0 = 73 km s−1 Mpc−1 , and C is the covariance matrix. The SNeIa data constraint contours are shown in Figs. 4–6. To constrain observables using BAO data we follow the procedure of [49]. With D V ( z) = [(1 + z)2 d2A cz/ H ( z)]1/3 (where d A is angular diameter distance), [49] measure

¯ V (0.275) D

−0.134  −0.255  Ωm h2 Ωb h2 Mpc. = (1104 ± 30) 0.02273 0.1326

We construct

(12)

2 χBAO = ( D¯ V − D V (0.275, H 0 , p))2 /σ D2¯ and use this V

to build the likelihood estimator LBAO with a Gaussian prior of Ωm h2 = 0.1326 ± 0.0063, and neglect the error for Ωb h2 as WMAP5 data constrains it to 0.5% [33]

Fig. 8. Joint H II galaxy and BAO data (solid lines) and BAO data only (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , w X ) plane for the spatially-flat XCDM parametrization. Conventions and notation are as in Fig. 2. The best-fit point with −2 log(Lmax ) = 53.5 is indicated by the solid black circle at Ωm = 0.25 and w x = −1.41.

LBAO (p) 2 2 2 χ2

− (Ωm h −Ω2m h ) − BAO 2σ e 2 d Ωm h2 = e

 e

− (Ωm h

2 −Ω h2 )2 m 2σ 2



d Ωm h2 .

The BAO data constraint contours are shown in Figs. 7–9.

(13)

D. Mania, B. Ratra / Physics Letters B 715 (2012) 9–14

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Fig. 9. Joint H II galaxy and BAO data (solid lines) and BAO data only (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , α ) plane for the spatially-flat φ CDM model. Conventions and notation are as in Fig. 3. The best-fit point with −2 log(Lmax ) = 55.6 is indicated by the solid black circle at Ωm = 0.27 and α = 0.

Fig. 11. Joint H II galaxy, SNeIa and BAO data (solid lines) and SNeIa and BAO data (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , w X ) plane for the spatially-flat XCDM parametrization. Conventions and notation are as in Fig. 2. The best-fit point for the first case with −2 log(Lmax ) = 609.4 is indicated by the solid black circle at Ωm = 0.26 and w x = −1.19 and for the second −2 log(Lmax ) = 555.3 is indicated as a diamond at Ωm = 0.27 and w x = −1.2.

Fig. 10. Joint H II galaxy, SNeIa and BAO data (solid lines) and SNeIa and BAO data (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , ΩΛ ) plane for the ΛCDM model. Conventions and notation are as in Fig. 1. The best-fit point for the first case with −2 log(Lmax ) = 611.0 is indicated by the solid black circle at Ωm = 0.26 and ΩΛ = 0.89 and for the second −2 log(Lmax ) = 555.9 is indicated as a diamond at Ωm = 0.27 and ΩΛ = 0.89.

Fig. 12. Joint H II galaxy, SNeIa and BAO data (solid lines) and SNeIa and BAO data (dashed lines) 1, 2, and 3σ confidence level contours in the (Ωm , α ) plane for the spatially-flat φ CDM model. Conventions and notation are as in Fig. 3. The best-fit point for the first case with −2 log(Lmax ) = 616.6 and for the second −2 log(Lmax ) = 562.0 is indicated by the solid black circle at Ωm = 0.26 and α = 0.

To derive joint H II galaxy and SNeIa (Figs. 4–6); H II and BAO (Figs. 7–9); and the combined HII, SNeIa and BAO constraints (Figs. 10–12) we maximize the products of likelihoods L(p) = LH II LSN , L(p) = LH II LBAO and L(p) = LH II LSN LBAO respectively

to get the best fit set of parameters p∗ , where LSN = e −χSN /2 and 2 LH II = e −χH II /2 . The 1, 2, and 3σ contours defined as points where the likelihood equals e −2.30/2 , e −6.17/2 , and e −11.8/2 of the maximum likelihood value. For comparison we have given SNIa and BAO data only constraints in Figs. 10–12. These figures illustrate that future improved H II data set can well complement SNeIa and BAO contours. Clearly, even adding currently available H II galaxy data to SNeIa or BAO data and to their combination makes the constraints tighter. It should be noted again that H II galaxy data itself is not yet of good enough quality and in the above analyses we are forced to ignore uncertainties on the Hubble constant. Therefore, contours appearing in the figures are tighter than they should be. 2

5. Conclusion We have used the starburst of [74] to constrain cosmological models. The resulting constraints that follow from other currently

galaxy luminosity distance data parameters in some dark energy are largely consistent with those available data although they are

not as restrictive as the constraints derived from SNeIa, BAO, and CMB anisotropy data. The [74] H II data are preliminary H II data. It is gratifying that they result in interesting constraints on cosmological parameters. We anticipate that new, near-future, H II galaxy data will provide very useful constraints on cosmological parameters that will complement those derived using other data. Acknowledgements D.M. thanks Manolis Plionis, Glenn Horton-Smith, and Yun Chen for valuable suggestions. In our numerical calculation we used Armadillo C++ linear algebra library [70]. We acknowledge support from DOE grant DEFG030-99EP41093, NSF grant AST-1109275, and from the SNSF (SCOPES grant No. 128040). References [1] [2] [3] [4] [5] [6] [7]

S.W. Allen, et al., MNRAS 383 (2008) 879. R. Amanullah, et al., Astrophys. J. 716 (2010) 712. P. Astier, et al., arXiv:1010.0509 [astro-ph.CO], 2010. M. Baldi, V. Pettorino, MNRAS 412 (2011) L1. M. Biesiada, A. Piórkowska, B. Malec, MNRAS 406 (2010) 1055. C. Blake, et al., arXiv:1108.2635 [astro-ph.CO], 2011. A. Blanchard, Astron. Astrophys. Rev. 18 (2010) 595.

14

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47]

D. Mania, B. Ratra / Physics Letters B 715 (2012) 9–14

M. Bonamente, et al., Astrophys. J. 647 (2006) 25. N. Brouzakis, et al., J. Cosmol. Astropart. Phys. 1103 (2011) 049. L. Campanelli, et al., arXiv:1110.2310 [astro-ph.CO], 2011. K.-H. Chae, et al., Astrophys. J. Lett. 607 (2004) L71. G. Chen, B. Ratra, Astrophys. J. 582 (2003) 586. G. Chen, B. Ratra, PASP 115 (2003) 1143. Y. Chen, B. Ratra, arXiv:1105.5660 [astro-ph.CO], 2011. Y. Chen, B. Ratra, Phys. Lett. B 703 (2011) 406. F.E.M. Costa, Phys. Rev. D 82 (2010) 103527. M.A. Dantas, et al., Phys. Lett. B 699 (2011) 239. T.M. Davis, et al., Astrophys. J. 666 (2007) 716. C. De Boni, et al., MNRAS 415 (2011) 2758. A. de Felice, S. Tsujikawa, Living Rev. Relativ. 13 (2010) 3. D.K. Erb, et al., Astrophys. J. 591 (2003) 110. D.K. Erb, et al., Astrophys. J. 646 (2006) 107. S. Ettori, et al., Astron. Astrophys. 501 (2009) 61. H. Farajollahi, et al., Astrophys. Space Sci. (2011) 563. E.J. Guerra, R.A. Daly, L. Wan, Astrophys. J. 544 (2000) 659. J. Guy, et al., Astron. Astrophys. 523 (2010) A7. T. Harko, F.S.N. Lobo, Eur. Phys. J. C 70 (2010) 373. T. Holsclaw, et al., Phys. Rev. Lett. 105 (2010) 241302. B. Jain, J. Khoury, Ann. Phys. 325 (2010) 1479. H.K. Jassal, J.S. Bagla, T. Padmanabhan, MNRAS 405 (2010) 2639. R. Jimenez, et al., Astrophys. J. 593 (2003) 622. Z. Keresztes, et al., Phys. Rev. D 82 (2010) 123534. E. Komatsu, et al., Astrophys. J., Suppl. Ser. 180 (2009) 330. G. La Vacca, S.A. Bonometto, Nucl. Phys. Proc. Suppl. 217 (2011) 68. S. Lee, K.-W. Ng, Phys. Rev. D 76 (2007) 043518. E.V. Linder, arXiv:1004.4646 [astro-ph.CO], 2010. D.-J. Liu, H. Wang, B. Yang, Phys. Lett. B 694 (2010) 6. J. Liu, M. Li, X. Zhang, J. Cosmol. Astropart. Phys. 1106 (2011) 028. J. Melnick, Astron. Astrophys. 70 (1978) 157. J. Melnick, ASP Conf. Proc. Ser. 297 (2003) 3. J. Melnick, R. Terlevich, M. Moles, MNRAS 235 (1988) 297. J. Melnick, R. Terlevich, E. Terlevich, MNRAS 311 (2000) 629. M.J. Mortonson, W. Hu, D. Huterer, Phys. Rev. D 83 (2011) 023015. B. Novosyadlyj, et al., Phys. Rev. D 82 (2010) 103008. N. Pan, et al., Class. Quantum Grav. 27 (2010) 155015. P.J.E. Peebles, Astrophys. J. 284 (1984) 439. P.J.E. Peebles, B. Ratra, Astrophys. J. Lett. 325 (1988) L17.

[48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86]

P.J.E. Peebles, B. Ratra, Rev. Mod. Phys. 75 (2003) 559. W.J. Percival, et al., MNRAS 401 (2010) 2148. L. Perivolaropoulos, J. Phys. Conf. Ser. 222 (2010) 012024. M. Pettini, et al., Astrophys. J. 554 (2001) 981. V. Pettorino, et al., Phys. Rev. D 82 (2010) 123001. N. Pires, Z.-H. Zhu, J.S. Alcaniz, Phys. Rev. D 73 (2006) 123530. M. Plionis, et al., AIP Conf. Proc. 1241 (2010) 267. M. Plionis, et al., MNRAS 416 (2011) 2981. M. Plionis, et al., J. Phys. Conf. Ser. 189 (2009) 012032. S. Podariu, P. Nugent, B. Ratra, Astrophys. J. 553 (2001) 39. S. Podariu, et al., Astrophys. J. 559 (2001) 9. S. Podariu, B. Ratra, Astrophys. J. 532 (2000) 109. B. Ratra, Phys. Rev. D 43 (1991) 3802. B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406. B. Ratra, M.S. Vogeley, PASP 120 (2008) 235. L. Samushia, G. Chen, B. Ratra, arXiv:0706.1963 [astro-ph], 2007. L. Samushia, et al., Phys. Lett. B 693 (2010) 509. L. Samushia, B. Ratra, Astrophys. J. Lett. 650 (2006) L5. L. Samushia, B. Ratra, Astrophys. J. Lett. 680 (2008) L1. L. Samushia, B. Ratra, Astrophys. J. 701 (2009) 1373. L. Samushia, B. Ratra, Astrophys. J. 714 (2010) 1347. L. Samushia, et al., MNRAS 410 (2011) 1993. C. Sanderson, Tech. rep., NICTA, 2010. D. Sapone, Int. J. Mod. Phys. A 25 (2010) 5253. B.E. Schaefer, Astrophys. J. 660 (2007) 16. A.A. Sen, R.J. Scherrer, Phys. Lett. B 659 (2008) 457. E.R. Siegel, et al., MNRAS 356 (2005) 1117. N. Suzuki, et al., Astrophys. J. 746 (2012) 85. R. Terlevich, J. Melnick, MNRAS 195 (1981) 839. E. Terlevich, R. Terlevich, J. Melnick, Rev. Mex. Astron. Astrof. (Ser. Conf.) 12 (2002) 272. M. Tong, H. Noh, Eur. Phys. J. C 71 (2011) 1586. S. Tsujikawa, arXiv:1004.1493 [astro-ph.CO], 2010. S. Wang, X.-D. Li, M. Li, Phys. Rev. D 83 (2011) 023010. Y. Wang, Phys. Rev. D 78 (2008) 123532. Y. Wang, Mod. Phys. Lett. A 25 (2010) 3093. Y. Wang, et al., MNRAS 409 (2010) 737. K.M. Wilson, G. Chen, B. Ratra, Mod. Phys. Lett. A 21 (2006) 2197. L. Xu, Y. Wang, J. Cosmol. Astropart. Phys. 1011 (2010) 014. Q.-J. Zhang, Y.-L. Wu, J. Cosmol. Astropart. Phys. 1008 (2010) 038.