Growth index of matter perturbations in the light of Dark Energy Survey
Abstract
We study how the cosmological constraints from growth data are improved by including the measurements of bias from Dark Energy Survey (DES). In particular, we utilize the biasing properties of the DES Luminous Red Galaxies (LRGs) and the growth data provided by the various galaxy surveys in order to constrain the growth index () of the linear matter perturbations. Considering a constant growth index we can put tight constraints, up to accuracy, on . Specifically, using the priors of the Dark Energy Survey and implementing a joint likelihood procedure between theoretical expectations and data we find that the best fit value is in between and . On the other hand utilizing the Planck priors we obtain and . This shows a small but nonzero deviation from General Relativity (), nevertheless the confidence level is in the range . Moreover, we find that the estimated mass of the darkmatter halo in which LRGs survive lies in the interval and , for the different bias models. Finally, allowing to evolve with redshift [Taylor expansion: ] we find that the parameter solution space accommodates the GR prediction at levels.
pacs:
98.80.k, 98.80.Bp, 98.65.Dx, 95.35.+d, 95.36.+x1 Introduction
The past and present analysis of various cosmological data (SNIa, Cosmic Microwave BackgroundCMB, Baryonic Acoustic OscillationsBAOs, Hubble parameter measurements etc) converge to the following cosmological paradigm, the observed Universe is spatially flat and the cosmic fluid consists of luminous (baryonic) matter, dark matter and some sort of dark energy (hereafter DE) which plays a key role in explaining the accelerated expansion of the universe (cf. Hicken2009 (); Komatsu2011 (); Blake2011 (); Hinshaw2013 (); Farooq2013 (); Ade2013 (); Aghanim:2018eyx () and references therein). Despite the fact that there is an agreement among the majority of cosmologists concerning the ingredients of the cosmic fluid however, there are different explanations regarding the physical mechanism which causes the accelerated expansion of the universe. In brief, the general avenue that one can design in order to study cosmic acceleration is to treat DE either as a new field in nature or as a modification of General Relativity (see for review Copeland2006 (); Caldwell2009 (); Amendola2010 ()).
It has been proposed Linder2004 (); Linder2007 (); Mar2014 (); Bas2016 () that in order to discriminate between scalar field DE and modified gravity one may utilize the evolution of the linear growth of matter fluctuations . In particular, we introduce the growth rate of clustering, which is given by , where is the linear growth factor (normalized to unity at the present epoch), is the scale factor of the universe, is the dimensionless matter density parameter and is the so called growth index Peebles1993 (); Wang1998 (). In fact the determination of the growth index is considered one of the main targets in these kind of studies because it can be used in order to test General Relativity (GR) on extragalactic scales, even in a model independent fashion Ness2015 (). Indeed, in the literature one may find a large family of studies in which the functional form of the growth index is given analytically for several cosmological models namely, scalar field DE Linder2007 (); Wang1998 (); Silveira1994 (); Nesseris2008 (); Basilakos2012 (), DGP Linder2007 (),Wei2008 (); Gong2008 (); Fu2009 (), Gannouji2009 (); Tsujikawa2009 (), BasFT () FinslerRanders Basilakos2013 (), running vacuum models Basola2015 (), clustered and Holographic dark energy Mehra2015 ().
From the view point of large scale structure, the study of the distribution of matter on extragalactic using different mass tracers (galaxies, AGNs, clusters of galaxies etc) provides important constraints on theories of structure formation. Specifically, owing to the fact that gravity reflects, via gravitational instability, on the physics of clustering Peebles1993 () it is natural to utilize the clustering/biasing properties of the extragalactic mass tracers in constraining cosmological models (see Matsubara2004 (); Basilakos2005 (); Basilakos2006 (); Krumpe2013 ()) as well as to test the validity of GR on cosmological scales (see Basilakosetal2012 (),Bean2013 ()). Following the above lines, in the current article we combine the linear bias data of Luminous Red Galaxies (hereafter LRGs; DES2017 ()), recently released by the DES group, with the growth rate data as provided by Sargedo et al. Sarg (), in order to place constraints on . Notice that is the dark matter halo in which the LRG live.
The structure of the paper is as follows. In section II we present the DES bias data and the growth data. In section III we provide the family of basic bias models, while in section IV we discuss the evolution of linear matter fluctuations. The outcome of our analysis is presented in section V, while our main conclusions can be found in section VI.
2 DESY1 Red Galaxies Bias Data and Growth data
In a sequence of previous theoretical articles we have proposed to use the biasing properties of extragalactic sources in order to constrain the growth index of matter fluctuations Basilakosetal2012 (). Therefore, in the light of recent Dark Energy Survey (DES) bias data, we attempt to compare the predictions of the most popular linear bias models (see below) with the data. Specifically, the DES bias data DES2017 () were extracted in the context of the angular correlation function (ACF) using the 1year DES sample of LRGs as tracers of the LSS. This population of galaxies can be observed in the redshift range . It is important to mention, that during the derivation of the bias data ElvinPoole et al. DES2017 () imposed the assumption of linear bias. Indeed, following the notations of Krause et. al. Krause2017 () the scale of Mpc, used by the DES team, ensures that the impact of nonlinear effects on biasing is practically negligible. In Table 1 we list the numerical values of the DES bias data with the corresponding errors. Regarding the cosmic expansion we restrict the present analysis to DES/Planck/JLA/BAO CDM cosmology, namely , , , and Abbot2017 (). In this context, the normalized Hubble parameter of the CDM model is written as
(2.1) 
Red. Range  Median Redshift  DESY1 bias 

In addition to DES bias data, we use in our analysis the growth data and the corresponding covariances as collected by Sargedo et al. Sarg () (see their Table I and references therein). This sample contains 22 entries for which the product is available as a function of redshift, where is the growth rate of clustering. It is well known that the product is almost a modelindependent parametrization of expressing the observed growth history of the universe Song09 ().
3 Bias Models
Let us first briefly present the main bias models. In particular, from the so called merging bias family we include here the models of Sheth, Mo & Tormen She2001 (), Jing Jing1998 ()and De Simeone et al. deSim2011 ().
For these models the bias factor is written as a function of the peakheight parameter, where is the linearly extrapolated density threshold above which structures collapse. Here we use the accurate fitting formula of Weinberg & Kamionkowski Wein () to estimate . Moreover, the mass variance is written as
(3.1) 
where is the tophat smoothing kernel with , is the halo mass and is the present value of the mean matter density, namely /Mpc. The quantity is the CDM linear power spectrum given by where is the spectral index of the primordial power spectrum and is the CDM transfer function provided by EisensteinHu ():
(3.2) 
with , , and with being is the shape parameter given by Sugiyama1995 ():
(3.3) 
Taking the aforementioned quantities into account and using Eq.(3.1) the normalization of the power spectrum becomes
(3.4) 
where .
Below we provide some details concerning the bias models.
3.1 SMT and JING
Sheth, Mo & Tormen (She2001 (), hereafter SMT) based on the ellipsoidal collapse model they found the following bias formula
(3.5) 
with
(3.6) 
Using Nbody simulations they evaluated the free parameters of the model,
Also, Jing Jing1998 () proposed the following bias form
(3.7) 
3.2 Dmr
De Simone et. al. deSim2011 () (hereafter DMR) generalized the original PressSchether formalism incorporating a nonMarkovian extension with a stochastic barrier. In this model, the critical value for spherical collapse was assumed to be a stochastic variable, whose scatter reflects a number of complicated aspects of the underlying dynamics. Therefore, the bias factor is
(3.8) 
3.3 Bpr
In addition to merging bias models we shall use the generalized model of Basilakos et al. Basilakosetal2012 () (hereafter BPR). This form of bias is valid for any dark energy model including those of modified gravity. In this case, using the hydrodynamic equations of motion, linear perturbation theory and the FriedmannLemaitre solutions a second differential equation of bias is derived Basilakosetal2012 (). The solution of the differential equation is given by:
(3.9) 
with where is the bias factor at the present time. The integration constants and can be found in Basilakosetal2012 (), namely
(3.10) 
and
(3.11) 
4 Evolution of liner growth
In this section we discuss the main points of the linear growth of matter fluctuations via which the growth index, , enters in the current analysis. Focusing on subhorizon scales the differential equation that governs the linear matter perturbations (Linder2004 (); Linder2007 (); Lue2004 (); Stabenau2006 (); Uzan2007 (); Tsujikawa2008 () and references therein) is given by
(4.1) 
where is the matter density, with being the Newton’s gravitational constant, while the effects of modified gravity are encapsulated in the quantity . Of course for those DE models which adhere to General Relativity reduces to , hence .
The solution of the aforementioned equation (4.1) is , where is the growth factor. For any type of gravity the growth rate of clustering is given by the following useful parametrization Peebles1993 (); Wang1998 (); Linder2007 ()
(4.2) 
and thus we have
(4.3) 
where and is the growth index. Notice that the growth factor is normalized to unity at the present epoch.
Now, inserting the operator and Eq.(4.2) into Eq.(4.1) we arrive at
(4.4) 
Considering the concordance CDM model, namely it is easy to show that
(4.5) 
where . In this case the Hubble parameter , where is given by Eq.(2.1) and is the Hubble constant^{1}^{1}1For the comoving distance and for the dark matter halo mass we use the traditional parametrization km/s/Mpc. Of course, when we treat the power spectrum shape parameter we utilize Abbot2017 ()..
Generally speaking the growth index may not be a constant but rather evolve with redshift; . In this framework, substituting Eq.(4.2) into Eq.(4.4) we find
(4.6) 
where the prime denotes derivative with respect to redshift. In the present work we restrict our analysis to the following two cases Basilakos2012 (); Polarski2008 (); Bal08 (); Belloso2009 ():
(4.7) 
Using the latter parametrization, which is nothing else but a Taylor expansion around , together with Eq.(4.6) evaluated at the present time (), we can write the parameter in terms of
(4.8) 
At large redshifts () the asymptotic value of the growth index becomes . In general, plugging into Eq. (4.8) we can define the constants as a function of . For examble, in the case of , , and , the above calculations give . .
5 The Likelihood analysis
In this section we provide the statistical method that we adopt in order to constrain the growth index, presented in the previous section. We implement a standard minimization analysis in order to constrains either the parameter space.Specifically, in our case the situation is as follows:
(1) For the DES biasing cosmological probe we use
(5.1) 
where the various forms of the theoretical are given in section III. Notice that , where and are the uncertainties of the observed bias and redshift respectively (see Table I).
(2) Regarding the analysis of the growthrate data we use
(5.2) 
where and is the inverse covariance matrix Sarg (). The theoretical growthrate is given by:
(5.3) 
The vectors and provide the free parameters that enter in deriving the theoretical expectations. The first vector includes the free parameters which are related to the expansion and the environment of the parent dark matter halo in which the LRGs DES galaxies live. Specifically, for constant we have , while for the case of evolving , the vector is defined as: . We remind the reader that the cosmological parameters are given in section II Abbot2017 ().
Since we wish to perform a joint likelihood analysis of the two cosmological probes and owing to the fact that likelihoods are defined as , the overall likelihood function becomes
(5.4) 
which is equivalent to:
(5.5) 
Based on the above we will provide our results for each free parameter that enters in the vector. The uncertainty of each fitted parameter will be estimated after marginalizing one parameter over the other, providing as its uncertainty the range for which .
As a further quality measure over the fits, we have used the AIC Akaike1974 () criterion, in a modified form that is appropriate for small data sets, Liddle:2007fy (). Considering Gaussian errors AIC is given by
(5.6) 
where is the total number of data and is the number of fitted parameters (see also Liddle:2007fy ()). Of course, a smaller value of AIC implies a better modeldata fit. In order to test the performance of the different bias models in fitting the data we need to utilize the model pair difference, namely . From one hand, the restriction indicates consistency between the two comparison models. On the other hand, the inequalities indicate a positive evidence against the model with higher value of Ann2002 (); Burham2004 (), while the condition suggests a strong such evidence.
5.1 Observational constraints
Below, we provide a qualitative discussion of our constraints, giving the reader the opportunity to appreciate the new results of the current study.
5.1.1 Constant growth index
Here we focus on the parametrization, which means that the parameter space contains the following free parameters . The presentation of our constraints is provided in Table II for the case of DES/Planck/JLA/BAO reference cosmology (see section II). The Table includes the goodness of fit statistics (, AIC), for the specific bias models. Also, in Figure 1 we present the 1, 2 and confidence contours in the plane.
In particular, we find:

For SMT model: (AIC=19.542), and .

For JING model: (AIC=20.475), and .

DMR model: (AIC=19.598), and .

For BPR model: (AIC=21.548), and .
We observe that the aforementioned bias models provide very similar results (within errors) as far as the growth index is concerned. The corresponding best fit values show a small but nonzero deviation from the theoretically predicted value of GR (see solid lines of Fig. 1), where the range of the confidence level is . Such a small discrepancy between the predicted and observationally fitted value of has also been discussed by other authors. For example recently, Zhao () found , while Yi () obtained . Also, similar results can be found in previous papers Other () in which the tension can reach to .
Furthermore, we find that the best bias model is the SMT, however the inequality AIC indicates that the SMT bias model is statistically equivalent with rest of the models. The second result is that the differences of the bias models are absorbed in the fitted value of the DM halo mass in which LRGs live, and which ranges from , for the different bias models and in the case of DESY1COSMO bias. As it can also be seen from Table II, our derived mass of the host DM halo mass is consistent with that of Papageorgiou et al. Pap2018 (), while Sawangwit et al. Sawangwit2011 () and Pouri et al. Pouri2014 () found .
CDM Expansion Model  Bias Model  AIC  

DES/Planck/JLA/BAO (Abbott et al. Abbot2017 ())  SMT  15.042  19.542  0  
JING  15.975  20.475  0.933  
DMR  15.096  19.598  0.056  
BPR  17.048  21.548  2.006  
Planck TT+TE+EE+low+lensing (Aghanim et al. Aghanim:2018eyx ())  SMT  15.057  19.557  0  
JING  15.952  20.452  0.895  
DMR  15.104  19.604  0.047  
BPR  16.947  21.447  18.49 
In order to complete the present investigation we repeat the likelihood procedure in the case of Planck TT+TE+EE+low+lensing CDM cosmology, hence , , , , and Aghanim:2018eyx (). Specifically, for the explored bias models we obtain (see also Table II):

For SMT model: (AIC=19.557), and .

For JING model: (AIC=20.452), and .

DMR model: (AIC=19.604), and .

For BPR model: (AIC=21.447), and .
Obviously, our statistical results remain quite robust (within ) against the choice of the undelying expansion Aghanim:2018eyx (), Abbot2017 (). Moreover, as it can be seen from Fig.2 the growth index of the Planck TT+TE+EE+low+lensing CDM cosmology deviates with respect to that of GR () at levels.
5.1.2 Constraints on
In this section we implement the overall likelihood procedure in the parameter space. Based on the considerations discussed in the previous section the statistical vector takes the form .
In Fig.3 we plot the results of our statistical analysis in the plane for the SMT bias model, since we have verified that using the other bias models we get similar contours. The predicted CDM values are indicated by the solid point, while the star corresponds to our best fit values. In brief for the DES/Planck/JLA/BAO cosmology we find , with (AIC=19.533), while in the case of Planck TT+TE+EE+low+lensing CDM cosmology we get , with (AIC=19.488). Notice that for the sake of simplicity we have marginalized the likelihood analysis over the LRG dark matter halo, namely and 13.07 respectively (see SMT model in Table II).
We conclude that the joint statistical analysis put tight constraints , however for the corresponding error bars remain quite large. Also the range of deviation from GR is . We argue that with the next generation of data (mainly from Euclid) we will be able to test whether the growth index of matter fluctuations depends on time.
6 Conclusions
Testing the validity of general relativity (GR) on extragalactic scales is considered one of the most important tasks in cosmological studies, hence it is crucial to minimize the amount of priors needed to successfully complete such an effort. One such prior is the growth index () of matter perturbations. It is well known that a necessary step toward testing GR is to measure at the accuracy level. Obviously, in order to control the systematic effects that possibly affect individual methods and tracers of the growth of matter perturbations we need to have independent estimations of .
In this article we used the biasing properties of the Luminous Red Galaxies, recently released by the group of Dark Energy Survey (DES), together with growth rate data in order to constrain the growth index of matter perturbations. Specifically, in the framework of concordance cosmology, we study the ability of four bias models to fit the DES bias data. Then we combined bias in a joint analysis with the growth rate of matter fluctuations to place constraints on the parameters.
Considering a constant growth index we placed constraints, up to accuracy, on the growth index. Specifically, using the priors of the Dark Energy Survey we found that the constraints remain mostly unaffected by using different forms of bias. In particular, we obtained , , and for SMT She2001 (), JING Jing1998 (), DMR deSim2011 () and BPR Basilakosetal2012 () bias models. Also utilizing the Planck priors we got , , and for the aforementioned bias factors. Obviously, we found a small but nonzero deviation from GR (), where the confidence level lies in the interval . Such a small discrepancy between the predicted and observationally fitted value of has also been reported in several studies Zhao (), Yi () and Other (). Moreover, the intrinsic differences of the bias models are absorbed in the fitted value of the darkmatter halo mass in which LRGs survive, and which belongs in the range .
Under the assumption that the growth index varies with time, namely , we showed that the parameter solution space accommodates the GR values at () level utilizing the DES/Planck/JLA/BAO (Planck) priors. Similar to previous studies, we placed tight constraints on , however the corresponding uncertainties of remain large. The next generation (mainly from Euclid) of dynamical data are expected to improve the constraints on , hence the validity of general relativity on extragalactic scales will be effectively checked.
Acknowledgments
Spyros Basilakos acknowledges support from the Cyprus Research Promotion Foundation in the context of the program âGRATOS: Graph Theoretical Tools for Sciencesâ (ref. number EXCELLENCE/216/0207) led by European University Cyprus.
References
 (1) M. Hicken et al., Astrophys. J., 700, 1097 (2009)
 (2) E. Komatsu et al., Astrophys. J. Supp., 192, 18 (2011)
 (3) C. Blake et al., Mon. Not. Roy. Soc., 418, 1707 (2011)
 (4) G. Hinshaw et al., Astrophys. J. Supp., 208, 19 (2013)
 (5) O. Farooq, D. Mania and B. Ratra, Astrophys. J., 764, 138 (2013).
 (6) Planck Collab. 2015, P.A.R. Ade et al., Astron. Astrophys. 594 (2016) A13.
 (7) N. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astroph.CO].
 (8) E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. of Mod. Phys. D., 15, 1753 (2006)
 (9) R. R. Caldwell and M. Kamionkowski, Ann. Rev. Nucl. Part. Sci., 59, 397 (2009)
 (10) L. Amendola and S. Tsujikawa, Dark Energy: Theory and Observations, Cambridge University Press, Cambridge UK (2010)
 (11) E. V. Linder, Phys. Rev. D., 70, 023511 (2004)
 (12) E. V. Linder and R. N. Cahn, Astrop. Phys., 28, 481 (2007)
 (13) H. Steigerwald, J. Bel and C. Marinoni, JCAP, 5, 42, 2014
 (14) S. Basilakos & S. Nesseris, Phys. Rev. D., 84, 123525 2016; Phys. Rev. D. 96, 063517 2017
 (15) P. J. E. Peebles, “Principles of Physical Cosmology”, Princeton University Press, Princeton New Jersey (1993)
 (16) L. Wang and P. J. Steinhardt, Astrophys. J., 508, 483 (1998)
 (17) S. Nesseris, D. Sapone, and J. GarciaBellido, Phys. Rev. D., 91, 023004 (2015)
 (18) V. Silveira and I. Waga, Phys. Rev. D., 50, 4890 (1994)
 (19) S. Nesseris and L. Perivolaropoulos, Phys. Rev. D., 77, 023504 (2008)
 (20) S. Basilakos, Intern. Journal of Modern Physics D, 21, 1250064 (2012); S. Basilakos and A. Pouri, Mon. Not. Roy. Soc., 423, 3761 (2012)
 (21) H. Wei, Phys. Lett. B., 664, 1 (2008)
 (22) Y. Gong, Phys. Rev. D., 78, 123010 (2008)
 (23) X. Fu, P. Wu and H. Yu, Phys. Lett. B, 677, 12 (2009)
 (24) R. Gannouji, B. Moraes and D. Polarski, JCAP, 2, 34 (2009)
 (25) S. Tsujikawa, R. Gannouji, B. Moraes and D. Polarski, Phys. Rev. D., 80, 084044 (2009)
 (26) S. Basilakos, Phys. Rev. D., 93, 083007 (2016)
 (27) S. Basilakos and P. Stavrinos, Phys. Rev. D., 87, 043506 (2013); G. Papagiannopoulos, S. Basilakos, A. Paliathanasis, S. Savvidou and P. C. Stavrinos, Class. Quant. Grav., 34, 225008 (2017)
 (28) S. Basilakos and J. Sola, Phys. Rev. D., 92, 123501 (2015)
 (29) A. Mehrabi, S. Basilakos and F. Pace, Mon. Not. R. Soc, 452, 2930 (2015); A. Mehrabi, S. Basilakos, M. Malekjani and Z. Davari, Phys. Rev. D. 92, 123513 (2015)
 (30) T. Matsubara, Astrophys. J., 615, 573 (2004)
 (31) S. Basilakos and M. Plionis, Mon. Not. Roy. Soc., 360, L35 (2005)
 (32) S. Basilakos and M. Plionis, Astrophys. J., 650, L1 (2006)
 (33) M. Krumpe, T. Miyaji and A. L. Coil, arXiv:1308.5976 (2013)
 (34) S. Basilakos, M. Plionis and A. Pouri, Phys. Rev. D., 83, 123525 (2011); S. Basilakos, J. B. Dent, S. Dutta, L. Perivolaropoulos and M. Plionis, Phys. Rev. D., 85, 123501 (2012); S. Basilakos and M. Plionis, Astrophys. J., 550, 522 (2001)
 (35) R. Bean et al., arXiv:1309.5385 (2013)
 (36) M. ElvinPoole, et al., Phys. Rev. D 98, 042006 (2018) (DES Y1COSMO)
 (37) Krause, E., et al. (DES Collaboration), submitted to Phys. Rev. D (2017), arXiv:1706.09359 [astroph.CO]
 (38) T. M. C. Abbott et al., Phys. Rev. D 98, 043526 (2018)
 (39) B. Sargedo, S. Nesseris and D. Sapone, Phys. Rev. D., 98, 083543 (2018)
 (40) YS. Song and W.J. Percival, JCAP, 10, 4, (2009)
 (41) , R. K. Sheth and G. Tormen, G. 1999, Mon. Not. R. Astron. Soc., 323 1 (2001)
 (42) Y. P. Jing, ApJ, 503, L9 (1998)
 (43) A. de Simone, M. Maggiore, A. Riotto, Mon. Not. R. Astron. Soc., 412, 2587 (2011)
 (44) N. N. Weinberg and M. Kamionkowski, Mon. Not. R. Astron. Soc., 341, 251 (2003)
 (45) D. J. Eisenstein and W. Hu, ApJ, 496, 605 (1998)
 (46) N. Sugiyama, ApJ, 100, 281 (1995)
 (47) A. Lue, R. Scoccimarro and G. D. Starkman, Phys. Rev. D., 69, 124015 (2004)
 (48) H. F. Stabenau and B. Jain, Phys. Rev. D., 74, 084007 (2006)
 (49) P. J. Uzan, Gen. Rel. Grav., 39, 307 (2007)
 (50) S. Tsujikawa, K. Uddin and R. Tavakol, Phys. Rev. D., 77, 043007 (2008)
 (51) D. Polarski and R. Gannouji, Phys. Lett. B., 660, 439 (2008)
 (52) G. Ballesteros and A. Riotto, Phys. Lett. B. 668, 171 (2008)
 (53) A. B. Belloso, J. GarciaBellido and D. Sapone, JCAP, 10, 10 (2011)
 (54) H. Akaike, A new look at the statistical model identification, IEEE Transactions on Automatic Control, 19, (1974) 716.
 (55) A. R. Liddle, Information criteria for astrophysical model selection, Mon. Not. Roy. Astron. Soc. 377, (2007) L74, [0701113].
 (56) K. P. Burnham, D. R. Anderson, Model selection and multimodel inference: a practical informationtheoretic approach, 2nd edn. Springer, New York (2002)
 (57) K. P. Burnham, D. R. Anderson, Multimodel inference: Understanding AIC and BIC in Model Selection, Sociol. Meth. & Res., 33, (2004) 261
 (58) MingMing Zhao, JingFei Zhang and Xin Zhang, Physics Lett. B 779 473 (2018)
 (59) ZhaoYu Yin and Hao Wei, (2019), [arXiv:1902.00289]
 (60) L. Xu, Phys. Rev. D 88, 084032 (2013); H. GilMarn, W. J. Percival, L. Verde, J. R. Brownstein, C. H. Chuang, F. S. Kitaura, S. A. RodrguezTorres and M. D. Olmstead, Mon. Not. Roy. Astron. Soc. 465, 1757 (2017); L. Samushia et al., Mon. Not. Roy. Astron. Soc. 429, 1514 (2013); F. Beutler et al. [BOSS Collaboration], Mon. Not. Roy. Astron. Soc. 443 1065 (2014); S. Basilakos, Mon. Not. Roy. Astron. Soc., 449, 2151 (2015)
 (61) A. Papageorgiou, S. Basilakos and M. Plionis, Mon. Not. R. Astron. Soc., 476, 2621 (2018)
 (62) U. Sawangwit, et al. Mon. Not. R. Astron. Soc. 416 3033
 (63) A. Pouri, S. Basilakos and M. Plionis, JCAP, J. Cosmol. & Astrop. Phys., 08, 042, (2014)