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tokens: δ m ( a i ) = 5 × 10 −5 . Additionally, we insert the initial conditions as follows (see also Batista Pace 2013 ; Malekjani et al. 2017 ) Using these initial values one can be confident that matter perturbations always remain in the linear regime. After finding the evolution of fluctuations (δ m , δ d ), we can calculate the growth rate of large-scale structures in the presence of oscillating DE models considered in this work. The growth rate function can be written as The value of σ 8 (), the matter fluctuation amplitude on scales of 8 Mpch −1 , can also be written as By multiplication of these two functions, one can calculate the value of f ()σ 8 () to compare it with f σ 8 observational data. The growth rate data set were obtained from redshift-space distortions (RSDs) from different galaxy surveys. These data points, when measured using RSDs, are degenerate with the Alcock–Paczynski (AP) effect (Alcock Paczynski 1979 ). Assuming an incorrect cosmological model for the coordinate transformation from redshift space to comoving space leads to residual geometric distortions known as the RSDs. These distortions are induced by the fact that measured distances along and perpendicular to the line of sight are fundamentally different. Measuring the ratio of galaxy clustering in the radial and transverse directions provides a probe of AP effect. However, given high precision clustering measurements over a wide range of scales, this degeneracy can be broken since RSD and AP have different scale dependences (Reid et al. 2012 ). In order to overcome the RSD problem, several methods have been used in the literature to apply the AP test to the large-scale structure (Song Percival 2009 ; Samushia, Percival Raccanelli 2012 ; Blake et al. 2012 ; Blake et al. 2013 ; Hudson Turnbull 2013 ; Feix, Nusser Branchini 2015 ; Howlett et al. 2015 ; Chuang et al. 2016 ; Huterer et al. 2017 ). Since we obtain our growth rate data from the above references; therefore, we can claim that the AP effect has been considered for the growth rate data which we used in our analysis. Nowadays there are more than 30 data points of recent f σ 8 measurements from different surveys. The information in some of these data points overlaps significantly with other data points in this collection. Some of them are updated version of previous measurements either with enhancements in the volume of the survey, during its scheduled run or with different methodologies by various groups. Therefore, the collection of these data points should not be used in its entirety. Therefore, in our analysis we use the ‘Gold-2017’ compilation of robust and independent f σ 8 measurements from different surveys constructed by Nesseris, Pantazis Perivolaropoulos ( 2017 ). We apply these data points to put constraints on model parameters. The likelihood function becomes where the statistical vector p is {Ω DM0 , Ω b0 , h , 0 , A, B , σ 8 } and σ 8 is the present value of σ 8 (). The best-fitting parameters and