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The stability condition of this future de Sitter solution[24] , f ′ ( R$_{1}$ ) > R$_{1}$f ′′ ( R$_{1}$ ), imposes the following constraint on x$_{1}$ . 

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which is stronger than any other constraint discussed above. For each n we can find x$_{1}$ which marginally satisfies (12) and gives the minimal allowed value of λ . Numerically we find ( n, x$_{1min}$, λ$_{min}$ ) = (2 , 1 . 267 , 0 . 9440) , (3 , 1 . 041 , 0 . 7259), and (4 , 0 . 9032 , 0 . 6081) for each n , respectively (if n = 2, the analytic expression for x$_{1min}$ is x 2 1min = √$_{13}$ − 2). For comparison, the analytic results for n = 1 are x$_{1min}$ = √$_{3}$ ≈ 1 . 732 , λ$_{min}$ = 8 / $_{(3}$√$_{3)}$ ≈ 1 . 540. 

We numerically solve evolution equation (7) using (6) to check numerical accuracy, taking t$_{i}$ at the epoch when matter density parameter took Ω$_{i}$ = 16 πGρ$_{i}$/ (16 πGρ$_{i}$ + λR$_{s}$ ) = 0 . 998. We determine the current epoch by the requirement that the value of Ω takes the ob- served central value Ω$_{0}$ = 0 . 27 and R$_{s}$ is fixed so that the current Hubble parameter H$_{0}$ = 72km/s/Mpc is reproduced. We find the ratio R$_{s}$/H 2 0 is well fit by a simple power- law R$_{s}$/H 2 0 = c$_{n}$λ − p$_{n}$ with ( n, c$_{n}$, p$_{n}$ ) = (2 , 4 . 16 , 0 . 953) , (3 , 4 . 12 , 0 . 837) , and (4 , 4 . 74 , 0 . 702), respectively, whereas in the ΛCDM limit it would behave as R$_{s}$/H 2 0 = 6(1 − Ω$_{0}$) /λ ≃ 4 . 38 λ − $^{1}$. 

Figures 1 depict evolution of w$_{DE}$ as a function of redshift z where phantom crossing is manifest. As expected, it approaches w$_{DE}$ = − 1 = constant as we increase λ for fixed n . For minimal allowed values of λ , deviations from w$_{DE}$ = − 1 are observed at ∼ 5% level in both directions for z ≲ 2 independently of n . Such behaviour of w$_{DE}$ is well admitted by all most recent observational data, see e.g. Ref. [2]. The average value of w$_{DE}$ over the interval 

