9 FIG. 1. Comparison of entropic acceleration and several ΛCDM models. The supernovae data points are plotted with error bars and the data is taken from [13]. The luminosity distance d$_{L}$ for the entropic models I (Eq. (16)) and II (Eq. (17)) are denoted by the dotted and dashed (blue) curves respectively. The theoretical predictions for ΛCDM are represented by the solid curves. where z is the redshift defined by z + 1 ≡ a$_{0}$/a . For ΛCDM, the luminosity distance can be written [1]: where, S ( x ) ≡ sin( x ) and κ = 1 − Ω$_{tot}$ for Ω$_{tot}$ > 1 while S ( x ) ≡ sinh( x ) with κ = 1 − Ω$_{tot}$ for Ω$_{tot}$ < 1 while S ( x ) ≡ x and κ = 1, for Ω$_{tot}$ = 1. For the ΛCDM models we take Ω$_{tot}$ = Ω$_{M}$ + Ω$_{Λ}$. Here we have defined Ω$_{M}$ ≡ ρ$_{M}$/ρ$_{c}$ = 8 πGρ$_{M}$/ 3 H 2 and Ω$_{Λ}$ = Λ / 3 H 2 where ρ$_{c}$ is the critical energy density. The results are that the entropic acceleration models we consider can provide excellent fits to the data as can be seen from Fig. (1). 5 The entropic models move smoothly from a decelerating to an accelerating phase sometime near a redshift of z = . 5, analogous to the ΛCDM models. We suspect a complete model may be further 5 In solving the equations for the fitting we have assumed the standard scaling behavior for matter and radiation. This scaling behavior is now determined only in part by the Friedmann-Lemaˆıtre equation and we do not propose any novel form for this constraint yet; rather, we assume that the standard scaling applies and should provide an adequate approximation for the purposes of Fig. (1). d$_{L}$ ( z ; Ω$_{M}$ , Ω$_{Λ}$ , H$_{0}$ ) = c (1 + z ) H$_{0}$ $^{√}$| κ | S ( $^{√}$| κ |$^{∫}$ z 0 dz ′ $^{[}$(1 + z $^{′}$) $^{2}$(1 + Ω$_{M}$ z $^{′}$) − z $^{′}$(2 + z $^{′}$)Ω$_{Λ}$ ] − 1 2 ) (20)