The proceeding comment raises a few points concerning our paper Dantchev textit{et al.}, Phys. Rev. E. {bf 89}, 042116 (2014). In this reply we stress that while Refs. Diehl textit{et al.} EPL {bf 100}, 10004 (2012) and Phys. Rev. E. {bf 89}, 062123
(2014) use three different models to study the the Casimir force for the $O(n rightarrow infty)$ model with free boundary conditions we study a single model over the entire range of temperatures, from above the bulk critical temperature, $T_c$, to absolute temperatures down to $T=0$. The use of a single model renders more transparent the crossover from effects dominated by critical fluctuations in the vicinity of the bulk transition temperature to effects controlled by Goldstone modes at low temperatures. Contrary to the assertion in the comment, we make no claim for the superiority of our model over any of those considered by Diehl textit{et al}. We also present additional evidence supporting our conclusion in Dantchev textit{et al.}, Phys. Rev. E. {bf 89}, 042116 (2014) that the temperature range in which our low-temperature analytical expansion for the Casimir force increases as $L$ grows and remains accurate for values of the ratio $T/T_c$ that become closer and closer to unity, while $T$ remains well outside of the critical region.
We study critical point finite-size effects on the behavior of susceptibility of a film placed in the Earths gravitational field. The fluid-fluid and substrate-fluid interactions are characterized by van der Waals-type power law tails, and the bounda
ry conditions are consistent with bounding surfaces that strongly prefer the liquid phase of the system. Specific predictions are made with respect to the behavior of $^3$He and $^4$He films in the vicinity of their respective liquid-gas critical points. We find that for all film thicknesses of current experimental interest the combination of van der Waals interactions and gravity leads to substantial deviations from the behavior predicted by models in which all interatomic forces are very short ranged and gravity is absent. In the case of a completely short-ranged system exact mean-field analytical expressions are derived, within the continuum approach, for the behavior of both the local and the total susceptibilities.
On the example of the spherical model we study, as a function of the temperature $T$, the behavior of the Casimir force in O(n) systems with a diffuse interface and slab geometry $infty^{d-1}times L$, where $2<d<4$ is the dimensionality of the system
. We consider a system with nearest-neighbor anisotropic interaction constants $J_parallel$ parallel to the film and $J_perp$ across it. The model represents the $ntoinfty$ limit of O(n) models with antiperiodic boundary conditions applied across the finite dimension $L$ of the film. We observe that the Casimir amplitude $Delta_{rm Casimir}(d|J_perp,J_parallel)$ of the anisotropic $d$-dimensional system is related to that one of the isotropic system $Delta_{rm Casimir}(d)$ via $Delta_{rm Casimir}(d|J_perp,J_parallel)=(J_perp/J_parallel)^{(d-1)/2} Delta_{rm Casimir}(d)$. For $d=3$ we find the exact Casimir amplitude $ Delta_{rm Casimir}= [ {rm Cl}_2 (pi/3)/3-zeta (3)/(6 pi)](J_perp/J_parallel)$, as well as the exact scaling functions of the Casimir force and of the helicity modulus $Upsilon(T,L)$. We obtain that $beta_cUpsilon(T_c,L)=(2/pi^{2}) [{rm Cl}_2(pi/3)/3+7zeta(3)/(30pi)] (J_perp/J_parallel)L^{-1}$, where $T_c$ is the critical temperature of the bulk system. We find that the effect of the helicity is thus strong that the Casimir force is repulsive in the whole temperature region.
