theoretical evaluations of ρ$_{G}$ reveals a striking difference between the cases of perfect and plasma mirrors. The slope β perf G obtained for perfect mirrors is larger than that β Gold G obtained for gold mirrors by a factor larger than 2 sion has been obtained for the force between two nanos- tructured surfaces made of real materials with arbitrary corrugation depth, corrugation width and distance [52]. In the plane-sphere geometry, it is also possible to use the general scattering formula (11) to obtain explicit eval- uations of the Casimir force. The reflection matrices may here be written in terms of Fresnel amplitudes on the plane mirror and of Mie amplitudes on the spherical one. The scattering formula is then obtained by writing also transformation formulas from the plane waves basis to the spherical waves basis and conversely. The result takes the form of a multipolar expansion with spherical waves labeled by quantum numbers ℓ and m ( | m | ≤ ℓ ). For doing the numerics, the expansion is truncated at some maximum value ℓ$_{max}$ , which restricts accurate evalua- tions to a domain x ≡ L/R > x$_{min}$ with x$_{min}$ propor- tional to 1 /ℓ$_{max}$ . Meanwhile, β Gold G is compatible with the experimental bound obtained in [44] (see [43]) whereas β perf G lies out- side this bound (see also [59]). The lesson to be learned from these results is that more work is needed to reach a reliable comparison of exper- iment and theory on the Casimir effect. Experiments are performed with large spheres for which the parame- ter L/R is smaller than 0.01, and efforts are devoted to calculations pushed towards this regime [67]. 5 β perf G ∼ − 0 . 48 , β Gold G ∼ − 0 . 21 (18) The plane-sphere geometry beyond PFA Meanwhile, the effect of temperature should also be correlated with the plane-sphere geometry. The first cal- culations accounting simultaneously for plane-sphere ge- ometry, temperature and dissipation have been published very recently [68] and they show several striking features. The factor of 2 between the long distance forces in Drude and plasma models is reduced to a factor below 3/2 in the plane-sphere geometry. Then, PFA underestimates the Casimir force within the Drude model at short dis- tances, while it overestimates it at all distances for the perfect reflector and plasma model. If the latter fea- ture were conserved for the experimental parameter re- gion R/L ( > 10 $^{2}$), the actual values of the Casimir force calculated within plasma and Drude model could turn out to be closer than what PFA suggests, which would diminish the discrepancy between experimental results and predictions of the thermal Casimir force using the Drude model. Such calculations have first been performed for per- fectly reflecting mirrors [59, 66]. It was thus found that the Casimir energy was smaller than expected from the PFA and, furthermore, than the result for electromag- netic fields was departing from PFA more rapidly than was expected from previously existing scalar calculations [40, 41]. It is only very recently that the same calcula- tions have been done for the more realistic case of metallic mirrors described by a plasma model dielectric function [43]. Results of these evaluations are expressed in terms of reduction factors defined for the force F or force gra- dient G with respect to the PFA expectations F PFA and G PFA respectively Examples of curves for ρ$_{F}$ and ρ$_{G}$ are shown on Fig.2 of [43] for perfect and plasma mirrors. Using these results, it is possible to compare the the- oretical evaluations to the experimental study of PFA in the plane-sphere geometry [44]. In this experiment, the force gradient is measured for various radii of the sphere and the results are used to obtain a constraint | β$_{G}$ | < 0 . 4 on the slope at origin β$_{G}$ of the function The authors thank I. Cavero-Pelaez, D. Dalvit, G.L. Ingold, M.-T. Jaekel and I. Pirozenkho for fruitful dis- cussions. A.C. and R.M. acknowledge support from the ESF Research Networking Programme CASIMIR (www.casimir-network.com). P.A.M.N. thanks CNPq, CAPES and Faperj for financial support. A.L. acknowl- edges support from the French Contract ANR-06-Nano- 062. Now the comparison of this experimental information to the slope obtained by interpolating at low values of x the Acknowledgments ρ$_{F}$ = F F PFA , ρ$_{G}$ = G G PFA (16) ρ$_{G}$ ( x ) = 1 + β$_{G}$x + O ( x $^{2}$) (17) [4] M. Bordag, U. Mohideen and V.M. Mostepanenko, Phys. Rep. 353 1 (2001) and references therein. [5] K.A. Milton, J. Phys. A20 4628 (2005). [6] R.S. Decca, D. L´ opez, E. Fischbach et al , Annals Phys. [1] H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51 793 (1948). [2] P.W. Milonni, The quantum vacuum (Academic, 1994). [3] S.K. Lamoreaux, Resource Letter in Am. J. Phys. 67 850 (1999).