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increase when T decreases would be a signature of the incipient long range amorphous order. Accordingly, χ$_{3}$ ( ω, T ) should display a peak for ωτ ( T ) ∼ 1 whose height grows as one approaches T$_{g}$ . Here τ ( T ) is the relaxation time of the supercooled liquid which increases rapidly as T decreases toward T$_{g}$ . Contrary to spin glasses, however, χ$_{3}$ ( ω = 0 , T ) should remain trivial since at long times glasses still behave as disordered liquids [13]. One expects that when τ ( T ) is large, χ$_{3}$ takes the following scaling form: 

<img src='content_image/344194.jpg'>

where N$_{corr}$ ( T ) is the T -dependent average number of dynamically correlated molecules, Δ χ$_{1}$ = χ$_{1}$ ( ω = 0) − χ$_{1}$ ( ω → ∞ ) is the part of the static linear susceptibility corresponding to the slow relaxation process we consider, a 3 the volume occupied by one molecule, and H a certain complex scaling function that goes to zero both for small and large arguments. The humped shape of |H ( ωτ ) | is a distinctive feature of the glassy correlations. Indeed, in the ‘no correlation case’ [14, 15], χ$_{3}$ ( ω, T ) is given by the prefactor of Eq. (1) times a function that, for all T , reaches its maximum value at ω = 0. Furthermore N$_{corr}$ ( T ) does not show any temperature dependence in that case. 

We have devised an experiment to measure, for the first time, χ$_{3}$ ( ω, T ) of a classic glass- former (glycerol), via third harmonics measurements. Our findings are in good agreement with the predictions of Eq. (1): the scaling, the growth of the number of dynamically corre- lated molecules with decreasing T , the humped shape of |H| and its power-law dependence for ωτ ≫ 1 are all confirmed experimentally. The anomalous increase of χ$_{3}$ is compatible with the existence of incipient critical fluctuations when T → T$_{g}$ , even if our measurements are too far from the possible critical point to be a clue of criticality. 

Our measurements were performed at temperatures between T$_{g}$ +4 K and T$_{g}$ +35 K, with T$_{g}$ ≈ 190 K. The linear susceptibility χ$_{1}$ ( ω ) = χ ′ $_{1}$( ω ) + iχ ′′ $_{1}$( ω ) quantifies the first harmonic response of polar molecules to a periodic excitation field Ee − $^{iωt}$. The polarization, P ( ω ) e − $^{iωt}$, of the system is given to first order by P ( ω ) = ϵ$_{0}$χ$_{1}$ ( ω ) E . We define the relaxation time τ ( T ) as the inverse of the frequency f$_{α}$ where χ ′′ 1 is maximum. The nonlinear response is dominated by χ$_{3}$ ( ω ) which gives to first order the magnitude of the third harmonics response to the field at pulsation ω [14, 16], P (3 ω ) e − 3 iωt = ϵ$_{0}$χ$_{3}$ ( ω )[ Ee − $^{iωt}$] 3 (the second harmonics is zero because of the E → − E symmetry). While χ$_{1}$ ( ω, T ) has been widely studied in 

