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In the general case we can still define times t for

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In the general case we can still define timesTfor the var-iousα, impose the monochromatic ansatz (147), and find thespatial solutionsψs, which in general will not be plane waves
11in anyX(b)variable independent ofα. The monochromaticsolutions can still be superposed into peaked wave-packets, asin (147). However it is important to realize that, as with anyother dispersive medium, the envelope of such packets moveswith a group speed that should not be confused with the phasespeed.Specifically, writing:ψs(b,α) = expbracketleftbiggi3Vcl2PP(b,α)bracketrightbigg(99)we identify dispersion relations:α·TP(b,α) = 0.(100)Assuming that the amplitudeA(α)is factorizable and suffi-ciently peaked aroundα0, we can expand:P(b,α) =P(b;α0) +summationdisplayi∂P∂αivextendsinglevextendsinglevextendsinglevextendsingleα0(αiαi0) +...(101)to find that the wave function factorizes as:ψei3Vcl2P(P(b;α0)α0·T)productdisplayiψi(b,Ti).(102)The first factor is the monochromatic (generally non-plane)wave centered onα0. The other factors describe envelopes ofthe form:ψi(b,Ti) =integraldisplayiA(αi)ei3Vcl2P(αiαi0)(Ti∂P∂αi)(103)which therefore move according to:Ti=∂P(b)∂αivextendsinglevextendsinglevextendsinglevextendsingleα0.(104)We can also dot this equation, to find the group speed on{b,Ti}space:cg=dbdTvextendsinglevextendsinglevextendsinglevextendsinglepeak=˙b˙Tvextendsinglevextendsinglevextendsinglevextendsinglepeak=12P∂α∂b.(105)The motion of these envelopes (and so of the peak of the dis-tribution) should agree with the classical equations of motion.We will show in the rest of this paper that indeed it does so,for coherent states, in a number of non-trivial situations (suchas for mixtures of fluids during transition periods when noneof them dominates, or for the sub-dominant clock).This obviously generalizes the construction for single flu-ids, for which a variableX(b)can be found such thatP=αX(b)for someα. Then, with a a suitable choice ofα(andcanonicalTα) we can always make the first term in the dis-persion relationsαTα(for example, in the case of Lambda byΛφ= 3/Λ,TΛTφ=TΛ2). They are “lineariz-ing” variables becauseclin=˙X/˙T= 1.IX.DEALING WITH CROSSOVER REGIONSAs it happens we are sitting right on a bounce inb. How dowe deal with such transitions? In this Section we show thatthe correct semi-classical limit is still obtainedassuming thewave function remains sharply peaked.What actually hap-pens to the wave function is left to future work [26]. We alsoinvestigate the fate of the minority clock (i.e. the radiation andLambda clocks in the Lambda and radiation epochs) once thehandover of clocks is completed. We will use as a workingmodel a mixture of radiation and Lambda, because the alge-bra is clearer, but generalizations to the more relevant case ofdust and Lambda behave in the same way.

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