Dynamical suppression of non-adiabatic modes

Recent analyses of the WMAP 5-year data constrain possible non-adiabatic contributions to the initial conditions of CMB anisotropies. Depending upon the early dynamics of the plasma, the amplitude of the entropic modes can experience a different suppression by the time of photon decoupling. Explicit examples of the latter observation are presented both analytically and numerically when the post-inflationary dynamics is dominated by a stiff contribution. Electronic address: [email protected] The initial conditions of the Boltzmann hierarchy can be usefully classified into adiabatic and non-adiabatic (i.e. entropic). The thermodynamic origin of this classification resides in the observation that the relative fluctuations of the specific entropy Sij do not necessarily vanish (as in the case of the adiabatic mode). The relative fluctuations of the specific entropy ςij can be written, in gauge-invariant terms, as Sij = δςij ςij ≡ −3(ζi − ζj), ζi = −Ψ + δi wi + 1 , (1) where wi is the barotropic index of the i-th species; in Eq. (1) Ψ represents the gaugeinvariant Bardeen potential and δi is the gauge-invariant density contrast. In the ΛCDM model (where Λ stands for the dark-energy component and CDM stands for the cold dark matter contribution) the indices i and j of Eq. (1) run over the four species of the plasma so that, in general, the initial conditions will contemplate one adiabatic mode and four nonadiabatic modes (see, for instance, [1, 2]). The initial conditions of the Boltzmann hierarchy can be set either by choosing only the adiabatic mode, or by selecting a combination of the adiabatic mode with one (or more) non-adiabatic modes. The obtained angular power spectra (both for temperature and polarization) can then be compared with the experimental data and interesting bounds can be set on the various combinations of the initial conditions [3, 4] (see also [1, 2]). Are there simple dynamical recipes able to suppress the entropic contributions? This is the basic question addressed in this paper. In the current framework, after inflation, the plasma was suddenly dominated by radiation. Absent the latter assumption, the nonadiabatic contribution to the pre-decoupling fluctuations of the spatial curvature will have a different relation to the entropic modes originally present right after inflation. While it is not mandatory to postulate different dynamical evolutions, it is useful to be aware of different possibilities which can help more dedicated scrutiny of the observational data. Prior to the radiation epoch, the plasma might have been expanding at a slower rate. This perspective was also invoked by Zeldovich who suggested that, prior to radiation dominance, the Universe was indeed quite stifff and characterized by a sound speed even comparable with the speed of light [5]. Post-inflationary phases stiffer than radiation can even lead to relic gravitons whose spectral energy density increases as a function of the comoving frequency [6]. If the inflaton field is identified with the quintessence field a stiff post-inflationary phase arises naturally [7] and this is what happens in the context of the so-called quintessential inflationary models [8] as well as in related contexts [9, 10]. Consider, for sake of simplicity, a post-inflationary plasma characterized by three distinct components so that the total energy density and the various pressures can be written as: ρt = ρm + ρr + ρS, pm = 0, pr = ρr 3 pS = wρS, (2) According to the present data, the adiabatic mode will have to be dominant in comparison with the remaining one (or more) entropic contributions. In the opposite case the (observed) anti-correlation peak in the temperature/polarization would not be correctly reproduced.