Scale factor, universal

As noted earlier in section A2.5.6.2. the assumption of homogeneity and tlie resnlting principle of two-scale-factor universality requires the amplitude coefficients to be related. In particnlar the following relations can be derived ... 

From the thermodynamic standpoint, the basic components of stars can be considered as photons, ions and electrons. The material particle gas (fermions) and the photon gas (bosons) react differently under compression and expansion. Put n photons and n material particles into a box. Let R be the size of the box (i.e. a characteristic dimension or scale factor). The relation between temperature and size is TR = constant for the photons and TR = constant for the particles. This difference of behaviour is very important in the Big Bang theory, for these equations show quite unmistakably that matter cools more quickly than radiation under the effects of expansion. Hence, a universe whose energy density is dominated by radiation cannot remain this way for long, in fact, no longer than 1 million years. 

Considering the inflation, we assume a universe with a critical density Qo = 1, k = 0 and qo = The universe is assumed to be matter dominated during all the epoch of interest. Then the scale factor Rit) of the universe expands as ... 

The expansion of the Universe in encoded into the time dependence of the scale factor ad ). The quantity t corresponds to the cosmic time. One can easily show that observers situated at xl = Const follow geodesics, therefore cosmic time corresponds to the proper time of such observers. It is assumed that any matter element is at rest or almost at rest with respect to the spatial part of the coordinate system. For this reason, the xl are called comoving coordinates, and any distance measured using line element ijdx dx3 is called comoving distance. 

The study of the inflationary perturbations that we have presented here is of course far from complete. We did not derive the calculation of the spectral indices as a function of the slow-roll parameters, nor of the running of the spectral indices. However we hope to have made it clear that this can been done analytically within the slow-roll approximation (as well as numerically, of course). Let us emphasize that many alternative models such as the ekpy-rotic universe or the pre Big-Bang scenario can also be studied within this framework, as the core ingredient (quantum fluctuations which are expelled from the Hubble radius) are present. What changes is the matter content of the Universe during this phase as well as the dynamics of the scale factor. 

Gravitational production of particles is an important phenomenon that is worth describing here. Consider a scalar field (particle) X of mass Mx in the expanding universe. Let r/ be the conformal time and a(r/) the time dependence of the expansion scale factor. Assume for simplicity that the universe is flat. 

Everything discussed so far has been geometrical , relying only on the form of the Robertson-Walker metric. To make further progress in understanding the evolution of the universe, it is necessary to determine the time dependence of the scale factor a(t). Although the scale factor is not an observable, the expansion rate, the Hubble parameter, H = H(t), is. 

The Friedmann equation (eq. 2.9) relates the time-dependence of the scale factor to that of the density. The Einstein equations yield a second relation among these which may be thought of as the surrogate for energy conservation in an expanding universe. 

This is a functional equation for g(,v) and the universal scale factor a. It is self-referential g(x) is defined in terms of itself. 

At this point, we mention a further consequence of the universality principle alluded to above. For each universality class (such as that of the Ising model or that of the XY model, etc.) not just the critical exponents are universal, but also the scaling function F(H), apart from non-universal scale factors for the occurring variables (a factor for H we have expressed via the ratio C/B in eq. (84), for instance). A necessary implication then is the universality of certain critical amplitude ratios, where all scale factors for the variables of interest cancel out. In particular, ratios of critical amplitudes of corresponding quantities above and below Tc, A+j A [eq. (7)], C+jC [eq. (6)] and f+/ [eq. (38)] are universal (Privman et al., 1991). A further relation exists between the amplitude D and B and C 1 Writing M H -> oo) = XHl/ cf. eqs. (87) and (91), the universality of M(H) states that X is universal. But since 0 = B tfM H) = B t PXH = B] SC S H] X, a comparison with eq. (45) yields... 

The law of corresponding states is based upon the hypothesis that, to an acceptable degree of accuracy, the intermolecular pair potentials for a number of fluids can be rendered conformal by the choice of two scaling factors, q for energy and collision cross sections for this universal potential are universal functions of the reduced temperature T = k T/ec... 

The potential energy of a pair of molecules can be represented by some universal function tp together with two scale factors r, e characteristic of the molecular species... 

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