Go to previous page Go up Go to next page

4.9 Isotropy: Applications of higher-power corrections

A simple model, which has played several important roles in this context, illustrates the basic features of higher-power corrections very well. This model is isotropic, spatially flat and sourced by a free, massless scalar. It was first looked at in the context of loop quantum cosmology in [89Jump To The Next Citation Point181] from the point of view of difference equations and boundary proposals. More recently, it was realized that the absence of a scalar potential allows the explicit derivation of the physical inner product and detailed numerical calculations of physical states [26Jump To The Next Citation Point], as well as the derivation of exact effective equations [70Jump To The Next Citation Point]. The relation to effective equations will be discussed in Section 6.3. For now, we can observe that the boundedness of functions such as sin(δc), which can arise from the discreteness directly, implies a lower bound to the volume of such models. In the effective Friedmann equation,
∘ --- 8πG p2 δ−2 sin2(δc) |p| = -------φ3∕2, (34 ) 6 |p |
the scalar momentum is constant due to the absence of a potential for the free scalar. Thus, − 2 2 p ∝ sin (δc) ≤ 1 shows that |p| must be bounded away from zero; the scale factor bounces instead of reaching a classical singularity at p = 0. The precise bounce scale depends on the form of δ(p).

One can translate this into a phenomenological Friedmann equation involving the scale factor and its time derivative. The Hamiltonian equation of motion for p implies

--- ˙p = 2δ− 1∘ |p |sin(δc)cos(δc)

and thus

( ) ( ) ( ) ˙a 2 p˙ 2 8 πG p2φ 8 πG p2φ 2 -- = 4 -- = --------3∕2 1 − --------3∕2pδ (35 ) a p 6 |p| 6 |p|
when this is inserted into Equation (34View Equation). (The right-hand side depends on the energy density but not on p, for δ(p) ∝ 1∕|p|1∕2, which is sometimes preferred [278Jump To The Next Citation Point].) The whole series of higher-power corrections thus implies only quadratic corrections for the energy density of a free scalar field [299Jump To The Next Citation Point278], which is usually easier and more insightful to analyze. One has to keep in mind, however, that this form of the equation is truly effective only in the absence of a scalar potential and of any deviations from isotropy. Otherwise there are further corrections, which prevent the equation from being brought into a simple quadratic energy-density form; see Section 6.

A similar form of effective equations can be used for a closed model, which, due to its classical recollapse now being combined with a quantum bounce, provides cyclic-universe models. These are no longer precise effective equations, but one can show that additional quantum corrections during the classical collapse phase are small. Thus, deviations from the phenomenological trajectory solving Equation (35View Equation) build up only slowly in time and require several cycles to become noticeable, as first analyzed and verified numerically in [28Jump To The Next Citation Point]. If one pretends that the same kind of quadratic corrections can be used for massive or self-interacting scalars, there is a rich phenomenology of cyclic universes, some of which has been analyzed in [281270271314307311287313151]. However, for such long evolution times it is crucial to consider all quantum effects, which for massive or self-interacting matter is not captured fully in the simple quadratic energy-density corrections.

Qualitatively, the appearance of bounces agrees with what we saw for effective densities, although the precise realization is different and occurs for different models. Moreover, if we bring together higher-power corrections, as well as effective densities, we see that, although individually they both have similar effects, together they can counteract each other. In the specific model considered here, using effective densities in the matter term and possibly the gravitational term results in an inequality of the form f(p) ∝ sin2(δc) ≤ 1 where f(p) replaces the classical p− 2 used above and has an upper bound. Thus, one cannot immediately conclude that p is bounded away from zero and rather has to analyze the precise numerical constants appearing in this relation. A conclusion about a bounce can then no longer be generic but will depend on initial conditions. This shows the importance of bringing all possible quantum corrections together in a consistent manner. It is also important to realize that, while Equation (34View Equation) is a precise effective equation for the dynamical behavior of a quantum state as described later, the inclusion of effective densities would imply further quantum corrections, which we have not described so far and which result from backreaction effects of a spreading state on its expectation values. These corrections would also have to be included for a complete analysis, which is still not finished.

Before all quantum corrections have been determined, one can often estimate their relative magnitudes. In the model considered here, this is possible when one uses the condition that a realistic universe must have a large matter content. Thus, pφ must be large, which affects the constants in the above relations to the extent that the maximum of f(p) will not be reached before sin(δc) reaches the value one. In this case there is thus a bounce independent of effective-density suppressions. Moreover, if one starts in a sufficiently semiclassical state at large volume, quantum backreaction effects will not change the evolution too much before the bounce is reached. Thus, higher power corrections are indeed dominant and can be used reliably. There are, however, several caveats: First, while higher-power corrections are relevant only briefly near the bounce, quantum backreaction is present at all times and can easily add up. Especially for systems with a different matter content such as a scalar potential, a systematic analysis has yet to be performed. Anisotropies and inhomogeneities can have a similar effect, but for inhomogeneities an additional complication arises: not all the matter is lumped into one single isotropic patch, but rather distributed over the discrete building blocks of a universe. Local matter contents are then much smaller than the total one, and the above argument for the dominance of higher-power corrections no longer applies. Finally, the magnitude of corrections in effective densities has been underestimated in most homogeneous studies so far because effects of lattice states were overlooked (see Appendix in [75Jump To The Next Citation Point]). They can thus become more important at small scales and possibly counteract a bounce, even if the geometry can safely be assumed to be nearly isotropic. Whether or not there is a bounce in such cases remains unknown at present.

  Go to previous page Go up Go to next page