Go to previous page Go up Go to next page

4.6 Isotropy: Applications of effective densities

There is now one characteristic correction of the matter Hamiltonian, coming directly from a loop quantization. Its implications can be interpreted as repulsive behavior on small scales and the exchange of friction and antifriction for matter, and it leads to many further consequences.

4.6.1 Collapsing phase

When the universe has collapsed to a sufficiently small size, repulsion becomes noticeable and bouncing solutions become possible, as illustrated in Figure 1View Image. Requirements for a bounce are that the conditions a˙= 0 and ¨a > 0 can be fulfilled at the same time, where the first can be evaluated with the Friedmann equation and the second with the Raychaudhuri equation. The first condition can only be fulfilled if there is a negative contribution to the matter energy, which can come from a positive curvature term k = 1 or a negative matter potential V (φ ) < 0. In these cases, there are classical solutions with a˙= 0, but they generically have ¨a < 0 corresponding to a recollapse. This can easily be seen in the flat case with a negative potential, where Equation (31View Equation) is strictly negative with d loga3d (a )j,l∕da ≈ 0 at large scales.

The repulsive nature at small scales now implies a second point where ˙a = 0 from Equation (30View Equation) at smaller a since the matter energy now also decreases as a → 0. Moreover, the Raychaudhuri equation (31View Equation) has an additional positive term at small scales such that ¨a > 0 becomes possible.

Matter also behaves differently through the Klein–Gordon equation (33View Equation). Classically, with a˙< 0, the scalar experiences antifriction and φ diverges close to the classical singularity. With the quantum correction, antifriction turns into friction at small scales, damping the motion of φ such that it remains finite. In the case of a negative potential [98Jump To The Next Citation Point], this allows the kinetic term to cancel the potential term in the Friedmann equation. With a positive potential and positive curvature, on the other hand, the scalar is frozen and the potential is canceled by the curvature term. Since the scalar is almost constant, the behavior around the turning point is similar to a de Sitter bounce [279303]. Further, more generic possibilities for bounces arise from other correction terms [147142Jump To The Next Citation Point].

View Image

Figure 1: Examples of bouncing solutions with positive curvature (left) or a negative potential (right, negative cosmological constant). The solid lines show solutions of equations with a bounce as a consequence of quantum corrections, while the dashed lines show classical solutions running into the singularity at a = 0 where φ diverges.

4.6.2 Expansion

Repulsion can not only prevent collapse but also accelerates an expanding phase. Indeed, using the behavior (27View Equation) at small scales in the Raychaudhuri equation (31View Equation) shows that ¨a is generically positive since the inner bracket is smaller than − 1∕2 for the allowed values 0 < l < 1. Thus, as illustrated by the numerical solution in the upper left panel of Figure 2View Image, inflation is realized by quantum gravity effects for any matter field irrespective of its form, potential or initial values [53Jump To The Next Citation Point]. The kind of expansion at early stages is generally super-inflationary, i.e., with equation of state parameter w < − 1. For free massless matter fields, w usually starts very small, depending on the value of l, but with a non-zero potential just as the mass term for matter inflation w is generally close to exponential: we ff ≈ − 1 for small p. This can be shown by a simple and elegant argument independently of the precise matter dynamics [148]; the equation of state parameter is defined as w = P∕ρ where P = − ∂E ∕∂V is the pressure, i.e., the negative change of energy with respect to volume, and ρ = E ∕V is the energy density. Using the matter Hamiltonian for E and 3∕2 V = |p|, we obtain

Peff = − 13|p|− 1∕2d′(p)p2φ − V (φ )

and thus, in the classical case,

12|p|−3p2φ − V (φ) w = 1---−3-2-------- 2|p| pφ + V (φ)

as usual. In loop cosmology, however, we have

1 −1∕2 ′ 2 w = − -3|p|----d-(p-)p-φ +-V-(φ) . eff 12|p|− 3∕2d(p)p2φ + V (φ)

(See [312] for a discussion of energy conditions in this context and [272] for an application to tachyon fields.)

View Image

Figure 2: Example of a solution of a(t) and φ(t) showing early loop inflation and later slow-roll inflation driven by a scalar that has pushed up its potential by loop effects. The left-hand side is stretched in time so as to show all details. An idea of the duration of different phases can be obtained from Figure 3Watch/download Movie.

Get Flash to see this player.

Figure 3: mpg-Movie (1154 KB) The initial push of a scalar φ up its potential and the ensuing slow-roll phase together with the corresponding inflationary phase of a.

In general we need to know the matter behavior to know w and weff. But we can get general qualitative information by treating pφ and V (φ) as unknowns determined by w and weff. In the general case there is no unique solution for p2φ and V (φ) since, after all, pφ and φ change with t. They are now subject to two linear equations in terms of w and we ff, whose determinant must be zero, resulting in

|p|3∕2(w + 1)(d(p) − 2|p|d′(p)) we ff = − 1 + --------------------33∕2------. 1 − w + (w + 1)|p| d (p)

Since for small p the numerator of the fraction approaches zero faster than the second part of the denominator, weff approaches − 1 at small volume except for the special case where w = 1, which is realized for V (φ ) = 0. Note that the argument does not apply to the case of vanishing potential since then p2 = const φ and V (φ ) = 0 presents a unique solution to the linear equations for w and w eff. In fact, this case leads in general to a much smaller 2 ′ we ff = − 3|p|d (p )∕d(p) ≈ − 1∕(1 − l) < − 1 [53Jump To The Next Citation Point]. Also at intermediate values of p, where the asymptotic argument does not apply, values of we ff that are smaller than − 1 are possible.

One can also see from the above formula that w eff, though close to − 1, is a little smaller than − 1 generally. This is in contrast to single field inflaton models where the equation of state parameter is a little larger than − 1. As we will discuss in Section 4.19, this opens the door to possible characteristic signatures distinguishing different models. However, here we refer to a regime where other quantum corrections are present and care in the interpretation of results is thus required.

Again, the matter behavior also changes, now with classical friction being replaced by antifriction [111Jump To The Next Citation Point]. Matter fields thus move away from their minima and become excited, even if they start close to a minimum (Figure 2View Image). Since this does not apply only to the homogeneous mode, it can provide a mechanism for structure formation as discussed in Section 4.19. But modified matter behavior also leads to improvements in combination with chaotic inflation as the mechanism to generate structure; if we now view the scalar φ as an inflaton field, it will be driven to large values in order to start a second phase of slow-roll inflation that is long enough. This “graceful entrance” [244Jump To The Next Citation Point] is satisfied for a large range of the ambiguity parameters j and l [97] (see also [195]), is insensitive to non-minimal coupling of the scalar [93], and can even leave signatures [295] in the cosmic microwave spectrum [194]. The earliest moments during which the inflaton starts to roll down its potential are not slow roll, as can also be seen in Figures 2View Image and 3Watch/download Movie, where the initial decrease is steeper. Provided the resulting structure can be seen today, i.e., there are not too many e-foldings from the second phase, this can lead to visible effects, such as a suppression of power. Whether or not those effects are to be expected, i.e., which magnitude of the inflaton is generally reached by the mechanism generating initial conditions, is to be investigated at the basic level of loop quantum cosmology. All this should be regarded only as initial suggestions, indicating the potential of quantum cosmological phenomenology, which have to be substantiated by detailed calculations, including inhomogeneities or at least anisotropic geometries. In particular, the suppression of power can be obtained by a multitude of other mechanisms.

4.6.3 Model building

It is already clear that there are different inflationary scenarios using effects from loop cosmology. A scenario without inflatons is more attractive since it requires less choices and provides a fundamental explanation for inflation directly from quantum gravity. However, it is also more difficult to analyze structure formation in this context, when there already are well-developed techniques in slow role scenarios.

In these cases, where one couples loop cosmology to an inflaton model, one still requires the same conditions for the potential, but generally gets the required large initial values for the scalar by antifriction. On the other hand, finer details of the results now depend on the ambiguity parameters, which describe aspects of the quantization also arising in the full theory.

It is possible to combine collapsing and expanding phases in cyclic or oscillatory models [215Jump To The Next Citation Point]. One then has a history of many cycles separated by bounces, whose duration depends on details of the model such as the potential. Here also, results have to be interpreted carefully since especially such long-term evolutions are sensitive to all possible quantum corrections, not just those included here. There can then be many brief cycles until eventually, if the potential is right, one obtains an inflationary phase, if the scalar has grown high enough. In this way, one can develop an idea of the history of our universe before the Big Bang. The possibility of using a bounce to describe the structure in the universe exists. So far this has only been described in models [200] using brane scenarios [219], in which the classical singularity has been assumed to be absent by yet-to-be-determined quantum effects. As it turns out, the explicit mechanism removing singularities in loop cosmology is not compatible with the assumptions made in those effective pictures. In particular, the scalar was supposed to turn around during the bounce, which is impossible in loop scenarios unless it encounters a range of positive potential during its evolution [98]. Then, however, an inflationary phase generally commences, as in [215244Jump To The Next Citation Point], which is then the relevant regime for structure formation. This shows how model building in loop cosmology can distinguish scenarios that are more likely to occur from quantum gravity effects.

Cyclic models can be argued to shift the initial moment of a universe in the infinite past, but they do not explain how the universe started. An attempt to explain this is the emergent universe model [160162], in which one starts close to a static solution. This is difficult to achieve classically, however, since the available fixed points of the equations of motion are not stable and thus a universe departs too rapidly. Loop cosmology, on the other hand, implies an additional fixed point of the effective equations, which is stable, and allows the universe to start in an initial phase of oscillations before an inflationary phase is entered [23861]. This presents a natural realization of the scenario in which the initial scale factor at a fixed point is automatically small so as to start the universe close to the Planck phase.

4.6.4 Stability

Cosmological equations displaying super-inflation or antifriction are often unstable in the sense that matter can propagate faster than light. This has been voiced as a potential danger for loop cosmology as well [139Jump To The Next Citation Point140Jump To The Next Citation Point]. An analysis requires inhomogeneous techniques, at least at an effective level, such as those described in Section 4.16. It has been shown that loop cosmology is free of this problem because the new behavior of the homogeneous mode of the metric and matter is not relevant for matter propagation [186Jump To The Next Citation Point]. The whole cosmological picture that follows from the effective equations is thus consistent.

  Go to previous page Go up Go to next page