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4.6 Isotropy: Applications

There is now one characteristic modification in 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 one can be evaluated with the Friedmann equation, and the second one 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 (f) < 0. In those 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 (30View Equation) is strictly negative with 3 d loga d(a)j,l/da ~~ 0 at large scales.

The repulsive nature at small scales now implies a second point where a = 0 from (29View Equation) at smaller a since the matter energy now decreases also for a --> 0. Moreover, the modified Raychaudhuri equation (30View Equation) has an additional positive term at small scales such that ¨a > 0 becomes possible.

Matter also behaves differently through the modified Klein-Gordon equation (32View Equation). Classically, with a < 0 the scalar experiences antifriction and f diverges close to the classical singularity. With the modification, antifriction turns into friction at small scales, damping the motion of f such that it remains finite. In the case of a negative potential [68Jump 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 [187203]. Further, more generic possibilities for bounces arise from other correction terms [10097].

View Image

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

4.6.2 Expansion

Repulsion can not only prevent collapse but also accelerates an expanding phase. Indeed, using the behavior (26View Equation) at small scales in the effective Raychaudhuri equation (30View 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 [45Jump To The Next Citation Point]. The kind of expansion at early stages is generically 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 such as a mass term for matter inflation w is generically close to exponential: weff ~~ - 1. This can be shown by a simple and elegant argument independently of the precise matter dynamics [101]: The equation of state parameter is defined as w = P/r where P = - @E/@V is the pressure, i.e., the negative change of energy with respect to volume, and r = E/V energy density. Using the matter Hamiltonian for E and 3/2 V = |p|, we obtain

1 - 1/2 ' 2 Peff = - 3|p| d (p)pf- V (f)

and thus in the classical case

1 -3 2 2|p|--pf---V-(f) w = 1|p|-3p2 + V (f) 2 f

as usually. In the modified case, however, we have

1| p |-1/2d'(p)p2+ V(f) weff = - 31------------f--------. 2|p|- 3/2d(p)p2f + V (f)

View Image

Figure 2: Example for a solution of a(t) and f(t), showing early loop inflation and later slow-roll inflation driven by a scalar that is 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.
Watch/download Movie

Figure 3: Movie showing the initial push of a scalar f 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 generic qualitative information by treating pf and V (f) as unknowns determined by w and weff. In the generic case, there is no unique solution for p2f and V(f) since, after all, pf and f change with t. They are now subject to two linear equations in terms of w and w eff, whose determinant must be zero resulting in
|p| 3/2(w + 1)(d(p) - 2|p| d'(p)) weff = - 1 + --------------------3--------. 1 - w + (w + 1)| p|3/2d(p)

Since for small p the numerator in the fraction approaches zero faster than the second part of the denominator, weff approaches minus one at small volume except for the special case w = 1, which is realized for V (f) = 0. Note that the argument does not apply to the case of vanishing potential since then 2 pf = const and V (f) = 0 presents a unique solution to the linear equations for w and weff. In fact, this case leads in general to a much smaller weff = - 23| p|d(p)'/d(p) ~~ -1/(1 - l) < - 1 [45Jump To The Next Citation Point].

One can also see from the above formula that weff, though close to minus one, is a little smaller than minus one generically. This is in contrast to single field inflaton models where the equation of state parameter is a little larger than minus one. As we will discuss in Section 4.15, this opens the door to characteristic signatures distinguishing different models.

Again, also the matter behavior changes, now with classical friction being replaced by antifriction [77Jump 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 only apply to the homogeneous mode, it can provide a mechanism of structure formation as discussed in Section 4.15. But also in combination with chaotic inflation as the mechanism to generate structure does the modified matter behavior lead to improvements: If we now view the scalar f as an inflaton field, it will be driven to large values in order to start a second phase of slow-roll inflation which is long enough. This is satisfied for a large range of the ambiguity parameters j and l [67] and can even leave signatures [197] in the cosmic microwave spectrum [134]: The earliest moments when 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 generically reached by the mechanism generating initial conditions, is currently being investigated at the basic level of loop quantum cosmology [27]. They should be regarded as first 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 inflaton is more attractive since it requires less choices and provides a fundamental explanation of inflation directly from quantum gravity. However, it is also more difficult to analyze structure formation in this context while there are already 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 generically 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 that also arise in the full theory.

It is also possible to combine collapsing and expanding phases in cyclic or oscillatory models [148Jump 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. 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 ideas for the pre-history of our universe before the Big Bang. There are also possibilities to use a bounce to describe the structure in the universe. So far, this has only been described in effective models [137] using brane scenarios [151] where 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 [68]. Then, however, generically an inflationary phase commences as in [148], 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 [110112] where 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 to start the universe in an initial phase of oscillations before an inflationary phase is entered [16053]. This presents a natural realization of the scenario where the initial scale factor at the 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, too [94Jump To The Next Citation Point95Jump To The Next Citation Point]. An analysis requires inhomogeneous techniques at least at an effective level, such as those described in Section 4.12. It has been shown that loop cosmology is free of this problem, because the modified behavior for the homogeneous mode of the metric and matter is not relevant for matter propagation [129Jump To The Next Citation Point]. The whole cosmological picture that follows from the effective equations is thus consistent.

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