That the Wheeler–DeWitt representation may not be the right choice is also indicated by the fact that its scale factor operator has a continuous spectrum, while quantum geometry which is a welldefined quantization of the full theory, implies discrete volume spectra. Indeed, the Wheeler–DeWitt quantization of full gravity exists only formally, and its application to quantum cosmology simply quantizes the classically reduced isotropic system. This is much easier, and also more ambiguous, and leaves open many consistency considerations. It would be more reliable to start with the full quantization and introduce the symmetries there, or at least follow the same constructions of the full theory in a reduced model. If this is done, it turns out that indeed we obtain a quantum representation inequivalent to the Wheeler–DeWitt representation, with strong implications in high energy regimes. In particular, just as the full theory such a quantization has a volume or
$p$
operator with a discrete spectrum, as derived in Section 5.2 .
4.4 Isotropy: Effective densities and equations
The isotropic model is thus quantized in such a way that the operator
$\hat{p}$
has a discrete spectrum containing zero. This immediately leads to a problem since we need a quantization of
$p{}^{3/2}$
in order to quantize a matter Hamiltonian such as ( 23 ) where not only the matter fields but also geometry are quantized. However, an operator with zero in the discrete part of its spectrum does not have a densely defined inverse and does not allow a direct quantization of
$p{}^{3/2}$
.
This leads us to the first main effect of the loop quantization: It turns out that despite the nonexistence of an inverse operator of
$\hat{p}$
one can quantize the classical
$p{}^{3/2}$
to a welldefined operator. This is not just possible in the model but also in the full theory where it even has been defined first [
193]
. Classically, one can always write expressions in many equivalent ways, which usually result in different quantizations. In the case of
$p{}^{3/2}$
, as discussed in Section 5.3 , there is a general class of ways to rewrite it in a quantizable manner [
41]
which differ in details but have all the same important properties. This can be parameterized by a function
$d(p{)}_{j,l}$
[
47,
50]
which replaces the classical
$p{}^{3/2}$
and strongly deviates from it for small
$p$
while being very close at large
$p$
. The parameters
$j\in \frac{1}{2}\mathbb{N}$
and
$0<l<1$
specify quantization ambiguities resulting from different ways of rewriting. With the function
$$\begin{array}{ccc}{p}_{l}\left(q\right)& =& \frac{3}{2l}{q}^{1l}(\frac{1}{l+2}\left((q+1{)}^{l+2}q1{}^{l+2}\right)\end{array}$$ 
(24)

$$\begin{array}{ccc}& & \frac{1}{l+1}q\left((q+1{)}^{l+1}sgn(q1\left)\rightq1{}^{l+1}\right))\end{array}$$  
we have
$$\begin{array}{c}d(p{)}_{j,l}:=p{}^{3/2}{p}_{l}\left(3\rightp/\gamma j{\ell}_{P}^{2}{)}^{3/(22l)},\end{array}$$ 
(25)

which indeed fulfills
$d(p{)}_{j,l}\sim p{}^{3/2}$
for
$\leftp\right\gg {p}_{*}:=\frac{1}{3}j\gamma {\ell}_{P}^{2}$
, but is finite with a peak around
${p}_{*}$
and approaches zero at
$p=0$
in a manner
$$\begin{array}{c}d(p{)}_{j,l}\sim {3}^{3(3l)/(22l)}(l+1{)}^{3/(22l)}\left(\gamma j{)}^{3(2l)/(22l)}{\ell}_{P}^{3(2l)/(1l)}\rightp{}^{3/(22l)}\end{array}$$ 
(26)

as it follows from
${p}_{l}\left(q\right)\sim 3{q}^{2l}/(1+l)$
. Some examples displaying characteristic properties are shown in Figure 9 in Section 5.3 .
The matter Hamiltonian obtained in this manner will thus behave differently at small
$p$
. At those scales also other quantum effects such as fluctuations can be important, but it is possible to isolate the effect implied by the modified density ( 25 ). We just need to choose a rather large value for the ambiguity parameter
$j$
such that modifications become noticeable already in semiclassical regimes. This is mainly a technical tool to study the behavior of equations, but can also be used to find constraints on the allowed values of ambiguity parameters.
We can thus use classical equations of motion, which are corrected for quantum effects by using the effective matter Hamiltonian
$$\begin{array}{c}{H}_{\phi}^{(eff)}(p,\phi ,{p}_{\phi}):=\frac{1}{2}d(p{)}_{j,l}{p}_{\phi}^{2}+p{}^{3/2}V\left(\phi \right)\end{array}$$ 
(27)

(see Section 5.5 for details on effective equations). This matter Hamiltonian changes the classical constraint such that now
$$\begin{array}{c}H=\frac{3}{8\pi G}\left({\gamma}^{2}\right(c\Gamma {)}^{2}+{\Gamma}^{2})\sqrt{\leftp\right}+{H}_{\phi}^{(eff)}(p,\phi ,{p}_{\phi})=0.\end{array}$$ 
(28)

Since the constraint determines all equations of motion, they also change: We obtain the effective Friedmann equation from
$H=0$
,
$$\begin{array}{c}{\left(\frac{\dot{a}}{a}\right)}^{2}+\frac{k}{{a}^{2}}=\frac{8\pi G}{3}\left(\frac{1}{2}\leftp{}^{3/2}d\right(p{)}_{j,l}{p}_{\phi}^{2}+V\left(\phi \right)\right)\end{array}$$ 
(29)

and the effective Raychaudhuri equation from
$\dot{c}=\{c,H\}$
,
$$\begin{array}{ccc}\frac{\ddot{a}}{a}& =& \frac{4\pi G}{3p{}^{3/2}}\left({H}_{matter}(p,\phi ,{p}_{\phi})2p\frac{\partial {H}_{matter}(p,\phi ,{p}_{\phi})}{\partial p}\right)\end{array}$$ 
(30)

$$\begin{array}{ccc}& =& \frac{8\pi G}{3}\left(\leftp{}^{3/2}d\right(p{)}_{j,l}^{1}{\dot{\phi}}^{2}\left(1\frac{1}{4}a\frac{dlog\left(\rightp{}^{3/2}d\left(p{)}_{j,l}\right)}{da}\right)V\left(\phi \right)\right).\end{array}$$ 
(31)

Matter equations of motion follow similarly as
$$\begin{array}{ccc}\dot{\phi}& =& \{\phi ,H\}=d(p{)}_{j,l}{p}_{\phi}\end{array}$$  
$$\begin{array}{ccc}{\dot{p}}_{\phi}& =& \{{p}_{\phi},H\}=\leftp{}^{3/2}{V}^{\prime}\right(\phi ),\end{array}$$  
which can be combined to the effective Klein–Gordon equation
$$\begin{array}{c}\ddot{\phi}=\dot{\phi}\dot{a}\frac{dlogd(p{)}_{j,l}}{da}\leftp{}^{3/2}d\right(p{)}_{j,l}{V}^{\prime}\left(\phi \right).\end{array}$$ 
(32)

Further discussion for different forms of matter can be found in [
186]
.
4.5 Isotropy: Properties and intuitive meaning
As a consequence of the function
$d(p{)}_{j,l}$
, the effective equations have different qualitative behavior at small versus large scales
$p$
. In the effective Friedmann equation ( 29 ), this is most easily seen by comparing it with a mechanics problem with a standard Hamiltonian, or energy, of the form
$$E=\frac{1}{2}{\dot{a}}^{2}\frac{2\pi G}{3{V}_{0}}{a}^{1}d(p{)}_{j,l}{p}_{\phi}^{2}\frac{4\pi G}{3}{a}^{2}V(\phi )=0$$
restricted to be zero. If we assume a constant scalar potential
$V\left(\phi \right)$
, there is no
$\phi $
dependence and the scalar equations of motion show that
${p}_{\phi}$
is constant. Thus, the potential for the motion of
$a$
is essentially determined by the function
$d(p{)}_{j,l}$
.
In the classical case,
$d\left(p\right)=p{}^{3/2}$
and the potential is negative and increasing, with a divergence at
$p=0$
. The scale factor
$a$
is thus driven toward
$a=0$
, which it will always reach in finite time where the system breaks down. With the effective density
$d(p{)}_{j,l}$
, however, the potential is bounded from below, and is decreasing from zero for
$a=0$
to the minimum around
${p}_{*}$
. Thus, the scale factor is now slowed down before it reaches
$a=0$
, which depending on the matter content could avoid the classical singularity altogether.
The behavior of matter is also different as shown by the effective Klein–Gordon equation (
32 ).
Most importantly, the derivative in the
$\dot{\phi}$
term changes sign at small
$a$
since the effective density is increasing there. Thus, the qualitative behavior of all the equations changes at small scales, which as we will see gives rise to many characteristic effects. Nevertheless, for the analysis of the equations as well as conceptual considerations it is interesting that solutions at small and large scales are connected by a duality transformation [
147]
, which even exists between effective solutions for loop cosmology and braneworld cosmology [
90]
.
We have seen that the equations of motion following from an effective Hamiltonian are expected to display qualitatively different behavior at small scales. Before discussing specific models in detail, it is helpful to observe what physical meaning the resulting modifications have.
Classical gravity is always attractive, which implies that there is nothing to prevent collapse in black holes or the whole universe. In the Friedmann equation this is expressed by the fact that the potential as used before is always decreasing toward
$a=0$
where it diverges. With the effective density, on the other hand, we have seen that the decrease stops and instead the potential starts to increase at a certain scale before it reaches zero at
$a=0$
. This means that at small scales, where quantum gravity becomes important, the gravitational attraction turns into repulsion. In contrast to classical gravity, thus, quantum gravity has a repulsive component which can potentially prevent collapse. So far, this has only been demonstrated in homogeneous models, but it relies on a general mechanism which is also present in the full theory.
Not only the attractive nature of gravity changes at small scales, but also the behavior of matter in a gravitational background. Classically, matter fields in an expanding universe are slowed down by a friction term in the Klein–Gordon equation (
32 ) where
$\dot{a}dlog{a}^{3}/da=3\dot{a}/a$
is negative. Conversely, in a contracting universe matter fields are excited and even diverge when the classical singularity is reached. This behavior turns around at small scales where the derivative
$dlogd(a{)}_{j,l}/da$
becomes positive. Friction in an expanding universe then turns into antifriction such that matter fields are driven away from their potential minima before classical behavior sets in. In a contracting universe, on the other hand, matter fields are not excited by antifriction but freeze once the universe becomes small enough.
These effects do not only have implications for the avoidance of singularities at
$a=0$
but also for the behavior at small but nonzero scales. Gravitational repulsion can not only prevent collapse of a contracting universe [
187]
but also, in an expanding universe, enhance its expansion.
The universe then accelerates in an inflationary manner from quantum gravity effects alone [
45]
.
Similarly, the modified behavior of matter fields has implications for inflationary models [
77]
.
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 1 . Requirements for a bounce are that the conditions
$\dot{a}=0$
and
$\ddot{a}>0$
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
$V\left(\phi \right)<0$
. In those cases, there are classical solutions with
$\dot{a}=0$
, but they generically have
$\ddot{a}<0$
corresponding to a recollapse. This can easily be seen in the flat case with a negative potential where ( 30 ) is strictly negative with
$dlog{a}^{3}d(a{)}_{j,l}/da\approx 0$
at large scales.
The repulsive nature at small scales now implies a second point where
$\dot{a}=0$
from ( 29 ) at smaller
$a$
since the matter energy now decreases also for
$a\to 0$
. Moreover, the modified Raychaudhuri equation ( 30 ) has an additional positive term at small scales such that
$\ddot{a}>0$
becomes possible.
Matter also behaves differently through the modified Klein–Gordon equation (
32 ). Classically, with
$\dot{a}<0$
the scalar experiences antifriction and
$\phi $
diverges close to the classical singularity. With the modification, antifriction turns into friction at small scales, damping the motion of
$\phi $
such that it remains finite. In the case of a negative potential [
68]
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 [
187,
203]
. Further, more generic possibilities for bounces arise from other correction terms [
100,
97]
.
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
$\phi $
diverges.
4.6.2 Expansion
Repulsion can not only prevent collapse but also accelerates an expanding phase. Indeed, using the behavior ( 26 ) at small scales in the effective Raychaudhuri equation ( 30 ) shows that
$\ddot{a}$
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
2 , inflation is realized by quantum gravity effects for any matter field irrespective of its form, potential or initial values [
45]
. The kind of expansion at early stages is generically superinflationary, 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 nonzero potential such as a mass term for matter inflation
$w$
is generically close to exponential:
${w}_{eff}\approx 1$
. 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/\rho $
where
$P=\partial E/\partial V$
is the pressure, i.e., the negative change of energy with respect to volume, and
$\rho =E/V$
energy density. Using the matter Hamiltonian for
$E$
and
$V=p{}^{3/2}$
, we obtain
$${P}_{eff}=\frac{1}{3}\leftp{}^{1/2}{d}^{\prime}\right(p){p}_{\phi}^{2}V(\phi )$$
and thus in the classical case
$$w=\frac{\frac{1}{2}p{}^{3}{p}_{\phi}^{2}V(\phi )}{\frac{1}{2}p{}^{3}{p}_{\phi}^{2}+V(\phi )}$$
as usually. In the modified case, however, we have
$${w}_{eff}=\frac{\frac{1}{3}\leftp{}^{1/2}{d}^{\prime}\right(p){p}_{\phi}^{2}+V(\phi )}{\frac{1}{2}\leftp{}^{3/2}d\right(p){p}_{\phi}^{2}+V(\phi )}.$$
Figure 2
: Example for a solution of
$a\left(t\right)$
and
$\phi \left(t\right)$
, showing early loop inflation and later slowroll 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 3 .
Figure 3
: Movie showing the initial push of a scalar
$\phi $
up its potential and the ensuing slowroll phase together with the corresponding inflationary phase of
$a$
.
In general, we need to know the matter behavior to know
$w$
and
${w}_{eff}$
. But we can get generic qualitative information by treating
${p}_{\phi}$
and
$V\left(\phi \right)$
as unknowns determined by
$w$
and
${w}_{eff}$
. In the generic case, there is no unique solution for
${p}_{\phi}^{2}$
and
$V\left(\phi \right)$
since, after all,
${p}_{\phi}$
and
$\phi $
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
$${w}_{eff}=1+\frac{\leftp{}^{3/2}\right(w+1\left)\right(d\left(p\right)\frac{2}{3}\leftp\right{d}^{\prime}\left(p\right))}{1w+(w+1)\leftp{}^{3/2}d\right(p)}.$$
Since for small
$p$
the numerator in the fraction approaches zero faster than the second part of the denominator,
${w}_{eff}$
approaches minus one at small volume except for the special case
$w=1$
, which is realized for
$V\left(\phi \right)=0$
. Note that the argument does not apply to the case of vanishing potential since then
${p}_{\phi}^{2}=const$
and
$V\left(\phi \right)=0$
presents a unique solution to the linear equations for
$w$
and
${w}_{eff}$
. In fact, this case leads in general to a much smaller
${w}_{eff}=\frac{2}{3}\leftp\rightd(p{)}^{\prime}/d(p)\approx 1/(1l)<1$
[
45]
. One can also see from the above formula that
${w}_{eff}$
, 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 [
77]
. Matter fields thus move away from their minima and become excited even if they start close to a minimum (Figure 2 ). 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
$\phi $
as an inflaton field, it will be driven to large values in order to start a second phase of slowroll 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 2 and 3 where the initial decrease is steeper. Provided the resulting structure can be seen today, i.e., there are not too many efoldings 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 welldeveloped 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 [
148]
. 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 prehistory 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 [
110,
112]
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 [
160,
53]
. 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 superinflation 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 [
94,
95]
. 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 [
129]
. The whole cosmological picture that follows from the effective equations is thus consistent.
4.7 Anisotropies
Anisotropic models provide a first generalization of isotropic ones to more realistic situations.
They thus can be used to study the robustness of effects analyzed in isotropic situations and, at the same time, provide a large class of interesting applications. An analysis in particular of the singularity issue is important since the classical approach to a singularity can be very different from the isotropic one. On the other hand, the anisotropic approach is deemed to be characteristic even for general inhomogeneous singularities if the BKL scenario [
31]
is correct.
A general homogeneous but anisotropic metric is of the form
$$d{s}^{2}=N(t{)}^{2}d{t}^{2}+{\sum}_{I,J=1}^{3}{q}_{IJ}(t){\omega}^{I}\otimes {\omega}^{J}$$
with leftinvariant 1forms
${\omega}^{I}$
on space
$\Sigma $
, which, thanks to homogeneity, can be identified with the simply transitive symmetry group
$S$
as a manifold. The leftinvariant 1forms satisfy the Maurer–Cartan relations
$$d{\omega}^{I}=\frac{1}{2}{C}_{JK}^{I}{\omega}^{J}\wedge {\omega}^{K}$$
with the structure constants
${C}_{JK}^{I}$
of the symmetry group. In a matrix parameterization of the symmetry group, one can derive explicit expressions for
${\omega}^{I}$
from the Maurer–Cartan form
${\omega}^{I}{T}_{I}={\theta}_{MC}={g}^{1}dg$
with generators
${T}_{I}$
of
$S$
.
The simplest case of a symmetry group is an Abelian one with
${C}_{JK}^{I}=0$
, corresponding to the Bianchi I model. In this case,
$S$
is given by
${\mathbb{R}}^{3}$
or a torus, and leftinvariant 1forms are simply
${\omega}^{I}=d{x}^{I}$
in Cartesian coordinates. Other groups must be restricted to class A models in this context, satisfying
${C}_{JI}^{I}=0$
since otherwise there is no Hamiltonian formulation. The structure constants can then be parameterized as
${C}_{JK}^{I}={\epsilon}_{JK}^{I}{n}^{\left(I\right)}$
.
A common simplification is to assume the metric to be diagonal at all times, which corresponds to a reduction technically similar to a symmetry reduction. This amounts to
${q}_{IJ}={a}_{\left(I\right)}^{2}{\delta}_{IJ}$
as well as
${K}_{IJ}={K}_{\left(I\right)}{\delta}_{IJ}$
for the extrinsic curvature with
${K}_{I}={\dot{a}}_{I}$
. Depending on the structure constants, there is also nonzero intrinsic curvature quantified by the spin connection components
$$\begin{array}{c}{\Gamma}_{I}=\frac{1}{2}\left(\frac{{a}_{J}}{{a}_{K}}{n}^{J}+\frac{{a}_{K}}{{a}_{J}}{n}^{K}\frac{{a}_{I}^{2}}{{a}_{J}{a}_{K}}{n}^{I}\right)\text{for}{\epsilon}_{IJK}=1.\end{array}$$ 
(33)

This influences the evolution as follows from the Hamiltonian constraint
$$\begin{array}{ccc}& & \frac{1}{8\pi G}({a}_{1}{\dot{a}}_{2}{\dot{a}}_{3}+{a}_{2}{\dot{a}}_{1}{\dot{a}}_{2}+{a}_{3}{\dot{a}}_{1}{\dot{a}}_{2}({\Gamma}_{2}{\Gamma}_{3}{n}^{1}{\Gamma}_{1}){a}_{1}({\Gamma}_{1}{\Gamma}_{3}{n}^{2}{\Gamma}_{2}){a}_{2}\end{array}$$  
$$\begin{array}{ccc}& & ({\Gamma}_{1}{\Gamma}_{2}{n}^{3}{\Gamma}_{3}){a}_{3})+{H}_{matter}\left({a}_{I}\right)=0.\end{array}$$ 
(34)

In the vacuum Bianchi I case the resulting equations are easy to solve by
${a}_{I}\propto {t}^{{\alpha}_{I}}$
with
${\sum}_{I}{\alpha}_{I}={\sum}_{I}{\alpha}_{I}^{2}=1$
[
135]
. The volume
${a}_{1}{a}_{2}{a}_{3}\propto t$
vanishes for
$t=0$
where the classical singularity appears. Since one of the exponents
${\alpha}_{I}$
must be negative, however, only two of the
${a}_{I}$
vanish at the classical singularity while the third one diverges. This already demonstrates how different the behavior can be from the isotropic one and that anisotropic models provide a crucial test of any mechanism for singularity resolution.
4.8 Anisotropy: Connection variables
A densitized triad corresponding to a diagonal homogeneous metric has real components
${p}^{I}$
with
$\left{p}^{I}\right={a}_{J}{a}_{K}$
if
${\epsilon}_{IJK}=1$
[
48]
. Connection components are
${c}_{I}={\Gamma}_{I}+\gamma {K}_{I}={\Gamma}_{I}+\gamma {\dot{a}}_{I}$
and are conjugate to the
${p}_{I}$
,
$\{{c}_{I},{p}^{J}\}=8\pi \gamma G{\delta}_{I}^{J}$
. In terms of triad variables we now have spin connection components
$$\begin{array}{c}{\Gamma}_{I}=\frac{1}{2}\left(\frac{{p}^{K}}{{p}^{J}}{n}^{J}+\frac{{p}^{J}}{{p}^{K}}{n}^{K}\frac{{p}^{J}{p}^{K}}{({p}^{I}{)}^{2}}{n}^{I}\right)\end{array}$$ 
(35)

and the Hamiltonian constraint (in the absence of matter)
$$\begin{array}{ccc}H& =& \frac{1}{8\pi G}\{\left[({c}_{2}{\Gamma}_{3}+{c}_{3}{\Gamma}_{2}{\Gamma}_{2}{\Gamma}_{3})(1+{\gamma}^{2}){n}^{1}{c}_{1}{\gamma}^{2}{c}_{2}{c}_{3}\right]\sqrt{\left\frac{{p}^{2}{p}^{3}}{{p}^{1}}\right}\end{array}$$  
$$\begin{array}{ccc}& & +\left[({c}_{1}{\Gamma}_{3}+{c}_{3}{\Gamma}_{1}{\Gamma}_{1}{\Gamma}_{3})(1+{\gamma}^{2}){n}^{2}{c}_{2}{\gamma}^{2}{c}_{1}{c}_{3}\right]\sqrt{\left\frac{{p}^{1}{p}^{3}}{{p}^{2}}\right}\end{array}$$  
$$\begin{array}{ccc}& & +\left[({c}_{1}{\Gamma}_{2}+{c}_{2}{\Gamma}_{1}{\Gamma}_{1}{\Gamma}_{2})(1+{\gamma}^{2}){n}^{3}{c}_{3}{\gamma}^{2}{c}_{1}{c}_{2}\right]\sqrt{\left\frac{{p}^{1}{p}^{2}}{{p}^{3}}\right}\}.\end{array}$$ 
(36)

Unlike in isotropic models, we now have inverse powers of
${p}^{I}$
even in the vacuum case through the spin connection, unless we are in the Bianchi I model. This is a consequence of the fact that not just extrinsic curvature, which in the isotropic case is related to the matter Hamiltonian through the Friedmann equation, leads to divergences but also intrinsic curvature. These divergences are cut off by quantum geometry effects as before such that also the dynamical behavior changes.
This can again be dealt with by effective equations where inverse powers of triad components are replaced by bounded functions [
62]
. However, even with those modifications, expressions for curvature are not necessarily bounded unlike in the isotropic case. This comes from the presence of different classical scales,
${a}_{I}$
, such that more complicated expressions as in
${\Gamma}_{I}$
are possible, while in the isotropic model there is only one scale and curvature can only be an inverse power of
$p$
, which is then regulated by effective expressions like
$d\left(p\right)$
.
4.9 Anisotropy: Applications
4.9.1 Isotropization
Matter fields are not the only contributions to the Hamiltonian in cosmology, but also the effect of anisotropies can be included in this way to an isotropic model. The late time behavior of this contribution can be shown to behave as
${a}^{6}$
in the shear energy density [
156]
, which falls off faster than any other matter component. Thus, toward later times the universe becomes more and more isotropic.
In the backward direction, on the other hand, this means that the shear term diverges most strongly, which suggests that this term should be most relevant for the singularity issue. Even if matter densities are cut off as discussed before, the presence of bounces would depend on the fate of the anisotropy term. This simple reasoning is not true, however, since the behavior of shear is only effective and uses assumptions about the behavior of matter. It can thus not simply be extrapolated to early times. Anisotropies are independent degrees of freedom which affect the evolution of the scale factor. But only in certain regimes can this contribution be modeled simply by a function of the scale factor alone; in general one has to use the coupled system of equations for the scale factor, anisotropies and possible matter fields.
4.9.2 Bianchi IX
Modifications to classical behavior are most drastic in the Bianchi IX model with symmetry group
$S\sim =SU\left(2\right)$
such that
${n}^{I}=1$
. The classical evolution can be described by a 3dimensional mechanics system with a potential obtained from ( 34 ) such that the kinetic term is quadratic in derivatives of
${a}_{I}$
with respect to a time coordinate
$\tau $
defined by
$dt={a}_{1}{a}_{2}{a}_{3}d\tau $
. This potential
$$\begin{array}{ccc}W\left({p}^{I}\right)& =& ({\Gamma}_{2}{\Gamma}_{3}{n}^{1}{\Gamma}_{1}){p}^{2}{p}^{3}+({\Gamma}_{1}{\Gamma}_{3}{n}^{2}{\Gamma}_{2}){p}^{1}{p}^{3}+({\Gamma}_{1}{\Gamma}_{2}{n}^{3}{\Gamma}_{3}){p}^{1}{p}^{2}\end{array}$$ 
(37)

$$\begin{array}{ccc}& =& \frac{1}{4}\left({\left(\frac{{p}^{2}{p}^{3}}{{p}^{1}}\right)}^{2}+{\left(\frac{{p}^{1}{p}^{3}}{{p}^{2}}\right)}^{2}+{\left(\frac{{p}^{1}{p}^{2}}{{p}^{3}}\right)}^{2}2({p}^{1}{)}^{2}2({p}^{2}{)}^{2}2({p}^{3}{)}^{2}\right)\end{array}$$  
diverges at small
${p}^{I}$
, in particular (in a direction dependent manner) at the classical singularity where all
${p}^{I}=0$
. Figure 4 illustrates the walls of the potential, which with decreasing volume push the universe toward the classical singularity.
Figure 4
: Movie illustrating the Bianchi IX potential ( 37 ) and the movement of its walls, rising toward zero
${p}^{1}$
and
${p}^{2}$
and along the diagonal direction, toward the classical singularity with decreasing volume
$V=\sqrt{\left{p}^{1}{p}^{2}{p}^{3}\right}$
. The contours are plotted for the function
$W({p}^{1},{p}^{2},{V}^{2}/({p}^{1}{p}^{2}\left)\right)$
.
As before in isotropic models, effective equations where the behavior of eigenvalues of the spin connection components is used do not have this divergent potential. Instead, if two
${p}^{I}$
are held fixed and the third approaches zero, the effective quantum potential is cut off and goes back to zero at small values, which changes the approach to the classical singularity. Yet, the effective potential is unbounded if one
${p}^{I}$
diverges while another one goes to zero and the situation is qualitatively different from the isotropic case. Since the effective potential corresponds to spatial intrinsic curvature, curvature is not bounded in anisotropic effective models. However, this is a statement only about curvature expressions on minisuperspace, and the more relevant question is what happens to curvature along trajectories obtained by solving equations of motion. This demonstrates that dynamical equations must always be considered to draw conclusions for the singularity issue.
The approach to the classical singularity is best analyzed in Misner variables [
157]
consisting of the scale factor
$\Omega :=\frac{1}{3}logV$
and two anisotropy parameters
${\beta}_{\pm}$
defined such that
$${a}_{1}={e}^{\Omega +{\beta}_{+}+\sqrt{3}{\beta}_{}},{a}_{2}={e}^{\Omega +{\beta}_{+}\sqrt{3}{\beta}_{}},{a}_{3}={e}^{\Omega 2{\beta}_{+}}.$$
The classical potential then takes the form
$$W(\Omega ,{\beta}_{\pm})=\frac{1}{2}{e}^{4\Omega}\left({e}^{8{\beta}_{+}}4{e}^{2{\beta}_{+}}cosh\left(2\sqrt{3}{\beta}_{}\right)+2{e}^{4{\beta}_{+}}(cosh(4\sqrt{3}{\beta}_{})1)\right),$$
which at fixed
$\Omega $
has three exponential walls rising from the isotropy point
${\beta}_{\pm}=0$
and enclosing a triangular region (Figure 5 ).
Figure 5
: Movie illustrating the Bianchi IX potential in the anisotropy plane and its exponentially rising walls. Positive values of the potential are drawn logarithmically with solid contour lines and negative values with dashed contour lines.
A cross section of a wall can be obtained by taking
${\beta}_{}=0$
and
${\beta}_{+}$
to be negative, in which case the potential becomes
$W(\Omega ,{\beta}_{+},0)\approx \frac{1}{2}{e}^{4\Omega 8{\beta}_{+}}$
. One thus obtains the picture of a point moving almost freely until it is reflected at a wall. In between reflections, the behavior is approximately given by the Kasner solution described before. This behavior with infinitely many reflections before the classical singularity is reached, can be shown to be chaotic [
32]
, which suggests a complicated approach to classical singularities in general.
Figure 6
: Approximate effective wall of finite height [
60]
as a function of
$x={\beta}_{+}$
, compared to the classical exponential wall (upper dashed curve). Also shown is the exact wall
$W({p}^{1},{p}^{1},(V/{p}^{1}{)}^{2})$
(lower dashed curve), which for
$x$
smaller than the peak value coincides well with the approximation up to a small, nearly constant shift.
With the effective modification, however, the potential for fixed
$\Omega $
does not diverge and the walls, as shown in Figure 6 , break down already at a small but nonzero volume [
60]
. As a function of densitized triad components the effective potential is illustrated in Figure 7 , and as a function on the anisotropy plane in Figure 8 . In this scenario, there are only finitely many reflections, which does not lead to chaotic behavior but instead results in asymptotic Kasner behavior [
61]
.
Figure 7
: Movie illustrating the effective Bianchi IX potential and the movement and breakdown of its walls. The contours are plotted as in Figure 4 .
Figure 8
: Movie illustrating the effective Bianchi IX potential in the anisotropy plane and its walls of finite height, which disappear at finite volume. Positive values of the potential are drawn logarithmically with solid contour lines and negative values with dashed contour lines.
Comparing Figure 5 with Figure 8 shows that in their center they are very close to each other, while strong deviations occur for large anisotropies. This demonstrates that most of the classical evolution, which mostly happens in the inner triangular region, is not strongly modified by the effective potential. Quantum effects are important only when anisotropies become too large, for instance when the system moves deep into one of the three valleys, or the total volume becomes small. In those regimes the quantum evolution will take over and describe the further behavior of the system.
4.9.3 Isotropic curvature suppression
If we use the potential for time coordinate
$t$
rather than
$\tau $
, it is replaced by
$W/\left({p}^{1}{p}^{2}{p}^{3}\right)$
, which in the isotropic reduction
${p}^{1}={p}^{2}={p}^{3}=\frac{1}{4}{a}^{2}$
gives the curvature term
$k{a}^{2}$
. Although the anisotropic effective curvature potential is not bounded it is, unlike the classical curvature, bounded from above at any fixed volume. Moreover, it is bounded along the isotropy line and decays when
$a$
approaches zero. Thus, there is a suppression of the divergence in
$k{a}^{2}$
when the closed isotropic model is viewed as embedded in a Bianchi IX model. Similarly to matter Hamiltonians, intrinsic curvature then approaches zero at zero scale factor.
This is a further illustration for the special nature of isotropic models compared to anisotropic ones. In the classical reduction, the
${p}^{I}$
in the anisotropic spin connection cancel such that the spin connection is a constant and no special steps are needed for its quantization. By viewing isotropic models within anisotropic ones, one can consistently realize the model and see a suppression of intrinsic curvature terms. Anisotropic models, on the other hand, do not have, and do not need, complete suppression since curvature functions can still be unbounded.
4.10 Anisotropy: Implications for inhomogeneities
Even without implementing inhomogeneous models the previous discussion allows some tentative conclusions as to the structure of general singularities. This is based on the BKL picture [
31]
whose basic idea is to study Einstein's field equations close to a singularity. One can then argue that spatial derivatives become subdominant compared to timelike derivatives such that the approach should locally be described by homogeneous models, in particular the Bianchi IX model since it has the most freedom in its general solution.
Since spatial derivatives are present, though, they lead to small corrections and couple the geometries in different spatial points. One can visualize this by starting with an initial slice which is approximated by a collection of homogeneous patches. For some time, each patch evolves independently of the others, but this is not precisely true since coupling effects have been ignored.
Moreover, each patch geometry evolves in a chaotic manner, which means that two initially nearby geometries depart rapidly from each other. The approximation can thus be maintained only if the patches are subdivided during the evolution, which goes on without limits in the approach to the singularity. There is, thus, more and more inhomogeneous structure being generated on arbitrarily small scales, which leads to a complicated picture of a general singularity.
This picture can be taken over to the effective behavior of the Bianchi IX model. Here, the patches do not evolve chaotically even though at larger volume they follow the classical behavior.
The subdivision thus has to be done also for the initial effective evolution. At some point, however, when reflections on the potential walls stop, the evolution simplifies and subdivisions are no longer necessary. There is thus a lower bound to the scale of structure whose precise value depends on the initial geometries. Nevertheless, from the scale at which the potential walls break down one can show that structure formation stops at the latest when the discreteness scale of quantum geometry is reached [
60]
. This can be seen as a consistency test of the theory since structure below the discreteness could not be supported by quantum geometry.
We have thus a glimpse on the inhomogeneous situation with a complicated but consistent approach to a general classical singularity. The methods involved, however, are not very robust since the BKL scenario, which even classically is still at the level of a conjecture for the general case [
32,
168]
, would need to be available as an approximation to quantum geometry. For more reliable results the methods need to be refined to take into account inhomogeneities properly.
4.11 Inhomogeneities
Allowing for inhomogeneities inevitably means to take a big step from finitely many degrees of freedom to infinitely many ones. There is no straightforward way to cut down the number of degrees of freedom to finitely many ones while being more general than in the homogeneous context. One possibility would be to introduce a smallscale cutoff such that only finitely many wave modes arise (e.g., through a lattice as is indeed done in some coherent state constructions [
184]
). This is in fact expected to happen in a discrete framework such as quantum geometry, but would at this stage of defining a model simply be introduced by hand.
For the analysis of inhomogeneous situations there are several different approximation schemes:

∙
Use only isotropic quantum geometry and in particular its effective description, but couple to inhomogeneous matter fields. Problems in this approach are that backreaction effects are ignored (which is also the case in most classical treatments) and that there is no direct way how to check modifications used in particular for gradient terms of the matter Hamiltonian. So far, this approach has led to a few indications of possible effects.

∙
Start with the full constraint operator, write it as the homogeneous one plus correction terms from inhomogeneities, and derive effective classical equations. This approach is more ambitious since contact to the full theory is realized. So far, there are not many results since a suitable perturbation scheme has to be developed.

∙
There are inhomogeneous symmetric models, such as Einstein–Rosen waves or the spherically symmetric one, which have infinitely many kinematical degrees of freedom but can be treated explicitly. Also here, contact to the full theory is present through the symmetry reduction procedure of Section 6 . This procedure itself can be tested by studying those models between homogeneous ones and the full theory, but results can also be used for physical applications involving inhomogeneities. Many issues that are of importance in the full theory, such as the anomaly problem, also arise here and can thus be studied more explicitly.
4.12 Inhomogeneous matter with isotropic quantum geometry
Inhomogeneous matter fields cannot be introduced directly to isotropic quantum geometry since after the symmetry reduction there is no space manifold left for the fields to live on. There are then two different routes to proceed: One can simply take the classical field Hamiltonian and introduce effective modifications modeled on what happens to the isotropic Hamiltonian, or perform a mode decomposition of the matter fields and just work with the spaceindependent amplitudes. The latter is possible since the homogeneous geometry provides a background for the mode decomposition.
The basic question, for the example of a scalar field, then is how the metric coefficient in the gradient term of Equation (
12 ),
${E}_{i}^{a}{E}_{i}^{b}/\sqrt{\leftdetE\right}$
, would be replaced effectively. For the other terms, one can simply use the isotropic modification, which is taken directly from the quantization. For the gradient term, however, one does not have a quantum expression in this context and a modification can only be guessed. The problem arises since the inhomogeneous term involves inverse powers of
$E$
, while in the isotropic context the coefficient just reduces to
$\sqrt{\leftp\right}$
, which would not be modified at all. There is thus no obvious and unique way to find a suitable replacement.
A possible route would be to read off the modification from the full quantum Hamiltonian, or at least from an inhomogeneous model, which requires a better knowledge of the reduction procedure.
Alternatively, one can take a more phenomenological point of view and study the effects of possible replacements. If the robustness of these effects to changes in the replacements is known, one can get a good picture of possible implications. So far, only initial steps have been taken and there is no complete programme in this direction.
Another approximation of the inhomogeneous situation has been developed in [
70]
by patching isotropic quantum geometries together to support an inhomogeneous matter field. This can be used to study modified dispersion relations to the extent that the result agrees with preliminary calculations performed in the full theory [
115,
3,
4,
181,
182]
even at a quantitative level. There is thus further evidence that symmetric models and their approximations can provide reliable insights into the full theory.
4.13 Inhomogeneity: Perturbations
With a symmetric background, a mode decomposition is not only possible for matter fields but also for geometry. The homogeneous modes can then be quantized as before, while higher modes are coupled as perturbations implementing inhomogeneities [
120]
. As with matter Hamiltonians before, one can then also deal with the gravitational part of the Hamiltonian constraint. In particular, there are terms with inverse powers of the homogeneous fields which receive modifications upon quantization. As with gradient terms in matter Hamiltonians, there are several options for those modifications which can only be restricted by relating them to the full Hamiltonian. This would require introducing the mode decomposition, analogously to symmetry conditions, at the quantum level and writing the full constraint operator as the homogeneous one plus correction terms.
An additional complication compared to matter fields is that one is now dealing with infinitely many coupled constraint equations since the lapse function
$N\left(x\right)$
is inhomogeneous, too. This function can itself be decomposed into modes
${\sum}_{n}{N}_{n}{Y}_{n}\left(x\right)$
, with harmonics
${Y}_{n}\left(x\right)$
according to the symmetry, and each amplitude
${N}_{n}$
is varied independently giving rise to a separate constraint.
The main constraint arises from the homogeneous mode, which describes how inhomogeneities affect the evolution of the homogeneous scale factors.
4.14 Inhomogeneous models
The full theory is complicated at several different levels of both conceptual and technical nature. For instance, one has to deal with infinitely many degrees of freedom, most operators have complicated actions, and interpreting solutions to all constraints in a geometrical manner can be difficult. Most of these complications are avoided in homogeneous models, in particular when effective classical equations are employed. These equations use approximations of expectation values of quantum geometrical operators which need to be known rather explicitly. The question then arises whether one can still work at this level while relaxing the symmetry conditions and bringing in more complications of the full theory.
Explicit calculations at a level similar to homogeneous models, at least for matrix elements of individual operators, are possible in inhomogeneous models, too. In particular the spherically symmetric model and cylindrically symmetric Einstein–Rosen waves are of this class, where the symmetry or other conditions are strong enough to result in a simple volume operator. In the spherically symmetric model, this simplification comes from the remaining isotropy subgroup isomorphic to U(1) in generic points, while the Einstein–Rosen model is simplified by polarization conditions that play a role analogous to the diagonalization of homogeneous models. With these models one obtains access to applications for black holes and gravitational waves, but also to inhomogeneities in cosmology.
In spherical coordinates
$x$
,
$\vartheta $
,
$\phi $
a spherically symmetric spatial metric takes the form
$$d{s}^{2}={q}_{xx}(x,t)d{x}^{2}+{q}_{\phi \phi}(x,t)d{\Omega}^{2}$$
with
$d{\Omega}^{2}=d{\vartheta}^{2}+{sin}^{2}\vartheta d{\phi}^{2}$
. This is related to densitized triad components by [
196,
136]
$$\left{E}^{x}\right={q}_{\phi \phi},({E}^{\phi}{)}^{2}={q}_{xx}{q}_{\phi \phi},$$
which are conjugate to the other basic variables given by the Ashtekar connection component
${A}_{x}$
and the extrinsic curvature component
${K}_{\phi}$
:
$$\left\{{A}_{x}\right(x),{E}^{x}(y\left)\right\}=8\pi G\gamma \delta (x,y),\left\{\gamma {K}_{\phi}\right(x),{E}^{\phi}(y\left)\right\}=16\pi G\gamma \delta (x,y).$$
Note that we use the Ashtekar connection for the inhomogeneous direction
$x$
but extrinsic curvature for the homogeneous direction along symmetry orbits [
75]
. Connection and extrinsic curvature components for the
$\phi $
direction are related by
${A}_{\phi}^{2}={\Gamma}_{\phi}^{2}+{\gamma}^{2}{K}_{\phi}^{2}$
with the spin connection component
$$\begin{array}{c}{\Gamma}_{\phi}=\frac{{E}^{x\prime}}{2{E}^{\phi}}.\end{array}$$ 
(38)

Unlike in the full theory or homogeneous models,
${A}_{\phi}$
is not conjugate to a triad component but to [
52]
$${P}^{\phi}=\sqrt{4({E}^{\phi}{)}^{2}{A}_{\phi}^{2}({P}^{\beta}{)}^{2}}$$
with the momentum
${P}^{\beta}$
conjugate to a U(1)gauge angle
$\beta $
. This is a rather complicated function of both triad and connection variables such that the volume
$V=4\pi \int \sqrt{\left{E}^{x}\right}{E}^{\phi}dx$
would have a rather complicated quantization. It would still be possible to compute the full volume spectrum, but with the disadvantage that volume eigenstates would not be given by triad eigenstates such that computations of many operators would be complicated [
74]
. This can be avoided by using extrinsic curvature which is conjugate to the triad component [
75]
. Moreover, this is also in accordance with a general scheme to construct Hamiltonian constraint operators for the full theory as well as symmetric models [
194,
42,
58]
.
The constraint operator in spherical symmetry is given by
$$\begin{array}{c}H\left[N\right]=(2G{)}^{1}{\int}_{B}dxN(x\left)\right{E}^{x}{}^{1/2}\left(({K}_{\phi}^{2}{E}^{\phi}+2{K}_{\phi}{K}_{x}{E}^{x})+(1{\Gamma}_{\phi}^{2}){E}^{\phi}+2{\Gamma}_{\phi}^{\prime}{E}^{x}\right)\end{array}$$ 
(39)

accompanied by the diffeomorphism constraint
$$\begin{array}{c}D\left[{N}^{x}\right]=\left(2G{)}^{1}{\int}_{B}{N}^{x}\right(x\left)\right(2{E}^{\phi}{K}_{\phi}^{\prime}{K}_{x}{E}^{x\prime}).\end{array}$$ 
(40)

We have expressed this in terms of
${K}_{x}$
for simplicity, keeping in mind that as the basic variable for quantization we will later use the connection component
${A}_{x}$
.
Since the Hamiltonian constraint contains the spin connection component
${\Gamma}_{\phi}$
given by ( 38 ), which contains inverse powers of densitized triad components, one can expect effective classical equations with modifications similar to the Bianchi IX model. However, the situation is now much more complicated since we have a system constrained by many constraints with a nonAbelian algebra. Simply replacing the inverse of
${E}^{\phi}$
with a bounded function as before will change the constraint algebra and thus most likely lead to anomalies. It is currently open if a more refined replacement can be done where not only the spin connection but also the extrinsic curvature terms are modified. This issue has the potential to shed light on many questions related to the anomaly issue. It is one of the cases where models between homogeneous ones, where the anomaly problem trivializes, and the full theory are most helpful.
4.15 Inhomogeneity: Results
There are some results obtained for inhomogeneous systems. We have already discussed glimpses from the BKL picture, which used loop results only for anisotropic models. Methods described in this section have led to some preliminary insights into possible cosmological scenarios.
4.15.1 Matter gradient terms and smalla effects
When an inhomogeneous matter Hamiltonian is available it is possible to study its implications on the cosmic microwave background with standard techniques. With modifications of densities there are then different regimes since the part of the inflationary era responsible for the formation of currently visible structure can be in the small
$a$
or large
$a$
region of the effective density.
The small
$a$
regime below the peak of effective densities has more dramatic effects since inflation can here be provided by quantum geometry effects alone and the matter behavior changes to be antifrictional [
45,
77]
. Mode evolution in this regime has been investigated for a particular choice of gradient term and using a powerlaw approximation for the effective density at small
$a$
, with the result that there are characteristic signatures [
130]
. As in standard inflation models the spectrum is nearly scale invariant, but its spectral index is slightly larger than one (blue tilt) as compared to slightly smaller than one (red tilt) for singlefield inflaton models. Since small scale factors at early stages of inflation generate structure which today appears on the largest scales, this implies that low multipoles of the power spectrum should have a blue tilt. The running of the spectral index in this regime can also be computed but depends only weakly on ambiguity parameters.
The main parameter then is the duration of loop inflation. In the simplest scenario, one can assume only one inflationary phase, which would require huge values for the ambiguity parameter
$j$
. This is unnatural and would imply that the spectrum is blue on almost all scales, which is in conflict with present observations. Thus, not only conceptual arguments but also cosmological observations point to smaller values for
$j$
, which is quite remarkable.
In order to have sufficient inflation to make the universe big enough one then needs additional stages provided by the behavior of matter fields. One still does not need an inflaton since now the details of the expansion after the structure generating phase are less important. Any matter field being driven away from its potential minimum during loop inflation and rolling down its potential thereafter suffices. Depending on the complexity of the model there can be several such phases.
4.15.2 Matter gradient terms and largea effects
At larger scale factors above the peak of effective densities there are only perturbative corrections from loop effects. This has been investigated with the aim of finding transPlanckian corrections to the microwave background, also here with a particular gradient term. In this model, cancellations have been observed that imply that corrections appear only at higher orders of the perturbation series and are too weak to be observable [
126]
.
A common problem of both analyses is that the robustness of the observed effects has not yet been studied. This is in particular a pressing problem since one modification of the gradient term has been chosen without further motivation. Moreover, the modifications in both examples were different. Without a more direct derivation of the modifications from inhomogeneous models or the full theory one can only rely on a robustness analysis to show that the effects can be trusted.
In particular the cancellation in the second example must be shown to be realized for a larger class of modifications.
4.15.3 Noninflationary structure formation
Given a modification of the gradient term, one obtains effective equations for the matter field, which for a scalar results in a modified Klein–Gordon equation. After a mode decomposition, one can then easily see that all the modes behave differently at small scales with the classical friction replaced by antifriction as in Section 4.5 . Thus, not only the average value of the field is driven away from its potential minimum but also higher modes are being excited. The coupled dynamics of all the modes thus provides a scenario for structure formation, which does not rely on inflation but on the antifriction effect of loop cosmology.
Even though all modes experience this effect, they do not all see it in the same way. The gradient term implies an additive contribution to the potential proportional to
${k}^{2}$
for a mode of wave number
$k$
, which also depends on the metric in a way determined by the gradient term modification. For larger scales, the additional term is not essential and their amplitudes will be pushed to similar magnitudes, suggesting scale invariance for them. The potential relevant for higher modes, however, becomes steeper and steeper such that they are less excited by antifriction and retain a small initial amplitude. In this way, the structure formation scenario provides a dynamical mechanism for a smallscale cutoff, possibly realizing older expectations [
165,
166]
.
4.15.4 Stability
As already noted, inhomogeneous matter Hamiltonians can be used to study the stability of cosmological equations in the sense that matter does not propagate faster than light. The modified behavior of homogeneous modes has led to the suspicion that loop cosmology is not stable [
94,
95]
since other cosmological models displaying superinflation have this problem. A detailed analysis of the loop equations, however, shows that the equations as they arise from modifications are automatically stable. While the homogeneous modes display superinflationary and antifrictional behavior, they are not relevant for matter propagation. Modes relevant for propagation, on the other hand, are modified differently in such a manner that the total behavior is stable [
129]
. Most importantly, this is an example where an inhomogeneous matter Hamiltonian with its modifications must be used and the qualitative result of stability can be shown to be robust under possible changes of the effective modification. This shows that reliable conclusions can be drawn for important issues without a precise definition of the effective inhomogeneous behavior.
4.16 Summary
Loop cosmology is an effective description of quantum effects in cosmology, obtained in a framework of a background independent and nonperturbative quantization. There is mainly one change compared to classical equations coming from modified densities in matter Hamiltonians or also anisotropy potentials. These modifications are nonperturbative as they contain inverse powers of the Planck length and thus the gravitational constant, but also perturbative corrections arise from curvature terms, which are now being studied.
The nonperturbative modification alone is responsible for a surprising variety of phenomena, which all improve the behavior in classical cosmology. Nevertheless, the modification had not been motivated by phenomenology but derived through the background independent quantization.
Details of its derivation in cosmological models and its technical origin will now be reviewed in Section
5 , before we come to a discussion of the link to the full theory in Section 6 .
5 Loop quantization of symmetric models

Analogies prove nothing, but they can make one feel more at home.
Sigmund Freud Introductory Lectures on Psychoanalysis
In full loop quantum gravity, the quantum representation is crucial for the foundation of the theory. The guiding theme there is background independence, which requires one to smear the basic fields in a particular manner to holonomies and fluxes. In this section, we will see what implications this has for composite operators and the physical effects they entail. We will base this analysis on symmetric models in order to be able to perform explicit calculations.
Symmetries are usually introduced in order to simplify calculations or make them possible in the first place. However, symmetries can sometimes also lead to complications in conceptual questions if the additional structure they provide is not fully taken into account. In the present context, it is important to realize that the action of a symmetry group on a space manifold provides a partial background such that the situation is always slightly different from the full theory. If the symmetry is strong, such as in homogeneous models, other representations such as the Wheeler–DeWitt representation can be possible even though the fact that a background has been used may not be obvious. While large scale physics is not very sensitive to the representation used, it becomes very important on the smallest scales, which we have to take into account when the singularity issue is considered.
Instead of looking only at one symmetric model, where one may have different possibilities to choose the basic representation, one should thus keep the full view on different models as well as the full theory. In fact, in loop quantum gravity it is possible to relate models and the full theory such that symmetric states and basic operators, and thus the representation, can be derived from the unique background independent representation of the full theory. We will describe this in detail in Section
6 , after having discussed the construction of quantum models in the present section.
Without making use of the relation to the full theory, one can construct models by
analogy. This means that quantization steps are modeled on those which are known to be crucial in the full theory, which starts with the basic representation and continues to the Hamiltonian constraint operator. One can then disentangle places where additional input as compared to the full theory is needed and which implications it has.
5.1 Symmetries and backgrounds
It is impossible to introduce symmetries in a completely background independent manner. The mathematical action of a symmetry group is defined by a mapping between abstract points, which do not exist in a diffeomorphism invariant setting (if one, for instance, considers only equivalence classes up to arbitrary diffeomorphisms).
More precisely, while the full theory has as background only a differentiable or analytic manifold
$\Sigma $
, a symmetric model has as background a symmetric manifold
$(\Sigma ,S)$
consisting of a differentiable or analytic manifold
$\Sigma $
together with an action of a symmetry group
$S:\Sigma \to \Sigma $
. How strong the additional structure is depends on the symmetry used. The strongest symmetry in gravitational models is realized with spatial isotropy, which implies a unique spatial metric up to a scale factor.
The background is thus equivalent to a conformal space.
All constructions in a given model must take its symmetry into account since otherwise its particular dynamics, for instance, could not be captured. The structure of models thus depends on the different types of background realized for different symmetry groups. This can not only lead to simplifications but also to conceptual differences, and it is always instructive to keep the complete view on different models as well as the full theory. Since the loop formalism is general enough to encompass all relevant models, there are many ways to compare and relate different systems. It is thus possible to observe characteristic features of (metric) background independence even in cases where more structure is available.
5.2 Isotropy
Isotropic models are described purely in terms of the scale factor
$a\left(t\right)$
such that there is only a single kinematical degree of freedom. In connection variables, this is parameterized by the triad component
$p$
conjugate to the connection component
$c$
.
If we restrict ourselves to invariant connections of a given form, it suffices to probe them with only special holonomies. For an isotropic connection
${A}_{a}^{i}=\stackrel{~}{c}{\Lambda}_{I}^{i}{\omega}_{a}^{I}$
(see Appendix 10.2 ) we can choose holonomies along one integral curve of a symmetry generator
${X}_{I}$
. They are of the form
$$\begin{array}{c}{h}_{I}=exp\int {A}_{a}^{i}{X}_{I}^{a}{\tau}_{i}=cos\frac{1}{2}\mu c+2{\Lambda}_{I}^{i}{\tau}_{i}sin\frac{1}{2}\mu c\end{array}$$ 
(41)

where
$\mu $
depends on the parameter length of the curve and can be any real number (thanks to homogeneity, path ordering is not necessary). Since knowing the values
$cos\frac{1}{2}\mu $
and
$sin\frac{1}{2}\mu c$
for all
$\mu $
uniquely determines the value of
$c$
, which is the only gauge invariant information contained in the connection, these holonomies describe the configuration space of connections completely.
This illustrates how symmetric configurations allow one to simplify the constructions behind the full theory. But it also shows which effects the presence of a partial background can have on the formalism [
15]
. In the present case, the background enters through the leftinvariant 1forms
${\omega}^{I}$
defined on the spatial manifold whose influence is contained in the parameter
$\mu $
. All information about the edge used to compute the holonomy is contained in this single parameter, which leads to degeneracies compared to the full theory. Most importantly, one cannot distinguish between the parameter length and the spin label of an edge: Taking a power of the holonomy in a nonfundamental representation simply rescales
$\mu $
, which could just as well come from a longer parameter length. That this is related to the presence of a background can be seen by looking at the roles of edges and spin labels in the full theory. There, both concepts are independent and appear very differently. While the embedding of an edge, including its parameter length, is removed by diffeomorphism invariance, the spin label remains welldefined and is important for ambiguities of operators. In the model, however, the full diffeomorphism invariance is not available such that some information about edges remains in the theory and merges with the spin label. Issues like that have to be taken into account when constructing operators in a model and comparing with the full theory.
The functions appearing in holonomies for isotropic connections define the algebra of functions on the classical configuration space which, together with fluxes, is to be represented on a Hilbert space. This algebra does not contain arbitrary continuous functions of
$c$
but only almost periodic ones of the form [
15]
$$\begin{array}{c}f\left(c\right)={\sum}_{\mu}{f}_{\mu}exp(i\mu c/2)\end{array}$$ 
(42)

where the sum is over a countable subset of
$\mathbb{R}$
. This is analogous to the full situation, reviewed in Section 3.4 , where matrix elements of holonomies define a special algebra of continuous functions of connections. As in this case, the algebra can be represented as the set of all continuous functions on a compact space, called its spectrum. This compactification can be imagined as being obtained from enlarging the classical configuration space
$\mathbb{R}$
by adding points, and thus more continuity conditions, until only functions of the given algebra survive as continuous ones. A wellknown example is the one point compactification, which is the spectrum of the algebra of continuous functions
$f$
for which
${lim}_{x\to \infty}f\left(x\right)={lim}_{x\to \infty}f\left(x\right)$
exists. In this case, one just needs to add a single point at infinity.
In the present case, the procedure is more complicated and leads to the Bohr compactification
${\overline{\mathbb{R}}}_{Bohr}$
, which contains
$\mathbb{R}$
densely. It is very different from the one point compactification, as can be seen from the fact that the only function which is continuous on both spaces is the zero function.
In contrast to the one point compactification, the Bohr compactification is an Abelian group, just like
$\mathbb{R}$
itself. Moreover, there is a onetoone correspondence between irreducible representations of
$\mathbb{R}$
and irreducible representations of
${\overline{\mathbb{R}}}_{Bohr}$
, which can also be used as the definition of the Bohr compactification. Representations of
${\overline{\mathbb{R}}}_{Bohr}$
are thus labeled by real numbers and given by
${\rho}_{\mu}:{\overline{\mathbb{R}}}_{Bohr}\to \mathbb{C},c\mapsto {e}^{i\mu c}$
. As with any compact group, there is a unique normalized Haar measure
$d\mu \left(c\right)$
given by
$$\begin{array}{c}{\int}_{{\overline{\mathbb{R}}}_{Bohr}}f\left(c\right)d\mu \left(c\right)={lim}_{T\to \infty}\frac{1}{2T}{\int}_{T}^{T}f\left(c\right)dc,\end{array}$$ 
(43)

where on the right hand side the Lebesgue measure on
$\mathbb{R}$
is used.
The Haar measure defines the inner product for the Hilbert space
${L}^{2}({\overline{\mathbb{R}}}_{Bohr},d\mu (c\left)\right)$
of square integrable functions on the quantum configuration space. As one can easily check, exponentials of the form
$\langle c\mu \rangle ={e}^{i\mu c/2}$
are normalized and orthogonal to each other for different
$\mu $
,
$$\begin{array}{c}\langle {\mu}_{1}{\mu}_{2}\rangle ={\delta}_{{\mu}_{1},{\mu}_{2}},\end{array}$$ 
(44)

which demonstrates that the Hilbert space is not separable.
Similarly to holonomies, one needs to consider fluxes only for special surfaces, and all information is contained in the single number
$p$
. Since it is conjugate to
$c$
, it is quantized to a derivative operator
$$\begin{array}{c}\hat{p}=\frac{1}{3}i\gamma {\ell}_{P}^{2}\frac{d}{dc},\end{array}$$ 
(45)

whose action
$$\begin{array}{c}\hat{p}\mu \rangle =\frac{1}{6}\gamma {\ell}_{P}^{2}\mu \mu \rangle =:{p}_{\mu}\mu \rangle \end{array}$$ 
(46)

on basis states
$\mu \rangle $
can easily be determined. In fact, the basis states are eigenstates of the flux operator, which demonstrates that the flux spectrum is discrete (all eigenstates are normalizable).
This property is analogous to the full theory with its discrete flux spectra, and similarly it implies discrete quantum geometry. We thus see that the discreteness survives the symmetry reduction in this framework [
37]
. Similarly, the fact that only holonomies are represented in the full theory but not connection components is realized in the model, too. In fact, we have so far represented only exponentials of
$c$
, and one can see that these operators are not continuous in the parameter
$\mu $
. Thus, an operator quantizing
$c$
directly does not exist on the Hilbert space.
These properties are analogous to the full theory, but very different from the Wheeler–DeWitt quantization. In fact, the resulting representations in isotropic models are inequivalent. While the representation is not of crucial importance when only small energies or large scales are involved [
18]
, it becomes essential at small scales which are in particular realized in cosmology.
5.3 Isotropy: Matter Hamiltonian
We now know how the basic quantities
$p$
and
$c$
are quantized, and can use the operators to construct more complicated ones. Of particular importance, also for cosmology, are matter Hamiltonians where now not only the matter field but also geometry is quantized. For an isotropic geometry and a scalar, this requires us to quantize
$p{}^{3/2}$
for the kinetic term and
$p{}^{3/2}$
for the potential term. The latter can be defined readily as
$\hat{p}{}^{3/2}$
, but for the former we need an inverse power of
$p$
. Since
$\hat{p}$
has a discrete spectrum containing zero, a densely defined inverse does not exist.
At this point, one has to find an alternative route to the quantization of
$d\left(p\right)=p{}^{3/2}$
, or else one could only conclude that there is no welldefined quantization of matter Hamiltonians as a manifestation of the classical divergence. In the case of loop quantum cosmology it turns out, following a general scheme of the full theory [
193]
, that one can reformulate the classical expression in an equivalent way such that quantization becomes possible. One possibility is to write, similarly to ( 13 )
$$d\left(p\right)={\left(\frac{1}{3\pi \gamma G}{\sum}_{I=1}^{3}tr\left({\tau}_{I}{h}_{I}\{{h}_{I}^{1},\sqrt{V}\}\right)\right)}^{6}$$
where we use holonomies of isotropic connections and the volume
$V=p{}^{3/2}$
. In this expression we can insert holonomies as multiplication operators and the volume operator, and turn the Poisson bracket into a commutator. The result