Perturbations in loop quantum cosmology
The era of precision cosmology has allowed us to accurately determine many important cosmological parameters, in particular via the CMB. Confronting Loop Quantum Cosmology with these observations provides us with a powerful test of the theory. For this to be possible we need a detailed understanding of the generation and evolution of inhomogeneous perturbations during the early, Quantum Gravity, phase of the universe. Here we describe how Loop Quantum Cosmology provides a completion of the inflationary paradigm, that is consistent with the observed power spectra of the CMB.
1 Open issues with inflation
The standard picture we have of early universe cosmology is that of inflation. This beautiful idea contains very few basic assumptions, yet it produces, with spectacular agreement, the power spectra of density fluctuations observed in the Cosmic Microwave Background (CMB). Nevertheless, inflation remains a paradigm in search of a model and there are several key questions that remain unanswered. Broadly speaking, these can be split into two categories: difficulties facing the particles physics interpretation of inflation, and those we expect to be related to quantum gravity. Examples of each type are given in Table (1).
Inflation is a period of quasi-de Sitter expansion and typically the assumption is that slow-roll inflation began with the quantum state describing perturbations in its ‘natural’ vacuum state: the Bunch-Davies vacuum. A priori this is an odd assumption, since it says that the quantum state is tuned to the subsequent (quasi-de Sitter) evolution of the geometry. Essentially this implies that the pre-inflationary dynamics of the universe and the ‘true’ initial state conspired in such a way as to ensure that we arrived at the onset of slow-roll inflation with no particles present (relative to the Bunch-Davies vacuum). The intuition behind this assumption was that even if there were particles present at the onset of inflation, the exponential expansion would rapidly dilute them and hence their consequences can safely be ignored. This intuition misses the important fact that quantum fields in a dynamical space-time experience both spontaneous and simulated creation of particles . The latter effect actually compensates for the exponential growth of the volume in such a way that the particle number density remains approximately constant.
In classical general relativity inflation is inevitably preceded by the Big-Bang and the only natural place to give initial conditions is at this singularity. An important open question then is: can one find a quantum gravity completion to the inflationary paradigm? To be viably, such a completion should be non-singular and agree with current observations. It would also open up the exciting possibility that we can directly observe the pre-inflationary universe. If it were possible to see observational consequences of the quantum state at the onset of slow-roll, we would be able to probe the dynamics of the quantum gravity era of the universe.
|Particle physics issues||Quantum gravity issues|
|What is the inflaton?||What are the initial conditions?|
|Why is the potential flat?||Is there a singularity?|
|How does it couple to the standard model?||Why is perturbation theory valid?|
|What interactions are present?||What happens when the frequencies|
2 Loop Quantum Cosmology
Loop Quantum Gravity is a particularly well developed approach to quantising gravity . It is a Hamiltonian quantisation that maintains the fundamental relationship between geometry and gravity and is fully non-perturbative. However a rigorous understanding of the dynamics of the theory are still lacking (see Madhavan Varadarajan’s talk in this session). One useful way to make progress is to consider simplified (truncated) systems of the full theory, which, on the one hand, can be completely understood, and on the other are physically interesting. This approach has been applied with great success to the study of black-holes (see Fernando Barbero’s talk in this session) and graviton propagators (see Carlo Rovelli’s talk in this session) within Loop Quantum Gravity. One can also consider the truncation of full general relativity to homogeneous systems and then use Loop Quantum Gravity techniques to quantise these. This leads to Loop Quantum Cosmology (LQC), which has turned out to be very successful at answering many of the difficulties facing classical cosmology. In particular, it has been shown in detail how the classical singularity is replaced by a ‘Big-Bounce’ and how the late time, low energy, limit reproduces the standard expectations of general relativity. It has also been shown that the probability of having a sufficiently long phase of (single scalar field driven) slow-roll inflation, within LQC, is very close to one .
However in order to make predictions about LQC effects on the CMB, one first has to extend the underlying approach to include (peturbative) inhomogeneities. There have been several promising attempts to include such inhomogeneities (see Jakub Mielczarek’s talk in this session) based on consistent alterations of the homogeneous formulation of LQC. Here we describe a systematic approach to extending the underlying formulation.
3 LQC and perturbations
The first step is to find a suitable truncation of the phase-space of classical general relativity that allows for cosmologies with perturbative inhomogeneities. Since we work in the Hamiltonian theory, we restrict our attention to cosmologies whose spatial slices are (flat) tori. One can then show that the full phase-space decomposes into homogeneous and purely inhomogeneous parts,
The canonical variables of the homogeneous phase-space () are , i.e. the scale factor and the scalar field , and their conjugate momenta. The canonical variables of the inhomogeneous phase-space () are the corresponding perturbations, , all of which are purely inhomogeneous.
All the canonical structures of the phase-space decompose in this way, in particular the symplectic structure and the Possion brackets factor. One can now consider the inhomogeneous fields as perturbations, expand the constraints and find gauge invariant degrees of freedom . In particular the gauge invariant scalar modes of the perturbations satisfy the constraint,
where and are canonically conjugate background variables, is the coordinate size of the 3-torus and are the gauge invariant, scalar degree of freedom (related to the Mukhanov variable). The important point is that gauge invariant scalar (and tensor) perturbations behave exactly as test scalar fields with a time dependent mass .
where is the Hamiltonian for the background geometry (and the potential term) and is the (time dependent) Hamiltonian for the perturbations. These are then promoted to operators and we deparameterise with respect to the scalar field , to find,
where is the operator that governs the LQC evolution of the background , and the right hand side has been approximated via a series expansion. This is the only approximation that is made in the procedure and it can be viewed as a test-field approximation. Essentially, one restrictions ones attention to those states in which the is dominant and does not alter the background dynamics222There are many important issues to do with ensuring that the integral over results in a well defined operator that are not being described here. See  for details.. The second term on the right hand side of Eq. (4) is precisely the Hamiltonian for the perturbations associated to the relational time defined by .
Finally, one decomposes the wave-function as a tensor product of back-ground and perturbation pieces,
and considers states for which is sharply peaked with respect to some particular classical background geometry at late times i.e. a semi-classical background geometry. Taking expectation values of Eq. (4) with respect to , one arrives the standard Hamiltonian for test quantum fields in a curved space-time, with the background scale factor replaced by .
4 Results and conclusions
In the previous section we briefly sketched how one can extend the formulation of LQC to include (perturbative) inhomogeneities. With this in hand, one can now look for consequences of the pre-inflationary era and in particular, we can look for deviations from the standard initial conditions for inflation. The standard approach in inflation is to assume the existence of a scalar field with a suitable potential, give oneself some initial conditions for the quantum state of the perturbations and hence calculate the late time power spectra. Here we will take exactly the same approach: we specify initial conditions at the Big-Bounce and calculate the resulting power-spectra (here we concentrate only on tensor perturbations). Since, at the bounce, the wavelength of all the observable modes is much smaller than the curvature scale, we take the initial state (for these modes) to be the Minkowski vacuum (see  for a discussion on this point).
Figure (1) shows the resulting late time (dimensionless) power spectra, , defined as,
where the variable is related to the the Mukhanov variable by , and represents the gravitational potential on constant hyper-surfaces.
The important result here is that provided the background scalar field, , at the bounce is large enough, i.e. , the power spectra is entirely consistent with that of standard inflation. Hence we have a completion of the inflationary paradigm, including the quantum gravity era, which agree with the observations. Note that there remains a (small but important) window () in which we may hope to see corrections to the largest scales (smallest ) of the observed power spectra, and hence directly observe pre-inflationary (i.e. quantum gravity) physics.
This work was supported in part by the NSF grants PHY0854743 and PHY1068743.
-  I. Agullo, A. Ashtekar, W. Nelson, In prep..
-  I. Agullo, L. Parker, Phys. Rev. D /bf 83 063526 (2011).
-  A. Ashtekar, M. Bojowald, J. Lewandowski, Adv. Theor. Math. Phys. 7 233-268 (2003), A. Ashtekar, J. Lewandowski, Class. Quant. Grav. 21 R53 (2004) .
-  A. Ashtekar, P. Singh, Class. Quant. Grav. 28 213001 (2011), M. Bojowald, Living Review 11 (2008) (cited on 20 March 2012) .
-  Big-bounce refs.
-  C. Germani, W. Nelson, M. Sakellariadou, Phys. Rev. D 76 043529 (2007), A. Ashtekar, D. Sloan, Gen. Rel. Grav. 43 3619-3656 (2011), A. Corichi, A. Karami, Phys. Rev. D 83 104006 (2011).
-  David Langlois, Class. Quantum. Grav. 11 387-407 (1994) .
-  A. Ashtekar, W. Kaminski, J. Lewandowski, Phys. Rev. D. 79 064030 (2009).