Bojowald.pdf

IGC–11/9–1

Quantum gravity in the very early universe

Martin Bojowald †

Institute for Gravitation and the Cosmos,

The Pennsylvania State University,

104 Davey Lab, University Park, PA 16802, USA

ABSTRACT

General relativity describes the gravitational eld geometrically and in a self-

interacting way because it couples to all forms of energy, including its own. Both

features make nding a quantum theory di cult, yet it is important in the high-

energy regime of the very early universe. This review article introduces some of the

results for the quantum nature of space-time which indicate that there is a discrete,

atomic picture not just for matter but also for space and time. At high energy

scales, such deviations from the continuum aect the propagation of matter, the

expansion of the universe, and perhaps even the form of symmetries such as Lorentz

or CP transformations. All these eects may leave traces detectable by sensitive

measurements, as pointed out here by examples.

Processes in the very early universe require for their description general relativity (space

is expanding) and quantum physics (the early universe is hot and dense). Sometimes, this

even involves quantum physics not just of matter but of gravity. Gravity is described by

the geometry of space-time, and so we need to quantize space and time. By experience

from quantum mechanics, one possible consequence is that elementary constituents, or

“atoms of space,” arise for space-time.

Dimensional arguments can be used to arrive at a rst estimate of direct eects. There is

a unique length parameter, the Planck length ℓ PL =

G/c 3 10 −35 m and a unique mass

parameter, the Planck mass M PL =

c/G 10 18 GeV 10 −6 g, that can be formed solely

by reference to the relevant fundamental constants, Newton’s gravitational constant G,

Planck’s constant , and the speed of light c. At those scales, or, perhaps more intuitively,

at the Planck density ρ PL = M PL /ℓ PL , quantum gravity becomes inevitable. Compared to

the current density of the universe, at about an atom per cubic meter, the Planck density

of roughly a trillion solar masses in the region of the size of a single proton, is huge. The

relevance of quantum gravity for current physics may thus be questioned.

However, dimensional arguments can be misleading when large dimensionless param-

eters are involved. In the context of quantum gravity, perhaps suggesting some kind of

Published as Nuclear Physics A862–863 (2011) 98–103

† e-mail address: bojowald@gravity.psu.edu

elementary atoms of space, such a parameter can easily be seen to arise: the number of

tiny, Planck-sized spatial atoms in a given macroscopic region under consideration. Simi-

lar questions, in which indirect evidence for phenomena on tiny scales can be found long

before the resolution of observations becomes good enough for direct tests, have played

important roles before in the history of physics. For instance, in 1905 Albert Einstein used

an analysis of Brownian motion to nd convincing evidence for atoms, and only fty years

later, in 1955, did Erwin Muller produce the rst direct image of atoms using eld ion

microscopy. By that time, the overwhelming majority of physicists was already convinced

of the reality of material atoms based on Einstein’s arguments.

Returning to quantum gravity, the best microscope we currently have to magnify and

probe the fundamental structures of space is the universe itself. By its own expansion,

it enlarges spatial regions and eventually translates their properties into visible large-

scale structures. This magnication process, of course, takes a long time and many other

processes happen throughout; no direct image can be obtained in this way and a great

deal of physics must be used to disentangle the form of original structures from what has

emerged in the meantime. With a good understanding of all the physics involved one can

begin to nd indirect evidence for eects controlled by quantum gravity.

The physics of quantum gravity is not well-understood at present, and no complete

theory is known. Nevertheless, several characteristic eects have been suggested which do

not so much depend on theoretical details but are rather based on general expectations from

fundamental properties of general relativity and quantum mechanics. One of the main such

suggestions is the atomic nature of space, and it is of relevance for early-universe cosmology.

An expanding discrete space grows not continuously but atom by atom. Implications are

weak for a large universe, but may be noticeable by sensitive observations of events that

happen su ciently early.

Observations which have a chance of providing su cient sensitivity with current tech-

nological means must rst be found by analyzing available theories. As an example one

may consider the abundances of light elements, which depend on the baryon-photon ratio

during big-bang nucleosynthesis, an early-universe process of proton-neutron interconver-

sion by the weak interaction. The baryon-photon ratio depends on the dilution behavior

of radiation and (relativistic) fermions. If discrete expansion leads to modications of the

dilution behavior, small changes in the abundance of light elements would be expected.

We will return to this example at a later stage.

Other examples for some chance of testing quantum gravity can be found in all the

phases included in the standard model of cosmology:

The big bang: an extreme phase starting with Planckian density preceded, in the clas-

sical understanding of general relativity, by a singularity 13.8 billion years ago.

Ination: an accelerated phase of expansion of currently unknown origin, happening at

an energy scale about 10 −10 ρ PL at which particle production seeds all matter as seen

in the cosmic microwave background (CMB) and the galaxy distribution.

Baryogenesis: the formation of baryons out of a primordial quark-gluon plasma, somehow

expected to lead to the matter/antimatter asymmetry of the present universe.

Nucleosynthesis: the generation of nuclei as bound states of the light baryons, arising

in relative quantities of about 75% hydrogen and deuterium, 25% helium, and just

trace amounts of other light elements.

CMB release: once atoms neutralize, the universe becomes transparent 380,000 years

after the big bang.

The succession of most of these phases is well supported by observations. However,

the picture is incomplete, for the story begins with a singularity at which the equations

of general relativity lose their meaning and unphysical conditions such as innite densities

and temperatures are reached. The singularity is a general consequence of the equations

that govern a classical universe, in the simplest case described by the Friedmann and

Raychaudhuri equations

= 8πG

a = − 4πG

ρ ,

(ρ + 3P)

(1)

for the scale factor a providing the distance measure of the universe, and with the energy

density ρ and pressure P of matter.

A simple singularity theorem can be obtained from this equation as follows: First, a

simple rewrite implies H = −4πG( 3 ρ+P) − H 2 for the Hubble parameter H = a/a. If we

assume the so-called strong energy condition ρ+3P 0 as a rather general requirement for

the matter ingredients, we obtain the inequality dH −1 /dt 1 and thus H −1 H −1

+t−t 0 .

If H −1

0 is negative, H −1 must be positive at t 1 = t 0 − H − 0 , and so H −1 = 0 at some time

when H ! 1 and ρ ! 1 diverge: a future singularity resulting from collapse. Similarly,

a past singularity is obtained if the universe is expanding at some time. More complicated

singularity theorems can be demonstrated under more general conditions, dropping the

symmetry assumption of an exactly isotropic universe and weakening the conditions posed

for matter. Thus, singularities are generic in space-time dynamics.

These conclusions lead to the identication of several shortcomings of the standard

model of cosmology despite its observational success: (i) Any singularity is unphysical

and must be eliminated by improving the theory. (ii) Ination assumes that matter starts

out in an initial vacuum state. Is this assumption appropriate, especially when the den-

sity and temperature diverge at the “initial” singularity? (iii) With current theories, the

matter/antimatter asymmetry that is supposed to form during baryogenesis is di cult to

explain. If there was a prehistory of the universe before the big bang, as one possible

scenario alternative to a singular one, more time existed for an asymmetry to build up.

(iv) The matter equation of state is important for some aspects of big-bang and other

phases, but is not well known for most of the scales between currently probed densities

and the Planck density. This lack of knowledge does not so much aect the singular nature

but plays a role for specic scenarios. To see what ingredients exactly we must bring un-

der better control to discuss possible improvements of the standard model, we need more

information about quantum gravity and the resulting space-time structure.

Gravity is “strongly interacting” at a fundamental, non-perturbative level. This state-

ment may come as a surprise, given that gravity is much weaker than the other fundamental

forces and can safely be ignored in particle interactions. However, particle physics already

provides indications for the special nature of gravity: Its well-known non-renormalizability

implies that gravity cannot be quantized as a weakly-interacting theory of gravitons on

some background space-time. The weak form of gravity should rather arise as the long-

range remnant of a more elementary theory. What exactly this elementary theory is is

di cult to extract from the long-range physics of gravity that we know. Theoretical mod-

els are based on suitable principles for their mathematical formulation, for which dierent

approaches exist, but no fully consistent version yet.

A quantization directly addressing the structure of space and time is loop quantum

gravity [1], based crucially on the principle of background independence. Some part of

the theory can be constructed by means analogous to those of lattice QCD, but with

one crucial dierence: General covariance implies that all states must be invariant under

deformations of space (dieomorphisms or coordinate changes). As a consequence, several

new features (and complications) compared to QCD arise: (i) Regular lattices are too

restrictive because they would be deformed when coordinates are changed. Instead lattices

are “oating;” they are not assigned a xed position in space. Only topological and

combinatorial properties of their linking and knotting behavior can be relevant for gravity.

(ii) No well-motivated restriction on the valence of lattice vertices exists (except the desired

but possibly deluding simplicity of their mathematical description). (iii) Superpositions

of dierent lattice states must be considered because the lattices correspond to states of

a fundamental quantum theory, not to an approximation of such a theory. (iv) States of

the continuum theory are described by lattices; they do not provide an approximation,

and no continuum limit is to be taken. In this way, one obtains a fundamental lattice

theory for quantum geometry. Geometrical excitations are, as we will see, generated by

creation operators for lattice links. Near the continuum, physics can only be described

by a highly excited many-particle state; in this sense the theory is “interacting”. So far,

the complicated resulting physics has mainly been analyzed in model systems, primarily

obtained by assuming spatial symmetries.

To provide more technical details, we describe space-time geometry by an su(2)-valued

“electric eld” E i and a “vector potential” ! i (using so-called Ashtekar–Barbero variables)

with the following meaning. 3

Electric eld: Geometrically called a densitized triad, it determines spatial distances and

angles by assigning three orthonormal vectors E i , i = 1,2,3, to each point in space.

Vector potential: ! i = ! i + γ ! i where ! i is related to the intrinsic curvature of space,

and ! i to extrinsic curvature of space in space-time. These two contributions are

3 In general relativity, we must distinguish between contravariant and covariant vector elds, denoted

here by arrows above or below the letter, respectively. On a metric manifold one can uniquely transform

between these two types of vector elds, but for gravity the metric follows from the fundamental elds.

It is not available before those elds are known. Keeping track of the metric dependences is crucial for a

background-independent formulation of quantum gravity.

added with a relative weighting γ, the real-valued Barbero–Immirzi parameter.

In addition to these geometrical properties and meanings of the elds, we have their

canonically conjugate nature: E i is the momentum of ! i , { ! i (x), E j (y)} = 8πγGδ ij ! δ(x,y)

(using the identity matrix ! ). We can thus proceed by attempting a canonical quantiza-

tion, with due observation of special properties resulting from symmetries of the theory, in

particular general covariance.

As in lattice gauge theories, we dene as basic variables holonomies h e = P exp( 2

e dλ ! j ·

t e σ j ) for the connection !

i along spatial curves e, with Pauli matrices σ j . However, we use

these objects in a way very dierent from lattice gauge theory; they will become creation

operators of quantum geometry. To that end, we dene a basic state ψ 0 by ψ 0 ( ! i ) = 1,

that is it is independent of the connection. 4 Excited states are then obtained by the ac-

tion of holonomies via multiplication in this connection representation. We present the

formulas only in a simplied U(1)-example where h e ( ! ) = exp(i

·t e ) are just phase

factors; SU(2) formulas as needed for gravity are analogous but more tedious. We thus

write all excited states obtained in this way as ψ e 1 ,k 1 ; ... ; e I ,k I = h k 1

e dλ !

· · · h k I

e I ψ 0 . A general state

is then labeled by a graph g, the collection of all curves used for holonomies to generate

the state, and integers k e as quantum numbers on the edges: ψ g,k ( ! ) =

e 1

e 2 g h e ( ! ) k E =

·t e ).

The Ashtekar–Barbero connection has momenta E i such that

e dλ !

e 2 g exp(ik e s

i E i · E i = (detq) −1 ·q

gives the inverse spatial metric q. Quantizing E i , or rather the uxes

S d 2 y !

·E i (with ! the

metric-independent co-normal to surfaces S), they naturally become derivative operators.

At the level of states, ux operators measure the excitation levels k e :

Eψ g,k = 8πγG

· δψ g,k

· ˆ

d 2 y !

δ ! (y) = 8πγℓ PL

k e Int(S,e)ψ g,k

(2)

e 2 g

with the intersection number Int(S,e). From this equation one readily concludes that the

ψ g,k are eigenstates of uxes, with eigenvalues given by 8πγℓ PL times an integer. Spatial

geometry is discrete: for gravity, uxes representing the spatial metric have discrete spectra,

and so do operators for area or volume constructed from them [3]. The Planck length ℓ PL =

G together with the Barbero–Immirzi parameter determines the elementary discreteness

scale. From computations of black-hole entropy one derives that γ is of the order one, but

somewhat smaller than one [4].

So far, the quantum geometry we developed is only of space, not space-time. In order

to see how the graph states evolve in time, possibly being reconnected and rened by the

creation of new vertices, we need to quantize the Hamiltonian. Schematically, it has the

form [5]

v,e K , V ])ψ g,k summing over

vertices v of the graph g and triples (IJK) of edges. As is clear from this expression, there

are creation operators (holonomies) as well as the volume operator V . At this fundamental

Hψ g,k =

v,IJK ǫ IJK tr(h v,e I h v + e I ,e J h −1

v + e J ,e I h −1

v,e J h v,e K [h −1

4 This state turns out to be normalizable by the inner product constructed via integration on spaces of

connections [2].

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