超弦理论与宇宙学

李淼

中国科学院理论物理研究所

String Theory and Cosmology

Miao Li

Institute of Theoretical Physics

Chinese Academy of Sciences

String theory is widely believed to be a theory of quantum gravity.

As such, it is usually formulated in a fixed background using the definition of scattering amplitudes

More precisely, we can study string theory in a time-independent background that has asymptotic geometry of either

(a) Minkowski

or

(b) Anti-de Sitter

String theory used to have

many different guises:

Open String, Closed String,

Heterotic String, Different

compactifications.

It is now understood

that they are manifestations

of a single grand theory,

M theory

Different string theories

are connected by duality

relations, the prototype

of this transformations is

the relation between a

electric charge and a

magnetic charge:

For example, in type IIB

theory, a string is

mapped to a D-string,

the new theory is D-

strings is again a type

IIB theory:

However, it has proven very difficult to study string theory in a time-varying background.

A universe with a startingpoint in time

De Sitter space with a bounce

The most used approach to cosmology in string theory is to

use adiabatic approximation. In such an approach, one uses

a collection of fields {F(t)} to describe the background at any

given time t, F(t) can be a scalar field, or the geometry

parameter. By adiabaticity, we mean that the physics of {F(t)}

is simply that of a fixed background with the same values of

these fields for all times.

However, this conservative, poor man’s approach

must miss some of most important ingredients of a

theory of quantum gravity.

One such ingredient is the so-called holography,

motivated by quantum physics of black holes.

A quantitative statement is that the entropy in a

region is bounded by the area of the surface

surrounding this region.

Bekenstein-Hawking formula:

This formula implies that, the physics of quantum gravity can

be utterly non-local and even a-causal.

In the context of cosmology, several people (Fischler,

Susskind, Bousso) proposed principle of

cosmological holography. Bousso’s covariant entropy

Bound:

So far, string theorists are faced with this very challenging problems:

(a)To formulate string theory on a time-varying background.

(b) To find a formalism reflecting directly the holographic principle.

Two major developments in

observational cosmology.

(1)Discovery of accelerating expansion.

(2) Detailed map of primordial perturbation constructed from the power spectrum of CMB

(cosmic microwave background)

(1) Dark energyReconstructed from data about supernov

ae type Ia

This results implies that there is dark energy in

our universe, or simply a cosmological constant.

According to these obsevations, our universe is

filled with relativistic matter and dark energy, the

latter is characterized by the equation of state

Furthermore, the dark energy density is very small

It is very important to determine the nature of the dark

energy through determining parameter w. For a cosmological

constant, w=-1.

Some of the most recent results are:

From astro-ph/0204512

The nonvanishing and a very small dark energy

density poses a serious challenge to string theory.

Since in string theory, as in a quantum field theory,

dark energy is understood as vacuum energy

generated by quantum fluctuations.

As such, the vacuum energy is always determined by

a characteristic energy scale.

The most natural scale is the Planck scale, at which a

particle will dress itself by a gravitational horizon:

So the largest theoretical dark energy value is

The ratio of the observed value to this theoretical

value is absurdly small

People have tried for several decades to understand

this problem and invented numerous ideas, by far not

a single idea is widely accepted as hopeful.

In the researcg community, one of the most popular

idea is the so-called quintessence model. In this

model, the dark energy comes from a scalar field Q,

a spatially homogeneous scalar field has the energy

density and pressure:

Thus

if

The quintessence model is at best a phenomenological

model, since, it is not yet possible to realize such a

model in string theory or a quantum field.

The essential difficulty is that in string theory, one

usually has super-symmetry, a kind of symmetry

relating bosons to fermions, and usually badly broken

in nature. When it is broken, we usually have a relation

The mass difference is too small to be consistent with

experiments.

Although string theory has not been able to resolve

this deep puzzle, string theory does hold the key to

understanding it. For instance, holography may imply

that dark energy related to the cosmic horizon. It has

been conjectured that in a holographic universe, dark

energy is given by

Where L is an infrared cut-off set by our universe.

More recently, it was argued that if L is the size of the

event horizon, then the present observational data

can be explained.

(2) Primordial perturbations.

A series of CMB experiments, in particular, the Wilkinson Microwave Anisotropy Probe (WMAP)experiment, has collected enough data to give a verydetailed map on the primordial perturbations generatedprior Big Bang. These perturbations are seeds of the structure (galaxies, clusters of galaxies, filaments,voids). From the data, many important cosmic parameters(age of universe, densities, Hubble constant…) are inferred.

The map of cosmic microwave background

fluctuations

Some of the cosmic parameters

Age of the universe 13.7 billion years old

Dark energy 73%, dark matter 23%, atoms 4%.

The Hubble constant was 71 +4/-3 km/s/Mpc .

The universe is flat.

……

More detailed data

More detailed results

Detailed results continued

Consistency with other experiments

These observations confirm the predictions of the

inflation scenario: prior big bang, there exists a very

short period during which the universe expands very

fast, and density perturbations are generated by

quantum fluctuations of the inflaton-a scalar field

driving inflation.

Inflation explores fundamental physics in at least two ways.

First, the inflaton potential is supposed to be very flat,

this is often referred to as a fine-tuning problem.

There is no natural way to construct such a potential

in a fundamental theory such as string theory.

Second, inflation greatly amplifies space. For

instance, the largest cosmic scale just entered our

horizon originated 60 e-foldings before the end of

inflation, thus, the ratio of amplification is

The Planck scale ended up to be about

larger than the size of atom.

Indeed, WMAP results indicate that the traditional

slow-roll inflation may not be good enough to explain

everything.

For example, the unexpected suppression of power

of low multi-pole correlation (if not due to systematic

error) certainly indicates that something unusual

happened 60 e-foldings before the end of inflation.

The running of spectral index of the primordial power

spectrum can not be explained by the usual inflation

too, it may not be to crazy to speculate that this is

really due to new effects in string theory, for instance,

non-commutative space-time.

Conclusions:

(1) We are in an exciting era of precision cosmological observation. Once in a while, new flux of experimental data comes to sight.

(2) A few serious challenges are awaiting fundamental theory such as string theory to meet.

(3) Many researchers in string theory are for the first time facing experiments, not just theoretic artifice.