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 starting point 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 energy Reconstructed from data about supernovae 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/

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 very detailed map on the primordial perturbations generated prior 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.