String cosmology
String theory |
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Fundamental objects |
Perturbative theory |
Non-perturbative results |
Phenomenology |
Mathematics |
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String cosmology is a relatively new field that tries to apply equations of string theory to solve the questions of early cosmology. A related area of study is brane cosmology.
Overview
This approach can be dated back to a paper by Gabriele Veneziano[1] that shows how an inflationary cosmological model can be obtained from string theory, thus opening the door to a description of pre-Big Bang scenarios.
The idea is related to a property of the
In the evolution of the universe, after the inflationary phase, the expansion observed today sets in that is well described by Friedmann equations. A smooth transition is expected between these two different phases. String cosmology appears to have difficulties in explaining this transition. This is known in the literature as the graceful exit problem.
An inflationary cosmology implies the presence of a scalar field that drives inflation. In string cosmology, this arises from the so-called dilaton field. This is a scalar term entering into the description of the bosonic string that produces a scalar field term into the effective theory at low energies. The corresponding equations resemble those of a Brans–Dicke theory.
Analysis has been worked out from a critical number of dimension (26) down to four. In general, one gets Friedmann equations in an arbitrary number of dimensions. The other way round is to assume that a certain number of dimensions is compactified producing an effective four-dimensional theory to work with. Such a theory is a typical Kaluza–Klein theory with a set of scalar fields arising from compactified dimensions. Such fields are called moduli.
Technical details
This section presents some of the relevant equations entering into string cosmology. The starting point is the Polyakov action, which can be written as
where is the
The above string action has a conformal invariance. This is a property of a two dimensional
and
The assumption that conformal invariance holds implies that
producing the corresponding equations of motion of low-energy physics. These conditions can only be satisfied perturbatively, but this has to hold at any order of perturbation theory. The first term in is just the anomaly of the bosonic string theory in a flat spacetime. But here there are further terms that can grant compensation of the anomaly also when , and from this cosmological models of a pre-big bang, scenario can be constructed. Indeed, this low energy equations can be obtained from the following action:
where is a constant that can always be changed by redefining the dilaton field. One can also rewrite this action in a more familiar form by redefining the fields (Einstein frame) as
and using one can write
where
This is the formula for the Einstein action describing a scalar field interacting with a gravitational field in D dimensions. Indeed, the following identity holds:
where is the Newton constant in D dimensions and the corresponding Planck mass. When setting in this action, the conditions for inflation are not fulfilled unless a potential or antisymmetric term is added to the string action,[3] in which case power-law inflation is possible.
Notes
References
- ISBN 978-0-521-63303-1.
- ISBN 978-0-521-63304-8.
- Lidsey, James D.; S2CID 119349072.
- Cicoli, Michele; Conlon, Joseph P; Maharana, Anshuman; Parameswaran, Susha; )