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| {{String theory|cTopic=Theory}}
| | The author's title is Andera and she believes it seems quite great. One of the issues she loves most is canoeing and she's been performing it for fairly a whilst. My wife and I reside in Mississippi and I adore each day residing here. Distributing manufacturing has been his profession for some time.<br><br>Also visit my blog - email psychic readings ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 check over here]) |
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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 [[physical cosmology|cosmology]]. A related area of study is [[brane cosmology]] .
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| This approach can be dated back to a paper by [[Gabriele Veneziano]]<ref name="Ven91">
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| {{cite journal|last=Veneziano|first=G.|authorlink=Gabriele Veneziano|title=Scale factor duality for classical and quantum strings|journal=[[Physics Letters B]]|volume=265|issue=3–4|pages=287|publisher=|location=|year=1991|url=|doi =10.1016/0370-2693(91)90055-U|id=|accessdate=|bibcode = 1991PhLB..265..287V }}</ref> 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.
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| The idea is related to a property of the [[bosonic string theory|bosonic string]] in a curve background, better known as [[nonlinear sigma model]]. First calculations from this model<ref name="Frie80">
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| {{cite journal|last=Friedan|first=D.|authorlink=Daniel Friedan|title=Nonlinear Models in 2+ε Dimensions|journal=[[Physical Review Letters]]|volume=45|issue=13|pages=1057|publisher=|location=|year=1980|url=http://www.physics.rutgers.edu/~friedan/papers/PRL_45_1980_1057.pdf|doi=10.1103/PhysRevLett.45.1057|id=|accessdate=|bibcode=1980PhRvL..45.1057F}}</ref> showed as the [[beta-function|beta function]], representing the running of the metric of the model as a function of an energy scale, is proportional to the [[Ricci tensor]] giving rise to a [[Ricci flow]]. As this model has [[conformal field theory|conformal invariance]] and this must be kept to have a sensible [[quantum field theory]], the [[beta-function|beta function]] must be zero producing immediately the [[Einstein field equations]]. While Einstein equations seem to appear somewhat out of place, nevertheless this result is surely striking showing as a background two-dimensional model could produce higher-dimensional physics. An interesting point here is that such a string theory can be formulated without a requirement of criticality at 26 dimensions for consistency as happens on a flat background. This is a serious hint that the underlying physics of Einstein equations could be described by an effective two-dimensional [[conformal field theory]]. Indeed, the fact that we have evidence for an inflationary universe is an important support to string cosmology.
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| 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 literature as the '''graceful exit problem'''.
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| An [[inflation (cosmology)|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 theory|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]].
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| 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 [[Compactification (physics)|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 [[Compactification (physics)|compactified]] dimensions. Such fields are called '''moduli'''.
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| ==Technical details==
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| This section presents some of the relevant equations entering into string cosmology. The starting point is the [[Polyakov action]], which can be written as:
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| :<math>S_2=\frac{1}{4\pi\alpha'}\int d^2z\sqrt{\gamma}\left[\gamma^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu+\alpha'\ ^{(2)}R\Phi(X)\right],</math>
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| where <math>\ ^{(2)}R</math> is the [[Ricci tensor|Ricci scalar]] in two dimensions, <math>\Phi</math> the [[dilaton]] field, and <math>\alpha'</math> the string constant. The indices <math>a,b</math> range over 1,2, and <math>\mu,\nu</math> over <math>1,\ldots,D</math>, where ''D'' the dimension of the target space. A further antisymmetric field could be added. This is generally considered when one wants this action generating a potential for inflation.<ref name="Wands96">
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| {{cite journal|last=Easther|first=R.|authorlink=Richard Eather|last2=Maeda|first2=Kei-ichi|authorlink2=Kei-hichi Maeda|first3=D.|last3=Wands|authorlink3=David Wands|title=Tree-level string cosmology|journal=[[Physical Review D]]|volume=53|issue=8|pages=4247|publisher=|location=|year=1996|url=|doi=10.1103/PhysRevD.53.4247|id=|accessdate=|arxiv = hep-th/9509074 |bibcode = 1996PhRvD..53.4247E }}</ref> Otherwise, a generic potential is inserted by hand, as well as a cosmological constant.
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| The above string action has a conformal invariance. This is a property of a two dimensional [[Riemannian manifold]]. At the quantum level, this property is lost due to anomalies and the theory itself is not consistent, having no [[unitarity]]. So it is necessary to require that [[conformal field theory|conformal invariance]] is kept at any order of [[perturbation theory]]. [[Perturbation theory]] is the only known approach to manage the [[quantum field theory]]. Indeed, the [[beta function (physics)|beta function]]s at two loops are
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| :<math>\beta^G_{\mu\nu}=R_{\mu\nu}+2\alpha'\nabla_\mu\Phi\nabla_\nu\Phi+O(\alpha'^2),</math>
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| :<math>\beta^{\Phi}=\frac{D-26}{6}-\frac{\alpha'}{2}\nabla^2\Phi+\alpha'\nabla_\kappa\Phi\nabla^\kappa\Phi+O(\alpha'^2).</math>
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| The assumption that [[conformal field theory|conformal invariance]] holds implies that
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| :<math>\beta^G_{\mu\nu}=\beta^\Phi=0,</math>
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| 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 <math>\beta^\Phi</math> is just the anomaly of the [[bosonic string theory]] in a flat spacetime. But here there are further terms that can grant a compensation of the anomaly also when <math>D\ne 26</math>, 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:
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| :<math>S=\frac{1}{2\kappa_0^2}\int d^Dx\sqrt{-G}e^{-2\Phi}\left[-\frac{2(D-26)}{3\alpha'}+R+4\partial_\mu\Phi\partial^\mu\Phi+O(\alpha')\right],</math>
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| where <math>\kappa_0^2</math> 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
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| :<math>\, g_{\mu\nu}=e^{2\omega}G_{\mu\nu}\!,</math>
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| :<math>\omega=\frac{2(\Phi_0-\Phi)}{D-2},</math>
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| and using <math>\tilde\Phi=\Phi-\Phi_0</math> one can write
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| :<math>S=\frac{1}{2\kappa^2}\int d^Dx\sqrt{-g}\left[-\frac{2(D-26)}{3\alpha'}e^{\frac{4\tilde\Phi}{D-2}}+\tilde R-\frac{4}{D-2}\partial_\mu\tilde\Phi\partial^\mu\tilde\Phi+O(\alpha')\right],</math>
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| where
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| :<math>\tilde R=e^{-2\omega}[R-(D-1)\nabla^2\omega-(D-2)(D-1)\partial_\mu\omega\partial^\mu\omega].</math>
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| 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:
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| :<math>\kappa=\kappa_0e^{2\Phi_0}=(8\pi G_D)^{\frac{1}{2}}=\frac{\sqrt{8\pi}}{M_p},</math>
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| where <math>G_D</math> is the Newton constant in D dimensions and <math>M_p</math> the corresponding Planck mass. When setting <math>D=4</math> in this action, the conditions for inflation are not fulfilled unless a potential or antisymmetric term is added to the string action,<ref name="Wands96"/> in which case power-law inflation is possible.
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| ==Notes==
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| {{Reflist|2}}
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| ==References==
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| * {{Cite book
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| | last=Polchinski
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| | first=Joseph
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| | authorlink=Joseph Polchinski
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| | year=1998a
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| | title=String Theory Vol. I: An Introduction to the Bosonic String
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| | publisher=[[Cambridge University Press]]
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| | isbn=0-521-63303-6
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| }}
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| *{{Cite book
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| | last=Polchinski
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| | first=Joseph
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| | authorlink=Joseph Polchinski
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| | year=1998b
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| | title=String Theory Vol. II: Superstring Theory and Beyond
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| | publisher=[[Cambridge University Press]]
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| | isbn=0-521-63304-4
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| }}
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| *{{cite journal
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| | last=Lidsey
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| | first=James D.
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| | authorlink=James D. Lidsey
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| | last2=Wands
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| | first2=David
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| | authorlink2=David Wands
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| | last3= Copeland
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| | first3= E. J.
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| | authorlink3=E. J. Copeland
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| | title=Superstring Cosmology
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| | journal=Physics Report
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| | volume=337
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| | issue=4–5
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| | pages=343
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| | year=2000
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| | arxiv=hep-th/9909061
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| | doi=10.1016/S0370-1573(00)00064-8
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| |bibcode = 2000PhR...337..343L }}
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| ==External links==
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=15204 String cosmology on arxiv.org]
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| *[http://www.ba.infn.it/~gasperin/ Maurizio Gasperini's homepage]
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| {{Astronomy subfields}}
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| {{DEFAULTSORT:String Cosmology}}
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| [[Category:String theory]]
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| [[Category:Physical cosmology]]
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| [[Category:General relativity]]
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The author's title is Andera and she believes it seems quite great. One of the issues she loves most is canoeing and she's been performing it for fairly a whilst. My wife and I reside in Mississippi and I adore each day residing here. Distributing manufacturing has been his profession for some time.
Also visit my blog - email psychic readings (check over here)