Non-BPS D-branes Non-BPS D-branes 　　　　 inin 　　 the Early the Early UniverseUniverse

JHEP 0603 (2006) 070 (hep-th/0601133)JHEP 0603 (2006) 070 (hep-th/0601133)

JHEP (2002) 072 (hep-th/0212063)JHEP (2002) 072 (hep-th/0212063)

JHEP (2003) 002 (hep-th/0303236)JHEP (2003) 002 (hep-th/0303236)

Prog. Theor. Phys. 112 (2004) 653 Prog. Theor. Phys. 112 (2004) 653 (hep-th/0403078)(hep-th/0403078)

　　　　　　　　　　　　 Hokkaido Univ.Hokkaido Univ. Kenji HottaKenji Hotta

1. Introduction1. Introduction

SuperstringSuperstring1-dim. extended object in 10-dim. spacetime1-dim. extended object in 10-dim. spacetime

DDpp-brane-brane (IIA (IIA p p : even, IIB : even, IIB p p : odd): odd)

pp-dim. extended object,-dim. extended object, ½ SUSY½ SUSY

hypersurface the ends ofhypersurface the ends of

open strings can attach toopen strings can attach to

Non-BPS DNon-BPS Dpp-brane, D-brane, Dp-p-DDp p Pair Pair (non-SUSY)(non-SUSY)

(IIA (IIA p p : odd, IIB : odd, IIB p p : even): even)

open string tachyon tachyon potentialopen string tachyon tachyon potential

Sen’s conjectureSen’s conjecture potential height=brane potential height=brane tensiontension

Tachyon potential of Tachyon potential of NN Non-BPS D Non-BPS Dpp--branes Based on BSFT branes Based on BSFT (BSFT: boundary (BSFT: boundary string field theory)string field theory)

Early Universe Early Universe (high temperature, high density)(high temperature, high density)

finite temperature system of unstable branesfinite temperature system of unstable branes finite temperature effective potential (BSFT)finite temperature effective potential (BSFT)

Non-BPS D9-branes and D9-D9 pairs become stable Non-BPS D9-branes and D9-D9 pairs become stable near the Hagedorn temperature. near the Hagedorn temperature. HottaHotta

Thermodynamic Balance on Non-BPS D9Thermodynamic Balance on Non-BPS D9(open string closed string)(open string closed string)

Open strings dominate the total energy of strings.Open strings dominate the total energy of strings.

Cosmological Model based on BSFT?Cosmological Model based on BSFT?Sen’s Born-Infeld type actionSen’s Born-Infeld type action

time evolution of universetime evolution of universe

in the presence of non-BPS D9-branes in the presence of non-BPS D9-branes brane brane inflation?inflation?

Decent RelationDecent RelationNon-BPS D9-branes or D9-D9 pairsNon-BPS D9-branes or D9-D9 pairs

tachyon condensationtachyon condensation

lower-dim. D-branes as topological defectslower-dim. D-branes as topological defectsex) D8-brane = kink solution on non-BPS D9-ex) D8-brane = kink solution on non-BPS D9-

branebrane

‘‘Brane World Formation Scenario’Brane World Formation Scenario’formation of our Brane World as a topological formation of our Brane World as a topological

defect in a cosmological contextdefect in a cosmological contextKKLT model, RS model, Brane Gas Cosmology, KKLT model, RS model, Brane Gas Cosmology,

ekpyrotic universeekpyrotic universe

We study the homogeneous and isotropic We study the homogeneous and isotropic tachyon condensation as a first step tachyon condensation as a first step towards ‘Brane World Formation Scenario’.towards ‘Brane World Formation Scenario’.

ContentsContents

1.1. Introduction Introduction 2.2. Phase Transition near the Hagedorn Phase Transition near the Hagedorn

TemperatureTemperature

3.3. Thermodynamic Balance on Non-BPS Thermodynamic Balance on Non-BPS D9-branesD9-branes

4.4. ActionAction

5.5. Constant Dilaton CaseConstant Dilaton Case

6.6. Conclusion and DiscussionConclusion and Discussion

2. Phase Transition near2. Phase Transition near the Hagedorn the Hagedorn TemperatureTemperature Hagedorn TemperatureHagedorn Temperature

maximum temperature for perturbative maximum temperature for perturbative stringsstrings

A single energetic string captures most of A single energetic string captures most of the energy.the energy.

Non-compact Flat BackgroundNon-compact Flat Background Non-BPS D9-branesNon-BPS D9-branes

term of finite temperature effective term of finite temperature effective potentialpotential

The coefficient vanishes whenThe coefficient vanishes when

Above becomes the potential Above becomes the potential minimum.minimum.

A phase transition occurs at A phase transition occurs at

and non-BPS D9-branes become and non-BPS D9-branes become stable.stable.

Non-BPS DNon-BPS Dpp-branes with -branes with No phase transition occurs.No phase transition occurs.

Toroidal Flat BackgroundToroidal Flat Background Non-BPS DNon-BPS Dpp-branes are extended in all the non--branes are extended in all the non-

compact directions.compact directions. A phase transition occurs.A phase transition occurs.

Non-BPS DNon-BPS Dpp-branes are not extended in all the -branes are not extended in all the non-compact directions.non-compact directions.

No phase transition occurs.No phase transition occurs.

DDp-p-DDp p PairsPairssimilar to the non-BPS D-brane casesimilar to the non-BPS D-brane case

The spacetime-filling branes are created near the The spacetime-filling branes are created near the Hagedorn temperature in all the cases.Hagedorn temperature in all the cases.

3. Thermodynamic Balance3. Thermodynamic Balance on Non-BPS D9- on Non-BPS D9-branesbranes

Thermodynamic Balance ConditionThermodynamic Balance Condition

open stringsopen stringsWe need an infinite energy to reach the Hagedorn temperature.We need an infinite energy to reach the Hagedorn temperature.

closed stringsclosed stringsWe can reach the Hagedorn temperature by supplying a finite We can reach the Hagedorn temperature by supplying a finite

energy.energy.

Energy flows from closed strings to open strings.Energy flows from closed strings to open strings.

Open strings dominate the total energyOpen strings dominate the total energy

4. Action4. Action

GravityGravityIIA SUGRA IIA SUGRA (closed string tree)(closed string tree)

Einstein gravityEinstein gravity

Open String Gas Open String Gas (high temperature)(high temperature)

Matsubara Method Matsubara Method (open string 1-loop)(open string 1-loop)

eq. of stateeq. of state

Non-BPS D9-brane Non-BPS D9-brane (zero temperature)(zero temperature)

Non-BPS D9-brane action Non-BPS D9-brane action (open string tree) (open string tree) (BSFT)(BSFT)

5. Constant Dilaton Case 5. Constant Dilaton Case

Open String Gas CaseOpen String Gas Caseeq. of motion eq. of motion (RW Spatially Flat Metric)(RW Spatially Flat Metric)

eq. of stateeq. of state

solutionsolution

initial singularity decelerationinitial singularity deceleration

low rolling low rolling tachyontachyon

Rolling Tachyon CaseRolling Tachyon Caseeq. of motion eq. of motion (RW Spatially Flat Metric)(RW Spatially Flat Metric)

Independent eqs. are two of three eqs.Independent eqs. are two of three eqs.

Numerical solutionNumerical solution

initial condition close to solutioninitial condition close to solution de Sitter solutionde Sitter solution

de Sitter decelerationde Sitter deceleration

tachyon mattertachyon matter

cf) Sugimoto-Terashima cf) Sugimoto-Terashima modelmodel

6. Conclusion and 6. Conclusion and DiscussionDiscussion

Time Evolution of the Universe in the Time Evolution of the Universe in the Presence of Non-BPS D9-branesPresence of Non-BPS D9-branesstring gas rolling tachyonstring gas rolling tachyon

Constant Dilaton CaseConstant Dilaton Casedeceleration inflation decelerationdeceleration inflation deceleration

Interpolation between High Case and Interpolation between High Case and CaseCasetachyon potential at intermediate temperaturetachyon potential at intermediate temperature

CorrectionCorrection correction, higher-loop correctioncorrection, higher-loop correction

‘‘Brane World Formation Scenario’Brane World Formation Scenario’Non-BPS D9-branes or D9-D9 pairsNon-BPS D9-branes or D9-D9 pairs

tachyon condensationtachyon condensation

lower-dim. D-branes as topological defectslower-dim. D-branes as topological defects Brane World?Brane World?inhomogeneous case, arbitrary matrixinhomogeneous case, arbitrary matrix

Closed String EmissionClosed String Emissionclosed string emission by D-brane decay closed string emission by D-brane decay Lambert-Liu-Lambert-Liu-

MaldacenaMaldacena

tachyon matter?tachyon matter?