rehan deen university of pennsylvania - cpe.vt. · pdf filebouncing cosmologiesbrane...

Bouncing cosmologies Brane world-volume action SUSY SUGRA Conclusions

Supergravitational Heterotic Galileons

Rehan Deen

University of Pennsylvania

String Pheno 2017, Virginia Tech

July 6, 2017

1 / 30


Introduction

Collaboration on bouncing cosmology with R.D., Burt Ovrut, Anna Ijjas, PaulSteinhardt, see also 1705.06729 [R.D., Burt Ovrut]

Bouncing cosmologies [review: Brandenberger, Peter ’16] are an alternative to inflationaryscenario : early universe is contracting (a < 0) then “bounces” and beginsexpanding (a > 0).

• Initial singularity problem avoided

• Bounce can be “classical” and avoid super-Planckian scales

Needs H > 0 - not satisfied by matter, radiation or CC dominated universe.

This requires matter which violates Null Energy Condition (NEC):

Tµνnµnν ≥ 0 , for nµn

µ = 0

ρ+ p ≥ 0

2 / 30


NEC violation - ghost condensate, P(X = (∂φ)2) theories, and Galileons

Galileons: higher derivative scalar theories with 2nd order e.o.m

L1 = π

L2 = −1

2(∂π)2

L3 = −1

2(∂π)2�π

L4 = −1

2(∂π)2

((�π)2 − π,µνπ,µν

)L5 = −

1

2(∂π)2

((�π)3 + 2π,µνπ,νρπ,ρµ − 3�ππ,µνπ,µν

)

3 / 30


NEC violation - ghost condensate, P(X = (∂φ)2) theories, and Galileons

Galileons: higher derivative scalar theories with 2nd order e.o.m

Galileon theories [Dvali, Gabadadze, Poratti; Nicolis, Rattazi, Tricherini; De Rham, Tolley; Deffayet et al.;

Trodden et al,; . . . ] can violate the NEC [ Khoury et al.; Koehn, Lehners, Ovrut], and give rise toa stable classical bounce [ Vikman et al.; Ijjas, Steinhardt; Koehn, Lehners, Ovrut; ]

Galileons arise as a description of the world-volume action of a probe brane inhigher dimensions[De Rham, Tolley; Goon, Hinterbichler, Trodden. . . ]

For instance

• “Regular” Galileons describe a probe 3-brane in 5d Minkowski space• Conformal Galileons describe a probe 3-brane in AdS5

They inherit a non-linearly realized symmetry from the higher dimensionalspace, e.g

Lconformal3 =

1

2(∂π)2�π − 1

4(∂π)4

Symmetry:

δπ = 1− xµ∂µπ, δµπ = 2xµ + x2∂µπ − 2xµxν∂ν π .

4 / 30


Can we incorporate this in a realistic string model?

Candidate: Heterotic M-theory

• Natural 5 dimensional setting

• Topological M5 branes wrapped on holomorphic curve in Calabi-Yau

• Choice of vector bundle on observable sector → MSSM + 3 R.H. ν

5 / 30


Outline for the rest of this talk

• Construct worldvolume action for probe 3-brane in heterotic M-theory a laGalileons

• Extend this result to N = 1 SUSY

• Extend to N = 1 SUGRA

• Conclusions

6 / 30


7 / 30


8 / 30


9 / 30


Bulk space is foliated by time-like hypersurfaces which are Gaussian normalwith respect to the bulk metric GAB (X ), [de Rham, Tolley; Trodden et al.]

GAB (X )dX AdX B = f (X 5)2gµν(X )dXµdXν + (dX 5)2

Kµν =∂X A

∂xµ∂X B

∂xν∇AnB gµν =

∂X A

∂xµ∂X B

∂xνGAB

Brane embedding coordinates X A(xµ) are five arbitrary functions of the worldvolumecoordinates xµ

Brane action must be invariant under arbitrary worldvolume diffeomorphisms:

δX A = ξµ∂µX A,

Implies that worldvolume action is composed entirely of the geometrical tensors

S =

∫d4x

√−g L

(gµν ,Kµν , ∇µ, Rαµβν

)

10 / 30


Demand: Action yields two-derivative equations of motion

We have a finite number of terms we can write down:

L1 =√−g

∫ π

dπ′f (π′)4,

L2 = −√−g ,

L3 =√−g K ,

L4 = −√−g R,

L5 =3

2

√−g KGB

with K = gµνKµν , R = gµν Rαµαν and KGB is a Gauss-Bonnet boundary termgiven by

KGB = −1

3K 3 + K 2

µνK −2

3K 3µν − 2

(Rµν −

1

2Rgµν

)Kµν

Naively, there are five scalar degrees of freedom - but one can use the gaugefreedom to set

Xµ(x) = xµ, X 5(x) = π(x).

Hence, there is really only a single scalar degree of freedom; the function π(x).

11 / 30


Heterotic M-theoryThe five dimensional metric takes the form [Lukas, Ovrut, Waldram]:

ds25 = e2A(y)ηµνdx

µdxν + e2B(y)dy 2

We can chose a gauge where the metric is [Ovrut, Stokes]:

ds25 = (1− 2αz)1/3ηαβdX

αdXβ + dz2 ⇒ f (z) = (1− 2αz)1/6

The coordinates are chosen so that:z = 0 corresponds to the observable wall

z = πρ corresponds to the hidden wall (ρ a fixed reference length,(πρ)−1 ∼ 1015 GeV)

The parameter α has mass dimension one ' 1014GeV,

α =π√2

(κ

4π

)2/31

v 2/3β

with κ the 11-d Newton constant and v is the CY reference volume

β =1

v 1/3

∫X

(c2(V (observable))− 1

2c2(TX )

)∧ ω ,

12 / 30


DBI Heterotic Galileons

The tadpole,√g ,√gK ,

√gR curvature terms give L =

4∑i=1

ciLi ,

L1 = −3

10α(1− 2απ)5/3

L2 = −(1− 2απ)2/3√

1 + (1− 2απ)−1/3(∂π)2

L3 = −α

3(1− 2απ)−1/3

[5− γ2

]− (1− 2απ)1/3�π + γ2[π3]

L4 = −γ(

[Π]2 − [Π2] + 2γ2(1− 2απ)−1/3[− [Π][π3] + [π4]

])+

10

3

α2

γ(1− 2απ)−4/3(−1 + γ2)

+2

3αγ(1− 2απ)−2/3

(− 4�π + γ2

[�π + 4(1− 2απ)−1/3[π3]

])+

2

3

α

γ(1− 2απ)−4/3

(1− 2γ2 + γ4

).

where γ ≡ 1/√

1 + (1− 2απ)−1/3(∂π)2, arbitrary coefficients c1, c2, c3, c4

have mass dimensions 5, 4, 3 and 2.

13 / 30


Derivative expansionConformal Galileons: total worldvolume Lagrangian isexpanded in powers of(∂/M)2, where R = 1/M is AdS5 radius

Terms of the same order are then grouped together, such that each collectionof terms becomes the n-th order conformal Galileon.

Due to the symmetry properties, one only needs to consider the expansion upto order (∂/M)8, higher order terms form total divergences.

Heterotic case: mass scale associated with the curvature of the five dimensionalspace is α -hence the appropriate expansion parameter will be (∂/α)2.

Truncate derivative expansion at a finite order in the expansion parameter.

Define the dimensionless field

π = απ

Scale the individual Lagrangians Li and coefficients ci as follows:

Li → α2−nLi , ci → αn−2ci

14 / 30


Analog of Galileons

Collecting terms of order (∂π/α)0, (∂π/α)2, (∂π/α)4 , (∂π/α)6 gives us:

L =∑4

i=1 LT ,1 where

LT,1 = −3

10c1(1− 2π)5/3 − c2(1− 2π)2/3 −

4

3c3(1− 2π)−1/3

LT,2 =

[−

1

2c2(1− 2π)1/3 − c3(1− 2π)−2/3 −

2

3c4(1− 2π)−5/3

](∂π

α

)2

LT,3 =

[−

1

2c3 − c4(1− 2π)−1

](∂π

α

)2 �π

α2

+

[1

8c2 +

1

3c3(1− 2π)−1 −

1

3c4(1− 2π)−2

](∂π

α

)4

LT,4 = −1

4c4(1− 2π)−1/3 ∂µ

α

(∂π

α

)2 ∂µ

α

(∂π

α

)2

+ c4(1− 2π)−1/3 �π

α2

π,µ

α

π,µν

α2

π,ν

α

−19

6c4(1− 2π)−4/3

(∂π

α

)4 �π

α2

+

[− c3(1− 2π)−1/3 −

11

3c4(1− 2π)−4/3

](∂π

α

)2 π,µ

α

π,µν

α2

π,ν

α

+

[−

1

16c2(1− 2π)−1/3 −

1

3(1− 2π)−4/3 −

9

4c4(1− 2π)−7/3

](∂π

α

)6

.

15 / 30


Expanding to first order in π (necessary for heterotic) yields:

L1 = −3

10c1 − c2 −

4

3c3 +

(c1 +

4

3c2 −

8

9c3

)π

L2 =

[−

1

2c2 − c3 −

2

3c4 +

(1

3c2 −

4

3c3 −

20

9c4

)π

](∂π

α

)2

L3 =

[−

1

2c3 − c4 − 2c4π

](∂π

α

)2�π

α2

+

[1

8c2 +

1

3c3 −

1

3c4 + (

2

3c3 −

4

3c4)π

](∂π

α

)4

L4 = −[

1

4c4 +

1

6c4π

]∂ν

α

(∂π

α

)2 ∂ν

α

(∂π

α

)2

+

[c4 +

2

3c4π

]�π

α2

π,µ

α

π,µν

α2

π,ν

α

−[

19

6c4 +

76

9c4π

](∂π

α

)4�π

α2

+

[− c3 −

11

3c4 + (−

2

3c3 −

88

9c4)π

](∂π

α

)2 π,µ

α

π,µν

α2

π,ν

α

+

[−

1

16c2 −

1

3c3 −

9

4c4 + (−

1

24c2 −

8

9c3 −

21

2c4)π

](∂π

α

)6

(ci ’s are dimension 4, mass scale set by α)

16 / 30


Analog of Galileons

Unlike 5d Minkowski or AdS5 cases, there is no non-linearly realized symmetryhere - the higher dimensional space is not maximally symmetric and has no’extra’ Killing vectors.

This means that there is nothing telling us how to organize the terms in thederivative expansion - the coefficients from the brane Lagrangian (the ci ’s)must be used.

As we will see, linearization and supersymmetrization will yield constraintsbetween the various ci ’s.

17 / 30


Extension to N = 1 SUSY

Since heterotic M-theory is a supersymmetric theory, the heterotic Galileonsmust be shown to have a supersymmetric completion

Similar work has been done in the case of the conformal Galileons has alreadybeen completed, see [Koehn, Lehners, Ovrut; Farakos et al.]

We use the superspace formalism to construct SUSY-invariant Lagrangianswhich contain the heterotic Galileons

Define a chiral multiplet P(x , θ, θ) , whose components are the complex scalar

A = 1√2(π + iχ), a Weyl fermion ψ and the complex scalar F .

We will only display the bosonic components to save space - already interestingeffects occur.

18 / 30


Extension to N = 1 SUSY

L1:

LSUSY1 = W (P)

∣∣∣∣θθ

+ W (P†)

∣∣∣∣θθ

L2:

LSUSY2 = K(P,P†)

∣∣∣∣θθθθ

K(P,P†) =(c2 + 2c3 + 4

3c4)

α2PP† +

1√

2

(− 13

c2 + 43

c3 + 209

c4)

α2(P2P† + PP†2)

L3:

LSUSY3 = LSUSY

3,1st term + LSUSY3,2nd term

LSUSY3,1st term =

1

16

[−

1√

2c3 −

√2

3c4 − 2c4(P + P†)

][DPDPD2P† + h.c.

]∣∣∣∣θθθθ

LSUSY3,2nd term =

1

4

[1

8c2 +

1

3c3 −

1

3c4 +

1√

2

(2

3c3 −

4

3c4

)P + P†

]·[

DPDPDP†DP†]∣∣∣∣θθθθ

19 / 30


L4:

LSUSY4 = LSUSY

4,1st term + LSUSY4,2nd term + LSUSY

4,3rd term + LSUSY4,4th term + LSUSY

4,5th term

LSUSY4, 1st term =

1

32

[1

4c4 +

1

6√

2c4(P + P)

]{D, D}(DPDP){D, D}(DP†DP†)


LSUSY4, 2nd term =

1

128

[c4 +

2

3√

2c4(P + P†)

]·[{D, D}(P + P†){D, D}(DPDP)D2P† + h.c.


LSUSY4, 3rd term =

1

32√

2

[19

6c4 +

76

9√

2c4(P + P†)

]·DPDPDP†DP†{D, D}{D, D}(P + P†)


LSUSY4, 4th term = −

1

128√

2

[− c3 −

11

3c4 + (−

2

3c3 −

88

9c4)(P + P†)

]·[{D, D}DPDPDPDP†{D, D}P + h.c.


LSUSY4, 5th term =

1

16

[−

1

16c2 −

1

3c3 −

9

4c4 +

1√

2(−

1

24c2 −

8

9c3 −

21

2c4)(P + P†)

]·DPDPDP†DP†{D, D}P{D, D}P†


20 / 30


Auxiliary field effects

21 / 30



where, for convenience, we define the mass dimension 2 parameters γ, δ, as

γ ≡(c2 + 2c3 + 4

3c4)

α2, δ ≡

1√

2

(− 13

c2 + 43

c3 + 209

c4)

α2.

22 / 30



In regular (two-derivative) SUSY, the field F appears quadratically in theLagrangian with no derivatives and acts as a Lagrange multiplier.

This means that we are able to use e.o.m to eliminate F , get a Lagrangianinvariant under SUSY on-shell. (Elimination works quantum-mechanically too.)

Here, we have two new effects:

• Quartic F -terms - (FF ∗)2

• Derivative terms in F , e.g. +F ∗∂F · (∂A− ∂A∗)Both of these have been explored elsewhere [Koehn et. al; Louis et al. ]

Attempt solution as follows - consider

LSUSY2 =

[−

1

2c2 − c3 −

2

3c4 + (

1

3c2 −

4

3c3 −

20

9c4)π

][(∂π

α)2 + (

∂χ

α)2 − 2

FF∗

α2

]

We will require that ∣∣∣∣Fα∣∣∣∣2 � 1 .

23 / 30



LSUSY(0)F = F (0) ∂W

∂A+ F∗(0) ∂W ∗

∂A+[γ + 2

√2δπ

]F (0)F∗(0)

F (0) = −1[

γ + 2√

2δπ] ∂W ∗

∂A

Choose as our superpotential

W (A) = β1A + β2A2 + β3A3

where βi ∈ R

V (π, χ) = −LSUSY(0)F

=β2

1

γ+

1√

2γ2

[− 4β2

1δ + 4β1β2γ

]π +

1

γ

[2β2

2 − 3β1β3

]χ2 +

9β23

4γχ4

+1√

2γ2

[12β1β3δ − 8β2

2δ + 6β2β3γ

]πχ2 +

9√

2

β23δ

γ2πχ4

In arriving at this expression, we have had to assume that∣∣∣∣ δγ∣∣∣∣ . 1

24 / 30


• Matching with L1 = − 310c1 − c2 − 4

3c3 +

(c1 + 4

3c2 − 8

9c3

)π tells us β1, β2

in terms of the c ′i s.

• Taking χ = ∂µχ = 0 ∀π

m2χ =

∂2V

∂χ2

∣∣∣∣χ=0

≥ 0 ,

⇒4β2

2 − 6β1β3

γ≥ 0 , 6β1β3δ − 4β2

2δ + 3β2β3γ ≥ 0

• For F (0) to be constant and small requires

2β1δ − β2γ = 0 ,

∣∣∣∣β1

γ

∣∣∣∣� 1

• Ghost free kinetic energy

−1

2c2 − c3 −

2

3c4 +

(−

1

2c2 −

4

3c3 −

2

3c4

)β21

γ2+(−

1

4c2 −

4

3c3 − 9c4

)β41

γ4< 0 ,

(1

3c2 −

4

3c3 −

20

9c4) +

(−

8

3c3 +

16

3c4

)β21

γ2+(−

1

6c2 −

32

9c3 − 42c4

)β41

γ4< 0

25 / 30


N = 1 SUGRA extensionThere is a known prescription for extending our SUSY action to N = 1 SUGRA[Wess and Bagger].

Higher derivative SUGRA work has been looked at before [Koehn, Lehners, Ovrut;

Baumann, Green; Farakos et al.; Ciupke] -

LSUGRA1 =

∫d2Θ 2EW (P) + h.c.

LSUGRA2 = M2

P

∫d2Θ 2E

[−

3

8(D2 − 8R)e−K(P,P†)/3M2

P

]+ h.c

LSUGRA3 = L3,I + L3,II

L3,I = −1

64

∫d2Θ 2E (D2 − 8R)

[−

1√

2c3 −

√2

3c4 − 2c4(P + P†)

]DPDPD2P†

+ h.c.

L3,II = −1

32

∫d2Θ 2E (D2 − 8R)

[1

8c2 +

1

3c3 −

1

3c4 +

1√

2(

2

3c3 −

4

3c4)(P + P†)

]·DPDPDP†DP†

+ h.c.

26 / 30


27 / 30


SUGRA extension

Auxiliary fields of supergravity - M, and bµ - can be integrated up to L3.

At LSUGRA4 , we will find higher orders in M, bµ as well as ∇ · b.

Coupling to curvature terms arise

LSUGRA4 ⊃

[c4 +

2

3√

2c4(A + A∗)

][17

4RFF∗∇ · ∇(A + A∗)

−9

8FF∗Rµν∇µA∇ν(A + A∗)

].

28 / 30


ConclusionsWe have found an N = 1 SUSY action for a probe brane in heterotic M-theory,by constructing the analog of Galileons.

• No symmetry, but truncated due to natural scale απρ.

• Linearization and supersymmetrization leads to constraints betweencoefficients.

Higher derivative lagrangian leads to interesting effects with auxiliary fields inboth SUSY and SUGRA - so far only perturbative approach has been taken

To do listSolve the equations of motion for π coming from the real scalar Lagrangian -sources a 4d aeff .

Examine full properties of propagating auxiliary fields

Include non-perturbative superpotential for branes in heterotic M-theory - thisgives rise (in 2-derivatives SUGRA) to an exponential potential a la ekpyrosis:

W ∼ exp(−T

2Y)

29 / 30


Thank you!

30 / 30

rehan deen university of pennsylvania - cpe.vt. · pdf filebouncing cosmologiesbrane...

Documents