R2 inflation with non-trivial superpotential couplings · 2014. 12. 17. · Introduction R2...

Introduction R2 supergravity Inclusion of Λ2 terms Inflation: numerical treatment The slow-roll parameters Conclusions

R2 inflation with non-trivial superpotentialcouplings

Peggy Kouroumalou

University of Athens

16-23 December, Miami 2014

hep-th 1411.5785

G.A. Diamandis,B.C.Georgalas, K.Kaskavelis, P.K,

A.B.Lahanas, G. Pavlopoulos

Peggy Kouroumalou University of Athens

R2 inflation with non-trivial superpotential couplings


1 Introduction

2 R2 supergravity

3 Inclusion of Λ2 terms

4 Inflation: numerical treatment

5 The slow-roll parameters

6 Conclusions




Introduction-Motivation

Recent observations coming from WMAP and PLANCK giveexperimental bounds both for spectral index ns = 0.9608± 0.0054and tensor to scalar ratio r < 0.11

BICEP2 experiment claims discovery of primordial gravitational wavesresulting to a value for tensor to scalar ratio: r = 0.2+0.07−0.05Starobinsky model of inflation which predicts r ' 0.004 is inagreement with PLANCK satellite data

Chaotic inflation seems to be closer to BICEP data

Supergravity models that incorporate R + R2 terms and reproduceStarobinsky’s inflation predictions for r , ns have received a lot ofattention recently [...]

Whether extensions of some of these models that include non trivialsuperpotential couplings can lead to succesful inflation as also underwhich, (if any) circumstances, agreement with BICEP2 data can beobtained, is an open question.




R2 supergravityR2 gravity supersymmetrized is equivalent to ordinary supergravity theory with 2chiral multiplets Λ,Φ [Whitt, Stelle]

R2 supergravity-Starobinsky model

LR =∫

d2Θ2E[− 18(D̄D̄ − 8R

)Ω(Φ, Φ̄) + Λ(Φ−R)] + h.c.m

LR2 =∫

d2Θ2E[− 18(D̄D̄ − 8R

)]R2

Generalization of LR

L =∫

d2Θ2E[− 18 (D̄D̄ − 8R) Ω′(Φ,Λ, Φ̄, Λ̄) + W (Φ,Λ)] + h.c.

W (Φ,Λ) = g(Λ)Φ + P(Φ,Λ)

Ω′(Φ,Λ, Φ̄, Λ̄) = Ω(Φ, Φ̄)− 12 (Λ + Λ̄), Ω(Φ, Φ̄) = ΦΦ̄

Departure from linearity of g(Λ) brings about new features that may affect thecosmological evolution




Superfield expansion: only bosonic terms

2E = e{1−ΘΘM̄} −→ chiral density

Φ = ϕ+ ΘΘ Fϕ , Λ = λ+ ΘΘFλ

R = − 16{

M + ΘΘ[− 12 R +

23 MM̄ +

13 bµb

µ − iDµbµ]}

g(Λ) = g(λ) + ΘΘ∂g∂λ

Fλ

W (Λ,Φ) = W (λ, ϕ) + ΘΘ (WλFλ + WϕFϕ)

Off-shell Lagrangian

Elimination of the auxiliary fields




Yields

LB”dual”←−−→ LR2 ∼

1144

Ωϕϕ̄∣∣∣∣∂g∂λ + Pϕλ + 2Ωϕ(∂2g∂λ2

ϕ + Pλλ)∣∣∣∣2

R2

In the general case no "pure"R2 theory, alsoR terms

Reduction to Starobinsky model

In the special case :Ω(Φ, Φ̄) = ΦΦ̄, g(Λ) = Λ, P(λ,Φ) = 0 } −→ ∼ (constant) R2 "pure"R2 theory

Neccessary condition for a possible R2 description is the existence of a scalar field λwith no kinetic term Ω(λ, λ̄) = 0




K = −3ln(−Ω′

3

), Ω′ = −3 + ϕϕ̄− ζ(ϕϕ̄)2 − 1

2(λ + λ̄)

Kähler function

stabilization term for the potential in the φ direction

For ζ = 0 −→ no-scale model

W (Φ,Λ) = g(Λ) Φ + α, g(Λ) = d + d1Λ +d2Λ2

quadratic term

4 real parameters α, d , d1, d2




If we express Kähler function and superpotential in terms of newsuperfields C, T

K = −3 ln (T + T̄ − CC̄ )W = 3M C (T − 1)

Λ = 6 T − 3Φ =√

3 C

with the following values for the parameters

α = d2 = 0, d = −3d1 ≡ −√

32

M, P(Φ, Λ) = 0

we retrieve Kallosh-Linde model (2013)With the choice α = 0 (potential minimum at ϕ = 0, concave up) therelevant Lagrangian for our model

λ = s + iσ

e−1L = −12

R − 34∂µs∂µs(s + 3)2

− 34∂µσ∂

µσ

(s + 3)2− 9 |g(s + iσ)|

2

(s + 3)2




e−1L = −12

R − 34∂µs∂µs(s + 3)2

− 34∂µσ∂

µσ

(s + 3)2− 9 |g(s + iσ)|

2

(s + 3)2

Scalar potential (Vmin. = 0)

Neglect the fluctuations of σ around zero

e−1L = −12

R − 34∂µs∂µs(s + 3)2

− 9 |g(s)|2

(s + 3)2




Figure : 3D plot of the scalar potential of the Lagrangian for values ofparameters d = −3×10−5, d1 = 10−5, and d2 = 5×10−10. The axesare along the real and the imaginary direction of the field λ ≡ s + i σ .




Potential in terms of new parametersDefine the field φ:

s = −3 + (3 + `) e√

23 φ

` =−d1 +

√d21 − 4 d d2

2 d2MP = 1

M2 = 12 |d1|2 (1− 4ab) ,

A =6b − 1 +

√1− 4ab

2√

1− 4aba =

dd1, b =

d2d1

inflation scale

CMB anisotropies→ M ' 10−5

Scalar potential

V (φ) =3M2

4( 1− e−

√23 φ )2 |1 + A (e

√23φ − 1)|2




Potential one minimum for A > 0, two minima for A < 0

A = 0 −→ Starobinsky modelWe consider A > 0 for which for values

φi <

√32

ln(

1 +1A

)

there is a plateau to sustain inflationFor larger values of φ, V exhibits a rapid exponential behaviour




Potential for A > 0 and A < 0

Figure : The general form of the scalar potential as function of φ for the cases A > 0(left panel) and A < 0 (right panel)




Inflaton potential




Solution of the differential equations

φ̈ + 3 H φ̇ + V ′(φ) = 0

3 H2 =φ̇2

2+ V (φ) ,

H = ȧ/a

with initial values

, a(t = 0) = 1,V (φP) = 1M = 10−5Mp

leading to

φP =

√38

ln(

43 A2 M2

)' 14.28− 1.22 ln A




Inflaton and cosmic scale factor evolutionAfter a sharp drop the inflaton follows a normal slow-roll evolutionThe exit from inflation occurs when Loga starts becoming almost constantThe damped oscillatory behaviour of φ as it drops within the minimum of thepotential

Figure : Evolution of the inflaton φ (left) and the logarithm of the cosmic scalefactor loga (right) with time. The time is taken in units of the scale M = 10−5Mp .The solid (Red) , dashed (Green) and dash - dot (Blue) lines correspond tovalues of A = 10−4, 2× 10−4, 4× 10−4 respectively.




Hubble parameter, number of e-foldings evolutionAfter a short drop-off H enters into the slow-roll era, during which it stays almostconstantThe number of e-foldings N(t) should be in the range is ∼ 50− 60 at a time t∗at which the inflaton receives the pivot value φ∗

Only values A ≤ 5× 10−4 allow for 50-60 e-foldings to the left of pivot value φ∗

Figure :

Evolution of the Hubble rate H (left) and the numberof e-foldings Nfold (right) with time




The slow-roll parameters

A denotes the departure from the Starobinsky model. In order to have50-60 e-foldings from φ∗ to φend A has to be small A ≤ 5× 10−4

In the slow-roll regime

�, η < 1

At the end of inflation � (and/or) η become of order 1The number of e-foldings can be calculated analytically

(k ≡A

1− A, x = exp

(−√

32φ

))

N(x) = −34

ln

[x (k + x2end )

xend (k + x2)

]+

3(1− k)4√

k

(arctan

xend√k− arctan

x√

k

)




constraints on A from slow-roll parameters A < 0.13constraints on A from sufficient number of e-foldings A < 5× 10−4

Figure :

The shadowed areas designates the region where slow-roll approximationholds. � < 1 ( in Blue ) allows for large A values while �( in Magenta )

requires A < .13.Peggy Kouroumalou University of Athens



Impact of A in �, r parameters�, as function of the inflaton field, or the field x is related to the one obtained byStarobinsky model �s

√� =

√�s +

2√

3

11 + (A−1 − 1) x

� parameter larger than �s by amounts controlled by the parameter AThat could in principle lead to larger values for � (and r).Analysis has to take into account the remaining cosmological data number ofe-foldings, which specifies the pivot scale, the spectral index nsIntersection of N and ns allows for values of r :0.003 < r < 0.0053.

A < 4.0 × 10−5, x ∼ 0.013

Within this region the allowed values for tensor to scalar ratio are

0.003 < r < 0.0053.




Figure : Regions for N, r , ns within the ranges designated at the borders of eachregion. The allowed r region by N and ns predicts r in the range 0.003 < r < 0.0053although the N and ns separately each allows for much higher values.




Conclusions

Generalization of the supergravity Starobinsky models allowing for superpotentialterms that are not linear in the superfield Λ that couples to the chiral R multiplet.2 parameters

M sets the scale of the inflationary potential

A deforms the Starobinsky potential

Value for tensor to scalar ratio,r,

r ' 0.005

Slightly larger than Starobinsky predicted value but not large enough tochallenge BICEP2Value of r could reach the upper limit of PLANCK data (r = 0.11) if we allow avalue for ns that exceeds its experimental bound by an amount of 15 per-cent.The allowed small value of A suggests that the departure from the linearity ispresumably due to quantum effects




Thank you!!This research has been co-financed by the European Union

(European Social Fund-ESF) and Greek national funds through theOperational Program " Education and Lifelong Learning" of the

National Strategic Reference Framework (NSRF)-Research FundingProgram: THALES.

Investing in knowledge society through the European Social Fund.



IntroductionR2 supergravityInclusion of 2 termsInflation: numerical treatmentThe slow-roll parametersConclusions

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