The Overshoot Problem in Inflation after TunnelingKoushik Dutta*

DESY, Theory Group, Notkestrasse 85, Bldg. 2a, D-22607 Hamburg, Germany

In the 'landscape' paradigm of String Theory, our observable Universe may originate via a tunneling event from a nearby metastable false vacuum, followed by sufficient amount of inflation. We discuss the overshoot problem in this set-up and show explicitly that there is no overshoot problem for higher order monomial potentials. The results substantially alleviate the initial value problem for "small-field" inflation models. 

*Work in collaboration with Pascal M. Vaudrevange and Alexander Westphal. Work to be published soon!

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!̈ + 3H!̇ = !V !(!) H2 = 13M2

P

!!̇2

2 + V (!)"

+ 1a2

The work is supported by the Impuls und Vernetzungsfond of the Helmholtz Association of German Research Centers and German Science Foundation (DFG).

String Theory Landscape

Large number of discrete vacuua separated by large potential barriers

Cosmology

‘Pocket Universe’ born out of tunneling event from a nearby metastable vacuua

Tunneling

Open FRW Universe with negative spatial curvature ( k = - 1)

Zero initial speed of the bubble at the nucleation

The Universe after Tunneling

Must be followed by ‘sufficient’ amount of inflation

Amount of inflation depends on the initial speed of the bubble when it reaches the plateau

Chances of Overshooting

How severe is the overshooting problem?

a(t) = t H = 1/t divergent initially and slows down the field!

!0 > MP No overshooting: similar to chaotic inflation!

Our Considerations !0 < MP

Model the problem:

Vexit ! !n Vplateau = V!(1!"

2!")

!̈ + 3t !̇ + (!1)n!n!1 = 0

Exiting part: Linear Exiting part: Quadratic

! = !0 + !V!8 t2 ! = 2!0

J1(mt)mt

!̇(! = 0) =!!("V!!0)/2 !̇(! = 0) = !0.21m!0

!(tc) = 34!

2

!"" !0 !(tc) = 3

4!

2

!"" 0.4!0

Exiting part: Quartic

The field reaches at the minimum at infinite time!!(t) = 8!08+!2

0t2 !̇(! = 0) = 0

Exiting part: Higher order No closed solution!

!̈ +"

t!̇ + t

!(m!1)!(m+3)2 !m = 0

C0 =12t!!1

!!̇2t2 + !̇!t("! 1)

"+

1m + 1

!!t

!!12

"m+1

! =m + 3m! 1

At t = 0, C0 = 0, therefore, at the bottom (! = 0), !̇ = 0

For a quartic and higher order monomials there is no overshoot problem!

First Integrals for the Modified Emden Equation q + alpha(t) + q n = 0, Journal of Mathematical Physics, 26 (10): 2510 - 2514, 1985 by P. G. L. LeachHandbook of Exact Solutions for Ordinary Differential Equations by A.D Polyanin and V. F. Zaitsev

1t2c

= V!3

For the quadratic case the field overshoots less than the linear potential.

Universal factor in front of the slope of the plateau! True for higher monomials also!

Plausibility argument:For higher order monomials the field falls down the hill relatively quickly with almost zero speed, and at the same time, the end part of the exiting potential is extremely shallow!

Exiting part: Qubic

!̈ +103t

!̇! !2 = 0

C0 =12!̇2t4 +

43!!̇t3 ! 1

3!3t4 +

29!2t2 ! 4

9!̇t! 28

27! !̇(! = 0) != 0

!̇lower(tf ) ! "0.03!""!0

!̈ +53t

!̇! "!2 = 0 !̇upper(tf ) ! "0.17!""!0

V! << V0

Kinetic energy < Curvature as long as !0 < MP

We have explicitly shown that curvature dominates in the left hand side of the monomial potential

More to think:Universal factor in front of the linear slope

Better understanding of qubic potential and polynomials