1. Inflation theory MASAHIDE YAMAGUCHI (Tokyo Institute of Technology)
Recontres du Vietnam

2. Contents Introduction General strategies to realize inflation
A symmetry like the shift symmetry
A diverging kinetic term
Attractor models (pole inflation)
k-inflation & higher derivative terms
Summary

3. Introduction

4. Generic predictions of inflation, which is an accelerated expansion in the early Universe
Spatially flat universe
Almost scale invariant, adiabatic, and
Gaussian primordial density fluctuations
Almost scale invariant and Gaussian
primordial tensor fluctuations
Generates anisotropy of CMBR. 4 4

5. Constraints on scalar and tensor perturbations from the PLANCK satellite
Observational constraints :
Theoretical predictions :
(95% CL TT,TE,EE+lowE+lensing)
Attractor models like
Starobinsky model
fit the data well.
(Starobinsky model)
Planck 2018 results. X

6. How to realize inflation
Next task is to identify the inflaton, a scalar field which caused inflation, that is, inflation model building

7. General strategies to realize inflation
Introduce a symmetry like the shift symmetry:
(Freese, Frieman, Olinto,
Kawasaki, MY, Yanagida.)
An action depends only on a kinetic term.
Then, a potential becomes flat, or even without a potential.
Consider a diverging kinetic term (pole inflation) :
(Galante et al., Broy et al.)
This is the opposite case, in which a kinetic term vanishes and a strong coupling problem happens.
Is there another generic strategy ?
As ρ approaches 0, the kinetic term diverges.
Then, after making the canonical normalization,
All of the coupling constants become effectively very weak.
 Asymptotically flat potential

8. Attractor models (pole inflation)

9. Starobinsky model (MG = 1) Conformal transformation with
Why is the potential of this model so flat ???
Asymptotically
flat potential

10. Conformal attractors Local (gauge) conformal symmetry :
(Kallosh & Linde, Ferrara et al., Kallosh et al. )
Local (gauge) conformal symmetry :
(global SO(1,1) symmetry for constant F(φ/χ) )
N.B. χ has wrong sign of kinetic term : compensator field
Gauge fixing with :
(Einstein frame)
If F is constant, the potential is simply C.C.
If F is smooth, the potential is stretched for large
Starobinsky model 

11. Conformal attractors II
Gauge fixing with :
(Jordan frame)
Conformal transformation with
(Einstein frame)
Same action
Due to the pole, the kinetic term diverges so that the potential becomes flat, or goes in to the weak coupling regime.
This is the opposite situation to the vanishing of the kinetic term, which leads to the string coupling regime.
The pole structure of the kinetic term stretch the potential effectively !!
α attractors :
(Conformal attractors including Strobinsky model correspond toα= 1.)

12. Pole inflation Primordial perturbations :
(Galante et al., Broy et al.)
K(ρ) has a pole at ρ= 0 in Laurent series :
canonical field φ
V(ρ) is regular at ρ= 0 :
(asymptotically flat)
Primordial perturbations :
α attractors  p=2, a2 = 3α/2
This kind of structure is discussed in the context of supergravity, but I do not know how generically this kind of structure appears in string theory.
ap dependence appears only in r.
Subleading terms yield higher order corrections.

13. Constraints on scalar and tensor perturbations from the PLANCK satellite
Observational constraints :
Theoretical predictions :
(95% CL TT,TE,EE+lowE+lensing)
Attractor models like
Starobinsky model
fit the data well.
(Starobinsky model)
Planck 2018 results. X

14. Pole structure of Higgs inflation
(Futamase & Maeda, Cervantes-Cota & Dehnen, Bezrukov & Shaposhnikov)
(ξ> 0)
This is the reason why Starobinsky model and Higgs inflation gives almost same predictions.
Leading term coincides with α attractors !!
(Density perturbations ξ~ 104  α ~ 1 )

15. Shift symmetry and extension

16. Shift symmetry and k-inflation
To end an inflation, slight
φ dependence is necessary.
A kinetic term of an inflaton is not necessarily canonical.
(k-inflation)
(Armendariz-Picon et.al. 1999)
In fact, an action may include even higher derivatives.
(Nicolis et.al. 2009)

17. The theory has Galilean shift symmetry in flat space :
Galileon field
Nicolis et al. 2009
Deffayet et al. 2009
The theory has Galilean shift symmetry in flat space :
Lagrangian has second order derivatives, but EOM is second order. 17

18. Why do we consider higher derivative terms ???

19. It is impossible to break null energy condition stably within k-inflation. That is, primordial tensor perturbations (GW) have always red spectrum. The equation of state, w = p /ρ, of dark energy is always larger than -1.

20. Null energy condition (NEC)
for any null vector ξμ.
This is the weakest among all of
the local classical energy conditions.
For a perfect fluid :
NEC   w ≧ -1
As long as NEC is conserved, even (eternal) inflation cannot solve the initial singularity problem.
As long as (and H > 0 for an expanding Universe like inflation) 20

21. Apparently, it looks that, if KX < 0, it can violate the NEC.
How robust is the NEC ?
Canonical kinetic term with potential:
(NEC is conserved)
How about k-inflation ?
(Armendariz-Picon, Damour, Mukhanov 1999)
Apparently, it looks that, if KX < 0, it can violate the NEC.
But, this is not the case. 21

22. Primordial density fluctuations
Garriga & Mukhanov 1999
Perturbed metric :
Comoving gauge :
Prescription:
Expand the action up to the second order
Eliminate α and β by use of the constraint equations
Obtain quadratic action for ζ
(sound velocities of
curvature perturbations)
In order to avoid the ghost and gradient instabilities, ε > 0 & cs2 > 0.
(Hsu et al. 2004)
(See also Dubovsky et al ) 22

23. Stable violation of NEC is impossible within k-inflation
It is impossible to break the NEC stably within k-inflation.
Background solutions can break NEC apparently.
But, the perturbations around them always become unstable
for such background solutions.
This is quite reasonable in some sense because violation of NEC must pay
some price. (see Sawicki & Vikman 2013, Easson, Sawicki, Vikman 2013)
e.g. An observer with almost speed of light observes arbitrary negative energy.
N.B. k-inflation is the most general action coming from
phi and its first derivatives.
One may wonder how about introducing higher order derivative terms. 23

24. Galilean Genesis
(Creminelli et al.,
Nicolis et al.,
Kobayashi et al. G-inflation)
(In the flat spacetime limit, this theory has conformal symmetry SO(4,2).)
Energy-momentum tensor :
A background solution, (t : -∞ -> 0 ) : Starts from Minkowski in infinite past.
(Actually, you can verify that H increases.)
( The NEC is violated !! ) 24

25. Primordial density fluctuations
Perturbed metric :
Comoving gauge :
In order to avoid the ghost and gradient instabilities, Gs > 0 & Fs > 0.
( The NEC is violated stably !! )
N.B.
A spectator field like curvaton is responsible for primordial density
perturbations because the genesis field predicts too blue (ns ~3)
perturbations in this simple model.
Primordial tensor perturbations are not generated at first order. 25

26. However, higher order derivative terms are generally dangerous !!

27. Ostrogradski’s theorem
(Ostrogradsky 1850)
Assume that and depends on :
(Non-degeneracy)
Canonical variables :
Non-degeneracy ⇔ ⇔
So, there is a several way to avoid this ghost. First one is to abandon the non-degeneracy condition.
Horndeski theory has second order differential EOM instead of fourth order differential EOM. So, it is clearly degenerate theory.
Hamiltonian:
p depends linearly on H so that no system of this form can be stable !!
N.B.
(propagators) 27 27

28. Loophole of Ostrogradski’s theorem
We can break the non-degeneracy condition
which requires that depends on ddot{q}.
In case Lagrangian depends on only a position q and
its velocity dot{q}, degeneracy implies that EOM is first order,
which represents not the dynamics but the constraint.
In case Lagrangian depends on q, dot{q}, ddot{q},
degeneracy implies that E-L EOM can be (more than) second order,
which can represent the dynamics. 28

29. A simple (degenerate) example
E-L eqs. ① ② If
① + ②
becomes ③ ③ ②
(Second order system for x & φ) 29

30. Generalized Galileon = Horndeski
Deffayet et al. 2009, 2011
Horndeski 1974
equivalence
Kobayashi, MY, Yokoyama 2011
This is the most general scalar tensor theory whose Euler-Lagrange EOMs are
up to second order though the action includes second derivatives.
Many of inflation and dark energy models can be understood in a unified manner.
Inmpose degeneracy condition on kinetic terms
Langlois and Noui proposed degenerate higher
order scalar-tensor (DHOST) theory.
e.g. 30

31. Third or even higher order derivatives ???
As far as I know, no (covariant) healthy scalar tensor theories
including third or even higher order derivatives have been found !!
You may wonder if no further interesting features would appear
even if one considers third or even higher order derivatives.
But, this might not be true because the structures of
(ghost) instabilities are different between a theory with
second order derivatives and one with third order derivatives.
pi_a (& P_a) are now expressed in terms of other variables.
(Motohashi, Suyama, MY)
We have only to remove linear
dependence of the momentum.
We need to remove linear dependences of
not only the momenta but also the canonical
variable corresponding to Q = ddot{q}.

32. Ostrogradski’s theorem
(Ostrogradsky 1850)
Assume that and depends on :
(Non-degeneracy)
Canonical variables :
Non-degeneracy ⇔ ⇔
So, there is a several way to avoid this ghost. First one is to abandon the non-degeneracy condition.
Horndeski theory has second order differential EOM instead of fourth order differential EOM. So, it is clearly degenerate theory.
Hamiltonian:
p depends linearly on H so that no system of this form can be stable !!
N.B.
(propagators) 32 32

33. Summary A symmetry like the shift symmetry or a pole structure of a
kinetic term is a key idea to realize inflation.
Is there yet another key idea to realize inflation naturally ?
Theories with second order derivatives can open a new
possibility of the predictions of inflation.
Unfortunately, no (covariant) healthy scalar tensor theories
including third or even higher order derivatives have been
found, which might open yet another new possibilities.