Curvature perturbations and non-gaussianity from waterfall phase transition. Hassan Firouzjahi. In collaborations with

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1 Curvature perturbations and non-gaussianity from waterfall phase transition Hassan Firouzjahi IPM, Tehran In collaborations with Ali Akbar Abolhasani, Misao Sasaki Mohammad Hossein Namjoo, Shahram Khosravi

2 Outline Waterfall dynamics in hybrid inflation Classifications of hybrid inflation parameters space Sharp phase transition during inflation and large spiky non- Gaussianity Models with a mild waterfall phase transition Conclusions

3 Hybrid Inflation: Hybrid inflation is a model consisting two scalar fields, the inflaton field and the waterfall field. V = m2 2 φ g2 φ 2 ψ 2 + λ 4 The waterfall field is very heavy during inflation, so it sits near its local minimum, ψ = 0 ( ψ 2 M 2 λ ) 2 ( The inflaton mass is given by m 2 ψ = 2 ψv = g 2 (φ 2 φ 2 c) φ c = M g During inflation, for φ < φ c, the potential is After the critical point, the waterfall field becomes tachyonic and rolls to its global minimum, ending inflation quickly.

4 Some Properties of Hybrid Inflation: The simple Z2 symmetric potential predicts domain walls formation at the end of inflation. To avoid this problem, we have to use a complex waterfall field which can couple to a U(1) gauge field. The simple hybrid potential predicts a blue-tilted spectral index which is nearly ruled out by cosmological observations. One has to use different variants of hybrid inflation. The simple hybrid potential is a small field model and it predicts a negligible gravitational wave amplitude. Hybrid infltaion can naturally be embedded in models of high energy physics. For example, brane & anti-brane inflation is a nice realization of hybrid inflation in string theory. At the end of brane inflation, cosmic superstrings are produced, which can have important observational predictions for string theory.

5 Waterfall dynamics in hybrid inflation It is useful to define the following dimensionless parameters α m2, β M 2 H 2 H 2 The evolution of the inflaton field is very simple during inflation n=-nc 0 nf where During inflation, the waterfall rapidly rolls to 0. The quantum fluctuations of waterfall is

6 After the phase transition, the waterfall fluctuations are We are interested in modes which become tachyonic. They are given by Imposing the initial Bunch-Davies vacuum, the waterfall fluctuations after the waterfall transition n >0 are given by δψ k exp ( ) 2 3 ɛ ψn 3/2 ɛ ψ 2βr We divide the tachyonic modes into two categories: 1- Large scales: these are modes which leave the horizon before the waterfall, k < kc, and are always tachyonic after the waterfall. 2- Small scales : these are modes which are tachyonic but remained sub-horizon till end of inflation.

7 In hybrid inflation the waterfall field is very massive so at the background ψ =0. However, the waterfall quantum fluctuations become highly tachyonic and behave as classical fields δψ I k(n) =δψ I k(0) exp ( ) 2 3 ɛ χn 3/2 We assume that the effective background trajectory is determined by One can check that In this view is averaged over a Hubble patch while δψl(n) 2 varies smoothly over super-horizon scales.

8 One can check that Therefore the small scale tachyonic modes have the dominant contribution in As a result Now we can calculate the time of end of inflation. This is when the waterfall back-reaction g 2 φ 2 δψ 2 on the inflaton dynamics induces an inflaton mass of order H Solving g 2 δψ 2 (n f ) = m2 2 The time of end of inflation is obtained to be n f = Γ ɛ 2/3 ψ, Γ ( ) 2/3 3 4 ln 8π2 α g 2 ɛ 4/3 ψ For example, in our numerical analysis n_f ~ 1.7 indicating that the waterfall phase transition is sharp.

9 Curvature perturbations from waterfall field Now comes the time to address our main question: What is the contributions of waterfall field to large scale curvature perturbations? Can it affect the standard single field predictions of hybrid inflation? Lyth, , Ali Akbar Abolhasani, H.F., We study this question using two methods: (a) (b) δn formalism. two-field adiabatic-entropic analysis Let s start from the δn formalism Fonseca, Sasaki, Wands, , Gong, Sasaki, Following Gong-Sasaki we have R c (x) 1 2ɛ ψ n 1/2 f ψ 2 (x, 0) δψ 2 (0) where the perturbations of waterfall field, after smoothing, is given by ψ 2 (x, 0) δψ 2 (x, 0) δψ 2 (0),

10 To calculate the power spectrum, we need to know the following integral convolution ( ) ( ) δψ 2 k δψ 2 k =2δ 3 (k + k ) d 3 q δψ q 2 δψ k q 2. Performing this integral one has P R (k) ɛ 22/3 ψ This indicates that the power spectrum induced from the waterfall field is highly blue-tilted, n R 4. Furthermore, it is peaked near kc. Knowing that KCMB / Kc ~ e^{-60} we obtain that P R (δψ) (k CMB ) P R (φ) (k CMB ) ( k k c ) 3 Furthermore, since the curvature perturbations are sourced by it is highly non-gaussian. The fact that the induced waterfall curvature perturbations peaks near the end of inflation and is highly non-gaussian can have important implications for primordial black-hole formations. The constraints from PBHs may be used to constraint the parameter space of hybrid inflation further. (δψ 2 (0)) k Lyth, ,

11 Extending the parameter space of hybrid inflation The standard hybrid inflation is parametrized by requiring φ c M P, β N e, ɛ ψ 1 A. Abolhasani, H.F. and M. Sasaki, S. Clesse, , We would like to relax these assumptions and see what happens. ( ) 1- φ 2 c > 12M 2 P, ɛ ψ 1 In this case inflation has three extended stages (a) before waterfall, (b) waterfall symmetry breaking completion (c) chaotic inflation 2- φ 2 c > 12M 2 P, ɛ ψ 1 In this case inflation has two extended stages separated by a short period of phase transition 3- φ 2 c < 12M 2 P ɛ ψ 1 ɛ ψ 1 : Standard Hybrid inflation :A mild phase transition with an extended period of inflation during waterfall.

12 Motivations for Local Features There seem to be glitches on CMB power spectrum at l ~ These may be interpreted as local features in primordial power spectrum. There are many models which can produce local features in curvature perturbation power spectrum during inflation. This includes: Particle Creations during Inflation, Fields Annihilation during Inflation, Rapid change in sound speed... Change in Inflaton Vacuum state (non- Bunch-Davies vacuum) The effects of heavy oscillatory fields Here we would like to employ the idea of sharp waterfall during early stage of inflation to create local features.

13 A model with local feature effect A. A. Abolhasani H.F., S. Khosravi, M. Sasaki Now we consider the interesting case where the system is basically a chaotic model with a rapid phase transition imposed on it φ 2 c > 12M 2 P, ɛ ψ 1 Inflation has two extended stages followed by a short period of waterfall transition In order to neglect the vacuum potential we require C g2 M 2 λm 1 2 Before the phase transition, the mass of inflaton is m. After the transition, the inflaton mass changes due to backreaction ( -Nc nf ne After the waterfall completion, n > nf, the waterfall settles to its instantaneous minima χ 2 (φ) = M 2 λ g2 λ φ2

14 As in standard hybrid inflation model we assume that the background waterfall trajectory is defined by δχ 2 To use delta N formalism to calculate curvature perturbations, we have to find the time of end of waterfall transition nf : Using delta N formalism we obtain Correspondingly, the curvature power spectrum is The inflaton power spectrum is easily obtained to be

15 The waterfall contribution to power spectrum and bi-spectrum is non-trivial now To find the waterfall contribution, we have to obtain the following convolution integral With some efforts, one can show that Where nt(k) represents the time when the mode k becomes tachyonic. As an order of magnitude estimate we have

16 Bi-spectrum and non-gaussianity We would like to calculate the curvature perturbations three-point function The first term can be interpreted as the intrinsic non-gaussianity. The second term can be interpreted as dynamical non-gaussianity. Define the intrinsic NG bi-spectrum The dynamical NG is decomposed into where

17 There are two categories of possible contractions in 1- Combining all 32 * 3 possibilities yield 2- Combining all 24 * 3 possibilities yield

18 After performing all possible contractions we obtain where Calculating the non-gaussianity parameter fnl we obtain For the dynamical non-gaussianity we have Due to sharp peak of waterfall field power spectrum, this is negligible unless two momenta are close to kmax using the delta N formula we obtain

19 The difficult task is to calculate the intrinsic non-gaussianity! For this purpose we have to obtain the following convolution integral With some efforts and approximations we obtain The resulting intrinsic non-gaussianity is Combining the dynamical and intrinsic non-gaussianities we have For example in our numerical example this gives

20 Super-Planckian Mild Phase transition Now we look in the limit where the phase transition is mild but with a super-planckian inflaton field: φ 2 c > 12M 2 P, ɛ ψ 1, β > 1 A. Abolhasani, H.F. and M. Sasaki, Nc 0 nf ne In this case inflation has three extended stages (a) before waterfall, (b) waterfall symmetry breaking completion (c) chaotic inflation As in standard Hybrid inflation model, our proposal is to use as the effective classical field. Using the delta N formalism for the curvature perturbations we obtain The curvature perturbations two-point function is κ ψ ɛ ψ

21 We have to obtain the following integral with a convolution This can be divided into UV and IR contributions of the convolution momenta and One can show the IR contribution dominates over the UV contribution and The spectral index and its running are n s 1= 4κ ψ ln(k/k c ) dn s d ln k = 4κ ψ

22 Problems with the COBE normalization With the curvature power spectrum P R one can not satisfy the COBE normalization P R What is the origin of this problem? In standard hybrid inflation with a sharp phase transition, this problem did not arise because the waterfall field power spectrum was highly blue-tilted, n R 4, and as we saw the waterfall field does not contribute into power spectrum. However, in the current model with an almost scale invariant waterfall power spectrum the waterfall fluctuations contributes to large scale curvature perturbations. Technically speaking, in simple models of inflation with a well-defined classical trajectory the ratio δφ/φ can be an arbitrary number to fit the COBE normalization. However, in our model there is no well-defined classical trajectory for waterfall and we defined as our classical trajectory. As a result, the ratio can not be made arbitrary small.

23 Conclusion All observations strongly support inflation as the leading theory of early universe and cosmological structure formation. Simple models of inflation predict almost Gaussian and almost scale invariant perturbations which are in very good agreements with data. Hybrid inflation is an interesting model of inflation. We conclude that for standard hybrid inflation with a very sharp phase transition, the waterfall dynamics induce no large scale curvature perturbations. However, these perturbations are highly blue-tilted and are highly non-gaussian. This may have interesting implications for primordial black hole formation. One can construct a chaotic model with a sharp waterfall phase transition on top of it. The waterfall quantum fluctuations produce a local feature in curvature perturbations and induce a very large spiky non-gaussianity.