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Inflaton oscillation&decay generating particle Inflaton oscillation&decay generating particle number asymmetry at the end of inflationnumber asymmetry at the end of inflation

Hiroyuki Takata, Tomsk State Pedagogical U.

Collaborated with Takuya Morozumi, Hiroshima u., Japan

Keiko Nagao, Okayama u. of science, Japan

Apriadi S. Adam, Hiroshima u., Japan

SSI2018

arXiv:1709.08781

Thank you for giving me a chance to talk..I'm going to talk by thie title about Generation of particle number asymmetry from inflaton .This talk is based on the collaboration with Morozumi and Apriadi from Hiroshima Uand Nagao from Okayama University of Science0:25

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1-1. Motivation and idea● We would like to know the origin of asymmetry between

particle and anti-particle.

● Seed of asymmetry possibly was at the preheating era; Inflaton oscillation possibly occurs together with particle creations, i.e. particle number asymmetry(PNA) also with..

● We would like to regard inflaton as origin of the asymmetry at the preheating era.

● Inflaton oscillation at the potential minumum transmits to oscillation of interacting fields with inflaton.

● Time evolution of number asymmetry of those fields can be explicitly calculated in quantum field theory.

I. IntroductionI. Introduction

early stage of reheating

First, Motivation and ideaWe would like to know the origin of asymmetry between particle anti-particle. Seed of asymmetry possibly exists at the preheating era. Preheating means early stage of reheating. At that time, Inflaton oscillation possibly occur together with particle creasion. So particle number asymmetry(PNA) also occure together.We would like to regard inflaton as origin of the asymmetery at the preheating era. Inflaton oscillation at the potential minumum transmits to oscillation of interacting fields with inflaton.Timevolution of number asymmetry of those fields can be explicitly calculated in quantum field theory.

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1-3. Preheating era after inflation

I. IntroductionI. Introduction

● Where is origin of PNA and how is the mechanism?● When did PNA start being created?

1-2. Number asymmetry between particle and anti-particle

possibly

Potential minimum

Inflaton

oscillate

Inflaton oscillates around the minimum

pic1

Particle

Anti-particle

Inflaton decays totwo other scalars and they interact

pic2

interact

more anti-particlesless particles

more particlesless anti-particles

oscillate

Particle number asymmetry oscillates

pic3

At preheating era, inflaton oscillates by remaining potential energy. pic1Inflaton decays to start creating particles-antiparticles

with PNA via their interactionpic2PNA also oscillates via oscillating inflaton background. pic3

(PNA)

Here are combined topics. Particle number asymmetry=PNA and inflation.First, number asymmetry between particle and anti-particle. We have more particle than antiparticle at present time. But particle and anti-particle are very similar, only difference is electric charge,opposite. So theory is naturaly constructed symmetric between them. In order to realize asymmetry, we put a seed/origin of aymmetry to model. Then we need clealify mechanism how asymmetry develops in time. At which time do we put seeds? When did asymmetry start being created? These are our natural question.We imagine it is at the preheating era just after inflation becase most of matter are created at reheating era. During Inflation there may be only inflatons. So the most simple economical scinario is the following.

See pic1. At preheating era, inflaton oscillates by remaining potential eneygy. See pic2. Inflatons decay to start creating particles-antiparticles with PNA via their interactionSee pic3. PNA also oscillates via oscillating inflaton background.1:42

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1-3. Strategy and Technique ● Model includes three scalar fields, one of them is identified as an inflaton.

● Friedmann-Robertson-Walker space-time with arbitrary scale factor.

● Model Includes both CP violating and particle number violating interaction.

● An Initial condition is given as density matrix of equilibrium at the end of inflation.

● Current of particle number asymmetry is calculated under oscillating inflaton background.

● Time development is treated by closed time path formalism in QFT.

I. IntroductionI. Introduction

How can we describe our idea in mathematical language. I will tell our strategy and technique here.Model includes three scalar fields, one of them is identified as an inflaton.We have gravity. It is treated as classical back ground, Friedmann-Robertson-Walker space-time with arbitrary time dependent scale factor.Model Includes both CP violating and particle number violating interaction. An Initial condition is given as density matrix of equilibrium at the end of inflation.Current of particle number asymmetry is calculated under oscillating inflaton background.Time development is treated by closed time path formalism in QFT.

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II.Contents of talkII.Contents of talk

I.I. IntroductionIntroduction

II.II. Contents of talkContents of talk

III.III. ModelModel

IV.IV. Current of Current of particle number asymmetryparticle number asymmetry

V.V. Hubble parameter approximationHubble parameter approximation

VI.VI. Numerical resultNumerical result

VII.VII. Summary and tasks Summary and tasks

THis is Contents ofg talkNext I present ,mode.Then analyutical result of current of particle number asymmetry will be shown.After approximating by weak Hubble parameter, newerical resuts will be shown.

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III. ModelIII. Model

: Fluctuation of inflaton around potential minimum

: order parameter of inflaton

Non-zero expectation value: OKPower index: 2 is not excluded (in LFM)* Slow roll condition: ? for the time being

Inflaton potential

*Ref: Martin and C. Ringeval, Phys.Rev. D82, 023511(2010), 1004.5525, “ First CMB Constraints on the Inflationary Reheating Temperature”

N.B. potential type: Large field model(LFM)/small field model/hybrid model is not specified.

・ Potential is assumed for slow-roll type.・ Non-zero expectation value of inflaton may lead to PNA.

3-1. Inflaton potential

To construct model, I explain about Inflaton potential and interaction. ♪The potential is assumed for slow-roll type ♪and has minimum at non-zero expectation value of inflaton. ♪We adopt this inflaton potential V(N). N is single neutral scalar field and is identified to inflaton.m_N is inflaton mass. Little v is order parameter of vacuum expectation value of inflaton. ♪If v is non-zero, inflaton has non-zero vacuum expectation value. ♪This power index 2 is chosen by simplicity. But in fact, by regarding this mode as, so called, large field model of inflation, this is still not excluded at least at the moment of this reference.But in our work, we will look at oscillating inflaton only at the bottom. We do not care, from where inflaton rolled. So we do not need to specify whether it is large field model, small field model or hybrid model.

♪One notation. Since inflaton N oscillate around v, N-v describes its fluctuation and is defined as φ3.

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III. ModelIII. Model3-2. Interaction to realize PNA

: Inflaton decays

Interaction between Inflaton N and complex scalar Φ

Mass difference between scalars Φ1 and Φ

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A, B: Seeds of Particle Number Asymmetry

Inflaton decays to scalars through interaction.Interaction vertex(A) must be complex to realize PNA.

Mass difference(B) between Φ1 and Φ

2 to realize PNA

What kind of interaction can realize PNA?First, interaction between inflaton N and complex scalar φ. ♪We adopt this interacting part of Lagrangian. ♪Φ1 and Φ2 are defined as real and imaginary part of complex scalar Φ. ♪Inflaton decays through this cubic interaction. A is Interaction vertex. It must be complex number to realize CP violation and so PNA.♪Next, mass term of Lagrangian is this. There must be mass difference(B) between Φ's to realize PNA.Clearly, A and B are seeds of PNA.

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3-3. Lagrangian defining modelIII. ModelIII. Model

: Neutral scalar(Inflaton) : Complex scalar

: Complex interaction vertex ….breaks CP, U(1) particle number: Real matter-curvature coupling ….. breaks particle number: Real mass difference for Φ ….. breaks particle number: Order parameter

: Fluctuation of inflaton around potential minimum

Gravity interacts only through scalar curvature.Metric(scale factor) : dynamical(Classical).

This is Lagrangian that defins our model. It includes inflaton N and one complex scalar Φ. They are treated as quantum fields. Gravity is classical.♪It is devided free part, interacting part and Einstein Lagrangian part.♪Free part includes kinetic term and mass term and term to copule gravity.♪Interaction is, as already explained, cubic type. Interaction with gravity is only through scalar curvature of metric.♪Its coupling is written as α2 and α3. In fact α2 also contribules to PNA

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IV. Current of IV. Current of particle number asymmetryparticle number asymmetry

: Particle number of

Caltzetta and Hu (1988)

suitable for non-equilibrium thermal field theory4-2. Current expectation in Closed Time Path(CTP) formalism

Particle number N in this model is a charge of U(1) of Non-zero U(1) current Jμ leads particle number breaking.

4-1. Particle number asymmetry

Here I show how to realize PNA in QFT. Since complex scalar Φ has U(1) charge and particle and anti-particle has opposite U(1) charge, particle number asynmetry is the same to U(1) charge asymmetry. U(1) charge is given by integration of U(1) current.Our main object to calculate is this U(1) current j for Φ. In QFT we calculate expectation value of the current j. In ordetr to know its time development within non-equiliblium thermal field theory, we use so called closed time path formalism. Furthermore, by using so called 2PI formalism, Green function takes part in dynamical onject as well as scalar field Φ. I do not explain those technical part in this talk.

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Intitial condition for Green function G

Inititial condition for field Φ's

We assume

We assume that after inflation universe is equilibrium (temperature T).Initial condition of density matrix is rewritten as that of Green function.

4-3. Initial condition at the end of inflation

IV. Current of IV. Current of particle number asymmetryparticle number asymmetry

Our model starts after inflation. So initial condition will be at the end of inflation or at the beginning of preheating. We use a kind of thermal filed theory, so we impose initial condition to density matrix. It is rewritten by initial condition of two point function, or Green function. It is given by this expression. ω is parameterized by mass and scale factor at initial time. Initial condition for inflaton and other scalar fields, are simply assumed as constant in space coordinate.

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Weak coupling expansion

・ Hat objects are re-scaled ones by scale factor.

Current in the first order of A.

4-4. Result of analytic expression

IV. Current of IV. Current of particle number asymmetryparticle number asymmetry

, ,

・ are given from free solution and are written by initial condition.

Current can be calculated with arbitrary scale factor. We expanded current by interaction vertex A up to first order.Hat objects are re-scaled ones by scale factor by these relations.G_free, Φ_free, K, E , are written by initial condition.♪Here is the result of analytical expression of current. v dependent part is omitted here.

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V. Habble parameter V. Habble parameter approximationapproximation 5-1. Space-time background

(space curvature k=0)

t0: time at the end of inflation

FRW universe of scale factor with arbitrary time dependence.

5-2. Weak Habble parameter approximation

We futher approximate, by Hubble parameter.Gravity is already chosn as FRW type with arbitraly time dependent scale factor. ♪From now, we expand it in tems of Hubble parameter H by this expression.t0 indicate time at the end of inflation and t is time describing development of PNA.

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V. Habble parameter V. Habble parameter approximationapproximation 5-3. PNA from various contribusions

5-4. Time development of PNAWe found time dependent upper and lower bounds of PNA. Bounds decrease as time.

PNA disappears after infinite time. However, PNA may remain at present time

Check by numerical calculation

proved analitically

by squeeze theorem,

We perform double expansion by interaction vertex A and Hubble parameter H and approximate up to the first order for both.♪There are no free part of current, 0th order of A. 1st order of A of PNA is divided to 4 parts. We find physical meanings for them.♪1st term is constant term means 0th order of Hubble parameter.2nd term is named as “Dilution effect”. It is caused by the increase of the volume of the universe.♪3rd term is named as “freezing interaction effect”. It is controlled by the strength of the interaction.♪4th term is named as “red shift effect”. It is through effective energy or momentum controlled by the scale factor.♪Next, we get simple property by analytical study. We found time dependent upper and lower bounds of PNA. Those bounds decrease as time. By squeeze theorem, we conclude that PNA disappears after infinite time. However, there is a possibility that PNA remains at present finite time. In order to see it we will perform numerical calculation.

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VI.Newmerical Results (v=0)VI.Newmerical Results (v=0)

End of inflationPreheating era Present toy universe

our scenario Curve running away from horizontal axis indicates particle number violating era of universe

6-3. Time development of current - Inflaton mass dependence

This is a numerical result. This is a result for v=0.Vertical axis is PNA and horizontal one is time.There are several curves, They show dependence of various inflaton mass. Curve running away from horizontal axis indicates particle number violating era of universeIn fact this curve is oscillates. Our scenario is following. Time zero is the end of inflation. Just after that PNA become nonzero at preheating era. And these large non zero part is at present time within our toy model.To be realistic, we expect that before PNA become zero, interaction with other various matter take part in the model and finite PNA remain until present time.

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VI.Newmerical Results (v=0)VI.Newmerical Results (v=0)

End of inflationPreheating era Present toy universe

our scenario

6-1. Time development of current - Temperature dependence

Curve running away from horizontal axis indicates particle number violating era of universe

This figure also has the same meaning. But these various curve describe dependence of initial temperature.

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VI.Newmerical Results (v=0)VI.Newmerical Results (v=0)

End of inflationPreheating era Present toy universe

our scenario

Curve running away from horizontal axis indicates particle number violating era of universe

6-2. Time development of current - Mass difference dependence

This figure has the same meaning. But these various curve describe dependence of mass difference B

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Summary●We consider a toy model at preheating era

in FRW background with arbitrary time dependent scale factor. ●The model includes interaction among an inflaton and two scalars. ●Inflaton potential has minimum at the non-zero expectation value.●Mass difference between two scalars and complex interaction coupling

lead to non-zero PNA.●Oscillation of inflaton transmits to oscillation of PNA.●Time development of PNA is explicitly calculated by using CTP in QFT●PNA is calculated in arbitrary scale factor

and also in weak Hubble parameter approximation.●Numerical calculation shows oscillating PNA in figures.

VII.Summary and tasksVII.Summary and tasks

Task●Clarify non-zero PNA mechanism and fit it to scenario of baryogenesis. ●Embed the model in standard inflation cosmology

as well as standard model of particle physics.●Constraints on inflation model from both side of PNA and cosmology.

SummaryWe consider a toy model at preheating era in FRW background with arbitrary time dependent scale factor. The model includes interaction among an inflaton and two scalars. Inflaton potential has minimum at the non-zero expectation value.Mass difference between two scalars and complex interaction coupling lead non-zero PNA.Oscillation of inflaton transmits to oscillation of PNA.Time development of PNA is explicitly calculated by using CTP in QFTPNA is calculated arbitrary scale factor and also in weak Hubble parameter approximation.Numerical calculation shows oscillating PNA in figures.

TaskClarify non-zero PNA mechanism and fit it to scenario of baryogenesis. Embed the model in standard inflation cosmology as well as standard model of particle physics.Constraints on inflation model from both side of PNA and cosmology.This is the end of my talk. Thank you for listening.