Quintom Bounce with a Quintom Bounce with a Galileon ModelGalileon Model

Chung-Yuan Christian University, Taiwan

&

Institute of High Energy Physics, Beijing

Based on 1108.0593

Collaborated with J. Evslin, Y. F. Cai, M. Z. Li, X. M. Zhang

Speaker: Taotao Qiu

OutlineOutlineWhy Quintom bounce?Quintom bounce of galileon model◦Background◦Perturbation

Conclusionoutlook

WHY QUINTOM WHY QUINTOM BOUNCE?BOUNCE?

Problems/constraints from theoretical/observational aspects: (such as BBN,CMB(COBE), etc)

Big Bang

Standard Models of the Early UniverseBig Bang Cosmology vs. Inflation Cosmology

The age of galaxies The redshift of the galactic spectrum

The He abundance The prediction of CMB temperature

Flatness problem Horizon problem

Singularity problem

Monopole problem

Structure formation problem

Inflation

Problems/constraints from theoretical/observational aspects: (such as BBN,CMB(COBE), etc)

Big Bang

Standard Models of the Early UniverseBig Bang Cosmology vs. Inflation Cosmology

The age of galaxies The redshift of the galactic spectrum

The He abundance The prediction of CMB temperature

Flatness problem Horizon problem

Singularity problem

Monopole problem

Structure formation problem

Inflation

The Alternatives of The Alternatives of InflationInflation

Pre-big bang ScenarioEkpyrotic ScenarioString gas/Hagedorn Scenario

Non-local SFT ScenarioBouncing Scenario

Ekpyrotic ModelThe collision of two M branes in 5D gives rise to a nonsingular cyclic universe, and the description of effective field theory in 4D is

1 DE domination2 decelerated expansion3 turnaround4 ekpyrotic contracting phase5 before big crunch6 a singular bounce in 4D7 after big bang8 radiation domination9 matter domination

J. Khoury, B. Ovrut, P. Steinhardt and N. Turok,Phys. Rev. D 64, 123522 (2001)

Ekpyrotic ModelThe collision of two M branes in 5D gives rise to a nonsingular cyclic universe, and the description of effective field theory in 4D is

1 DE domination2 decelerated expansion3 turnaround4 ekpyrotic contracting phase5 before big crunch6 a singular bounce in 4D7 after big bang8 radiation domination9 matter domination

Failure of effective field theory description, uncertainty involved in perturbations.

Contraction: Expansion:

BouncingPoint: Nearby:

In order to connect this process to the observable universe (radiation dominant, matter dominant, etc), w goes to above -1

Y. Cai, T. Qiu, Y. Piao, M. Li and X. Zhang, JHEP 0710:071, 2007

(Non-singular) Bounce Cosmology expansioncontraction IR size with Low energy scale

Singularity problem avoided!

( )

Formalism:

So w crosses -1, namely Quintom bounce!

If w>-1 at the beginning, w will cross twice.

Realization of a Quintom Bounce

As for any kind of matter, which is (1) in 4D classical Einstein Gravity, (2)described by

single simple component (either perfect fluid or single scalar field with lagrangian as

), and (3) coupled minimally to Gravity or other matter, its

Equation of State can never cross the cosmological constant boundary (w=-1).

Quintom realization: No-Go theorem

To realize Quintom, one of the conditions should be violatedi) Double field Quintom bounce:

ii) Single field Quintom bounce with higher derivative term:

Y. Cai, T. Qiu, R. Brandenberger, Y. Piao, X. Zhang, JCAP 0803:013,2008; Y. Cai, T. Qiu, J. Xia, X. Zhang, Phys.Rev.D79:021303,2009.

(also known as Lee-Wick Bounce)

Y. Cai, T. Qiu, R. Brandenberger, X. Zhang, Phys.Rev.D80:023511,2009; J. Karouby, T. Qiu, R. Brandenberger, Phys.Rev.D84:043505,2011.

Bo Feng et al., Phys. Lett. B 607, 35 (2005);A. Vikman, Phys. Rev. D 71, 023515 (2005);Gong-Bo Zhao et al., Phys. Rev. D 72, 123515 (2005);J. Xia, Y. Cai, T. Qiu, G. Zhao and X. Zhang, Int.J.Mod.Phys.D17:1229-1243,2008.

Galileon Galileon TheoriesTheories

Galileon Models: Lagrangian with higher derivative operator, but the equation of motion remains second order, so the model can have w cross -1 without ghost mode.

Basically 5 kinds of Galileon model:

But can be generalized…

Usually, both of the two cases have more than two DYNAMICAL degrees of freedom, which will contain ghost modes.

Problem with Quintom bounce:

Recently: a kind of Galileon theory has been proposed!

A. Nicolis et al., Phys.Rev.D79:064036,2009;C. Deffayet et al., Phys.Rev.D79:084003,2009.

C. Deffayet et al., arXiv:1103.3260 [hep-th]

Cosmological Applications of Cosmological Applications of Galileon TheoriesGalileon Theories

Galileon as dark energy models:R. Gannouji,M. Sami, Phys.Rev.D82:024011,2010. A. De Felice, S. Tsujikawa, Phys.Rev.Lett.105:111301,2010. C. Deffayet,O. Pujolas,I. Sawicki, A. Vikman, JCAP 1010:026,2010.

Galileon as inflation and slow expanstion models:P. Creminelli, A. Nicolis, E. Trincherini, JCAP 1011:021,2010.T. Kobayashi,M. Yamaguchi,J. Yokoyama, Phys.Rev.Lett.105:231302,2010. C. Burrage,C. de Rham,D. Seery,A. Tolley, JCAP 1101:014,2011. K. Kamada, T. Kobayashi, M. Yamaguchi, J. Yokoyama, Phys.Rev.D83:083515,2011.Z. Liu, J. Zhang, Y. Piao, arXiv:1105.5713 [astro-ph.CO]

Observational constraints on Galileon models:S. Nesseris,A. De Felice, S. Tsujikawa, Phys.Rev.D82:124054,2010A. Ali,R. Gannouji, M. Sami, Phys.Rev.D82:103015,2010.

Galileon as spherically symmetric models:D. Mota, M. Sandstad,T. Zlosnik, JHEP 1012:051,2010.… … … … … …

Can Galileon be used as bounce models???

QUINTOM BOUNCE WITH A QUINTOM BOUNCE WITH A GALILEON MODELGALILEON MODEL

BACKGROUNDBACKGROUNDPERTURBATIONPERTURBATION

Our New Bounce Model with Galileon

The action:

Stress energy tensor:

From which we get energy density and pressure:

which was also used in arXiv: 1007.0027 for “Galileon Genesis”.

where

Considering , and thus is monotonic increasing, so the first term in H, is always larger than 0.

Solution for Bounce to Happen

So we get one property of the field: evolve as a monotonic function!

In order to have bounce, H must reach 0, so negative branch is chosen.

Reality of square root:

From the Friedmann Equation we get the Hubble parameter:

where

Asymptotic solution of Our Model

with

Equation of motion:

Hubble parameter:

In contracting phase:

Analysis of the asymptotic behavior when

Terms in EoM has different orders of t inconsistent !

EoM becomes:

inconsistent !inconsistent !

consistent !

The only consistent solution has a radiation dominant behavior!

iii)

I.II. i)

ii)

Numerical Plots of Our Model (1)Plots of Hubble parameter and scale factor in our model:

Parameter choice:

Bounce can happen naturally in our model around t=30.

Reheating?

Plots of field and EoS w in our model:

behaves as a monotonic function, and the equation of state is approximately 1/3 (radiation-dominant like) in contracting phase, and cross -1 before bounce in our model.

Numerical Plots of Our Model (2)

QUINTOM BOUNCE WITH A QUINTOM BOUNCE WITH A GALILEON MODELGALILEON MODEL

BACKGROUNDBACKGROUNDPERTURBATIONPERTURBATION

Perturbation Theoryy

Theoretical aspects: stability must be guaranteed!

Observational aspects: should obtain a (nearly) scale-invariant power spectrum and small tensor-to-scalar ratio

Why perturbations?

Primordial perturbations provide seeds for structure formation and explains why our current universe is not complete isotropic.

Two constraints for linear perturbations:

Perturbations of Our Bounce Model

Perturbed metric in ADM form:

Perturbed action:

Gauge: uniform

lapse function

inverse shift vector

Constraint equations:

Solution:

is positive definite: no ghost instability!is model dependent: have to be checked numerically.

Up to second order

Ghost instability: Gradient instability:

In our model,

There are two kinds of instabilities at linear level:

Stability of Perturbation of Our Model

Stability of Perturbation of Our Model

Numeric plots for and

Both and are positive all over the bouncing process, and we have which also behaves like radiation!

Spectrum of Perturbation of Our Model

Blue spectrum inconsistent with observational data!

Equation of motion: set

In radiation dominant phase:

Effective mass

Solution:

like a massless scalar field!

Mechanism of Getting Scale Invariant Power Spectrum

D. Lyth and D. Wands, Phys.Lett.B524:5-14,2002.

An alternative: Curvaton Mechanism

Curvaton: a light scalar field other than inflaton to produce curvature perturbation.The simplest curvaton model:

Curvature perturbation:

For Gaussian part:

where

The equation of motion:

with

Solution: Power spectrum:

Curvaton Mechanism in Our Model

Our curvaton action:

The general solution:

Our model: due to the background, in order to have scale invariant power spectrum, curvaton have to couple kinetically to the Galileon field

The equation of motion:

In radiation dominant phase:

Scale Invariant Power Spectrum from Curvaton

growing constant

constantdecaying

Superhorizon solution:

There are two cases of getting scale invariant power spectrum:

Subhorizon solution:

From matching condition: and are independent of k!

q=2:

q=-4:

Back reaction of the Curvaton Field

The energy density of :

In contracting phase where the universe is radiation-like:

From the equation of motion:

In order for not to destroy the background evolution: one needs

Question: will the growth of energy density of destroy the process of bounce?

In our case which can produce scale invariant power spectrum:

q=2: Safe from back reaction of q=-4: Needs severe fine-tuning.

Tensor Perturbation of Our ModelPerturbed metric:

Perturbed action (up to second order):

Expand the tensor perturbation:

Equation of motion:

Tensor Perturbation of Our Model

In radiation dominant phase:

Solution:

Tensor spectrum:

Spectrun index:

In observable region we have , namely the spectrum is severely suppressed, so the tensor-to-scalar ratio

like a massless scalar field!

Blue spectrum!

WMAP data predicts quite small r, so is consistent with our model!D. Larson et al. [WMAP collaboration], arXiv:1001.4635 [astro-ph.CO].

Conclusion Bounce needs equation of state w cross -1, namely Quintom Bounce can be in form of galileon, where there are only two

dynamical degrees of freedom and ghost can be eliminated. Quintom bounce in Galileon form:

◦ Background behavior: Radiation-dominant like.◦ Perturbation1: free from instability but cannot provide scale

invariant power spectrum◦ Perturbation2: The way of providing scale-invariant power

spectrum is curvaton. In our model there are two cases. ◦ Perturbation3: The back reaction is small in one case, but the other

case needs fine tuning.◦ Perturbation4: The tensor spectrum is blue and the tensor-to-

scalar ratio is small.

Outlook

Final state of reheating

Nongaussianities

Reheating

Through reheating, inflaton decay to matter and radiation after inflation

In galileon cosmology, reheating can help avoid divergenceL. Levasseur, R. Brandenberger, A. Davis, arXiv:1105.5649

In bouncing cosmology, reheating is also important and maybe different from normal inflation

T. Qiu, K. Yang, JCAP 1011:012,2010. Y. Cai, R. Brandenberger, X. Zhang, arXiv:1105.4286 

What is the reheating process like of our Galileon bounce model???

Reheating is important in inflationary scenario!

Reheating mechanisms:

The motivation of reheating in our bounce model:

Normal: L. Kofman, A. Linde, A. Starobinsky, Phys.Rev.Lett.73:3195-3198,1994. Geometric: B. Bassett, S. Liberati, Phys.Rev.D58:021302,1998.Curvaton: Bo Feng, Ming-zhe Li, Phys.Lett.B564:169-174,2003. ……

Non-gaussianities

WMAP-7 data:

Planck data:

E. Komatsu et al., arXiv:1001.4538.

Planck collaboration, astro-ph/0604069.

For canonical single scalar field inflation:X. Chen, M. Huang, S. Kachru, G. Shiu, JCAP 0701:002,2007.

For bounce cosmology: new shape with sizable amplitude and

Y. Cai, W. Xue, R. Brandenberger, X. Zhang, JCAP 0905:011,2009.

Non-Gaussianities is important for 1) meeting the more and more accurate observational data and 2) distinguishing the over models for early universe

Definition:

Observational constraints:

Theoretical results:

What will the non-Gaussianities behave like for our Galileon bounce model???

THANKS FOR THANKS FOR ATTENTION!ATTENTION!

谢谢！谢谢！

Galileon genesisGalileon genesis

Our bounce model “Galileon genesis”, 1007.0027