Cosmology with Axionic-­‐Quintessence

Coupled with Dark Ma<er

Anjan Ananda Sen Centre For TheoreAcal Physics Jamia Millia Central University

New Delhi

FTAG, IIT Gandhinagar, 5th September, 2013

Plan •  Brief introduction on accelerating Universe

and dark energy •  Scalar Field Models •  Thawing Vs. Tracking •  Thawing Models •  Axion in string theory as a quintessence

model. •  Model with Axion-Dark-Matter coupling. •  Observational constraint for both

uncoupled and coupled scenario. •  Conclusion.

Based On •  G. Gupta, S. Panda and AAS, Phys. Rev.D 85, 023501 (2012)

•  S. Kumar, S. Panda, AAS, Class. Quant. Grav. 30, 155011 (2013)

Cosmic Archeology

Cosmic ma)er structures: less direct probes of expansion

Supernovae: direct probe of cosmic expansion

CMB: direct probe of quantum fluctua<ons

Explaining The Late Time Accelera<on

•  There are essenAally two ways to explain this late Ame acceleraAon assuming FRW metric:

Modified Gravity Inclusion of New Field Topic of this talk

Different Models………..

1)  Cosmological Constant 2)  Dark energy w = pressure/energy density = constant 3)  Dark energy w = w(z) 4)  Freezing or tracker quintessence 5)  Thawing quintessence 6)  Phantom model w < -­‐1 7)  Scalar-­‐Tensor models 8)  Coupled Quintessence 9)  K-­‐Essence 10) Chaplygin and Generalized Chaplygin Gas 11)  F ( R ) and f(G) models 12) DGP model 13) Cardassian model………and MANY MORE

Current Constraint on Dark Energy Equation of State

-1.6 -1.4 -1.2 -1 -0.8 -0.6w0

-3

-2

-1

0

1

wa

Assuming w = w0 + wa (1-­‐a), CPL ParametrizaAon, CMB(Planck)+SN+Bao+HST

What’s the Problem with Cosmological Constant?

Fine Tuning Problem !!

•  Why now?

ρ ~ R-3

Vacuum Energy: ρ ~ constant

Two problems:

Cosmic Coincidence Problem !!

Quintessence

•  Quintessence is model of Dark Energy involving standard scalar fields.

•  The basic idea is same as Infla<on, only the energy scale involved is much lower than that of Infla<on, and also there is a large ma)er component present.

Energy density : Pressure :

Accelera<on occurs, if equa<on of state ra<o

Poten<al energy dominates (slow roll): V >> K .E ⇒

Kine<c energy dominates (fast roll): K.E >> V ⇒

The Energy Density :

Important QuanAAes:

Quintessence The Lagrangian

Dynamics of Quintessence

Equa<on of mo<on of scalar field

•  driven by steepness of poten<al •  slowed by Hubble fric<on

Broad categoriza<on -­‐-­‐ which term dominates:

•  field rolls but decelerates as dominates energy

•  field starts frozen by Hubble drag and then rolls

Trackers vs. Thawers

Caldwell & Linder 2005

Thawing Vs Tracking

Thawing Models: Same as inflaton. IniAally frozen (w=-­‐1)at the flat part of the potenAal due to large Hubble Damping. Later on as the Hubble damping decreases, the field slowly thaws away from w= -­‐1.

Tracking Models: Field fast rolls iniAally mimicking the background (w= 0 or 1/3). Later on as the slope of the potenAal changes, it exits the tracking regime and start behaving like a dark energy (w<-­‐2/3).

Quintessence Model

3H2 = 8πG(ρm + ρφ)

φ + 3Hφ + dVdφ = 0

aa = − 4πG

3 (ρm + ρφ + 3pφ)

ρφ = 12 φ2 + V (φ) pφ = 1

2 φ2 − V (φ)

System of equaAons (Assuming FRW background):

Quintessence Model

We can define:

à d/dlog(a)

Autonomous System of Equa<on

One can solve with ini<al condi<ons

Scherrer and Sen PRD, 2008

Slow Roll Parameters

Thawing Models •  The model is very close to the cosmological constant

•  Assume Power PotenAals:

•  Can be related to (at present).

•  The crucial quanAty is . If this quanAty is very small, all the potenAals behaves similarly and the evoluAon is nearly same as LCDM.

•  For ~ 1, different potenAals behave differently and also deviate from the LCDM behavior.

γi = 1 + wi ≈ 0

V (φ) ∼ φn → Γ = − 1n + 1

Ωφi Ωφ

λi

λi

Thawing Models

V (φ) = φ−2, eφ,φ2

λi = 1, 23 , 1

3

•  We want to study models close to Λ behaviour. •  Scalar field with iniAal value φi in a nearly flat potenAal

V(φ) saAsfying the Slow-­‐Roll CondiAons: One can find the universal behaviour (irrespecAve of the form for

potenAal):

Models Close to w = -­‐1 (assuming Thawing Model)

γ = 1 + w = λ2i3

1√Ωφ− 1

2

1

Ωφ− 1

Log

1+√

Ωφ

1−√

Ωφ

2

Ωφ =1 + (Ω−1

φ0 − 1)(a/a0)−3−1

Scherrer and Sen, PRD 2008

Quintessence from String Theory

•  We should keep in mind, all these potenAals are arbitrary, unAl and unless fundamental physics selects a potenAal.

•  Good News: the recent paper by Panda, Sumitomo and Trivedi (PST hereaker): Construc<on of a quintessence model in a string theory set up (arXiv:1011.5877, PRD, 2011)

PotenAal obtained : Linear: V (φ) ∼ φ

Important Features of PST Model for Quintessence

•  The model is based on the idea of McAllister-­‐Silverstein-­‐Westphal which was earlier used for large field inflaAon.

•  The model provides a quintessence candidate, the axion field that appears in the R-­‐R sector in the Type-­‐IIB string theory.

•  Axion field field is symmetric under a constant shik of the field (shik symmetry).

•  To generate the poten<al for the axion field, one has to break this shik symmetry. And in the PST model, it is broken in presence of branes in highly warped regions or throats of the compact space.

•  The resulAng form of the potenAal is a simple Linear func<on of the field. •  Choosing the field from the R-­‐R sector of theory, ensures that the field

does not couple to any other field in the perturbaAve sector of the string theory (which includes all fields present in the standard model). This ensures that the potenAal does not get corrected due to the super-­‐symmetry breaking and preserves the flatness of the poten<al.

•  But the field may couple to the dark ma)er sector which may arise in the non-­‐perturbaAve sector of the string theory.

Cosmology with PST model •  Coupled Quintessence Scenario: Axion is coupled with Dark Ma<er sector

but with the Baryonic Ma<er Sector. •  Due to our lack of understanding about dark ma<er sector in String

theory, and its possible interacAon with Axion field, we take this coupling in a phenomenological way. We follow the standard pracAce in Coupled-­‐Quintessence scenario that is available in the literature.

•  The relevant equaAons are:

•  C = 0, we recover the standard uncoupled case. For simplicity, we assume C = constant in our invesAgaAon. This is similar to what one gets in the Einstein’s frame in Scalar-­‐Tensor Theory.

φ + dVdφ + 3Hφ = C(φ)ρd

ρd + 3Hρd = −C(φ)ρdφ

ρb + 3Hρb = 0

3H2 = κ

2 (ρb + ρd + ρφ)

Cosmology with PST model •  Form of the potenAal:

•  Where ,

•  Here

•  is the string coupling constant, is the volume of the internal space in the ten dimensional space <me. is related to the warp factor at the loca<on of the brane and is the radius of the ADS-­‐like throat. It can be show that term dominates over the term. Hence we shall consider only term.

•  is a posiAve numerical constant of O(1).

V (φ) = µ4

faφ

f2a = g2

sM2pl

6L4 µ4 = µ1 + µ2

µ1 = e4A0

(2π)5gsα2 µ2 = cM4SBe2A0

R2

α2L4

gs V1 = L6α2

A0

Rµ2 µ1

µ2

c

Dynamical System

For Linear PotenAal: IniAal CondiAons: , . For , it is chosen such that,

Ωφ = W

3γΩφ(1− Ωφ − z2) + 3Ωφ(1− Ωφ)(1− γ)

γ = W

3γΩφ

(1− Ωφ − z2)(2− γ) + λ

3γΩφ(2− γ)

z = − 32zΩφ(1− γ)

λ =

3γΩφλ2(1− Γ)

γ = 1 + wφz = κ√

ρb√3H

λ = −1κV

dVdφ , Γ =

V d2Vdφ2

( dVdφ )2 W = C

κ2

Γ = 0

Ωφi λi zi Ωb0 = 0.05

Cosmology with PST model •  First of all the cosmological evoluAon does not depend whether the scalar

field is frozen iniAally (e.o.s = -­‐1) or is fast rolling to mimic a dust (e.o.s=0).

0.0 0.2 0.4 0.6 0.8 1.0!1.0

!0.5

0.0

0.5

1.0

a

"#

•  Behavior of the equaAon of state as funcAon of redshik for different couplings, W (Bo<om is for uncoupled, and W increases from bo<om to top):

Cosmology with PST model

!7 !6 !5 !4 !3 !2 !1 0!1.0

!0.5

0.0

0.5

1.0

log!a "

"#

λi = 0.7

•  Behavior of the equaAon of state as funcAon of redshik for different iniAal slopes of the potenAal (Increasing from bo<om to top):

Cosmology with PST model λi

!7 !6 !5 !4 !3 !2 !1 0!1.0

!0.5

0.0

0.5

1.0

log!a "

"#

W = 0.03

Cosmology with PST model •  Behaviour of the density parameters:

!"

!d

!b

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

a

!λi = 0.7, W = 0.03

Linear Perturbation •  Interested in Sub-­‐Horizon PerturbaAon in the Newtonian Limit. •  Can safely assume the axion field does not cluster •  The relevant equaAons are:

•  Growth FuncAon:

δ

d +1 + H

H− 2βdx

δ

d − 32 ((1 + 2W 2)δdΩd + δbΩb) = 0

δ

b +1 + H

H− 2βbx

δ

b − 32 (δdΩd + δbΩb) = 0

βb = 0, βd = W =c

κ

Total PerturbaAon: δt = δdΩd+δbΩbΩd+Ωb

f =dlogδt

dloga

Behavior of the Growth Function

Top to Bo<om: LCDM, W=0,0.03,0.06

!2.5 !2.0 !1.5 !1.0 !0.5 0.00.5

0.6

0.7

0.8

0.9

1.0

ln !a "

f

Observational Constraints

Data Used from: •  Union 2.1 compila<on for SnIa: 580 data points

•  Measurement of Hubble parameter at different redshio: 18 data points

•  Measurements of the angular scale of BAO peak in LSS: from SDSS, 6df and Wiggle-­‐z surveys.

•  Measurements of the growth func<on at different redshios.

Results

0.20 0.22 0.24 0.26 0.28 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

!d0

" i

0.20 0.22 0.24 0.26 0.28 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

!d0

" i

0.20 0.22 0.24 0.26 0.28 0.30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

!d0

" i

W=0

W=0.03

W=0.06

Results

0.2 0.4 0.6 0.8 1.0

!1.02

!1.01

!1.00

!0.99

!0.98

a

!"

0.2 0.4 0.6 0.8 1.0

!1.00

!0.95

!0.90

!0.85

a

!"

W=0

W=0.01

W=0.06

0.2 0.4 0.6 0.8 1.0

!1.015

!1.010

!1.005

!1.000

!0.995

!0.990

!0.985

a

!"

Constraints on String Parameters Top to Bo<om: W = 0, 0.01, 0.03, 0.06

2 4 6 8 10

136

138

140

142

144

!

log ee"2A 0 #

β = R2

gsαL2

MSB = α TeV

Conclusions •  PST model is probably the first thoroughly constructed and

controlled example of cosmological quintessence scalar field model in string theory.

•  In this set up, the quintessence field (the axion) does not couple to the standard model field by construc<on. But it may couple to the dark ma)er sector.

•  The cosmological evoluAon of the model where the axion is coupled to the dark ma<er but not with the baryon has been studied.

•  Behaviour of the growth funcAon for the ma<er overdensity has also been studied.

•  ObservaAonal constraints shows that, for the uncoupled case, the model is restricted to evolve very close to the cosmological constant. But for the coupled, one can get substan<al devia<on fro C.C depending on the strength of the coupling.

Thank You

Cosmic Coincidence Problem

•  More about Coincidence Problem: The fracAon of Energy density stored in Cosmological Constant compared to the Total Energy density: Let us assume today and call the transiAon the period from to . This corresponds to change of e-­‐folding of the order of 2 or the expansion of the Universe by the factor of 7. Compare this with the expansion of the Universe since decoupling which is by the factor of 1100.

ΩΛ(a) = ρΛρt

= ΩΛ0ΩΛ0+(1−ΩΛ0)e−3N N = Log(a/a0)

ΩΛ0 = 0.73 ΩΛ = 0.05ΩΛ = 0.95