Dark Matter Dark Energy Interactions · 2017. 10. 4. · Dark Matter – Dark Energy Interactions Emmanuel N. Saridakis Physics Department, National and Technical University of Athens,

Dark Matter – Dark Energy

Interactions

Emmanuel N. Saridakis

Physics Department, National and Technical University of Athens, Greece

Physics Department, Baylor University, Texas, USA

E.N.Saridakis – 9th Aegean, Sifnos. Sept 2017

2

Goal

We investigate cosmological scenarios in a universe where dark sectors are allowed to mutually interact

Note:

A consistent or interesting cosmology is not a proof for the consistency of the underlying gravitational theory


Why Modification?

Knowledge of Physics: Standard Model


Why Modification?

Knowledge of Physics: Standard Model + General Relativity


Why Modification? Universe History:


6

Modified Gravity

Non-minimal gravity-matter coupling


(Gen. Proca)

7

Scalar-Tensor Theories

Add a scalar field:

Conformal Transf. to Jordan frame:

,)()(2)()(16

1ghLsRfgL m

ggh )(


8

8


Add a scalar field:

Conformal Transf. to Jordan frame:

Redefinition of :

Brans-Dicke for

GR for

,)()(2)()(16

1ghLsRfgL m

ggh )(

,)(2

)(

16

1gLVRgL m

0., Vconst

.,0/', 2 constV

[Brans,Dicke, PR 124] [Santos,Gregory, Annals Phys. 258]


9


Field equations:

□

For Brans-Dicke:

PPN parameters:

Newton’s constant: with

TgVG 8)(

2

2

)32(

400002

1,1

PPNPPN

1

23

24

G

TVV 8'24)(' 2

112107.1 yrG

G

[D.F. Toress, PRD 66]


10

Brans-Dicke Cosmology

Friedmann-Robertson-Walker metric:

Friedmann equations:

Scalar-field equation:

Matter equation:

ji

ij dxdxtadtds )(222

2

22

63

8

HH m

HpHH m 2

28

132

22

0332

83

mm pH

3

V

V

d

dVV2

32

2

03 mmm pH


11

Inflation in Brans-Dicke Cosmology

[La,Steinhardt PRL 62], [Green, Liddle PRD 54]


12

Dark Energy in Brans-Dicke Cosmology

Effective Dark Energy sector:

2

68

3 HDE 8

V

HpDE 2

28

1 2

8

V

DE

DEDE

pw

2

0)(

V

V

[D.F. Toress, PRD 66] E.N.Saridakis – 9th Aegean, Sifnos. Sept 2017

13

13

Horndeski Theories

Most general 4D scalar-tensor theories having second-order field equations:

5

2i

iH LL

),(][2 XKKL

),(][ 333 XGGL

2

,4444 ),(][ XGRXGGL

2)(36

1),(][

3

,5555 XGGXGGL

[G. Horndeski, Int. J. Theor. Phys. 10 ]

2/ X


14

14

Horndeski Theories


[Nicolis, Rattazzi, Trincherini, PRD 79]

5

2i

iH LL

),(][2 XKKL

),(][ 333 XGGL

2

,4444 ),(][ XGRXGGL

2)(36

1),(][

3

,5555 XGGXGGL


Coincides with Generalized Galileon theories

bc ,

2/ X


[Deffayet, Esposito-Farese, Vikman PRD 79]

15

15

Horndeski Cosmology (background)

Field Equations:

In flat FRW:

with

SHRSHL ....

mXXXX

XXXXXX

XGGXHXGGXH

GHGHXXGGXHGHXGHGXKXK

)23(6)25(2

612)(246262

,5,5

2

,5,5

3

,4,4,4,4

2

4

2

,3,3,

mX

XXXX

XXXXXX

pGHXGXHXHXHGHXXHX

GXHGHHHHXGHXXGGH

GXHXXGHGXHXGHGHHGGXK

,5,5

2

,5

,5

22

,5

23

,4,4,4

,4,4,4,4

2

4

2

,3,3

43)(22)(4

4)322(2)2(44)2(2

88412)23(2)(2

PJadt

d

a)(

1 3

3

)(6)23(212)2(626 ,5,5

2

,5,5

3

,4,4,4

2

,3,3, XXXXXXXXXX XGGHXGGXHHXGXGGHGHXGKJ

XXX GXHXGHGHXXHGHHGGXKP ,5

3

,5

2

,4,4

2

,3,3, 26)2(6)2(6)(2

[De Felice,Tsujikawa JCAP 1202]


16

16

Horndeski Cosmology (perturbations)

Scalar perturbations:

No-ghost condition:

No Laplacian instabilities condition:

with


ji

ij dxdxadtds )21()21( 222

0

3

942

2

2

2311

w

wwwwQS

094

6244232

2311

2

12

2

11214

2

22

2

12

wwww

pwwwwwwwwHwwc mm

S

,5,5,441 222w GHGXXGG XX

,5,5,5,5,5

22

,5

2

,4

2

,4

2

,4,4,44,4

3

,3,3,3,3,,3

18152713626

21675418

63623w

GGHXGGXHGXXGHXH

GXGHXXHGGGXHGGHXH

HGGXGHGXXXKKX

XXXXXXXXX

XXXXXXXXX

XXXXXXX

SHRSHL ....

XXGXGG ,5,544 222w

22

,5,5,5,5

2

,4,4,4,4

2

4,32

45628

2441642w

HXGHGGHXHGX

GXHGGHGXHGXG

XXXX

XXXXX


17

17

Inflation in Horndeski Theories

0,),(),(),( 543

33 GGX

M

cXGVXXK

22

2

1)( mV 4

4

1)( V

[Ohashi,Tsujikawa, JCAP 1210]


18

18

Inflation in Horndeski Theories

G-Inflation (Shift-symmetric):

0,),(),(),( 543

33 GGX

M

cXGVXXK

22

2

1)( mV 4

4

1)( V

[Ohashi,Tsujikawa, JCAP 1210]

0,1

),(,2

),( 54333

2

GGXM

XGM

XXXK

17.0r

[Kobayashi,Yamaguchi,Yokoyama PRL 105] [Banerjee, Saridakis PRD 95] E.N.Saridakis – 9th Aegean, Sifnos. Sept 2017

19

19

Dark Energy in Horndeski Theories

Background evolution: Universe thermal history

554332 ,1,),(,),( cGGcXGXcXK

[Ali,Gannouji,Sami PRD 82] [Leon, Saridakis JCAP 1303]


20

20

Dark Energy in Horndeski Theories

Background evolution: Universe thermal history

Perturbations:

with

Clustering growth rate:

γ(z): Growth index.

554332 ,1,),(,),( cGGcXGXcXK

[Ali,Gannouji,Sami PRD 82]

),,,,( 543 GGGKGG effeff

mmeffmm GH 42

)(ln

lna

ad

dm

m

[Leon, Saridakis JCAP 1303]


21

21

Fab Four

ringogeorgepauljohnFF LLLLL

GVL johnjohn )(

PVL paulpaul )(

RVL georgegeorge )(

GVL ringoringoˆ)(

][][][ 22

RRRRP

24ˆ RRRRRG

[Charmousis,Copeland,Padilla,Saffin PRL 108]

[Copeland,Padilla,Saffin JCAP 1212] E.N.Saridakis – 9th Aegean, Sifnos. Sept 2017

22

22

Nonminimal Derivative Coupling

In flat FRW:

rm SSVGgRG

gxdS

)()(

2

1

16

14

rmVH

GH

)(91

23

8 22

2

rm ppV

HHHGHH )(

4321

2832 2

22

[Saridakis,Suskov PRD 81]


23

23

Nonminimal Derivative Coupling – Dark Energy

In flat FRW:

rm SSVGgRG

gxdS

)()(

2

1

16

14

rmVH

GH

)(91

23

8 22

2

rm ppV

HHHGHH )(

4321

2832 2

22

[Saridakis,Suskov PRD 81]

[Dent,Dutta,Saridakis,Xia JCAP 1311]

eVV 0)( nVV 0)(


24

24

Nonminimal Derivative Coupling - Inflation

New Higgs Inflation:

[Skugoreva,Sushkov,Toporensky PRD 88]

[Dalianis,Koutsoumbas,Ntrekis,Papantonopoulos JCAP 1702]

[Germani,Kehagias PRL 105] 05.0r

2

0)( VV


25

25

Beyond Horndeski Theories

Beyond Horndeski, free from Ostrogradski instabilities but still propagating 2+1 dof’s:

with

Primary constraint prevents the propagation of extra degrees of freedom

5

2i

iBH LL

][ 222 ALL H

][]2[ ,32,3333 XCLXCCLL H

X

H 1

2

,444

,42,443444

2][]2[][ galXH

X

HH LX

XBABXCLXCCLBLL

2

21 XLgal

[Gleyzes,Langlois,Piazza,Vernizzi, PRL 114], [Crisostomi,Hull,Koyama,Tasinato, JCAP 1603 ]

2/ X

2

2/5

5,5

,52,553544553

3][]2[][][ galXH

X

HHH LX

AXBXDLXDDLCLGLL

2

23

23

2

32 XLgal

),( XAA ii

),( XBB ii

dXXAC 2/3

33 )(2

1

dXXBC 2/1

,44 )(

dXXBXC 2/3

,55 )(4

1

dXXCD 2/1

,55 )( dXXBG X

2/1

,55 )(


26

26

Bi-scalar Theories

Modified gravity propagating 2+2 dof’s

For

RRRfgxdS ,)(, 24

RRRQRRKRRRf 222 )(,)(,,)(,

geBBGGBKK 32

2,,,,

[Naruko,Yoshida,Mukohyama CQG 33 ]

3

23

23

223

24

4

1ˆ2

1

4

1ˆ

6

1ˆ

2

1ˆ2

1eQeKeQgegRgxdS


27

27

Bi-scalar Theories

Modified gravity propagating 2+2 dof’s

For

eg.:

[Saridakis,Tsoukalas PRD 93 ]

RRRfgxdS ,)(, 24

RRRQRRKRRRf 222 )(,)(,,)(,

3

23

23

223

24

4

1ˆ2

1

4

1ˆ

6

1ˆ

2

1ˆ2

1eQeKeQgegRgxdS

geBBGGBKK 32

2,,,,

BBGBK

,,2

,

HeeDE 66218

1

2

1 33/23/222

663

121

8

1

2

1 23/23/222 eepDE

[Naruko,Yoshida,Mukohyama CQG 33 ]


28

28

Dark Matter – Dark Energy Interaction

Theoretical argument: In principle, since the underlying theory and the microphysics of both dark energy and dark matter is unknown, possible mutual interactions cannot be excluded.


29

29

Dark Matter – Dark Energy Interaction

Theoretical argument: In principle, since the underlying theory and the microphysics of both dark energy and dark matter is unknown, possible mutual interactions cannot be excluded.

Phenomenological argument: Alleviate the coincidence problem: Why are the DE and DM densities nearly equal today, although they scale independently through the expansion history

[Mimoso, Nunes, Pavon, PRD 73] [Billyard, Coley, PRD 61]

[Wang, Gong, Abdalla, PLB 624] [Chen, Gong, Saridakis JCAP 0904]

[Caldera-Cabral, Maartens, Urena-Lopez, PRD 79] [Clifton, Barrow, PRD 73]


30

30

DM – DE Interaction

Assume that DE and DM are effectively described by perfect fluids.

DMSSRG

gxdS

16

14

bS

DMDE

GH

3

82


DMDMDEDE ppGH 4

31

31


Assume that DE and DM are effectively described by perfect fluids.

Equations give only the total conservation, namely

If we assume DM conservation, i.e then DE is also conserved:

DMSSRG

gxdS

16

14

0)()()( DM

ab

DE

ab

btot

ab

b TTT

bS

DMDE

GH

3

82

DMDMDEDE ppGH 4

0)( DM

ab

bT 0)( DE

ab

bT

03 DMDMDM pH

03 DEDEDE pH


32

32


However, it is not forbidden to assume DM – DE interaction by arbitrarily splitting as:

with a phenomenological descriptor of the interaction (positive

corresponds to energy transfer from DE to DM and vice versa).

a

DM

ab

b QT )(

a

DE

ab

b QT )(

aQ aQ


33

33


However, it is not forbidden to assume DM – DE interaction by arbitrarily splitting as:

with a phenomenological descriptor of the interaction (positive

corresponds to energy transfer from DE to DM and vice versa).

Despite possible pathologies (curvature perturbation blowing up in super-Hubble scales [Valiviita,Majerotto,Maartens, JCAP 0807]) it leads to interesting cosmological behavior.

a

DM

ab

b QT )(

a

DE

ab

b QT )(

aQ


34

34

Phenomenological Models

I)

II)

III)

etc…

DMDMDEDEHQQ 30

DMQQ 0

nnn

DMHQQ 232

0


35

35

Phenomenological Models

I)

II)

III)

etc…

Obtain late time attractors with

DMDMDEDEHQQ 30

1~/ DMDER

DMQQ 0

nnn

DMHQQ 232

0

[Valiviita,Majerotto,Maartens, MNRAS 402] [Chen, Gong, Saridakis JCAP 0904]

[Caldera-Cabral, Maartens, Urena-Lopez, PRD 79]


36

36

More general phenomenological models

with . known

Solve coincidence problem, can lead to intermediate acceleration

DEaHQ )(3 aa 0)( )(aDE

[Chen, Gong, Saridakis JCAP 0904]


37

37

Observational constraints

Impose SNIa, BAO and CMB observational constraints

Incorporate relativistic effects in the large-scale power spectrum.

[Clemson, Koyama, Zhao, Maartens, Valiviita PRD 85]


[Duniya, Bertacca, Maartens, PRD 91]

38

38

Another approach to phenomenological models

If Q=0 then . Instead of imposing Q one can parametrize its effect assuming:

(perturbations can also be studied; obtain matter overdensity)

3

0 / aDMDM

3

0 /aDMDM [Wang, Meng CQG 22]


39

39


If Q=0 then . Instead of imposing Q one can parametrize its effect assuming:

(perturbations can also be studied; obtain matter overdensity)

H0+SNIa+BAO+CMB

Slight tendency towards interacting DE

δ<0 implies energy flow DM -> DE

3

0 / aDMDM

3

0 /aDMDM [Wang, Meng CQG 22]

[Nunes, Pan, Saridakis PRD 94]


40

40

Lagrangian? Covariant formulation?

Microscopic Lagrangian of DM-DE interaction is unknown. Effective Lagrangians are also absent.


41

41



Two interacting fluids:

Covariant approach (two “not-tilted” fluids, i.e with common 4-velocity):

is a current energy density that describes the energy transfer between the fluids

(time dependent due to spacial isotropy)

[Faraoni, Dent Saridakis PRD 90]

QpH 111 3

QpH 222 3

abbaabbaab uquqgpuupT 111

)1(


)2(

cc utq )(


42

42



Two interacting fluids:

Covariant approach (two “not-tilted” fluids, i.e with common 4-velocity):

is a current energy density that describes the energy transfer between the fluids

(time dependent due to spacial isotropy)

Imperfect fluids with

Hence, not a robust Lagrangian description for imperfect fluids


QpH 111 3

QpH 222 3


)1(


)2(

cc utq )(

23)( ii

i pT

b

b

aiiab

b

iiiai

b

baib

b

a

i

ab

b uupuuppuupuuT 222)(


43

43


Inspired by the Lagrangian formulation of classical dissipative oscillator we can remove the “imperfectness” by transforming the metric as:

baabab uugg 2


44

44


Inspired by the Lagrangian formulation of classical dissipative oscillator we can remove the “imperfectness” by transforming the metric as:

Hence:

Describes a perfect fluid with and in spacetime metric

: Lagrangian description in a fictitious metric that depends on the fluid

Still not ideal for multiple fluids.

baabab uugg 2

abbaab gpuuppT 22

22 p pp abg

0 ab

bT

pgL



45

45


Matter fluid:

are Lagrange multipliers, and are the Lagrange coordinates of the fluid

vector-density particle-number flux

Dark Energy is described by a scalar field:

A

AM sJsngL ,,,),(

A ,, A

J

)(

2

1

VgL


46

46


Matter fluid:

are Lagrange multipliers, and are the Lagrange coordinates of the fluid

vector-density particle-number flux

Dark Energy is described by a scalar field:

DM-DE interaction:

Algebraic coupling:

Derivative Coupling:

Al. coupl.:

Der. Coupl.:

Perturbations, structure formation, quasi-static limit etc

A

AM sJsngL ,,,),(

[Koivisto, Saridakis, Tamanini JCAP 1509]

A ,, A

J

)(

2

1

VgL

A

AM sJsngLL ,,,int ),,(

A

AM sJJsnfsngLL

,,,int ),,(),(

)()( dmTQT

),(nQ

u

n

nfnQ

),(2


47

47

Dark energy - dark matter interaction/unification from generalized Galileons


[Nicolis,Rattazzi,Trincherini, PRD 79]

5

2i

iH LL

),(][2 XKKL

),(][ 333 XGGL

2

,4444 ),(][ XGRXGGL

2)(36

1),(][

3

,5555 XGGXGGL


Coincides with Generalized Galileon theories

bc ,

2/ X


[Deffayet, Esposito-Farese, Vikman PRD 79]

48

48


Field Equations In flat FRW:

with

mXXXX

XXXXXX

XGGXHXGGXH

GHGHXXGGXHGHXGHGXKXK

)23(6)25(2

612)(246262

,5,5

2

,5,5

3

,4,4,4,4

2

4

2

,3,3,

mX

XXXX

XXXXXX

pGHXGXHXHXHGHXXHX

GXHGHHHHXGHXXGGH

GXHXXGHGXHXGHGHHGGXK

,5,5

2

,5

,5

22

,5

23

,4,4,4

,4,4,4,4

2

4

2

,3,3

43)(22)(4

4)322(2)2(44)2(2

88412)23(2)(2

PJadt

d

a)(

1 3

3

)(6)23(212)2(626 ,5,5

2

,5,5

3

,4,4,4

2

,3,3, XXXXXXXXXX XGGHXGGXHHXGXGGHGHXGKJ

XXX GXHXGHGHXXHGHHGGXKP ,5

3

,5

2

,4,4

2

,3,3, 26)2(6)2(6)(2



49

49


In flat FRW:

GXXRgxdS22

1 53

2

2

4

0933 2

53

2

2

2 XHXXXH

023232 2

53

2

2

2 XHXHHXXXHH

[Koutsoumbas,Ntrekis,Papantonopoulos,Saridakis, 1704.08640]

02

369236

23 23

523

XXHXXHXXHHX

XHX


50

50


We can rewrite as:

with

Klein-Gordon becomes:

Define Equation-of-State parameter:

3

82 GH

pGH 4


03 pH

XHXXX 2

53

2

2 93

XHXHHXXXp 232 2

53

2

2

/pw


51

51


Shift symmetry allows to write:

with and

)(9)()27(6))(2(3672

)()32()(6)27(2)(12)(

32

5

2

525522

2

2

32

2

2

5252

2

2

fff

fffp


2

525 )3(123)( f3 /pw


52

52


Shift symmetry allows to write:

with and

Allows for a unified description of universe evolution. (Generalized) Chaplygin gas:

)(9)()27(6))(2(3672

)()32()(6)27(2)(12)(

32

5

2

525522

2

2

32

2

2

5252

2

2

fff

fffp


2

525 )3(123)( f3

/Ap

03 pH

11

)1(3

000

1

a

AA

1

)1(3

0000

1

a

AAAp

/pw

a

az 01


Simplest case:

Model I :

53 53



)(362

1

)(366

1

)(2

2

2

2

2

2

zH

zHzw

0,0 52 1w

0,0,0 52


54

54


Simplest case:

Model I :

we demand and


)(362

1

)(366

1

)(2

2

2

2

2

2

zH

zHzw

0,0 52 1w

0,0,0 52

-0.70)w(z 0H0)H(z


Model II :

55 55



)(6)(9

)(15)(

2

5

2

5

2

2

5

zHzH

zHzw

0,0,0 52

we demand and -0.70)w(z 0H0)H(z


Model II :

56 56



)(6)(9

)(15)(

2

5

2

5

2

2

5

zHzH

zHzw

0,0,0 52



Model II :

57 57



)(6)(9

)(15)(

2

5

2

5

2

2

5

zHzH

zHzw

0,0,0 52

580 SN Ia data points


Model II :

58 58



)(6)(9

)(15)(

2

5

2

5

2

2

5

zHzH

zHzw

0,0,0 52



59

59


Scalar perturbations:

No-ghost condition:

No Laplacian instabilities condition:

with


ji

ij dxdxadtds )21()21( 222

0

3

942

2

2

2311

w

wwwwQS

094

6244232

2311

2

12

2

11214

2

22

2

12

wwww

pwwwwwwwwHwwc mm

S

,5,5,441 222w GHGXXGG XX

,5,5,5,5,5

22

,5

2

,4

2

,4

2

,4,4,44,4

3

,3,3,3,3,,3

18152713626

21675418

63623w

GGHXGGXHGXXGHXH

GXGHXXHGGGXHGGHXH

HGGXGHGXXXKKX

XXXXXXXXX

XXXXXXXXX

XXXXXXX

SHRSHL ....

XXGXGG ,5,544 222w

22

,5,5,5,5

2

,4,4,4,4

2

4,32

45628

2441642w

HXGHGGHXHGX

GXHGGHGXHGXG

XXXX

XXXXX


Model II :

60 60



0,0,0 52


Healthy scalar perturbations. Necessary to see tensor perturbations, and the speed of gravitational waves.

61

Conclusions

i) Modification of our knowledge is probably required for the explanation of cosmological evolution.

ii) There is a huge variety of modifications.

iii) Dark Energy (or Modified Gravity) - Dark Matter interaction cannot be excluded, and it can alleviate the coincidence problem.

iv) Many phenomenological approaches. Can become Covariant. A full Lagrangian description is still missing.

v) DE - DM interaction/unification from generalized Galileons with shift-symmetry. Unified universe evolution.

vi) SN Ia data OK. Necessary: Confront with CMB, BAO, and LSS data. Need to add baryonic matter separately. Perform full perturbation analysis, confront with data.


62

THANK YOU!


Documents

Dark Matter Dark Energy Interactions · 2017. 10. 4. · Dark Matter – Dark Energy Interactions Emmanuel N. Saridakis Physics Department, National and Technical University of Athens,