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Cosmological Scenarios in Horndeski Theory
Sergey Sushkov
Spontaneous Workshop on Cosmology XI\Hot Topics in Modern Cosmology"
IESC, Carg�ese, France, May 4, 2017
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Based on
S. Sushkov, Phys.Rev. D80 (2009) 103505
E. Saridakis, SVS, Phys.Rev. D81 (2010) 083510
S. Sushkov, Phys.Rev. D85 (2012) 123520
M. Skugoreva, SVS, A. Toporensky, Phys.Rev. D88 (2013) 083539
J. Matsumoto, SVS, JCAP 1511, 047 (2015)
A. Starobinsky, SVS, M. Volkov, JCAP 1606, 007 (2016)
J. Matsumoto, SVS, arXiv:1703.04966
Collaborators:E. Saridakis, J. Matsumoto, M. Skugoreva,A. Toporensky, M. Volkov, A. Starobinsky
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Plan
Plan of the talk
Motivation
Scalar �elds in gravitational physics
Scalar �elds minimally coupled to gravity
Scalar �elds nonminimally coupled to gravity
Scalar �elds with nonminimal derivative coupling
Horndeski model
Cosmological models with nonminimal derivative coupling
No potential
Cosmological constant
Power-law potential
Higgs-like potential
Role of matter
The screening Horndeski cosmologies
Summary
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Scalar �elds in gravitational physics
Scalar �elds in gravitational physics:
gravitational potential in Newtonian gravity
variation of \fundamental" constants
Brans-Dicke theory initially elaborated to solve the Mach problem
various compacti�cation schemes
the low-energy limit of the superstring theory
scalar �eld as in aton
scalar �eld as dark energy and/or dark matter
fundamental Higgs bosons, neutrinos, axions, . . .
etc. . .
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Scalar �elds minimally coupled to gravity
S =
Zd4xp�g�LGR + LS
�
LGR { gravitational Lagrangiangeneral relativity; LGR = Rsquare gravity; LGR = R+ cR2
f(R)-theories; LGR = f(R)etc...
LS { scalar �eld Lagrangian;ordinary STT; LS = ��(r�)2 � 2V (�)� = +1 { canonical scalar �eld� = �1 { phantom or ghost scalar �eld
with negative kinetic energyV (�) { potential of self-action
K-essence; Ls = K(X) [X = (r�)2]etc...
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Scalar �elds nonminimally coupled to gravity
Bergmann-Wagoner-Nordtvedt scalar-tensor theories
S =
Zd4xp�g�f(�)R� h(�)(r�)2 � 2V (�)
�f(�)R =) nonminimal coupling between � and R
Conformal transformation to the Einstein frame (Wagoner, 1970):
~g�� = f(�)g�� ;d�
d = f
�����fh+ 3
2
�df
d�
�2������1=2
S =
Zd4x
p�~g
h~R� �( ~r )2 � 2U( )
i
=) new scalar �eld � = sign
�fh+ 3
2
�dfd�
�2�U( ) =) new e�ective potential
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Scalar �elds nonminimally coupled to gravity
Bergmann-Wagoner-Nordtvedt scalar-tensor theories
S =
Zd4xp�g�f(�)R� h(�)(r�)2 � 2V (�)
�f(�)R =) nonminimal coupling between � and R
Conformal transformation to the Einstein frame (Wagoner, 1970):
~g�� = f(�)g�� ;d�
d = f
�����fh+ 3
2
�df
d�
�2������1=2
S =
Zd4x
p�~g
h~R� �( ~r )2 � 2U( )
i
=) new scalar �eld � = sign
�fh+ 3
2
�dfd�
�2�U( ) =) new e�ective potential
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Scalar �elds nonminimally coupled to gravity
General scalar-tensor theories
S =
Zd4xp�g�F (�;R)� (r�)2 � 2V (�)
�F (�;R) =) generalized nonminimal coupling between � and R
Conformal transformation to the Einstein frame (Maeda, 1989):
~g�� = 2g�� ;2
16������@F (�;R)@R
����
S =
Zd4x
p�~g
"~R
16�� h(�) �1( ~r�)2 � 3
32� �2( ~r )2 + U(�; )
#
� 2 =) new (second!) scalar �eldU(�; ) =) new e�ective potential
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Scalar �elds nonminimally coupled to gravity
General scalar-tensor theories
S =
Zd4xp�g�F (�;R)� (r�)2 � 2V (�)
�F (�;R) =) generalized nonminimal coupling between � and R
Conformal transformation to the Einstein frame (Maeda, 1989):
~g�� = 2g�� ;2
16������@F (�;R)@R
����
S =
Zd4x
p�~g
"~R
16�� h(�) �1( ~r�)2 � 3
32� �2( ~r )2 + U(�; )
#
� 2 =) new (second!) scalar �eldU(�; ) =) new e�ective potential
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Scalar �elds nonminimally coupled to gravity
Some remarks:
A nonminimal scalar �eld is conformally equivalent to the minimalone possessing some e�ective potential V (�)
A behavior of the scalar �eld is \encoded" in the potential V (�)
The potential V (�) is a very important ingredient of variouscosmological models: a slowly varying potential behaves like ane�ective cosmological constat providing one or more than onein ationary phases.An appropriate choice of V (�) is known as a problem of �ne tuningof the cosmological constant.
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Nonminimal derivative coupling generalization
S =
Zd4xp�g�R� g���;��;� � 2V (�)
�
. &
F (�;R;R�� ; : : : )
nonminimal couplinggeneralization!
K(�;�; �;�� ; : : : ; R;R�� ; : : : )
nonminimal derivative couplinggeneralization!
& .
Theories with higher derivatives!
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Nonminimal derivative coupling generalization
S =
Zd4xp�g�R� g���;��;� � 2V (�)
�.
&
F (�;R;R�� ; : : : )
nonminimal couplinggeneralization!
K(�;�; �;�� ; : : : ; R;R�� ; : : : )
nonminimal derivative couplinggeneralization!
& .
Theories with higher derivatives!
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Nonminimal derivative coupling generalization
S =
Zd4xp�g�R� g���;��;� � 2V (�)
�. &
F (�;R;R�� ; : : : )
nonminimal couplinggeneralization!
K(�;�; �;�� ; : : : ; R;R�� ; : : : )
nonminimal derivative couplinggeneralization!
& .
Theories with higher derivatives!
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Nonminimal derivative coupling generalization
S =
Zd4xp�g�R� g���;��;� � 2V (�)
�. &
F (�;R;R�� ; : : : )
nonminimal couplinggeneralization!
K(�;�; �;�� ; : : : ; R;R�� ; : : : )
nonminimal derivative couplinggeneralization!
& .
Theories with higher derivatives!
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Horndeski theory
In 1974, Horndeski derived the action of the most general scalar-tensortheories with second-order equations of motion[G.Horndeski, Second-Order Scalar-Tensor Field Equations in aFour-Dimensional Space, IJTP 10, 363 (1974)]
Horndeski Lagrangian:
LH =p�g (L2 + L3 + L4 + L5)
L2=G2(X;�) ;
L3=G3(X;�)�� ;
L4=G4(X;�)R+ @XG4(X;�) ����� r�
��r��� ;
L5=G5(X;�)G��r���� 16 @XG5(X;�) �
����� r�
��r���r
�� ;
where X = � 12 (r�)2, and Gk(X;�) are arbitrary functions,
and ����� = 2! ��[����], ������� = 3! ��[��
���
��]
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Fab Four subclass of the Horndeski theory
There is a special subclass of the theory, sometimes called Fab Four (F4),for which the coe�cients are chosen such that the Lagrangian becomes
LF4 =p�g (LJ + LP + LG + LR � 2�)
with
LJ=VJ(�)G��r��r�� ;
LP=VP (�)P����r��r��r��� ;
LG=VG(�)R ;LR=VR(�) (R����R
���� � 4R��R�� +R2):
Here the double dual of the Riemann tensor is
P���� = �1
4��� ����� R
�� � = �R��
�� + 2R�[��
��] � 2R�
[����] �R��[����] ;
whose contraction is the Einstein tensor, P���� = G�
� .
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Fab Four subclass of the Horndeski theory
Fab Four Lagrangian:
LF4 =p�g (LJ + LP + LG + LR � 2�)
The Fab Four model is distinguished by the screening property { it isthe most general subclass of the Horndeski theory in which atspace is a solution, despite the presence of the cosmological term �.
This property suggests that � is actually irrelevant and hence thereis no need to explain its value.
Indeed, however large � is, Minkowski space is always a solution andso one may hope that a slowly accelerating universe will be asolution as well.
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Theory with nonminimal kinetic coupling
Action:
S =1
2
Z �M2
PlR� (� G�� + " g��)r��r��� 2V (�)�p�g d4x+ Sm
Field equations:
M2PlG�� = � T�� + � T (�)
�� + T (m)��
[�g�� + �G�� ]r�r�� = V 0�
T (�)�� =�r��r��� 1
2�g��(r�)2 � g��V (�);
T��=� 12r��r��R+ 2r��r(��R
��) � 1
2(r�)2G�� +r��r��R����
+r�r��r�r���r�r����+ g���� 1
2r�r��r�r��+
12(��)2
�r��r��R���
T (m)�� =(�+ p)U�U� + pg�� ;
Notice: The �eld equations are of second order!
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Cosmological models: General formulas
Ansatz:
ds2 = �dt2 + a2(t)dx2;
� = �(t)
a(t) cosmological factor, H = _a=a Hubble parameter
Field equations:
3M2PlH
2 =1
2_�2��� 9�H2
�+ V (�);
M2Pl(2
_H + 3H2) = �1
2_�2h�+ �
�2 _H + 3H2 + 4H �� _��1
�i+ V (�);
d
dt
�(�� 3�H2)a3 _�
�= �a3 dV (�)
d�
V (�) � const =) _� =Q
a3(�� 3�H2)Q is a scalar charge
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Cosmological models: General formulas
Ansatz:
ds2 = �dt2 + a2(t)dx2;
� = �(t)
a(t) cosmological factor, H = _a=a Hubble parameter
Field equations:
3M2PlH
2 =1
2_�2��� 9�H2
�+ V (�);
M2Pl(2
_H + 3H2) = �1
2_�2h�+ �
�2 _H + 3H2 + 4H �� _��1
�i+ V (�);
d
dt
�(�� 3�H2)a3 _�
�= �a3 dV (�)
d�
V (�) � const =) _� =Q
a3(�� 3�H2)Q is a scalar charge
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Cosmological models: I. No potential V (�) � 0
Trivial model without kinetic coupling, i.e. � = 0
S =
Zd4xp�g �M2
PlR� (r�)2�
Solution:
a0(t) = t1=3; �0(t) =1
2p3�
ln t
ds20 = �dt2 + t2=3dx2
t = 0 is an initial singularity
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Cosmological models: I. No potential V (�) � 0
Trivial model without kinetic coupling, i.e. � = 0
S =
Zd4xp�g �M2
PlR� (r�)2�
Solution:
a0(t) = t1=3; �0(t) =1
2p3�
ln t
ds20 = �dt2 + t2=3dx2
t = 0 is an initial singularity
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Cosmological models: II. No potential V (�) � 0
Model without free kinetic term, i.e. � = 0
S =
Zd4xp�g �M2
PlR� �G���;��;��
Solution:
a(t) = t2=3; �(t) =t
2p3�j�j ; � < 0
ds20 = �dt2 + t4=3dx2
t = 0 is an initial singularity
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Cosmological models: II. No potential V (�) � 0
Model without free kinetic term, i.e. � = 0
S =
Zd4xp�g �M2
PlR� �G���;��;��
Solution:
a(t) = t2=3; �(t) =t
2p3�j�j ; � < 0
ds20 = �dt2 + t4=3dx2
t = 0 is an initial singularity
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Cosmological models: III. No potential V (�) � 0
Model for an ordinary scalar �eld (� = 1) withnonminimal kinetic coupling � 6= 0
S =
Zd4xp�g �M2
PlR� (g�� + �G��)�;��;�
Asymptotic for t!1:
a(t) � a0(t) = t1=3; �(t) � �0(t) =1
2p3�
ln t
Notice: At large times the model with � 6= 0 has the same behavior likethat with � = 0
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Cosmological models: III. No potential V (�) � 0
Model for an ordinary scalar �eld (� = 1) withnonminimal kinetic coupling � 6= 0
S =
Zd4xp�g �M2
PlR� (g�� + �G��)�;��;�
Asymptotic for t!1:
a(t) � a0(t) = t1=3; �(t) � �0(t) =1
2p3�
ln t
Notice: At large times the model with � 6= 0 has the same behavior likethat with � = 0
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Cosmological models: III. No potential V (�) � 0
Asymptotics for early times
The case � < 0:
at!0 � t2=3; �t!0 � t
2p3�j�j
ds2t!0 = �dt2 + t4=3dx2
t = 0 is an initial singularity
The case � > 0:
at!�1 � eH�t; �t!�1 � Ce�t=p�
ds2t!�1 = �dt2 + e2H�tdx2
de Sitter asymptotic with H� = 1=p9�
Sergey Sushkov Horndeski Cosmologis 18 / 47
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���
Cosmological models: III. No potential V (�) � 0
Plots of � = ln a in case � 6= 0, � = 1, V = 0.
(a) � < 0;� = 0;�1;�10;�100
(b) � > 0;� = 0; 1; 10; 100
De Sitter asymptotics: �(t) =tp9�
) H =1p9�
Notice: In the model with nonmnimal kinetic coupling oneget de Sitter phase without any potential!
Sergey Sushkov Horndeski Cosmologis 19 / 47
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���
Cosmological models: IV. Cosmological constant
Models with the constant potential V (�) = M2
Pl� = const
S =
Zd4xp�g �M2
Pl(R� 2�)� [�g�� + �G�� ]�;��;��
There are two exact de Sitter solutions:
I. �(t) = H�t; �(t) = �0 = const;
II. �(t) =tp3j�j ; �(t) =MPl
����3�H2� � 1
�
����1=2
t;
H� =p�=3
Sergey Sushkov Horndeski Cosmologis 20 / 47
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���
Cosmological models: IV. Cosmological constant
Models with the constant potential V (�) = M2
Pl� = const
S =
Zd4xp�g �M2
Pl(R� 2�)� [�g�� + �G�� ]�;��;��
There are two exact de Sitter solutions:
I. �(t) = H�t; �(t) = �0 = const;
II. �(t) =tp3j�j ; �(t) =MPl
����3�H2� � 1
�
����1=2
t;
H� =p�=3
Sergey Sushkov Horndeski Cosmologis 20 / 47
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���
Cosmological models: IV. Cosmological constant
Plots of �(t) in case � > 0, � = 1, V =M2Pl�
(a) H2� < _�2 < 1=9� (b) 1=9� < _�2 < 1=3� < H2
�
De Sitter asymptotics:�1(t) = H�t (dashed),
�2(t) = t=p9� (dash-dotted),
�3(t) = t=p3� (dotted).
Sergey Sushkov Horndeski Cosmologis 21 / 47
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���
Cosmological models: IV. Cosmological constant
Plots of �(t) in cases � > 0, � = �1 and � < 0, � = 1
(a) � < 0, � = 1 (b) � > 0, � = �1
De Sitter asymptotic:
�1(t) = H�t (dashed).
Sergey Sushkov Horndeski Cosmologis 22 / 47
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���
Role of potential
S =
Zd4xp�g �M2
PlR� [g�� + �G�� ]�;��;� � 2V (�)
%
What a role does a potential play in cosmologicalmodels with the nonminimal kinetic coupling?
Sergey Sushkov Horndeski Cosmologis 23 / 47
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���
Power-law potential V (�) = V0�N
Skugoreva, Sushkov, Toporensky, PRD 88, 083539 (2013)
Models with the quadratic potential V (�) = 1
2m2�2
Primary (early-time) \kinetic" in ation:
Ht!�1 � 1p9�
(1 + 12�m
2)
Late-time cosmological scenarios:Oscillatory asymptotic or \graceful" exit from in ation
Ht!1 � 2
3t
�1� sin 2mt
2mt
�
quasi-de Sitter asymptotic or secondary in ation
Ht!1 � 1p3�
�1�
q16�m
2
�
Sergey Sushkov Horndeski Cosmologis 24 / 47
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���
Cosmological models: Power-law potential
Initial conditions�0 = _�0
Initial conditions�0 = � _�0
De Sitter asymptotics: Ht!�1 � 1=p9�(1 + 1
2�m2),
Ht!1 � 1=p3��1�
q16�m2
�.
Sergey Sushkov Horndeski Cosmologis 25 / 47
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���
Higgs potential V (�) = �4(�
2� �20)
2
Matsumoto, Sushkov, JCAP2015
Higgs �eld: � ' 0:14, �0 ' 246 GeV
Field equations
H2 =12_�2 + V (�)
3(M2Pl +
32�
_�2)
�� =[1 + 12�� _�2 + 96�2�2 _�4 + 8��V (�)(12�� _�2 � 1)]�1
��� 2p3� _�[1 + 8�� _�2 � 8��V (�)]
q[ _�2 + 2V (�)](12�� _�2 + 1)� (12�� _�2 + 1)(4�� _�2 + 1)V�
�
Sergey Sushkov Horndeski Cosmologis 26 / 47
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���
Higgs potential: Dynamical system
Dimensionless variables and parameters:
x =�
�0; y =
p8�G� _�; � = �0t; V0 = 2�G���40; � G�20
Autonomous dynamical system:
dx
d�=
r�
4V0y;
dy
d�=
1
�
��q3� �V �10 y
�1 + y2 � V0(x2 � 1)2
��q[y2 + 2V0(x2 � 1)2]
�32y
2 + 1�
� 2p�V0
�32y
2 + 1� �
12y
2 + 1�x(x2 � 1)
�;
where � = 1 + 32y
2 + 32y
4 + V0(x2 � 1)2
�32y
2 � 1�:
Sergey Sushkov Horndeski Cosmologis 27 / 47
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���
Higgs potential: Stationary points and phase portrait
Stationary points:(�1; 0) global minima of V (�); � = ��0, V (��0) = 0(0; 0) local maximum of V (�); � = 0, V (0) = V0 =
�4�
40
(�1; 0) \wings" of V (�); �! �1Phase diagrams:
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
x
y
1
4���40 �M2
Pl
(0; 0) { saddle point
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
x
y
1
4���40 > M2
Pl
(0; 0) { stable node!
Sergey Sushkov Horndeski Cosmologis 28 / 47
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���
Accelerated cosmological scenarios: Quasi-de Sitter
Quasi-de Sitter scenario
t!1 distant future asymptotic
Τ = 0
Τ = 5.5
Τ = 15
Τ = 0
Τ = 6.5
Τ = 17
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
x
y
�! 0; _�! 0
V (�)! V0 = ��40=4 (maximum)
0 5 10 15 20 25 30
0.85
0.90
0.95
1.00
ΤΚH[Τ]
H(t)! H1 =q
23�G��
40=const
Sergey Sushkov Horndeski Cosmologis 29 / 47
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���
Accelerated cosmological scenarios: Big Rip
Big Rip scenario
t! t� �nite time asymptotic
Τ = 0
Τ = 1
Τ = 2.5
Τ = 0
Τ = 1.4
2 3 4 5 6 7 8
0
1
2
3
4
5
6
x
y
�(t) 'r
392�
�
1
(t� � t)2�!1; _�!1V (�) ' ��4=4!1
0 1 2 3 4 50
2
4
6
8
10
12
ΤΚH[Τ]
H2(t) ' 49
9
1
(t� � t)2 !1
Sergey Sushkov Horndeski Cosmologis 30 / 47
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���
Accelerated cosmological scenarios: Little Rip
Little Rip scenario
t!1 distant future asymptotic
Τ = 0
Τ = 1.3
Τ = 0
Τ = 1.2
Τ = 8Τ = 8
3.5 4.0 4.5 5.0
-0.5
0.0
0.5
1.0
x
y
�(t) ' (2=3�3G3��2)1=8 t1=4
�!1; _�! 0
V (�) ' ��4=4!1
0 2 4 6 8 101
2
3
4
5
6
Τ
ΚH[Τ]
H(t) ' (8�=27�G�2)1=4 t1=2 !1
Sergey Sushkov Horndeski Cosmologis 31 / 47
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���
Cosmological scenarios with nonminimal kinetic couplingand matter Sushkov, PRD 85 (2012) 123520
Role of matter?
S =
Zd4xp�g �M2
Pl(R� 2�)� [g�� + �G�� ]�;��;�+ Smatter
Stress-energy tensor: T(m)�� = diag(�; p; p; p)
Field equations:
3M2PlH
2 =1
2_�2�1� 9�H2
�+M2
Pl� + �;
M2Pl(2
_H + 3H2) = �1
2_�2h1 + �
�2 _H + 3H2 + 4H �� _��1
�i+M2
Pl�� p
d
dt
�(1� 3�H2)a3 _�
�= 0
=) _� =Q
a3(1� 3�H2)
Sergey Sushkov Horndeski Cosmologis 32 / 47
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���
Cosmological scenarios with nonminimal kinetic couplingand matter Sushkov, PRD 85 (2012) 123520
Role of matter?
S =
Zd4xp�g �M2
Pl(R� 2�)� [g�� + �G�� ]�;��;�+ Smatter
Stress-energy tensor: T(m)�� = diag(�; p; p; p)
Field equations:
3M2PlH
2 =1
2_�2�1� 9�H2
�+M2
Pl� + �;
M2Pl(2
_H + 3H2) = �1
2_�2h1 + �
�2 _H + 3H2 + 4H �� _��1
�i+M2
Pl�� p
d
dt
�(1� 3�H2)a3 _�
�= 0
=) _� =Q
a3(1� 3�H2)
Sergey Sushkov Horndeski Cosmologis 32 / 47
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���
Cosmological scenarios with nonminimal kinetic couplingand matter Sushkov, PRD 85 (2012) 123520
Role of matter?
S =
Zd4xp�g �M2
Pl(R� 2�)� [g�� + �G�� ]�;��;�+ Smatter
Stress-energy tensor: T(m)�� = diag(�; p; p; p)
Field equations:
3M2PlH
2 =1
2_�2�1� 9�H2
�+M2
Pl� + �;
M2Pl(2
_H + 3H2) = �1
2_�2h1 + �
�2 _H + 3H2 + 4H �� _��1
�i+M2
Pl�� p
d
dt
�(1� 3�H2)a3 _�
�= 0 =) _� =
Q
a3(1� 3�H2)
Sergey Sushkov Horndeski Cosmologis 32 / 47
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���
Cosmological scenarios with nonminimal kinetic couplingand matter
Modi�ed Friedmann equation:
H2 = H20
��0 +
m0
a3+
�0(1� 9�H2)
a6(1� 3�H2)2
�
Constraint for parameters:
�0 +m0 +�0(1� 9�H2
0 )
(1� 3�H20 )
2= 1
Universal asymptotic:
H ! H� = 1=p9� at a! 0
Notice: The asymptotic H � H� at early cosmological timesis only determined by the coupling parameter � and does not
depend on other parameters!
Sergey Sushkov Horndeski Cosmologis 33 / 47
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���
Cosmological scenarios: Numerical solutions
Scale factor a(t)
Hubble parameter H(a) Acceleration parameter q
Sergey Sushkov Horndeski Cosmologis 34 / 47
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���
Cosmological scenarios: Estimations
H�tf � 60 e-foldstf ' 10�35 sec the end of initial in ationary stage
) H� = 1=p9� ' 6� 1036 sec�1
� ' 10�74 sec2 or l� = �1=2 ' 10�27 cm
H0 � 70 (km=sec)=Mpc � 10�18 sec�1 Present Hubble parameter = 3�H2
0 ' 10�109 Extremely small!
H2 = H20
��0 +
m0
a3+
�0(1� 9�H2)
a6(1� 3�H2)2
�) �0 +m0 +�0 � 1
�0 = 0:73;�0 = 0:23;m0 = 0:04 ) q0 = 0:25
Sergey Sushkov Horndeski Cosmologis 35 / 47
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���
Cosmological scenarios: Estimations
H�tf � 60 e-foldstf ' 10�35 sec the end of initial in ationary stage
) H� = 1=p9� ' 6� 1036 sec�1
� ' 10�74 sec2 or l� = �1=2 ' 10�27 cm
H0 � 70 (km=sec)=Mpc � 10�18 sec�1 Present Hubble parameter = 3�H2
0 ' 10�109 Extremely small!
H2 = H20
��0 +
m0
a3+
�0(1� 9�H2)
a6(1� 3�H2)2
�) �0 +m0 +�0 � 1
�0 = 0:73;�0 = 0:23;m0 = 0:04 ) q0 = 0:25
Sergey Sushkov Horndeski Cosmologis 35 / 47
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���
Screening properties of Horndeski model
The FLRW ansatz for the metric:
ds2 = �dt2 + a2(t)
�dr2
1�Kr2 + r2(d#2 + sin2 #d'2)
�;
a(t) cosmological factor, H = _a=a Hubble parameter
Gravitational equations:
�3M2Pl
�H2 +
K
a2
�+
1
2" 2 � 3
2� 2
�3H2 +
K
a2
�+ �+ � = 0;
�M2Pl
�2 _H + 3H2 +
K
a2
�� 1
2" 2 � � 2
�_H +
3
2H2 � K
a2+ 2H
_
�+ �� p = 0:
The scalar �eld equation:
1
a3d
dt
�a3�3�
�H2 +
K
a2
�� "
�
�= 0;
where = _�, and � = �(t) is a homogeneous scalar �eld
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���
Screening properties of Horndeski model
The �rst integral of the scalar �eld equation:
a3�3�
�H2 +
K
a2
�� "
� = Q;
where Q is the Noether charge associated with the shift symmetry�! �+ �0.
Let Q = 0. One �nds in this case two di�erent solutions:
GR branch: = 0 =) H2 +K
a2=
�+ �
3M2Pl
Screening branch: H2 +K
a2=
"
3�=) 2 =
� (� + �)� "M2Pl
� ("� 3� K=a2)
NOTICE: The role of the cosmological constant in the screening solutionis played by "=3� while the �-term is screened and makes no contributionto the universe acceleration.
Note also that the matter density � is screened in the same sense.
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���
Screening properties of Horndeski model
Let Q 6= 0, then
=Q
a3�3� (H2 + K
a2 )� "� ;
and the modi�ed Friedmann equation reads
3M2Pl
�H2 +
K
a2
�=Q2
�"� 3�
�3H2 + K
a2
��2a6
�"� 3� (H2 + K
a2 )�2 + �+ �:
Introducing dimensionless values and density parameters
H2 = H20 y; a = a0 a ; �cr = 3M2
PlH20 ; � =
"
3� H20
;
0 =�
�cr; 2 = � K
H20a
20
; 6 =Q2
6� a60H20 �cr
; � = �cr
�4
a4+
3
a3
�
gives
the master equation:
y = 0 +2
a2+
3
a3+
4
a4+
6
�� � 3y + 2
a2
�a6�� � y + 2
a2
�2Sergey Sushkov Horndeski Cosmologis 38 / 47
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���
Asymptotical behavior: Late time limit a!1
GR branch:
y = 0 +2
a2+
3
a3+
4
a4+
(� � 30) 6
( 0 � �)2 a6 +O�
1
a7
�=) H2 ! �=3
Notice: The GR solution is stable (no ghost) if and only if � > 0.
Screening branches:
y� = � +2
a2� �
( 0 � �) a3 �26
�a5� 6(� � 30)� 3�
2(0 � �)2 a6 +O�
1
a7
�
=) H2 ! "=3�
Notice: The screening solutions are stable (no ghost) if and only if0 < � < 0.
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Asymptotical behavior: The limit a! 0
GR branch:
y =4
a4+
3
a3+
24 � 36
4a2+
336
4a+O(1)
Notice: The GR solution is unstable
Screening branch:
y+=36
4 a2� 336
24 a
+5
3� +
3623 + 92
6
34
+O(a);
y�=�
3+
4 �2
27 6
�4 a
2 +3 a3�+O(a4)
Notice: Both screening solutions are stable
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���
Global behavior
y = 0 +2
a2+
3
a3+
4
a4+
6
�� � 3y + 2
a2
�a6�� � y + 2
a2
�2
�
0.0 0.5 1.0 1.5 2.0 2.5 3.00.8
1.0
1.2
1.4
1.6
1.8
2.0
Solutions y(a) for 0 = 6 = 1, 2 = 0, 3 = 4 = 0 and for � = 6
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���
Global behavior
y = 0 +2
a2+
3
a3+
4
a4+
6
�� � 3y + 2
a2
�a6�� � y + 2
a2
�2
�
�
��
0 1 2 3 4 50.0
0.2
0.4
0.6
0.8
1.0
Solutions y(a) for 0 = 6 = 1, 2 = 0, 3 = 4 = 0, � = 0:2
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���
Global behavior
y = 0 +2
a2+
3
a3+
4
a4+
6
�� � 3y + 2
a2
�a6�� � y + 2
a2
�2
�
�
�
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
Solutions y(a) for 0 = 6 = 1, 3 = 5, 4 = 0, � = 0:2. One has 2 = 0.
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Summary. I.
The nonminimal kinetic coupling provides an essentially newin ationary mechanism which does not need any �ne-tunedpotential.
At early cosmological times the coupling �-terms in the �eldequations are dominating and provide the quasi-De Sitter behaviorof the scale factor: a(t) / eH�t with H� = 1=
p9� and � ' 10�74
sec2 (or l� � �1=2 ' 10�27 cm)
The model provides a natural mechanism of epoch change withoutany �ne-tuned potential.
The nonminimal kinetic coupling crucially changes a role of thescalar potential. Power-law and Higgs-like potentials with kineticcoupling provide accelerated regimes of the Universe evolution.
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Summary. II.
The theory with nonminimal kinetic coupling admits variouscosmological solutions.
Ghost-free solutions exist if � � 0 and " � 0.
The no-ghost conditions eliminate many solutions, as for examplethe bounces or the \emerging time" solutions.
For � > 0 there exists a ghost-free solution. It describes a universewith the standard late time dynamic dominated by the �-term,radiation and dust. At early times the matter e�ects are totallyscreened and the universe expands with a constant Hubble ratedetermined by "=�. Since it contains two independent parameters �and 0 � � in the asymptotics, this solution can have an hierarchybetween the Hubble scales at the early and late times. However, atlate times it is not screening and dominated by �, thus invokingagain the cosmological constant problem.
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Conclusions
For 0 < � < 0 there exist two ghost-free solutions, A and B. Thesolution A is sourced by the scalar �eld, with or without the matter,while the solution B exists only when the matter is present. Theyboth show the screening because their late time behaviour iscontrolled by � � "=� and not by �. Therefore, they could inprinciple describe the late time acceleration while circumventing thecosmological constant problem, and one might probably �ndarguments justifying that "=� should be small. At the same time,these solutions cannot describe the early in ationary phase. Indeed,the near singularity behaviour of the solution B does not correspondto in ation, while the solution A does show an in ationary phase,but with essentially the same Hubble rate as at late times, hencethere is no hierarchy between the two Hubble scales.
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THANKS FOR YOUR ATTENTION!
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