Cosmologyffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/VIKMAN_Alexander.pdf · 2014. 7. 22. ·...
Transcript of Cosmologyffp14.cpt.univ-mrs.fr/DOCUMENTS/SLIDES/VIKMAN_Alexander.pdf · 2014. 7. 22. ·...
Cosmology
with Mimetic Matter
Alexander Vikman
18.07.14
Monday, July 21, 14
Mimetic MatterChamseddine, Mukhanov (2013)
Monday, July 21, 14
Mimetic MatterChamseddine, Mukhanov (2013)
One can encode the conformal / scalar part of the physical metric in a scalar field : gµ⌫ �
gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
Monday, July 21, 14
Mimetic MatterChamseddine, Mukhanov (2013)
auxiliary metric
One can encode the conformal / scalar part of the physical metric in a scalar field : gµ⌫ �
gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
Monday, July 21, 14
Mimetic MatterChamseddine, Mukhanov (2013)
auxiliary metric
One can encode the conformal / scalar part of the physical metric in a scalar field : gµ⌫ �
gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
S [gµ⌫ ,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m)
◆�
gµ⌫=gµ⌫(g,�)
Monday, July 21, 14
Mimetic MatterChamseddine, Mukhanov (2013)
auxiliary metric
The theory becomes invariant with respect to Weyl transformations:
gµ⌫ ! ⌦2 (x) gµ⌫
One can encode the conformal / scalar part of the physical metric in a scalar field : gµ⌫ �
gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
S [gµ⌫ ,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m)
◆�
gµ⌫=gµ⌫(g,�)
Monday, July 21, 14
Mimetic MatterChamseddine, Mukhanov (2013)
auxiliary metric
The theory becomes invariant with respect to Weyl transformations:
gµ⌫ ! ⌦2 (x) gµ⌫
The scalar field obeys a constraint (Hamilton-Jacobi equation):
gµ⌫ @µ�@⌫� = 1
One can encode the conformal / scalar part of the physical metric in a scalar field : gµ⌫ �
gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
S [gµ⌫ ,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m)
◆�
gµ⌫=gµ⌫(g,�)
Monday, July 21, 14
S [gµ⌫ ,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m)
◆�
gµ⌫=gµ⌫(g,�)
with gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
Monday, July 21, 14
S [gµ⌫ ,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m)
◆�
gµ⌫=gµ⌫(g,�)
with gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
is not in the Horndeski construction,see talks of Deffayet, Sivanesan, Vernizzi, Piazza
Monday, July 21, 14
S [gµ⌫ ,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m)
◆�
gµ⌫=gµ⌫(g,�)
with gµ⌫ (g,�) = gµ⌫ g↵� @↵�@��
is not in the Horndeski construction,see talks of Deffayet, Sivanesan, Vernizzi, Piazza
But it is still a system with one degree of freedom!
Monday, July 21, 14
Mimetic Dark MatterChamseddine, Mukhanov; Golovnev; Barvinsky (2013)
Lim, Sawicki, Vikman; (2010)
Monday, July 21, 14
Mimetic Dark MatterChamseddine, Mukhanov; Golovnev; Barvinsky (2013)
Lim, Sawicki, Vikman; (2010)
Weyl-invariance allows one to fix gµ⌫ = gµ⌫
Monday, July 21, 14
Mimetic Dark MatterChamseddine, Mukhanov; Golovnev; Barvinsky (2013)
Lim, Sawicki, Vikman; (2010)
Weyl-invariance allows one to fix gµ⌫ = gµ⌫
one implements constraint through � (gµ⌫@µ�@⌫�� 1)
S [gµ⌫ ,�,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m) + � (gµ⌫@µ�@⌫�� 1)
◆
Monday, July 21, 14
Mimetic Dark MatterChamseddine, Mukhanov; Golovnev; Barvinsky (2013)
Lim, Sawicki, Vikman; (2010)
Weyl-invariance allows one to fix gµ⌫ = gµ⌫
one implements constraint through � (gµ⌫@µ�@⌫�� 1)
S [gµ⌫ ,�,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m) + � (gµ⌫@µ�@⌫�� 1)
◆
The system is equivalent to standard GR + irrotational dust, moving along timelike geodesics with the velocity and energy density uµ = @µ� ⇢ = �
Monday, July 21, 14
Mimetic Dark MatterChamseddine, Mukhanov; Golovnev; Barvinsky (2013)
Lim, Sawicki, Vikman; (2010)
“Cold Dark Matter”?
Weyl-invariance allows one to fix gµ⌫ = gµ⌫
one implements constraint through � (gµ⌫@µ�@⌫�� 1)
S [gµ⌫ ,�,�,�m] =
Zd4x
p�g
✓�1
2R (g) + L (g,�m) + � (gµ⌫@µ�@⌫�� 1)
◆
The system is equivalent to standard GR + irrotational dust, moving along timelike geodesics with the velocity and energy density uµ = @µ� ⇢ = �
Monday, July 21, 14
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Monday, July 21, 14
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !V (�)
Monday, July 21, 14
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !V (�)
gµ⌫ @µ�@⌫� = 1 Convenient to take as time �
Monday, July 21, 14
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !V (�)
gµ⌫ @µ�@⌫� = 1 Convenient to take as time �
Adding a potential = adding a function of time in the equation 2H + 3H2 = V (t)
Monday, July 21, 14
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !V (�)
gµ⌫ @µ�@⌫� = 1 Convenient to take as time �
Enough freedom to obtain any cosmological evolution!
Adding a potential = adding a function of time in the equation 2H + 3H2 = V (t)
Monday, July 21, 14
Mimicking any cosmological evolution, But always with zero sound speed
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Just add a potential !V (�)
gµ⌫ @µ�@⌫� = 1 Convenient to take as time �
V (�) =1
3
m4�2
e� + 1In particular gives the same cosmological
inflation as potential in the standard case 1
2m2�2
Enough freedom to obtain any cosmological evolution!
Adding a potential = adding a function of time in the equation 2H + 3H2 = V (t)
Monday, July 21, 14
Perturbations IEven with potential, the field
still moves along the timelike geodesics
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Monday, July 21, 14
Perturbations IEven with potential, the field
still moves along the timelike geodesics
cS = 0
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
Monday, July 21, 14
Perturbations IEven with potential, the field
still moves along the timelike geodesics
cS = 0
Lim, Sawicki, Vikman; (2010)Chamseddine, Mukhanov, Vikman (2014)
� = C1 (x)
✓1� H
a
Zadt
◆+
H
aC2 (x)
Here on all scales but in the usual cosmology it is an approximation for superhorizon scales
Monday, July 21, 14
How to give a sound speed to Mimetic Matter?
Chamseddine, Mukhanov, Vikman (2014)
Monday, July 21, 14
How to give a sound speed to Mimetic Matter?
Chamseddine, Mukhanov, Vikman (2014)
Just add higher derivatives ! 1
2� (⇤�)2
Monday, July 21, 14
How to give a sound speed to Mimetic Matter?
Chamseddine, Mukhanov, Vikman (2014)
The sound speed is c2s =�
2� 3�
Just add higher derivatives ! 1
2� (⇤�)2
Monday, July 21, 14
How to give a sound speed to Mimetic Matter?
Chamseddine, Mukhanov, Vikman (2014)
The sound speed is c2s =�
2� 3�
Back to waves, oscillators and normal quantum fluctuations!
Just add higher derivatives ! 1
2� (⇤�)2
Monday, July 21, 14
The scalar field still obeys a constraint (Hamilton-Jacobi equation) gµ⌫ @µ�@⌫� = 1
Monday, July 21, 14
The scalar field still obeys a constraint (Hamilton-Jacobi equation) gµ⌫ @µ�@⌫� = 1
Higher time derivatives can be eliminated just by the differentiation of this equation
Monday, July 21, 14
The scalar field still obeys a constraint (Hamilton-Jacobi equation) gµ⌫ @µ�@⌫� = 1
Higher time derivatives can be eliminated just by the differentiation of this equation
There are only minor changes (rescaling) in the background evolution equations e.g.
2H + 3H2 =2
2� 3�V (t)
Monday, July 21, 14
Perturbations HDChamseddine, Mukhanov, Vikman (2014)
Monday, July 21, 14
Perturbations HDChamseddine, Mukhanov, Vikman (2014)
��+H��� c2sa2
���+ H �� = 0
c2s =�
2� 3�with
Monday, July 21, 14
Perturbations HDChamseddine, Mukhanov, Vikman (2014)
��+H��� c2sa2
���+ H �� = 0
c2s =�
2� 3�with
� = ��
Monday, July 21, 14
Quantization
Monday, July 21, 14
Quantization
S = �1
2
Zd⌘d
3x
✓�
c
2s
��
0���
0 + ...
◆the action
Monday, July 21, 14
Quantization
S = �1
2
Zd⌘d
3x
✓�
c
2s
��
0���
0 + ...
◆the action
��k ⇠r
cs�
k�3/2
short wavelength quantum fluctuations
match with the long-wave-length limit
Monday, July 21, 14
Perturbations in Mimetic InflationChamseddine, Mukhanov, Vikman (2014)
� ' k�1on scale
Monday, July 21, 14
Perturbations in Mimetic InflationChamseddine, Mukhanov, Vikman (2014)
� ' k�1on scale
Newtonian potential �� 'p
cs/� ·Hcsk'H
Monday, July 21, 14
Perturbations in Mimetic InflationChamseddine, Mukhanov, Vikman (2014)
� ' k�1on scale
Newtonian potential �� 'p
cs/� ·Hcsk'H
for cS ⌧ 1 �` ⇠ c�1/2s Hcsk⇠Ha
Monday, July 21, 14
Perturbations in Mimetic InflationChamseddine, Mukhanov, Vikman (2014)
� ' k�1on scale
Newtonian potential �� 'p
cs/� ·Hcsk'H
for cS ⌧ 1 �` ⇠ c�1/2s Hcsk⇠Ha
�` ⇠ c1/2s Hcsk⇠HacS � 1for
Monday, July 21, 14
Perturbations in Mimetic InflationChamseddine, Mukhanov, Vikman (2014)
� ' k�1on scale
Gravitational Waves are unchanged h� ' Hk'H
Newtonian potential �� 'p
cs/� ·Hcsk'H
for cS ⌧ 1 �` ⇠ c�1/2s Hcsk⇠Ha
�` ⇠ c1/2s Hcsk⇠HacS � 1for
Monday, July 21, 14
Perturbations in Mimetic InflationChamseddine, Mukhanov, Vikman (2014)
� ' k�1on scale
Gravitational Waves are unchanged h� ' Hk'H
nS � 1 = nT spectral indices
Newtonian potential �� 'p
cs/� ·Hcsk'H
for cS ⌧ 1 �` ⇠ c�1/2s Hcsk⇠Ha
�` ⇠ c1/2s Hcsk⇠HacS � 1for
Monday, July 21, 14
Perturbations in Mimetic InflationChamseddine, Mukhanov, Vikman (2014)
� ' k�1on scale
Gravitational Waves are unchanged h� ' Hk'H
nS � 1 = nT spectral indices
Newtonian potential �� 'p
cs/� ·Hcsk'H
�� � h�
But it seems that there is no usual Non-Gaussianity!
small sound speed
for cS ⌧ 1 �` ⇠ c�1/2s Hcsk⇠Ha
�` ⇠ c1/2s Hcsk⇠HacS � 1for
Monday, July 21, 14
Conclusions
Monday, July 21, 14
New large class of scalar-tensor theories beyond (but not in contradiction with) Horndeski
Conclusions
Monday, July 21, 14
New large class of scalar-tensor theories beyond (but not in contradiction with) Horndeski
Can unite part of DM with DE strongly, decouples equation of state from the sound speed
Conclusions
Monday, July 21, 14
New large class of scalar-tensor theories beyond (but not in contradiction with) Horndeski
Can unite part of DM with DE strongly, decouples equation of state from the sound speed
New class of inflationary models with suppressed gravity waves and seemingly low non-Gaussianity
Conclusions
Monday, July 21, 14
!anks a lot for a"ention!
New large class of scalar-tensor theories beyond (but not in contradiction with) Horndeski
Can unite part of DM with DE strongly, decouples equation of state from the sound speed
New class of inflationary models with suppressed gravity waves and seemingly low non-Gaussianity
Conclusions
Monday, July 21, 14