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