New Perspectives on Cosmology Mimetic Cosmology Alexander Vikman Institute of Physics of the Czech Academy of Sciences 07.01.2016
This talk is mostly based on arxiv: 1512.09118, K. Hammer, A. Vikman arxiv: 1412.7136, JCAP 1506 (2015) 06, 028 L. Mirzagholi, A. Vikman arxiv: 1403.3961, JCAP 1406 (2014) 017 A. H. Chamseddine and V. Mukhanov, A. Vikman arxiv: 1003.5751, JCAP 1005 (2010) 012 I. Sawicki, E. Lim, A. Vikman
Plan of the talk
Plan of the talk Introduction
Plan of the talk Introduction What is mimetic construction?
Plan of the talk Introduction What is mimetic construction? Mimetic DM
Plan of the talk Introduction What is mimetic construction? Mimetic DM Imperfect DM with one scalar field and vorticity
Plan of the talk Introduction What is mimetic construction? Mimetic DM Imperfect DM with one scalar field and vorticity Mimetic Inflation
Plan of the talk Introduction What is mimetic construction? Mimetic DM Imperfect DM with one scalar field and vorticity Mimetic Inflation Open questions and conclusions
SM 5%
SM DM 5% 27%
SM DM 5% DE 27% 68%
SM DM 5% DE 27% 68% we feel them through gravity only!
Mimetic Matter Chamseddine, Mukhanov (2013)
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field:
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g µ g @ @
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g µ g @ @ physical metric of free fall
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g µ g @ @ physical metric of free fall auxiliary metric, dynamical variable
Mimetic Matter Chamseddine, Mukhanov (2013) One can encode the conformal part of the physical metric in a scalar field: g µ = g µ g @ @ physical metric of free fall auxiliary metric, dynamical variable S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) matter
Mimetic Matter Chamseddine, Mukhanov (2013)
Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g µ g @ @ physical metric of free fall auxiliary metric, dynamical variable
Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g µ g @ @ physical metric of free fall auxiliary metric, dynamical variable The theory becomes invariant with respect to Weyl transformations: g µ! 2 (x) g µ
Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g µ g @ @ physical metric of free fall auxiliary metric, dynamical variable The theory becomes invariant with respect to Weyl transformations: g µ! 2 (x) g µ The scalar field obeys the relativistic Hamilton-Jacobi equation: g µ @ µ @ =1
Mimetic Matter Chamseddine, Mukhanov (2013) g µ = g µ g @ @ physical metric of free fall auxiliary metric, dynamical variable The theory becomes invariant with respect to Weyl transformations: g µ! 2 (x) g µ The scalar field obeys the relativistic Hamilton-Jacobi equation: g µ @ µ @ =1 g µ = g µ g @ @ 1
the Hamilton-Jacobi equation g µ @ µ @ =1
the Hamilton-Jacobi equation g µ @ µ @ =1 corresponding four-velocity u µ = @ µ is geodesic
the Hamilton-Jacobi equation g µ @ µ @ =1 corresponding four-velocity u µ = @ µ is geodesic a µ = u r u µ = r (r r µ )= 1 2 @ µ (@ ) 2 =0
Modification of the Einstein equation
Modification of the Einstein equation S g µ
Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T (g)) @ µ @ =0
Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T (g)) @ µ @ =0 c.f. Einstein equations with dust or DM: G µ = T µ + u µ u = G T
Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T (g)) @ µ @ =0 c.f. Einstein equations with dust or DM: G µ = T µ + u µ u = G T
Modification of the Einstein equation S g µ G µ (g) T µ (g) (G (g) T (g)) @ µ @ =0 c.f. Einstein equations with dust or DM: G µ = T µ + u µ u = G T Dark Matter energy density: 4-velocity: u µ = @ µ
S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g µ g @ @
S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g µ g @ @ S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X L m ( g,, m), where X = 1 2 g,,
S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g µ g @ @ S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X Brans-Dicke! = 3/2 L m ( g,, m), where X = 1 2 g,,
S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g µ g @ @ S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X Brans-Dicke! = 3/2 L m ( g,, m), where X = 1 2 g,, is not in the Horndeski (1974) construction of the most general scalar-tensor theory with second order equations of motion
S [ g µ,, m] = Z d 4 x apple p 1 g 2 R (g)+l (g, m) g µ =g µ ( g, ) with g µ ( g, )= g µ g @ @ S 0 [ g, '] = Z d 4 x p g XR( g)+ 3 2 g X, X, X Brans-Dicke! = 3/2 L m ( g,, m), where X = 1 2 g,, is not in the Horndeski (1974) construction of the most general scalar-tensor theory with second order equations of motion But it is still a system with one degree of freedom + standard two polarizations for the graviton!
Lagrange Multiplier and Weyl-Invariance Hammer, Vikman (2015)
Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,,
Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,, Weyl-invariance: g µ! 2 (x) g µ,!, X! 2 (x) X,! 2 (x),
Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,, Weyl-invariance: g µ! 2 (x) g µ,!, X! 2 (x) X,! 2 (x), gauge invariant variables g µ =(2X) 1 g µ, ' = ', X = X, =2X,
Lagrange Multiplier and Weyl-Invariance S 0 [ g, ', X, ]= Z d 4 x p g apple XR( g)+ 3 2 g X, X X + Hammer, Vikman (2015) X 1 2 g,, Weyl-invariance: g µ! 2 (x) g µ,!, X! 2 (x) X,! 2 (x), gauge invariant variables g µ =(2X) 1 g µ, ' = ', X = X, =2X, S 0 [g, ', ] = Z d 4 x p g apple 1 2 R (g)+ 2 g ', ', 1.
Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010);
Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010); constraint via Lagrange multiplier Z S [g,,, SM] = d 4 x p g (@ ) 2 1 1 2 R + 1 2 (@ ) 2 1 + L SM
Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010); constraint via Lagrange multiplier Z S [g,,, SM] = d 4 x p g (@ ) 2 1 1 2 R + 1 2 (@ ) 2 1 + L SM T µ = u µ u
Mimetic Dark Matter via Lagrange Multiplier Chamseddine, Mukhanov; Golovnev; Barvinsky (2013) Lim, Sawicki, Vikman (2010); constraint via Lagrange multiplier Z S [g,,, SM] = d 4 x p g (@ ) 2 1 1 2 R + 1 2 (@ ) 2 1 + L SM T µ = u µ u Dark Matter Lagrange multiplier is the energy density u µ = @ µ
dust / DM via Lagrange multiplier (@ ) 2 1
dust / DM via Lagrange multiplier (@ ) 2 1 Cosmological Constant / DE via Lagrange multiplier (r µ V µ 1) M. Henneaux and C. Teitelboim (1989)
Disformal Transformation Nathalie Deruelle and Josephine Rua (2014), Domènech et al. (2015) One obtains the same dynamics (the same Einstein equations), if instead of varying the Einstein-Hilbert action with respect to the metric g µ one plugs in a disformal transformation (Bekenstein 1993) g µ = F (,w) `µ + H (,w)@ µ @ w = `µ @ µ @ w 2 F @ with and H + F @w w 6= 0 and varies with respect to `µ,
Disformal Transformation Nathalie Deruelle and Josephine Rua (2014), Domènech et al. (2015) One obtains the same dynamics (the same Einstein equations), if instead of varying the Einstein-Hilbert action with respect to the metric g µ one plugs in a disformal transformation (Bekenstein 1993) g µ = F (,w) `µ + H (,w)@ µ @ w = `µ @ µ @ w 2 F @ with and H + F @w w 6= 0 and varies with respect to `µ, Mimetic gravity is an exception! And it does provide new dynamics!
Mimicking any cosmological evolution Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010)
Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010)
Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V
Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V g µ @ µ @ =1 Convenient to take as time
Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V g µ @ µ @ =1 Convenient to take as time potential provides time-dependent pressure
Mimicking any cosmological evolution Just add a potential V ( )! Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) T µ = u µ u + g µ V g µ @ µ @ =1 Convenient to take as time potential provides time-dependent pressure Enough freedom to obtain any cosmological evolution!
In particular inflation as V ( )= 1 3 1 2 m2 2 m 4 2 e +1 gives the same cosmological potential in the standard case
Perturbations I Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) Even with potential, the energy still moves along the timelike geodesics
Perturbations I Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) Even with potential, the energy still moves along the timelike geodesics c S =0
Perturbations I Chamseddine, Mukhanov, Vikman (2013) Lim, Sawicki, Vikman; (2010) Even with potential, the energy still moves along the timelike geodesics c S =0 Newtonian potential: Z H =C 1 (x) 1 adt + H a C 2 (x) a Here on all scales but in the usual cosmology it is an approximation for superhorizon scales
Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013)
Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives
Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives Z r µ r r µ r is not that useful: d 4 x p Z g ;µ; ;µ; = d 4 x p g ( ) 2 R µ ;µ ;
Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives There are no new degrees of freedom, because higher time derivatives can be eliminated by differentiating the Hamilton-Jacobi equation. Z r µ r r µ r is not that useful: d 4 x p Z g ;µ; ;µ; = d 4 x p g ( ) 2 R µ ;µ ;
Next term in the gradient expansion Chamseddine, Mukhanov, Vikman (2013) ( ) 2 the unique quadratic term with higher derivatives There are no new degrees of freedom, because higher time derivatives can be eliminated by differentiating the Hamilton-Jacobi equation. Z r µ r r µ r is not that useful: d 4 x p Z g ;µ; ;µ; = d 4 x p g ( ) 2 R µ ;µ ; expansion = = r µ u µ
CHARGE CONSERVATION
CHARGE CONSERVATION no potential! + c symmetry
CHARGE CONSERVATION no potential! + c symmetry r µ J µ =0 Noether current:
CHARGE CONSERVATION no potential! + c symmetry r µ J µ =0 Noether current: charge density n / a 3
Imperfection
Imperfection J µ shift-charge current Eckart frame
Imperfection J µ shift-charge current Energy flow: timelike eigenvector of energy-momentum tensor Eckart frame Landau-Lifshitz frame
Imperfection / 2 J µ shift-charge current Energy flow: timelike eigenvector of energy-momentum tensor Eckart frame Landau-Lifshitz frame
Imperfection in the Noether current J µ = u µ,µ
Imperfection in the Noether current J µ = u µ,µ expansion = = r µ u µ
Imperfect Dark Matter Mirzagholi, Vikman (2014)
Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ
Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ? µ = g µ u µ u expansion = r µ u µ
Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r? µ = g µ u µ u expansion = r µ u µ
Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r energy density " = 1 2 2? µ = g µ u µ u expansion = r µ u µ
Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r energy density " = pressure p = 1 2 2 + 1 2 2? µ = g µ u µ u expansion = r µ u µ
Imperfect Dark Matter (no potential) Mirzagholi, Vikman (2014) T µ = "u µ u p? µ +q µ u + q u µ energy flow q µ =? µ r energy density " = pressure p = 1 2 2 + 1 2 2? µ = g µ u µ u expansion = r µ u µ = u µ r µ
Vorticity for a single dof DM!
Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J
Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J Vorticity vector: µ (V )= 1 2 " µ V V ; ' 2 2 " µ,,,
Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J = = r µ u µ Vorticity vector: µ (V )= 1 2 " µ V V ; ' 2 2 " µ,,,
Vorticity for a single dof DM! in the frame moving with the charges (Eckart frame) V µ = J µ p J J = = r µ u µ Vorticity vector: µ (V )= 1 2 " µ V V ; ' 2 2 " µ,,, the circulation is conserved up to O 2
Perturbations II
Perturbations II + H c 2 s a 2 + Ḣ =0 with the sound speed c 2 s = 2 3
Perturbations II + H c 2 s a 2 + Ḣ =0 with the sound speed c 2 s = 2 3 Newtonian potential: =
Perturbations II + H c 2 s a 2 + Ḣ =0 with the sound speed c 2 s = 2 3 Ramazanov,Capela (2014) Newtonian potential: = c S 10 5
Background cosmology
Background cosmology " = 2 2 3 n +3c2 S ext p =3c 2 SP ext
Background cosmology " = 2 2 3 n +3c2 S ext p =3c 2 SP ext DM n / a 3
Background cosmology " = 2 2 3 n +3c2 S ext p =3c 2 SP ext DM n / a 3 G e = G N 1+3c 2 S
Background cosmology " = 2 2 3 n +3c2 S ext p =3c 2 SP ext DM n / a 3 G e = G N 1+3c 2 S G N bounds are mild: 3 c 2 S matter c 2 S radiation. 0.066 ± 0.039 Narimani, Scott, Afshordi(2014)
mimetic construction and inflation n / a 3
mimetic construction and inflation n / a 3 shift-symmetry breaking is needed for mimetic DM!
Generating shift-charge (DM) during radiation domination époque
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) H 1
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1 =3H
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr )
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq )
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) ' acr a eq ' z eq z cr
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) acr ' a eq ' z eq z cr apple
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) acr ' a eq ' z eq z cr apple apple 10 10
Generating shift-charge (DM) during radiation domination époque t[7]= ( ) r µ J µ = 1 2 0 ( ) 2 H 1 n (t cr ) / a 3 Z =3H dt 0 a 3 H 2 ' rad (t cr ) at DM / radiation equality DM (z eq )= rad (z eq ) acr ' a eq ' z eq z cr apple apple 10 10 T cr ' T eq & 10 GeV
Back to inflation with potential and ( ) 2
Quantization
Quantization the action Z S = 1 2 d d 3 x c 2 s 0 0 +...
Quantization the action Z S = 1 2 d d 3 x c 2 s 0 0 +... short wavelength quantum fluctuations r cs ~ k 3/2 k ' match with the long-wave-length limit
Perturbations in Mimetic Inflation
Perturbations in Mimetic Inflation on scale ' k 1
Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h
Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h
Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 S H c S k'h
Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 Gravitational Waves are unchanged h ' H k'h S H c S k'h
Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 Gravitational Waves are unchanged h ' H k'h S H c S k'h spectral indices n S 1=n T
Perturbations in Mimetic Inflation on scale ' k 1 Newtonian potential ' p c s / H cs k'h for c S 1 ' c 1/2 H S cs k'h for c S 1 ' c 1/2 Gravitational Waves are unchanged h ' H k'h S H c S k'h spectral indices n S 1=n T small sound speed h But the non-gaussianity is vanishingly small!
Further Directions
Further Directions Caustics?
Further Directions Caustics? Galactic Halos?
Further Directions Caustics? Galactic Halos? Gravitational collapse?
Further Directions Caustics? Galactic Halos? Gravitational collapse? Accretion of the mimetic DM?
Further Directions Caustics? Galactic Halos? Gravitational collapse? Accretion of the mimetic DM? Bullet Cluster?
Further Directions Caustics? Galactic Halos? Gravitational collapse? Accretion of the mimetic DM? Bullet Cluster? Quantum corrections?
Conclusions
Conclusions New large class of Weyl-invariant scalar-tensor theories
Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom.
Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom. Imperfect Dark Matter, with a small sound speed and vorticity. Only one free parameter for the late Universe.
Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom. Imperfect Dark Matter, with a small sound speed and vorticity. Only one free parameter for the late Universe. New class of inflationary models with suppressed gravity waves and low non-gaussianity.
Conclusions New large class of Weyl-invariant scalar-tensor theories Unification of Dark Matter with Dark Energy in one degree of freedom. Imperfect Dark Matter, with a small sound speed and vorticity. Only one free parameter for the late Universe. New class of inflationary models with suppressed gravity waves and low non-gaussianity.!anks a lot for a"ention!