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Models of Coupled Dark Matter to Dark Energy

Models of coupled dark to dark

Alkistis Pourtsidou

Department of & , Bologna University, Italy

[Work in collaboration with Ed Copeland and Costas Skordis, Nottingham, UK] Motivation

I The Dark . What is ? What is dark matter?

I ΛCDM fits the data extremely well, but suffers from two fundamental problems: fine-tuning & coincidence. Alternatives: e.g. , modified

I DM and DE might be coupled. This can solve the coincidence problem.

I Plethora of coupled models in the literature → the form of the coupling is chosen phenomenologically at the level of the field equations

I What is the most general phenomenological model we can construct?

I We present three distinct classes (Types) of coupled DE models, introducing the coupling at the level of the action.

Alkistis Pourtsidou Models of coupled dark matter to dark energy Theories of coupled DM/DE

I Consider Dark Energy (DE) coupled to (c) [e.g. Kodama & Sasaki ’84, Ma & Bertschinger ’95] (c) (DE) I T and T are not separately conserved:

(c)µ (DE)µ ∇µT ν = −∇µT ν = Jν

I Various forms of coupling have been considered. Examples: Jν = Cρc∇ν φ [Amendola ’00] (c) Jν = Γρcuν [Valiviita,Majerotto,Maartens ’08] I FRW background with J¯0 (J¯i = 0 because of isotropy)

ρ˙c + 3Hρc = −J¯0

ρ˙DE + 3HρDE = J¯0 ¯˙ I In previous examples: J¯0 = Cρ¯cφ, J¯0 = aΓ¯ρc I Many models in the literature, and their cosmological implications (CMB, P(k), Supernovae, growth, non-linear perturbations, N-body simulations, instabilities...) I In order to make further progress, we need to construct general models.

Alkistis Pourtsidou Models of coupled dark matter to dark energy Fluids in

I The fluid pull-back formalism is a description of relativistic fluids at the level of the action [see review by Andersson & Comer ’08 and references therein]

I The action for GR coupled to an adiabatic fluid is taken to be 1 Z √ Z √ S = d4x −gR − d4x −gf(n), 16πG

where n is the fluid/particle number density.

I f(n) is (in principle) an arbitrary function, whose form determines the and speed of sound of the fluid

I For pressureless matter f = f0n

I Vary the action S to get field and fluid equations

Alkistis Pourtsidou Models of coupled dark matter to dark energy Coupled CDM/scalar field φ

I We want to construct a model where the adiabatic fluid is explicitly coupled to a DE field φ 1 µν µ I Invariants: n, Y = 2 g ∇µφ∇ν φ, Z = u ∇µφ I The Lagrangian has the form

L = L(n, Y, Z, φ).

I GR+quintessence+fluid is

L = Y + V (φ) + f(n)

I k-essence is L = F (Y, φ) + f(n)

I Our total (general) action is 1 Z √ Z √ S = d4x −gR − d4x −gL(n, Y, Z, φ) 16πG

I We can now consider different classes of theories

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 1

I Type-1 models are classified via

L(n, Y, Z, φ) = F (Y, φ) + f(n, φ)

with f(n) = g(n)eα(φ)

I Notation: φµ ≡ ∇µφ, fn ≡ df/dn, FY ≡ ∂F/∂Y etc.

I These models describe a K-essence scalar field coupled to matter. If F = Y + V (φ), we describe coupled quintessence models. φ I Field energy- tensor Tµν = FY φµφν − F gµν

I Fluid energy-momentum tensor with ρ = f and P = nfn − f

I The evolution of the fluid ρ is given by

µ µ u ∇µρ + (ρ + P )∇µu = Zραφ

I Coupling current Jµ = −ραφφµ

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 2

I Type-2 models are classified via

L(n, Y, Z, φ) = F (Y, φ) + f(n, Z)

I Field energy-momentum tensor same as Type-1

I Fluid energy-momentum tensor with ρ = f − ZfZ and P = nfn − f

I CDM has P = 0 which means f = nh(Z)

I We parameterize h(Z) in terms of an integral of a new function β(Z) (this simplifies the equations)

I The evolution of the fluid energy density ρ is given by

µ µ µ u ∇µρ + ρ∇µu = −Z∇µ(ρβu )

ν I Coupling current Jµ = ∇ν (ρβu )φµ

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 3

I Type-3 models are classified via

L(n, Y, Z, φ) = F (Y, Z, φ) + f(n)

φ I Field energy-momentum tensor Tµν = FY φµφν − F gµν − ZFZ uµuν I Fluid energy-momentum tensor same as Type-1

I Evolution of the fluid energy density ρ same as Type-1 (but different momentum transfer) ν ˜ ν I Coupling current Jµ = ∇ν (FZ u )φµ + FZ DµZ + ZFZ u ∇ν uµ

I Type 3 is very special → J¯0 = 0: no coupling at the background field equations!

I Furthermore, the energy-conservation equation remains uncoupled even at the linear level

I Type-3 is a pure momentum-transfer theory

Alkistis Pourtsidou Models of coupled dark matter to dark energy : Type 1

For Type-1, coupled quintessence is described by F = Y + V (φ). CDM fluid means f = neα(φ). We investigate the background FRW space-time and linear perturbations (dots are derivatives wrt confromal time τ)

1 φ¯˙2 1 φ¯˙2 ρ¯φ = + V (φ), P¯φ = − V (φ) 2 a2 2 a2

¯¨ ¯˙ 2 2 I Background Klein Gordon: φ + 2Hφ + a Vφ = −a ρα¯ φ ¯˙ I Evolution of CDM density: ρ¯˙ + 3¯ρH =ρα ¯ φφ with solution −3 α(φ) ρ¯ = ρ0a e

I Synchronous gauge: 2 2 2 2  1  i j ds = −a dτ + a (1 + 3 h)γij + Dijν dx dx I Perturbed Klein-Gordon: 1 ϕ¨ + 2Hϕ˙ + k2 + a2V  ϕ + φ¯˙h˙ = −a2α ρδ¯ φφ 2 φ

Alkistis Pourtsidou Models of coupled dark matter to dark energy Cosmology: Type 1

I The perturbed CDM equations are 1 δ˙ = −k2θ − h˙ + α ϕ˙ 2 φ

˙ ¯˙ h (φ)i θ = −Hθ − αφφ θ − θ

with θ(φ) = ϕ/φ¯˙

I We can now study the effect of the coupling to the evolution of the background as well as to the CMB temperature and matter power spectra. We use a modified version of CAMB and compare with the uncoupled case. −λφ I We choose the 1EXP quintessence potential V (φ) = V0e

I We choose α(φ) = α0φ, with α0 a constant. This model has been studied extensively [e.g. Xia ’09, Tarrant et al. ’12]

I We keep λ = 1.22 (fixed) and vary V0 and ρ¯c,0 so that each cosmology (coupled and uncoupled) evolves to the cosmological parameters values [Ade et al. ’13]

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 1: Evolution of Ωcdm

1

α = 0 0 α = 0.15 0.8 0

0.6 c Ω

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 a

The CDM density is higher at early times for the couple case, in order to evolve to the same cosmological parameters today. The matter-radiation equality occurs earlier in the coupled case.

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 1: Matter power spectra

α =0 10000 0 α =0.15 0 )

3 100 Mpc -3 (k) (h 0

P 1

0.01

0.0001 0.01 1 100 -1 k (h Mpc )

P (k) affected on small scales. There is more dark matter at early times, matter-radiation equality earlier. Only small scale perturbations have time to enter the horizon and grow during radiation-dominated era. The turnover happens on smaller scales. The growth is enhanced, small scale power increases, larger σ8.

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 1: CMB temperature spectra

Small scale anisotropies decrease. The locations of the CMB peaks shift towards smaller scales. Large-scale anisotropies (ISW effect) decrease.

Alkistis Pourtsidou Models of coupled dark matter to dark energy Cosmology: Type 2

φ φ I ρ¯ and P¯ same as Type 1     ρβ¯ Z ¯¨ ¯˙ ¯˙ 2 I Background KG: 1 − 1+Zβ¯ φ − Hφ + 3Hφ + a Vφ = 0

I We choose a sub case with β(Z) = β0/Z

I The background CDM equation is solved to give

β −3 0 ρ¯c =ρ ¯c,0a Z¯ 1−β0

¯˙ ¯˙ ¯ I Since Z¯ = −φ/a, ρ¯c depends on the time derivative φ instead of φ itself which is a notable difference from the Type-1 case.

I We also derive the perturbed KG and perturbed CDM equations

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 2: Evolution of Ωcdm

1

β0 = 0 β = 1/11 0.8 0

0.6 c Ω

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 a

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 2: Matter power spectra

β =0 10000 0 β0=1/11 )

3 100 Mpc -3 (k) (h 0

P 1

0.01

0.0001 0.01 1 100 -1 k (h Mpc )

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 2: CMB temperature spectra

Alkistis Pourtsidou Models of coupled dark matter to dark energy Cosmology: Type 3

I We consider F = Y + V (φ) + γ(Z) ˙2 ˙ ˙2 φ 1 φ¯ φ¯ ¯φ 1 φ¯ I ρ¯ = 2 a2 + a γZ + γ + V and P = 2 a2 − γ − V ¯¨ ¯˙  2 I Background KG: (1 − γZZ )(φ − Hφ) + 3aH γZ − Z¯ + a Vφ = 0 2 I We choose a sub case with γ(Z) = γ0Z

I Type 3 is special → no coupling appears at the background level fluid equations

I We also derive the perturbed KG and perturbed CDM equations — note the δ evolution is the same as in the uncoupled case. This is a pure momentum-transfer coupling up-to linear order.

I This case has ghost/strong coupling problems for γ0 ≥ 1/2. Introducing the coupling at the level of the action helps identify pathologies/instabilities.

Alkistis Pourtsidou Models of coupled dark matter to dark energy Hubble parameter H(z)

240 no coupling

210 Type 1, α0 = 0.15

Type 2, β0 = 1/11

180 Type 3, γ0 = 0.15 Simon et al. (2005) 150

120

H(z) (km/s/Mpc) 90

60

30

0 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 z

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 3: Matter power spectra

γ =0 10000 0 γ0=0.15 )

3 100 Mpc -3 (k) (h 0

P 1

0.01

0.0001 0.01 1 100 -1 k (h Mpc )

Alkistis Pourtsidou Models of coupled dark matter to dark energy Type 3: CMB temperature spectra

Alkistis Pourtsidou Models of coupled dark matter to dark energy Conclusions

I We have presented a novel method to construct general interacting dark energy models

I We have constructed 3 general classes of theories

I To study cosmological implications, we used a specific form for the coupling function and a specific potential. Observational signatures depend heavily on these choices.

I However, it is possible to parameterize the field equations by introducing all possible type of terms that can appear in a coupled theory using the PPF approach.

I This can pave the way for model-independent approach to test coupled theories, by reducing the large parameter space of free functions and coupling types to a small set of constants

Alkistis Pourtsidou Models of coupled dark matter to dark energy

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