Models of coupled dark matter to dark energy · There is more dark matter at early times,...

Models of coupled dark matter to dark energy

Alkistis Pourtsidou

Department of Physics & Astronomy, Bologna University, Italy

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

Motivation

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

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

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

I Plethora of coupled models in the literature → the form of thecoupling is chosen phenomenologically at the level of the fieldequations

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 Cold Dark Matter (c) [e.g.Kodama & Sasaki ’84, Ma & Bertschinger ’95]

I T (c) and T (DE) are not separately conserved:

∇µT (c)µν = −∇µT (DE)µ

ν = Jν

I Various forms of coupling have been considered. Examples:Jν = Cρc∇νφ [Amendola ’00]

Jν = Γρcu(c)ν [Valiviita,Majerotto,Maartens ’08]

I FRW background with J0 (Ji = 0 because of isotropy)

ρc + 3Hρc = −J0

ρDE + 3HρDE = J0

I In previous examples: J0 = Cρc˙φ, J0 = aΓρc

I Many models in the literature, and their cosmological implications(CMB, P(k), Supernovae, growth, non-linear perturbations, N-bodysimulations, instabilities...)

I In order to make further progress, we need to construct generalmodels.


Fluids in General Relativity

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

I The action for GR coupled to an adiabatic fluid is taken to be

S =1

16πG

∫d4x√−gR−

∫d4x√−gf(n),

where n is the fluid/particle number density.

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

I For pressureless matter f = f0n

I Vary the action S to get field and fluid equations


Coupled CDM/scalar field φ

I We want to construct a model where the adiabatic fluid is explicitlycoupled to a DE field φ

I Invariants: n, Y = 12gµν∇µφ∇νφ, Z = uµ∇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 isL = F (Y, φ) + f(n)

I Our total (general) action is

S =1

16πG

∫d4x√−gR−

∫d4x√−gL(n, Y, Z, φ)

I We can now consider different classes of theories


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. IfF = Y + V (φ), we describe coupled quintessence models.

I Field energy-momentum tensor Tφµν = FY φµφν − FgµνI Fluid energy-momentum tensor with ρ = f and P = nfn − fI The evolution of the fluid energy density ρ is given by

uµ∇µρ+ (ρ+ P )∇µuµ = Zραφ

I Coupling current Jµ = −ραφφµ


Type 2


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

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

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

uµ∇µρ+ ρ∇µuµ = −Z∇µ(ρβuµ)

I Coupling current Jµ = ∇ν(ρβuν)φµ


Type 3


L(n, Y, Z, φ) = F (Y, Z, φ) + f(n)

I Field energy-momentum tensor Tφµν = FY φµφν − Fgµν − ZFZuµuνI Fluid energy-momentum tensor same as Type-1

I Evolution of the fluid energy density ρ same as Type-1 (but differentmomentum transfer)

I Coupling current Jµ = ∇ν(FZuν)φµ + FZDµZ + ZFZu

ν∇νuµI Type 3 is very special → J0 = 0: no coupling at the background

field equations!

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

I Type-3 is a pure momentum-transfer theory


Cosmology: Type 1

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

ρφ =1

2

˙φ2

a2+ V (φ), Pφ =

1

2

˙φ2

a2− V (φ)

I Background Klein Gordon: ¨φ+ 2H ˙φ+ a2Vφ = −a2ραφ

I Evolution of CDM density: ˙ρ+ 3ρH = ραφ˙φ with solution

ρ = ρ0a−3eα(φ)

I Synchronous gauge:ds2 = −a2dτ2 + a2

[(1 + 1

3h)γij +Dijν]dxidxj

I Perturbed Klein-Gordon:

ϕ+ 2Hϕ+(k2 + a2Vφφ

)ϕ+

1

2˙φh = −a2αφρδ


Cosmology: Type 1

I The perturbed CDM equations are

δ = −k2θ − 1

2h+ αφϕ

θ = −Hθ − αφ ˙φ[θ − θ(φ)

]with θ(φ) = ϕ/ ˙φ

I We can now study the effect of the coupling to the evolution of thebackground as well as to the CMB temperature and matter powerspectra. We use a modified version of CAMB and compare with theuncoupled case.

I We choose the 1EXP quintessence potential V (φ) = V0e−λφ

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

I We keep λ = 1.22 (fixed) and vary V0 and ρc,0 so that eachcosmology (coupled and uncoupled) evolves to the PLANCK

cosmological parameters values [Ade et al. ’13]


Type 1: Evolution of Ωcdm

0 0.2 0.4 0.6 0.8 1a

0

0.2

0.4

0.6

0.8

1

Ωc

α0 = 0

α0 = 0.15

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


Type 1: Matter power spectra

0.0001 0.01 1 100

k (h Mpc-1

)

0.01

1

100

10000

P0(k

) (h

-3 M

pc3

)

α0=0

α0=0.15

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


Type 1: CMB temperature spectra

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


Cosmology: Type 2

I ρφ and Pφ same as Type 1

I Background KG:(

1− ρβZ1+Zβ

)(¨φ−H ˙φ

)+ 3H ˙φ+ a2Vφ = 0

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

I The background CDM equation is solved to give

ρc = ρc,0a−3Z

β01−β0

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

I We also derive the perturbed KG and perturbed CDM equations


Type 2: Evolution of Ωcdm

0 0.2 0.4 0.6 0.8 1a

0

0.2

0.4

0.6

0.8

1

Ωc

β0 = 0

β0 = 1/11



0.0001 0.01 1 100

k (h Mpc-1

)

0.01

1

100

10000P

0(k

) (h

-3 M

pc3

) β

0=0

β0=1/11


Cosmology: Type 3

I We consider F = Y + V (φ) + γ(Z)

I ρφ = 12

˙φ2

a2 +˙φaγZ + γ + V and Pφ = 1

2

˙φ2

a2 − γ − VI Background KG: (1− γZZ) ( ¨φ−H ˙φ) + 3aH

(γZ − Z

)+ a2Vφ = 0

I We choose a sub case with γ(Z) = γ0Z2

I Type 3 is special → no coupling appears at the background levelfluid 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 apure 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 identifypathologies/instabilities.


Hubble parameter H(z)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2z

0

30

60

90

120

150

180

210

240H

(z)

(km

/s/M

pc)

no coupling

Type 1, α0 = 0.15

Type 2, β0 = 1/11

Type 3, γ0 = 0.15

Simon et al. (2005)



0.0001 0.01 1 100

k (h Mpc-1

)

0.01

1

100

10000P

0(k

) (h

-3 M

pc3

) γ

0=0

γ0=0.15


Conclusions

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

I We have constructed 3 general classes of theories

I To study cosmological implications, we used a specific form for thecoupling function and a specific potential. Observational signaturesdepend heavily on these choices.

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

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


Models of coupled dark matter to dark energy · There is more dark matter at early times,...

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