models of coupled dark matter to dark energy · there is more dark matter at early times,...
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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]
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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
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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.
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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
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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
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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
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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 − 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
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Type 3
I Type-3 models are classified via
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
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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
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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]
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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.
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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.
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Type 1: CMB temperature spectra
Small scale anisotropies decrease. The locations of the CMB peaks shifttowards smaller scales. Large-scale anisotropies (ISW effect) decrease.
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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
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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
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Type 2: Matter power spectra
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
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Type 2: CMB temperature spectra
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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.
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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)
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Type 3: Matter power spectra
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
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Type 3: CMB temperature spectra
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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
Alkistis Pourtsidou Models of coupled dark matter to dark energy