The Dark Degeneracy: the Dark Fluid and Interacting Cosmic Components

The Dark Degeneracy: The Dark Fluid and Interacting Cosmic Components

Jorge Cervantes In collaboration with Alejandro Avilés

National Nuclear Research Institute (ININ), Mexico

Cosmology at the Beach, 2012 Standard Model of Cosmology

2 2 2 i i Homogeneus and isotropic (and ﬂat) spacetime:d s = −dt + a (t)δij dx dx .

2 2 a˙ 8πG H ≡ = (ρT ) a 3

a¨ 4πG a = − (ρT + 3PT ) a 3 where ρT = ργ + ρν + ρb + ρdm + ρde, which comply with

ρ˙rel + 4Hρrel = 0, ρ˙nonrel + 3Hρnonrel = 0, ρde = const. ··· 2 Two main properties are relevant: clustering dark matter( cs = 0) and accelerating dark energy (PT < −1/3). Standard Model of Cosmology

2 2 2 i i Homogeneus and isotropic (and ﬂat) spacetime:d s = −dt + a (t)δij dx dx .

2 2 a˙ 8πG H ≡ = (ρT ) a 3

a¨ 4πG a = − (ρT + 3PT ) a 3 where ρT = ργ + ρν + ρb + ρdm + ρde, which comply with

2 2 2 i i Homogeneus and isotropic (and ﬂat) spacetime:d s = −dt + a (t)δij dx dx .

2 2 a˙ 8πG H ≡ = (ρT ) a 3

a¨ 4πG a = − (ρT + 3PT ) a 3 where ρT = ργ + ρν + ρb + ρdm + ρde, which comply with

ρ˙rel + 4Hρrel = 0, ρ˙nonrel + 3Hρnonrel = 0, ρde = const.

··· 2 Two main properties are relevant: clustering dark matter( cs = 0) and accelerating dark energy (PT < −1/3). Standard Model of Cosmology

2 2 2 i i Homogeneus and isotropic (and ﬂat) spacetime:d s = −dt + a (t)δij dx dx .

2 2 a˙ 8πG H ≡ = (ρT ) a 3

a¨ 4πG a = − (ρT + 3PT ) a 3 where ρT = ργ + ρν + ρb + ρdm + ρde, which comply with

ρ˙rel + 4Hρrel = 0, ρ˙nonrel + 3Hρnonrel = 0, ρde = const. ··· 2 Two main properties are relevant: clustering dark matter( cs = 0) and accelerating dark energy (PT < −1/3). Supernova Cosmology Project Knop et al. (2003) ΩΜ , ΩΛ 0.25,0.75 0.25, 0 24 1, 0

22 Supernova Cosmology Project B m

effective 18 Calan/Tololo & CfA

14 1.0

0.5 ΩΜ , ΩΛ

0.25,0.75 0.0 0.25, 0

0.5 1, 0 mag. residual from empty cosmology 1.0 0.0 0.2 0.4 0.6 0.8 1.0

redshift z Dark ﬂuids or dark geometry? Dark Fluid Components 1 obs dark Gµν = T + T 8πG µν µν Any selection of the dark energy-momentum tensor is arbitrary.

Dark Geometry 1 dark obs (Gµν + G ) = T 8πG µν µν Any selection of the dark Einstein tensor is arbitrary.

···

A popular choice (historical reasons!):

dark dm de Tµν = Tµν + Tµν

with: dm Tµν = ρuµuν a perfect ﬂuid of CDM,

de Tµν = Λ gµν the cosmological constant, a very successful model , but there are many possibilities. Dark ﬂuids or dark geometry? Dark Fluid Components 1 obs dark Gµν = T + T 8πG µν µν Any selection of the dark energy-momentum tensor is arbitrary.

Dark Geometry 1 dark obs (Gµν + G ) = T 8πG µν µν Any selection of the dark Einstein tensor is arbitrary.

···

A popular choice (historical reasons!):

dark dm de Tµν = Tµν + Tµν

with: dm Tµν = ρuµuν a perfect ﬂuid of CDM,

Dark Geometry 1 dark obs (Gµν + G ) = T 8πG µν µν Any selection of the dark Einstein tensor is arbitrary.

···

A popular choice (historical reasons!):

dark dm de Tµν = Tµν + Tµν

with: dm Tµν = ρuµuν a perfect ﬂuid of CDM,

Dark Geometry 1 dark obs (Gµν + G ) = T 8πG µν µν Any selection of the dark Einstein tensor is arbitrary.

···

A popular choice (historical reasons!):

dark dm de Tµν = Tµν + Tµν

with: dm Tµν = ρuµuν a perfect ﬂuid of CDM,

de Tµν = Λ gµν the cosmological constant, a very successful model , but there are many possibilities. Alternative models to ΛCDM

I k-essence

I phantom energy

I strings’s motived models: ghost condensates, UV corrections to curvature, conformal anomalies.

I Non-minimal couplings f (φ)R

I dm-de interacting models

I Chameleons

I Galileons, etc, etc

...and more

I MOND

I backreaction of perturbations α I Chaplygin gas, p = 1/ρ

I Brane worlds: RS, DPG, Gauss Bonnet

I Loop Quantum Cosmology

I Infrared corrections to curvatures: f(R)

I Cosmic voids, archipelago.

...Let’s look at a possible model. Alternative models to ΛCDM

I k-essence

I phantom energy

I strings’s motived models: ghost condensates, UV corrections to curvature, conformal anomalies.

I Non-minimal couplings f (φ)R

I dm-de interacting models

I Chameleons

I Galileons, etc, etc

...and more

I MOND

I backreaction of perturbations α I Chaplygin gas, p = 1/ρ

I Brane worlds: RS, DPG, Gauss Bonnet

I Loop Quantum Cosmology

I Infrared corrections to curvatures: f(R)

I Cosmic voids, archipelago.

...Let’s look at a possible model. Alternative models to ΛCDM

I k-essence

I phantom energy

I strings’s motived models: ghost condensates, UV corrections to curvature, conformal anomalies.

I Non-minimal couplings f (φ)R

I dm-de interacting models

I Chameleons

I Galileons, etc, etc

...and more

I MOND

I backreaction of perturbations α I Chaplygin gas, p = 1/ρ

I Brane worlds: RS, DPG, Gauss Bonnet

I Loop Quantum Cosmology

I Infrared corrections to curvatures: f(R)

I Cosmic voids, archipelago.

...Let’s look at a possible model. Dark matter from scalar ﬁeld-baryonic couplings

Plethora of couplings: Conformal, trace, non-minimal...

Z 4 p R 1 ,α S = d x −g − φ φ,α − V (φ) + S + Sm, 16πG 2 int √ with Lint = −gA(φ)T .

2 δ(S + Sm) T = − √ int gµν . −g δgµν

(φ) α(φ) (m) Gµν = 8πG(Tµν + e Tµν )

dm z}|{ φ int b α(φ) Tµν = Tµν + Tµν +Tµν , T = e ρ | {z } dark Tµν

and 0 0 α(φ) φ − V (φ) = α (φ)e ρ, Dark matter from scalar ﬁeld-baryonic couplings

Plethora of couplings: Conformal, trace, non-minimal...

Z 4 p R 1 ,α S = d x −g − φ φ,α − V (φ) + S + Sm, 16πG 2 int √ with Lint = −gA(φ)T .

2 δ(S + Sm) T = − √ int gµν . −g δgµν

(φ) α(φ) (m) Gµν = 8πG(Tµν + e Tµν )

dm z}|{ φ int b α(φ) Tµν = Tµν + Tµν +Tµν , T = e ρ | {z } dark Tµν

and 0 0 α(φ) φ − V (φ) = α (φ)e ρ, Dark matter from scalar ﬁeld-baryonic couplings

Plethora of couplings: Conformal, trace, non-minimal...

Z 4 p R 1 ,α S = d x −g − φ φ,α − V (φ) + S + Sm, 16πG 2 int √ with Lint = −gA(φ)T .

2 δ(S + Sm) T = − √ int gµν . −g δgµν

(φ) α(φ) (m) Gµν = 8πG(Tµν + e Tµν )

dm z}|{ φ int b α(φ) Tµν = Tµν + Tµν +Tµν , T = e ρ | {z } dark Tµν

and 0 0 α(φ) φ − V (φ) = α (φ)e ρ, Dark matter from scalar ﬁeld-baryonic couplings

Plethora of couplings: Conformal, trace, non-minimal...

Z 4 p R 1 ,α S = d x −g − φ φ,α − V (φ) + S + Sm, 16πG 2 int √ with Lint = −gA(φ)T .

2 δ(S + Sm) T = − √ int gµν . −g δgµν

(φ) α(φ) (m) Gµν = 8πG(Tµν + e Tµν )

dm z}|{ φ int b α(φ) Tµν = Tµν + Tµν +Tµν , T = e ρ | {z } dark Tµν

and 0 0 α(φ) φ − V (φ) = α (φ)e ρ, Background

2 8πG 1 ˙ 2 α(φ) I H = 3 2 φ + V (φ)+(e − 1)ρ + ρ ¨ ˙ 0 0 α(φ) 0 −3 I φ + 3Hφ + V (φ) + α (φ)e ρba = 0

1 V (φ) = 1 m2 φ2, eα = 1 + 1 φ2 wdark = − (in slow roll) 2 φ 2 α(φ)− 1+ e 1 ρ0a−3 V (φ) b

(0) α(φ)− Ω C = e 1 ρ ≈ DM ≈ 0.3, 1 V (φ) (0) Ω z=0 DE (0) Ω C = eα(φ) − 1 ≈ DM ≈ 4.7 2 (0) Ω z=0 b Background

2 8πG 1 ˙ 2 α(φ) I H = φ + V (φ)+(e − 1)ρ + ρ V (φ) = 1 m2 φ2, eα = 1 + 1 φ2 3 2 2 φ 2 ¨ ˙ 0 0 α(φ) 0 −3 I φ + 3Hφ + V (φ) + α (φ)e ρba = 0

1.0 1 wdark = − (in slow roll) eα(φ)−1 1+ ρ0a−3 0.5 V (φ) b

(0) dark 0.0 α(φ)− Ω Ω C = e 1 ρ ≈ DM ≈ 0.3, 1 V (φ) (0) -0.5 Ω z=0 DE (0) Ω α(φ) -1.0 C = e − 1 ≈ DM ≈ 4.7 0.0 0.5 1.0 1.5 2.0 2 (0) Ω a z=0 b

Union2 0.40 Union2 data

44 0.35 42

1 0.30 40 Μ C 38

0.25 36

0.20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 4.2 4.4 4.6 4.8 5.0 5.2 z C2 Linear Perturbations

Β = 0.01 -0.002

200 -0.003

150 -0.004 Y -0.005 ∆ 100

-0.006 50 -0.007 0.00 0.02 0.04 0.06 0.08 0.10 0 0.00 0.02 0.04 0.06 0.08 0.10 mΦ t

0.0002 φ˙ δφ˙ − φ˙ 2Ψ − m2 φ δφ 0.0001 2 0 0 φ 0 c = s φ 2 s 0.0000 φ˙ δφ˙ − φ˙ 2Ψ + m2 φ δφ + 1 φ2(δ + 0 δφ) c 0 0 0 φ 2 0 1+ 1 φ 2 -0.0001 2 0

-0.0002

0.00 0.02 0.04 0.06 0.08 0.10 (A. Aviles and J.L. Cervantes-Cota, 2011a) Towards a single ﬂuid description

The elements: locally clustering (perturbations) and globally accelerating (background).

Astrophysics At very large scales (> 100’s Mpc) the Universe is homogeneous and isotropic: ρ(t,~x) = ρ(t)

Matter perturbations: δ(t,~x) = δρ/ρ

00 0 2 2 δ + 2Hδ + cs kphys − 4πGρ δ = 0,

where a prime stands for a cosmic time derivative.

r π I l > lJ Growth Jeans length lJ = cs Gρ I l < lJ No growth and we infer non-baryonic structure growth from dwarf galaxies to cosmological scales.

2 We deﬁne the Dark Fluid (d) as a barotropic ﬂuid with cs = 0. 1 Pd (ρ) = wd (ρ)ρd −→ wd ∝ −→ Pd = const. ρd

Astrophysical observations constrain Pd ρastro, but no necessarily equal to zero, and this will be important to interpret the single ﬂuid as dark energy too. Towards a single ﬂuid description

The elements: locally clustering (perturbations) and globally accelerating (background).

Astrophysics At very large scales (> 100’s Mpc) the Universe is homogeneous and isotropic: ρ(t,~x) = ρ(t)

Matter perturbations: δ(t,~x) = δρ/ρ

00 0 2 2 δ + 2Hδ + cs kphys − 4πGρ δ = 0,

where a prime stands for a cosmic time derivative.

r π I l > lJ Growth Jeans length lJ = cs Gρ I l < lJ No growth and we infer non-baryonic structure growth from dwarf galaxies to cosmological scales.

2 We deﬁne the Dark Fluid (d) as a barotropic ﬂuid with cs = 0. 1 Pd (ρ) = wd (ρ)ρd −→ wd ∝ −→ Pd = const. ρd

Astrophysical observations constrain Pd ρastro, but no necessarily equal to zero, and this will be important to interpret the single ﬂuid as dark energy too. Towards a single ﬂuid description

The elements: locally clustering (perturbations) and globally accelerating (background).

Astrophysics At very large scales (> 100’s Mpc) the Universe is homogeneous and isotropic: ρ(t,~x) = ρ(t)

Matter perturbations: δ(t,~x) = δρ/ρ

00 0 2 2 δ + 2Hδ + cs kphys − 4πGρ δ = 0,

where a prime stands for a cosmic time derivative.

r π I l > lJ Growth Jeans length lJ = cs Gρ I l < lJ No growth and we infer non-baryonic structure growth from dwarf galaxies to cosmological scales.

2 We deﬁne the Dark Fluid (d) as a barotropic ﬂuid with cs = 0. 1 Pd (ρ) = wd (ρ)ρd −→ wd ∝ −→ Pd = const. ρd

Astrophysical observations constrain Pd ρastro, but no necessarily equal to zero, and this will be important to interpret the single ﬂuid as dark energy too. Towards a single ﬂuid description

The elements: locally clustering (perturbations) and globally accelerating (background).

Astrophysics At very large scales (> 100’s Mpc) the Universe is homogeneous and isotropic: ρ(t,~x) = ρ(t)

Matter perturbations: δ(t,~x) = δρ/ρ

00 0 2 2 δ + 2Hδ + cs kphys − 4πGρ δ = 0,

where a prime stands for a cosmic time derivative.

r π I l > lJ Growth Jeans length lJ = cs Gρ I l < lJ No growth and we infer non-baryonic structure growth from dwarf galaxies to cosmological scales.

2 We deﬁne the Dark Fluid (d) as a barotropic ﬂuid with cs = 0. 1 Pd (ρ) = wd (ρ)ρd −→ wd ∝ −→ Pd = const. ρd

Astrophysical observations constrain Pd ρastro, but no necessarily equal to zero, and this will be important to interpret the single ﬂuid as dark energy too. Background Cosmology

2 2 2 a˙ 8πGa H ≡ = (ργ + ρν + ρ + ρ ), a 3 b d

ρ˙rel + 4Hρrel = 0, ρ˙nonrel + 3Hρnonrel = 0, ρ˙d + 3H(1 + wd )ρd = 0.

ρ K ρ ρ = d0 1 + , P = − d0 d 1 + K a3 d 1 + K

ρd > 0 −→ Pd < 0.

The single ﬂuid behaves as dark matter and dark energy! Background Cosmology

2 2 2 a˙ 8πGa H ≡ = (ργ + ρν + ρ + ρ ), a 3 b d

ρ˙rel + 4Hρrel = 0, ρ˙nonrel + 3Hρnonrel = 0, ρ˙d + 3H(1 + wd )ρd = 0.

ρ K ρ ρ = d0 1 + , P = − d0 d 1 + K a3 d 1 + K

ρd > 0 −→ Pd < 0.

The single ﬂuid behaves as dark matter and dark energy! The Dark Degeneracy

1 Equation of state parameter of the dark ﬂuid: wd = − 1+Ka−3

Equation of state parameter of the dark components in the ΛCDM model: P i wi ρi 1 wdark ≡ P ρ = − Ω i i 1+ dm a−3 ΩΛ

Under the identiﬁcations

Ωd = Ωdm + ΩΛ, K = Ωdm/ΩΛ the background evolution in both models is exactly the same!

I The Dark Degeneracy (M. Kunz 2009). Any collection of ﬂuids whose total equation of state parameter is equal to wdark presents the degeneracy. The background evolution is the same as in the ΛCDM model.

I The fundamental reason is that we deﬁne

1 Equation of state parameter of the dark ﬂuid: wd = − 1+Ka−3

Equation of state parameter of the dark components in the ΛCDM model: P i wi ρi 1 wdark ≡ P ρ = − Ω i i 1+ dm a−3 ΩΛ

Under the identiﬁcations

Ωd = Ωdm + ΩΛ, K = Ωdm/ΩΛ the background evolution in both models is exactly the same!

I The fundamental reason is that we deﬁne

1 Equation of state parameter of the dark ﬂuid: wd = − 1+Ka−3

Equation of state parameter of the dark components in the ΛCDM model: P i wi ρi 1 wdark ≡ P ρ = − Ω i i 1+ dm a−3 ΩΛ

Under the identiﬁcations

Ωd = Ωdm + ΩΛ, K = Ωdm/ΩΛ the background evolution in both models is exactly the same!

I The fundamental reason is that we deﬁne

dark 1 obs T = Gµν − T . µν 8πG µν Any split of the the dark energy momentum tensor is arbitrary, but the particular given properties of the single ﬂuid are important to exactly emulate the ΛCDM model. Beyond the background Perturbations to the background metric in the Newtonian Conformal gauge 2 2 2 i j ds = a (τ) − (1 + 2Ψ)dτ + (1 − 2Φ)δij dx dx , Perturbations to the matter ﬁelds.

0 i i i i i T 0 = −ρ(1 + δ), T 0 = −(ρ + P)v , T j = (P + δP)δ j + PΠ j .

2 i For the dark ﬂuid, δPd = cs δρd = 0, and being a perfect ﬂuid Π j = 0, the perturbed eqs. are:

˙ ˙ δd = −(1 + wd )(θd − 3Φ) + 3Hwd δd , ˙ 2 θd = −Hθd + k Ψ.

But these are exactly the same as the usual Cold Dark Matter equations, provided that

I ρd δd = ρdmδdm

I ρd (1 + wd )θd = ρdmθdm The ﬂuid equations are supplemented with the Einstein’s equations:

2 2 X k Φ = −4πGa ρi ∆i , i 2 2 X k (Φ − Ψ) = 12πGa (ρi + Pi )σi . i

θi the sum runs over all fluid contributions and ∆i = δi + 3H(1 + wi ) the rest fluid energy density. k2 Beyond the background Perturbations to the background metric in the Newtonian Conformal gauge 2 2 2 i j ds = a (τ) − (1 + 2Ψ)dτ + (1 − 2Φ)δij dx dx , Perturbations to the matter fields.

0 i i i i i T 0 = −ρ(1 + δ), T 0 = −(ρ + P)v , T j = (P + δP)δ j + PΠ j .

2 i For the dark ﬂuid, δPd = cs δρd = 0, and being a perfect ﬂuid Π j = 0, the perturbed eqs. are:

˙ ˙ δd = −(1 + wd )(θd − 3Φ) + 3Hwd δd , ˙ 2 θd = −Hθd + k Ψ.

But these are exactly the same as the usual Cold Dark Matter equations, provided that

I ρd δd = ρdmδdm

I ρd (1 + wd )θd = ρdmθdm The ﬂuid equations are supplemented with the Einstein’s equations:

2 2 X k Φ = −4πGa ρi ∆i , i 2 2 X k (Φ − Ψ) = 12πGa (ρi + Pi )σi . i

0 i i i i i T 0 = −ρ(1 + δ), T 0 = −(ρ + P)v , T j = (P + δP)δ j + PΠ j .

2 i For the dark ﬂuid, δPd = cs δρd = 0, and being a perfect ﬂuid Π j = 0, the perturbed eqs. are:

˙ ˙ δd = −(1 + wd )(θd − 3Φ) + 3Hwd δd , ˙ 2 θd = −Hθd + k Ψ.

But these are exactly the same as the usual Cold Dark Matter equations, provided that

I ρd δd = ρdmδdm

I ρd (1 + wd )θd = ρdmθdm The ﬂuid equations are supplemented with the Einstein’s equations:

2 2 X k Φ = −4πGa ρi ∆i , i 2 2 X k (Φ − Ψ) = 12πGa (ρi + Pi )σi . i

0 i i i i i T 0 = −ρ(1 + δ), T 0 = −(ρ + P)v , T j = (P + δP)δ j + PΠ j .

2 i For the dark ﬂuid, δPd = cs δρd = 0, and being a perfect ﬂuid Π j = 0, the perturbed eqs. are:

˙ ˙ δd = −(1 + wd )(θd − 3Φ) + 3Hwd δd , ˙ 2 θd = −Hθd + k Ψ.

But these are exactly the same as the usual Cold Dark Matter equations, provided that

I ρd δd = ρdmδdm

I ρd (1 + wd )θd = ρdmθdm The ﬂuid equations are supplemented with the Einstein’s equations:

2 2 X k Φ = −4πGa ρi ∆i , i 2 2 X k (Φ − Ψ) = 12πGa (ρi + Pi )σi . i

θi the sum runs over all ﬂuid contributions and ∆i = δi + 3H(1 + wi ) the rest ﬂuid energy density. k2 -3 0.6 HaL HcL -4 0.5

D -5 Α < 1 6 ∆DM

- 0.4 -6 10

Α > 1 ∆ 0.3

´ -7 @

-8 0.2 F ∆ -9 d 0.1 Α grows -10 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.6 HbL -0.2 HdL

0.5 D -0.4 3

Α > 1 - -0.6

0.4 10 b

∆ -0.8 ´

Α < 1 @ Α < 1 0.3 -1.0 Κ

Α > 1 Θ -1.2

0.2 -1.4

0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 a a

ρd δd = αρdmδdm, ρd (1 + wd )θd = αρdmθdm

Beyond linear order...

ΛCDM d ΛCDM (i) d (i) 1. If Tµν = Tµν then Tµν = Tµν (0) (1) (2) Tµν = T + T + T + ··· ΛCDM (i) d (i) ΛCDM (i+1) d (i+1) µν µν µν 2. Tµν = Tµν does not imply Tµν = Tµν

(A. Aviles and J.L. Cervantes-Cota, 2011b) -3 0.6 HaL HcL -4 0.5

D -5 Α < 1 6 ∆DM

- 0.4 -6 10

Α > 1 ∆ 0.3

´ -7 @

-8 0.2 F ∆ -9 d 0.1 Α grows -10 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.6 HbL -0.2 HdL

0.5 D -0.4 3

Α > 1 - -0.6

0.4 10 b

∆ -0.8 ´

Α < 1 @ Α < 1 0.3 -1.0 Κ

Α > 1 Θ -1.2

0.2 -1.4

0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 a a

ρd δd = αρdmδdm, ρd (1 + wd )θd = αρdmθdm

Beyond linear order...

ΛCDM d ΛCDM (i) d (i) 1. If Tµν = Tµν then Tµν = Tµν (0) (1) (2) Tµν = T + T + T + ··· ΛCDM (i) d (i) ΛCDM (i+1) d (i+1) µν µν µν 2. Tµν = Tµν does not imply Tµν = Tµν

(A. Aviles and J.L. Cervantes-Cota, 2011b) The averaging problem

The smoothness of the Universe at large scales allows us to expand the energy-momentum tensor:

(0) (0) Tµν = hTµν i, Gµν (g ) = hTµν i.

But this is not true in general due to the nonlinear character of gravity, it is well known that backreaction effects contribute to this last equation. (S.R. Green and R.M. Wald, 2011)

In fact, hPi = P(hρi) is not necessarily true and non-linear instabilities occur (P. P. Avelino et al 2004, 2007, 2008).

However, there are two exceptions: the equation of state are P = constant and P = wρ, with w a constant. Ours is the former model and we are safe from this averaging problem. The averaging problem

The smoothness of the Universe at large scales allows us to expand the energy-momentum tensor:

(0) (0) Tµν = hTµν i, Gµν (g ) = hTµν i.

But this is not true in general due to the nonlinear character of gravity, it is well known that backreaction effects contribute to this last equation. (S.R. Green and R.M. Wald, 2011)

In fact, hPi = P(hρi) is not necessarily true and non-linear instabilities occur (P. P. Avelino et al 2004, 2007, 2008).

However, there are two exceptions: the equation of state are P = constant and P = wρ, with w a constant. Ours is the former model and we are safe from this averaging problem. The averaging problem

The smoothness of the Universe at large scales allows us to expand the energy-momentum tensor:

(0) (0) Tµν = hTµν i, Gµν (g ) = hTµν i.

But this is not true in general due to the nonlinear character of gravity, it is well known that backreaction effects contribute to this last equation. (S.R. Green and R.M. Wald, 2011)

In fact, hPi = P(hρi) is not necessarily true and non-linear instabilities occur (P. P. Avelino et al 2004, 2007, 2008).

However, there are two exceptions: the equation of state are P = constant and P = wρ, with w a constant. Ours is the former model and we are safe from this averaging problem. The averaging problem

The smoothness of the Universe at large scales allows us to expand the energy-momentum tensor:

(0) (0) Tµν = hTµν i, Gµν (g ) = hTµν i.

But this is not true in general due to the nonlinear character of gravity, it is well known that backreaction effects contribute to this last equation. (S.R. Green and R.M. Wald, 2011)

In fact, hPi = P(hρi) is not necessarily true and non-linear instabilities occur (P. P. Avelino et al 2004, 2007, 2008).

However, there are two exceptions: the equation of state are P = constant and P = wρ, with w a constant. Ours is the former model and we are safe from this averaging problem. Multiple Interacting Cosmic Components ! µν X µν µν µν G = 8πG Ti + TSM with ∇µTSM = 0. i i labels the different dark components.

µν ν X ν ν 1 ν ν ∇µT = Q −→ Q = 0, Q = (q + δq)u + f i i i a i

ρ˙i + 3H(1 + wi )ρi = qi

˙ ˙ δPi qi δqi δi + (1 + wi )(θi − 3Φ) + 3H − wi δi + (δi − Ψ) − = 0 δρi ρi ρi

˙ 2 ˙ qi wi wi δPi /δρi 2 2 k fi θi + H 1 − 3wi + θi + θi − k δi − k Ψ + = 0 H(1+wi )ρi 1+wi 1+wi ρi (1+wi )

1 P 1 P Deﬁning δT = ρi δi , θT = ρi (1 + wi )θi . ρT i ρT (1+wT ) i

˙ ˙ δPT δT + (1 + wT )(θT − 3Φ) + 3H − wT δT = 0 δρT and ˙ ˙ wT δPT /δρT 2 2 2 θT + H(1 − 3wT )θT + θT − k δT − k Ψ + k σT = 0. 1 + wT 1 + wT

P i wi ρi 1 2 I Taking, wT ≡ P ρ = − Ω and imposing csT = 0, we obtain the dark ﬂuid. i i 1+ dm a−3 ΩΛ Multiple Interacting Cosmic Components ! µν X µν µν µν G = 8πG Ti + TSM with ∇µTSM = 0. i i labels the different dark components.

µν ν X ν ν 1 ν ν ∇µT = Q −→ Q = 0, Q = (q + δq)u + f i i i a i

ρ˙i + 3H(1 + wi )ρi = qi

˙ ˙ δPi qi δqi δi + (1 + wi )(θi − 3Φ) + 3H − wi δi + (δi − Ψ) − = 0 δρi ρi ρi

˙ 2 ˙ qi wi wi δPi /δρi 2 2 k fi θi + H 1 − 3wi + θi + θi − k δi − k Ψ + = 0 H(1+wi )ρi 1+wi 1+wi ρi (1+wi )

1 P 1 P Deﬁning δT = ρi δi , θT = ρi (1 + wi )θi . ρT i ρT (1+wT ) i

˙ ˙ δPT δT + (1 + wT )(θT − 3Φ) + 3H − wT δT = 0 δρT and ˙ ˙ wT δPT /δρT 2 2 2 θT + H(1 − 3wT )θT + θT − k δT − k Ψ + k σT = 0. 1 + wT 1 + wT

µν ν X ν ν 1 ν ν ∇µT = Q −→ Q = 0, Q = (q + δq)u + f i i i a i

ρ˙i + 3H(1 + wi )ρi = qi

˙ ˙ δPi qi δqi δi + (1 + wi )(θi − 3Φ) + 3H − wi δi + (δi − Ψ) − = 0 δρi ρi ρi

˙ 2 ˙ qi wi wi δPi /δρi 2 2 k fi θi + H 1 − 3wi + θi + θi − k δi − k Ψ + = 0 H(1+wi )ρi 1+wi 1+wi ρi (1+wi )

1 P 1 P Deﬁning δT = ρi δi , θT = ρi (1 + wi )θi . ρT i ρT (1+wT ) i

˙ ˙ δPT δT + (1 + wT )(θT − 3Φ) + 3H − wT δT = 0 δρT and ˙ ˙ wT δPT /δρT 2 2 2 θT + H(1 − 3wT )θT + θT − k δT − k Ψ + k σT = 0. 1 + wT 1 + wT

µν ν X ν ν 1 ν ν ∇µT = Q −→ Q = 0, Q = (q + δq)u + f i i i a i

ρ˙i + 3H(1 + wi )ρi = qi

˙ ˙ δPi qi δqi δi + (1 + wi )(θi − 3Φ) + 3H − wi δi + (δi − Ψ) − = 0 δρi ρi ρi

˙ 2 ˙ qi wi wi δPi /δρi 2 2 k fi θi + H 1 − 3wi + θi + θi − k δi − k Ψ + = 0 H(1+wi )ρi 1+wi 1+wi ρi (1+wi )

1 P 1 P Deﬁning δT = ρi δi , θT = ρi (1 + wi )θi . ρT i ρT (1+wT ) i

˙ ˙ δPT δT + (1 + wT )(θT − 3Φ) + 3H − wT δT = 0 δρT and ˙ ˙ wT δPT /δρT 2 2 2 θT + H(1 − 3wT )θT + θT − k δT − k Ψ + k σT = 0. 1 + wT 1 + wT

P i wi ρi 1 2 I Taking, wT ≡ P ρ = − Ω and imposing csT = 0, we obtain the dark ﬂuid. i i 1+ dm a−3 ΩΛ Interactions to the standard model

Couplings between cosmic components: baryons (b), photons (γ), dark ﬂuid (d) ...

I Thomson scattering (up to quadrupole moment):

δQb = δQγ = δQd = 0 2 2 2 k fγ = −k fb = −ργ (1 + wγ )axeneσT (θb − θγ ), k fd = 0

Now we look for interactions between the dark sector and the standard model...... to break the dark degeneracy?

I Electromagnetic inspired (ΣI ): ρ f = −f = ρ (1 + w )a d0 Σ (θ − θ )/k 2 d b d d 3 I b d mpa

2 2 2 n I Chameleon gravity inspired (ΣII ): keff = k g(ρ) = k /a

ρd0 2 fd = −fb = ρd (1 + wd )ΣII (θb − θd )/k mp Interactions to the standard model

Couplings between cosmic components: baryons (b), photons (γ), dark ﬂuid (d) ...

I Thomson scattering (up to quadrupole moment):

δQb = δQγ = δQd = 0 2 2 2 k fγ = −k fb = −ργ (1 + wγ )axeneσT (θb − θγ ), k fd = 0

Now we look for interactions between the dark sector and the standard model...... to break the dark degeneracy?

I Electromagnetic inspired (ΣI ): ρ f = −f = ρ (1 + w )a d0 Σ (θ − θ )/k 2 d b d d 3 I b d mpa

2 2 2 n I Chameleon gravity inspired (ΣII ): keff = k g(ρ) = k /a

ρd0 2 fd = −fb = ρd (1 + wd )ΣII (θb − θd )/k mp Interactions to the standard model

Couplings between cosmic components: baryons (b), photons (γ), dark ﬂuid (d) ...

I Thomson scattering (up to quadrupole moment):

δQb = δQγ = δQd = 0 2 2 2 k fγ = −k fb = −ργ (1 + wγ )axeneσT (θb − θγ ), k fd = 0

Now we look for interactions between the dark sector and the standard model...... to break the dark degeneracy?

I Electromagnetic inspired (ΣI ): ρ f = −f = ρ (1 + w )a d0 Σ (θ − θ )/k 2 d b d d 3 I b d mpa

2 2 2 n I Chameleon gravity inspired (ΣII ): keff = k g(ρ) = k /a

ρd0 2 fd = −fb = ρd (1 + wd )ΣII (θb − θd )/k mp CAMB and COSMOMC

6000 -6 6000 YI = 0.1 x 10 mT YII = 10 x mT YI = 0.01 YII = 5 YI = 0.001 YII = 0 5000 YI = 0 5000 YII = -5 YII = -10

4000 4000 2 2 K) K) µ µ ( ( l l

C 3000 C 3000 +1) +1) l l ( ( l l 2000 2000

1000 1000

0 0 10 100 1000 10 100 1000 l l

The effective thermalized cross section for 2 5 interaction II is a Σ ∼ Σ at z ∼ 10 . Before 3 II I −22 ΣI −22 cm /s 3.58 × 10 < < 8.68 × 10 . this epoch interaction I dominates. Just before 2 md GeV/c recombination, at z ' 1100, interactions I and II 3 −13 ΣII −13 cm /s are smaller in strength than the Thomson −1.94 × 10 < < −1.05 × 10 . 2 interaction by about 9 and 5 orders of md GeV/c magnitude, respectively. -6

0.01 Y0.02 I [ × 100.03 mT ]0.04 0.05 0.06 0.021 0 Using 0.023 1 b h Σ 2 I Σ + 0.025 II

-6

0.005 Y [ × 10 m ] 0.015 0.02 I 0.01 T 0 0.021 1 Σ b I h only 2 0.023 YII [ mT ] -10 -8 -6 -4 -2 0 0 0.01 0.02 Y I [ × 0.03 10 -6

m 0.04 T ] 0.05 0.06 0.135 3.4

0.13 3.3 0.125

0.12

] 3.2 s

2 0.115 A h 0.11 10 3.1 DM

1 0.105

log[10 3 0.1

0.095 2.9 0.09

0.085 2.8 0.022 0.025 0.92 0.96 1 1.04 h2 n 1b s

Red lines are for a model with ΣI and ΣII , blue for a model with only ΣI , and black for ΛCDM. 6000 YI + YII YI 5000 RCDM WMAP7yr 2

K) 4000 µ ( l

C 3000 +1)

l 2000 ( l 1000

300 6 -500

1 10 100 1000 l

CMB power spectrum for the mean estimated values stemming from CosmoMC. Table: Summary of constraints. The upper panel contains the parameter spaces explored with MCMC for each one of the three models. The bottom panel contains derived parameters. The data used are the WMAP seven years data, Union 2 compilation and HST.

a a a Parameter Model ΣI + ΣII Model ΣI Model ΛCDM

2Ω 2 . +0.066 . +0.056 . +0.053 10 bh 2 420−0.064 2 219−0.056 2 243−0.053 Ω 2 . +0.0061 . +0.0047 . +0.0041 c h 0 1114−0.0060 0 1046−0.0049 0 1089−0.0041 θ . +0.003 . +0.003 . +0.003 1 039−0.003 1 037−0.003 1 039−0.003 τ . +0.00565 . +0.0061 . +0.00618 0 08712−0.00721 0 08646−0.0067 0 08797−0.00627 8Σ b . +1.169 . +0.2980 10 I 2 910−1.229 0 4845−0.3824 −− Σ b − . +2.218 II 7 169−1.959 −− −− . +0.0192 . +0.0135 . +0.0123 ns 0 9869−0.0184 0 9551−0.0137 0 9651−0.0124 [ 10 ] . +0.051 . +0.040 . +0.031 log 10 As 3 118−0.051 3 039−0.040 3 070−0.033 c ASZ 1.054 ± 0.578 0.9544 ± 0.5911 1.040 ± 0.574 Ω . +0.033 . +0.031 . +0.027 d 0 952−0.033 0 956−0.030 0 955−0.027 K . +0.042 . +0.034 . +0.034 0 296−0.041 0 270−0.036 0 291−0.032 . +0.12 . +0.11 . +1.16 t0 13 64−0.13 Gyr 13 85−0.12 Gyr 13 79−1.15 Gyr Ω . +0.024 . +0.022 . +0.019 Λ 0 734−0.024 0 754−0.021 0 740−0.020 d . +1.86 . +2.09 . +1.71 H0 71 55−1.91 71 95−1.96 71 14−1.85 Notes. a. The mean values of the posterior distribution for each parameter. The quoted errors show the 68% conﬁdence levels. −25 2 b. ΣI and ΣII are given in units of the Thomson scattering cross section times the speed of light, σT = 6.65 × 10 cm and c = 1. c. The quoted errors in Asz are the standard deviations of the distributions. d. H0 is given in Km/s/Mpc. Matter Power spectrum

Y1 Y1 + Y2, set 2 Y2 Y1 + Y2, set 1 RCDM RCDM, set 2 DR7 LRG DR7 LRG ] ] 3 3 10000 Mpc Mpc

-3 10000 -3 [ h [ h P(k) P(k)

1000 0.01 0.1 0.01 0.1 k [ h Mpc-1 ] k [ h Mpc-1 ]

Set 1 includes WMAP7, HST, and UNION 2. Set 2 includes WMAP7, HST, UNION 2, and SDSS DR7 LRG 2 1DM h 0.08 0.09 0.11 0.12 0.13 0.14 0.15 0.1 0.02 0.021 0.022 e 1 Set 1 0.023 b h 2 0.024 0.025 0.026

2 1DM h 0.08 0.09 0.11 0.12 0.13 0.14 0.15 0.1 0.02 e 2 Set 1 0.023 b h 2 breaking degeneracies?

But these interactions do not necessarily break the dark degeneracy. They can be understood as interactions between dark matter and baryons:

˙ ˙ δqdm δdm = −θdm + 3Φ + , ρdm 2 ˙ 2 k fdm θdm = −Hθdm + k Ψ + , ρdm with

δqdm = δqd and fdm = fd . Summary