Testing dark energy as a function of scale

Ignacy SawickiAIMS

arXiv:1305.008, 1210.0439 (PRD) + 1208.4855 (JCAP)Together with: L. Amendola, M. Kunz, M. Motta, I.Saltas.

The Bygone Era of Easy Choices

Λ

Dark Energy

• 𝑤 = −1

• 𝑤 ≠ −1

“Modified gravity”

• 𝑤 =/≠ −1• 𝑐s

2 = 1• 𝜂 ≠ 1

k-essence• 𝑤 =/≠ −1• 𝑐s

2 ≠ 1• 𝜂 = 1

15 November 2013 AIMS, Muizenberg

Managing the Model Bestiary

Slow-Rolling 𝝓𝟐 ≪ 𝛀𝑿𝑯

𝟐

Fast-Rolling 𝝓𝟐 ∼ 𝛀𝑿𝑯

𝟐

Acceleration effectively from Λ

𝑐s2 = 1

Non-minimal coupling gives fifth force

Chameleon screening & Compton scale

(coupled) Quintessence, 𝒇 𝑹 , Brans-Dicke

Acceleration from kinetic condensate

Can describe hydrodynamics (incl. imperfect corrections)

Realistically should be nearly shift-symmetric

Non-trivial acoustic metric

Screening through Vainsteinmechanism

k-essence, KGB, galileons, shift-symmetric Horndeski


What you get depends on what you put in

PlanckAde et al. (2013)

SDSS-III DR9Anderson et al. (2012)


In this talk…

What properties can we actually observe without having assumed a model first? Only 𝐻(𝑧) not 𝑤 Only potentials Φ, Ψ, not e.g. DM growth rate

Can we measure properties of DE in a model-independent way? Not all, but can form null tests from data which can eliminate

model classes

Fundamental reason: dark degeneracy between dark matter and dark energy All cosmological probes are only sensitive to geodesics


Our Limited Eyes

Galaxies P(k): BAO/RSD

Galaxy Shapes:Lensing

Supernovae:𝑑L


The Best-Case Scenario

as little as feasibleAssume

• FRW + (scalar) linear perturbations

• Matter & light move on geodesics of some metric

• Linear density bias 𝛿gal = 𝑏(𝑘, 𝑎)𝛿m• (Equivalence principle/Universality of couplings)

build Super-EuclidInfinite €$£¥

• Desired precision for position and redshift

• SNe

• lensing

• counting galaxies


LSS: Galaxy Power Spectrum

Baryon Acoustic Oscillations is a fixed ruler

use to measure distance if same physical size


SDSS III, Anderson et al. (2012)

Background

• 𝐻0𝐷 𝑧 =1

−Ω𝑘0sinh −Ω𝑘0

𝐻0d𝑧

𝐻(𝑧)

SNe, ⊥ BAO, CMB peak

• 𝐻 𝑧 =Δ𝑧

𝑠 𝑧∥ BAO

• Observables are 𝐻(𝑧)/𝐻0, Ω𝑘0• Not𝑤 𝑧 or Ωm

In principle


Dark Degeneracy

In principle no way of measuring split between DE and DM

Only choice of parameterisation breaks degeneracy

e.g. constant 𝑤

Kunz (2007)

Ω𝑋 = 1 −𝐻02

𝐻2Ω𝑘0𝑎

−2 + Ωm0𝑎−3

Anderson et al. (2012)


Natural EoS for Quintessence


Huterer and Peiris (2006)

𝑤 = 𝑤0 + 𝑤𝑎 1 − 𝑎 ?

Perturbations

Want to measure 𝐺effand 𝜂 to determine DE model

Can we actually do this?

Remember: 𝐺eff and 𝜂hide dynamics No reason for them to be

simple

3 Φ′ −Ψ + 𝑘2Φ =3

2Ωm𝛿m +

𝟑

𝟐𝛀𝑿𝜹𝑿

Φ+Ψ = 𝜹𝝅 = 𝜎Ω𝑋𝛿𝑋

𝑘2Ψ = −3

2𝑮𝐞𝐟𝐟 𝒌, 𝒂 Ωm𝛿m

Φ+Ψ = 1 − 𝜼(𝒌, 𝒂) Ψ

d𝑠2 = − 1 + 2Ψ d𝑡2 + 𝑎2 1 + 2Φ d𝒙𝟐

𝛿m′′ + 2 +

𝐻′

𝐻𝛿m′ −

3

2𝑮𝐞𝐟𝐟 𝒌, 𝒂 𝜹𝐦 = 0


Is dark energy smooth?

• 𝜂 = 1

• 𝐺eff = 1

Λ: of course

• 𝑐s2 = 1

• 𝜂 = 1

• 𝐺eff → 1 +𝛼

𝑐s2𝑘2

Quintessence: more or less

• 𝑐s2 = 1

• 𝜂 =1

2

• 𝐺eff =4

3

𝑓(𝑅): not at all

𝛿𝜌𝑋 = −1

3𝛿𝜌m


LSS: Measure Galaxy Shapes

Weak lensing Gravity from DM and DE

changes path of light, distorting galaxy shapes

Can invert this shear to measure the gravitational potential

𝐿 = 𝑘2 Φ−Ψ

Measure distribution of potential not of DM


LSS: Galaxy Power Spectrum

Amplitude: related to dark matter through bias𝛿gal = 𝑏 𝑘, 𝑧 𝛿m 𝑏 can only be measured

when you know what DE is

𝜎8 is not an observable


SDSS III, Anderson et al. (2012)

LSS: Redshift-Space Distortions Real Space

Redshift Space

Measure peculiar velocity of galaxies, 𝜃gal


Hawkins et al (2002)

𝛿gal𝑧 𝑘, 𝑧, cos2𝛼 = 𝛿gal 𝑘, 𝑧 − cos2𝛼

𝜃gal 𝑘, 𝑧

𝐻

How are RSD (ab)used?

BOSS DR9 + WiggleZ, SDSS LRG, 2dFRGS Samushia et al. (2012)


• Only measuring velocities of galaxies… everything else is our interpretation

• Non-linearity important at early times. How do you set the initial conditions?

Continuity for DM

𝛿m′ + 𝜃m ≈ 0

• If 𝜃m = 𝜃gal then can measure

dark matter growth rate

𝛿m′ ≡ 𝑓𝛿m = 𝑓𝜎8

From acceleration measure force

𝛿gal𝑧 𝑘, 𝑧, cos2𝛼 = 𝛿gal 𝑘, 𝑧 − cos2𝛼

𝜃gal 𝑘, 𝑧

𝐻

Galaxies move on geodesics

(𝑎2𝜃gal)′ =𝑘2

𝐻Ψ


𝑘2Ψ = −𝑅′ − 𝑅 2 +𝐻′

𝐻

𝐴(𝑘, 𝑧) 𝑅(𝑘, 𝑧)

𝑘2 Φ−Ψ = 𝐿

Reconstruction of Metric

Ratios of potentials always observable

We measure power spectra of potentials, not dark matter

−Φ

Ψ= 𝜂

Ψ′

Ψ= 1 + Γ


What about 𝐺eff?

Dark degeneracy strikes back

No way of measuring 𝐺eff without a model

Would somehow need to weigh DM and separated from DE

𝐺eff′

𝐺eff+ 𝐺eff

Ωm0 1 + 𝜂

𝐿/𝑅= Γ


So what?

Full constraints on particular models of course are perfectly fine Expensive and non-generic: how to anoint the particular

model? Initial conditions?

In practice, we use parameterisations which represent parts of model space Are they consistent? Do they say anything about my model? Do they allow us to unambiguously see the things my

model can’t do?


The model space

If 𝑋 small, then nothing new

Quintessence𝑓 𝑅Brans-Dicke

If 𝑋 large, then any term can be important

The background is a path across the 4D operator space


ℒ ∼ 𝐾 𝑋,𝜙 + 𝐺3 𝑋, 𝜙 ⧠𝜙 +

+𝐺4 𝑋, 𝜙 𝛻𝜇𝛻𝜈𝜙2+ 𝐺5 𝑋, 𝜙 𝛻𝜇𝛻𝜈𝜙

3+ grav

Horndeski (1974)Nicolis, Ratazzi, Tricherini (2009

Deffayet, Gao, Steer, Zahariade (2011)

ℒ ≈ 𝑋 + 𝑉 𝜙 + 𝑓(𝜙)𝑅

2𝑋 ≡ 𝜕𝜇𝜙2

What can we actually say?

On FRW, get corrections to perfect fluid that go as 𝑘2

𝑇𝜇𝜈𝜙= 𝑇𝜇𝜈

perf+ 𝜅3𝑘

2𝜇𝜈 + 𝜅4𝑘

2𝜇𝜈

Alternative: e.g. braneworld models: corrections go as 𝑘 Lorentz-violating: higher powers of 𝑘


𝑆2(𝑘) = d𝑡𝑎3𝜅perf 𝑡 𝒪perf 𝑡, 𝑘2 + 𝜅3 𝑡 𝒪3 𝑡, 𝑘2 +

+𝜅4 𝑡 𝒪4 𝑡, 𝑘2 + 𝜅5(𝑡)𝒪5(𝑡, 𝑘2)

Measure DE properties fromscale dependence

on the realised background

Amin, Wagoner, Blandford (2007)

𝐺eff 𝜂, 𝐺effJeans

Blas, Sibiryakov (2011)

Creminelli, Luty, Nicolis, Senatore (2006)IS, Saltas, Amendola, Kunz (2012)

Gleyzes, Piazza, Vernizzi (2013)

Is it any scalar at all?

𝛿𝑇00 ⊃ 𝛿𝜙, 𝛿𝜙, 𝛿m 𝛿𝑇𝑖

0 ⊃ 𝛿𝜙, 𝜹𝝓, 𝜃m 𝛿𝑇𝑗𝑖 ⊃ 𝜹𝝓

𝛿𝑇𝑖𝑖 ⊃ 𝛿𝜙 , 𝛿𝜙, 𝜹𝝓 𝛿𝜙 = EoM

Φ′′

Ψ+ 𝛼1

Φ′

Ψ+ 𝛼2

Ψ′

Ψ+ 𝛼3 + 𝛼4𝑘

2Φ

Ψ+ 𝛼5 + 𝛼6𝑘

2 Ψ = Ωm𝛼7𝜃m

Γ(𝑘, 𝑧) 𝑅′/𝑅

Fix 𝛼𝑖(𝑧)

𝜂(𝑘, 𝑧)

2 October2013 NYU Abu Dhabi

𝑓(𝑅): one param 𝑚C(𝑧)

The Takeaway

In principle, we can reconstruct the evolution of the metric We cannot get the split between DE and DM without assuming

some class of models

Generically, DE models predict a change in the power law for Ψ as a function of scale Different frameworks give you different scale dependence: could

potentially eliminate scalars completely

If I told you today that the background was inconsistent with 𝑤 = −1, what have you learned? If that happens, we’ll have to be more sophisticated about

interpreting the data


Testing dark energy as a function of scale

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