Testing dark energy as a function of scale
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Ignacy SawickiAIMS
arXiv:1305.008, 1210.0439 (PRD) + 1208.4855 (JCAP)Together with: L. Amendola, M. Kunz, M. Motta, I.Saltas.
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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
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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
15 November 2013 AIMS, Muizenberg
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What you get depends on what you put in
PlanckAde et al. (2013)
SDSS-III DR9Anderson et al. (2012)
15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
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15 November 2013 AIMS, Muizenberg
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Our Limited Eyes
Galaxies P(k): BAO/RSD
Galaxy Shapes:Lensing
Supernovae:𝑑L
15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
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LSS: Galaxy Power Spectrum
Baryon Acoustic Oscillations is a fixed ruler
use to measure distance if same physical size
15 November 2013 AIMS, Muizenberg
SDSS III, Anderson et al. (2012)
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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
15 November 2013 AIMS, Muizenberg
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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)
15 November 2013 AIMS, Muizenberg
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Natural EoS for Quintessence
15 November 2013 AIMS, Muizenberg
Huterer and Peiris (2006)
𝑤 = 𝑤0 + 𝑤𝑎 1 − 𝑎 ?
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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
15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
SDSS III, Anderson et al. (2012)
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LSS: Redshift-Space Distortions Real Space
Redshift Space
Measure peculiar velocity of galaxies, 𝜃gal
15 November 2013 AIMS, Muizenberg
Hawkins et al (2002)
𝛿gal𝑧 𝑘, 𝑧, cos2𝛼 = 𝛿gal 𝑘, 𝑧 − cos2𝛼
𝜃gal 𝑘, 𝑧
𝐻
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How are RSD (ab)used?
BOSS DR9 + WiggleZ, SDSS LRG, 2dFRGS Samushia et al. (2012)
15 November 2013 AIMS, Muizenberg
• 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
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From acceleration measure force
𝛿gal𝑧 𝑘, 𝑧, cos2𝛼 = 𝛿gal 𝑘, 𝑧 − cos2𝛼
𝜃gal 𝑘, 𝑧
𝐻
Galaxies move on geodesics
(𝑎2𝜃gal)′ =𝑘2
𝐻Ψ
15 November 2013 AIMS, Muizenberg
𝑘2Ψ = −𝑅′ − 𝑅 2 +𝐻′
𝐻
𝐴(𝑘, 𝑧) 𝑅(𝑘, 𝑧)
𝑘2 Φ−Ψ = 𝐿
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Reconstruction of Metric
Ratios of potentials always observable
We measure power spectra of potentials, not dark matter
−Φ
Ψ= 𝜂
Ψ′
Ψ= 1 + Γ
15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
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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?
15 November 2013 AIMS, Muizenberg
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15 November 2013 AIMS, Muizenberg
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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
15 November 2013 AIMS, Muizenberg
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
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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
15 November 2013 AIMS, Muizenberg
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)
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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)
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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
15 November 2013 AIMS, Muizenberg