- The document discusses methods for characterizing dark energy and modified gravity models in a model-independent way using cosmological observations. - Due to the "dark degeneracy" between dark matter and dark energy, it is not possible to separately measure the properties of dark matter and dark energy without assuming a specific model class. - Observables like the Hubble parameter H(z) and gravitational potentials can be reconstructed from the data, but this does not break the degeneracy between dark matter and dark energy contributions. - The scale-dependence of quantities like the gravitational potentials and growth rate can be used to test and constrain broad classes of dark energy and modified gravity models in a more model-independent way.

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

2. The Bygone Era of Easy Choices
Λ
•
𝑤 = −1
Dark Energy
•
𝑤 ≠ −1
“Modified
gravity”
•
•
•
k-essence
𝑤 =/≠ −1
2
𝑐s = 1
𝜂≠1
15 November 2013
AIMS, Muizenberg
•
•
•
𝑤 =/≠ −1
2
𝑐s ≠ 1
𝜂=1

3. Managing the Model Bestiary




Acceleration effectively from Λ
2
𝑐s = 1
Non-minimal coupling gives fifth
force
Chameleon screening & Compton
scale






(coupled) Quintessence, 𝒇 𝑹 ,
Brans-Dicke
Slow-Rolling
𝝓𝟐 ≪ 𝛀𝑿 𝑯𝟐
15 November 2013

Acceleration from kinetic
condensate
Can describe hydrodynamics
(incl. imperfect corrections)
Realistically should be nearly
shift-symmetric
Non-trivial acoustic metric
Screening through Vainstein
mechanism
k-essence, KGB, galileons,
shift-symmetric Horndeski
Fast-Rolling
𝝓𝟐 ∼ 𝛀𝑿 𝑯𝟐
AIMS, Muizenberg

4. What you get depends on what
you put in
Planck
Ade et al. (2013)
SDSS-III DR9
Anderson et al. (2012)
15 November 2013
AIMS, Muizenberg

5. 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

6. 15 November 2013
AIMS, Muizenberg

7. Our Limited Eyes
Supernovae:
𝑑L
15 November 2013
Galaxies P(k):
BAO/RSD
AIMS, Muizenberg
Galaxy Shapes:
Lensing

8. The Best-Case Scenario
Assume
•
•
•
•
as little as feasible
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)
Infinite €$£¥ build Super-Euclid
• Desired precision for position and redshift
• SNe
• lensing
• counting galaxies
15 November 2013
AIMS, Muizenberg

9. 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)
15 November 2013
AIMS, Muizenberg

10. Background
SNe, ⊥ BAO,
CMB peak
• 𝐻0 𝐷 𝑧 =
∥ BAO
• 𝐻 𝑧 =
In principle
15 November 2013
1
−Ω 𝑘0
sinh
−Ω 𝑘0
𝐻0 d𝑧
𝐻(𝑧)
Δ𝑧
𝑠 𝑧
• Observables are 𝐻(𝑧)/𝐻0 , Ω 𝑘0
• Not 𝑤 𝑧 or Ωm
AIMS, Muizenberg

11. Dark Degeneracy
2
𝐻0
Ω 𝑋 = 1 − 2 Ω 𝑘0 𝑎−2 + Ωm0 𝑎−3
𝐻

In principle no way of
measuring split
between DE and DM

Only choice of
parameterisation
breaks degeneracy
 e.g. constant 𝑤
15 November 2013
Anderson et al. (2012)
AIMS, Muizenberg
Kunz (2007)

12. Huterer and Peiris (2006)
Natural EoS for Quintessence
𝑤 = 𝑤0 + 𝑤 𝑎 1 − 𝑎
15 November 2013
AIMS, Muizenberg
?

13. Perturbations
d𝑠 2 = − 1 + 2Ψ d𝑡 2 + 𝑎2 1 + 2Φ d𝒙 𝟐
′
2
3
Ω 𝛿
2 m m
3 Φ −Ψ + 𝑘 Φ=
Φ + Ψ = 𝜹𝝅 = 𝜎Ω 𝑋 𝛿 𝑋
+
𝟑
𝟐
𝛀𝑿 𝜹𝑿


Want to measure 𝐺eff
and 𝜂 to determine DE
model
Can we actually do this?
3
𝑘 2 Ψ = − 𝑮 𝐞𝐟𝐟 𝒌, 𝒂 Ωm 𝛿m
2
Φ + Ψ = 1 − 𝜼(𝒌, 𝒂) Ψ

Remember: 𝐺eff and 𝜂
hide dynamics
 No reason for them to be
′′
𝛿m
𝐻′
+ 2+
𝐻
15 November 2013
′
𝛿m
3
− 𝑮 𝐞𝐟𝐟 𝒌, 𝒂 𝜹 𝐦 = 0
2
AIMS, Muizenberg
simple

14. Is dark energy smooth?
• 𝜂=1
• 𝐺eff = 1
Λ:
of course
2
• 𝑐s = 1
• 𝜂=1
• 𝐺eff → 1 +
2
• 𝑐s = 1
𝛼
2
𝑐s 𝑘 2
Quintessence:
more or less
• 𝜂=
1
2
• 𝐺eff =
4
3
𝑓(𝑅):
not at all
1
𝛿𝜌 𝑋 = − 𝛿𝜌m
3
15 November 2013
AIMS, Muizenberg

15. 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

15 November 2013
AIMS, Muizenberg
𝐿 = 𝑘2 Φ − Ψ
Measure
distribution of
potential not of DM

16. 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

15 November 2013
AIMS, Muizenberg
𝐿 = 𝑘2 Φ − Ψ
Measure
distribution of
potential not of DM

17. LSS: Galaxy Power Spectrum

Amplitude: related to
dark matter through
bias
𝛿gal = 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)
15 November 2013
AIMS, Muizenberg

18. LSS: Redshift-Space Distortions
Hawkins et al (2002)


15 November 2013
𝑘,
𝑧, cos 2
𝛼 = 𝛿gal 𝑘, 𝑧 −
AIMS, Muizenberg
Redshift Space

𝑧
𝛿gal
Real Space
Measure peculiar
velocity of galaxies,
𝜃gal
cos 2
𝜃gal 𝑘, 𝑧
𝛼
𝐻

19. How are RSD (ab)used?
 Continuity for DM
′
𝛿m + 𝜃m ≈ 0
•
If 𝜃m = 𝜃gal then can measure
dark matter growth rate
′
𝛿m ≡ 𝑓𝛿m = 𝑓𝜎8
•
•
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?

20. From acceleration measure force
𝑧
𝛿gal
𝑘,
𝑧, cos 2
𝛼 = 𝛿gal 𝑘, 𝑧 −
cos 2
𝐴(𝑘, 𝑧)

𝜃gal 𝑘, 𝑧
𝛼
𝐻
𝑅(𝑘, 𝑧)
Galaxies move on geodesics
2
(𝑎 𝜃gal )′ =
𝑘2
Ψ
𝐻
𝐻′
𝑘 2 Ψ = −𝑅′ − 𝑅 2 +
𝐻
𝑘2 Φ − Ψ = 𝐿
15 November 2013
AIMS, Muizenberg

21. Reconstruction of Metric

Ratios of potentials always observable
Φ
− = 𝜂
Ψ

Ψ′
=1+Γ
Ψ
We measure power spectra of potentials,
not dark matter
15 November 2013
AIMS, Muizenberg

22. What about 𝐺eff ?
′
𝐺eff
Ωm0 1 + 𝜂
+ 𝐺eff
=Γ
𝐺eff
𝐿/𝑅

Dark degeneracy strikes back

No way of measuring 𝐺eff without a model
 Would somehow need to weigh DM and
separated from DE
15 November 2013
AIMS, Muizenberg

23. 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

24. 15 November 2013
AIMS, Muizenberg

25. Horndeski (1974)
Nicolis, Ratazzi, Tricherini (2009
Deffayet, Gao, Steer, Zahariade (2011)
The model space
ℒ ∼ 𝐾 𝑋, 𝜙 + 𝐺3 𝑋, 𝜙 ⧠𝜙 +
+𝐺4 𝑋, 𝜙

𝛻𝜇 𝛻 𝜈 𝜙
2
+ 𝐺5 𝑋, 𝜙
𝛻𝜇 𝛻 𝜈 𝜙
If 𝑋 small, then nothing new
Quintessence
ℒ ≈ 𝑋 + 𝑉 𝜙 + 𝑓(𝜙)𝑅
𝑓 𝑅
Brans-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
3
+ grav
2𝑋 ≡
𝜕𝜇 𝜙
2

26. What can we actually say?
d𝑡𝑎3
𝑆2 (𝑘) =
Creminelli, Luty, Nicolis, Senatore (2006)
IS, Saltas, Amendola, Kunz (2012)
Gleyzes, Piazza, Vernizzi (2013)
𝜅perf 𝑡 𝒪perf 𝑡, 𝑘 2 + 𝜅3 𝑡 𝒪3 𝑡, 𝑘 2 +
+𝜅4 𝑡 𝒪4 𝑡, 𝑘 2 + 𝜅5 (𝑡)𝒪5 (𝑡, 𝑘 2 )
On FRW, get corrections to perfect fluid that go as 𝑘 2
𝜙
𝑇 𝜇𝜈
=
perf
𝑇 𝜇𝜈
Jeans
+ 𝜅3 𝑘 2
+ 𝜅4 𝑘 2
𝜇𝜈
𝐺eff
𝜇𝜈
𝜂, 𝐺eff
Alternative:
 e.g. braneworld models: corrections go as 𝑘
 Lorentz-violating: higher powers of 𝑘
Measure DE properties from
scale dependence
on the realised background
15 November 2013
AIMS, Muizenberg
Amin, Wagoner, Blandford (2007)
Blas, Sibiryakov (2011)

27. Is it any scalar at all?
0
𝛿𝑇0 ⊃ 𝛿𝜙, 𝛿𝜙, 𝛿m
𝛿𝑇𝑖 𝑖 ⊃ 𝛿𝜙 , 𝛿𝜙, 𝜹𝝓
𝛿𝑇𝑖0 ⊃ 𝛿𝜙, 𝜹𝝓, 𝜃m
𝛿𝜙 = EoM
Φ′′
Φ′
Ψ′
Φ
+ 𝛼1
+ 𝛼2
+ 𝛼3 + 𝛼4 𝑘 2
+ 𝛼5 + 𝛼6 𝑘 2
Ψ
Ψ
Ψ
Ψ
Γ(𝑘, 𝑧)
𝜂(𝑘, 𝑧)
Ψ = Ωm 𝛼7 𝜃m
𝑅′ /𝑅
Fix 𝛼 𝑖 (𝑧)
𝑓(𝑅): one param 𝑚C (𝑧)
2 October2013
𝛿𝑇𝑗 𝑖 ⊃ 𝜹𝝓
NYU Abu Dhabi

28. 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