To Lambda or not to Lambda? - Indian Statistical Institutesupratik/talkaapcos15.pdf ·...

To Lambda or not to Lambda?

Supratik Pal

Indian Statistical Institute Kolkata

October 17, 2015

Conclusion

We don’t know :)

Partly based on my works with

Dhiraj Hazra, Subha Majumdar, Sudhakar Panda, Anjan Sen,Debabrata Adak, Debasish Majumdar

Thirty Meter Telescope’s Science Report 2015 together withPlanck 2015 paper [1505.01195]

Some random questions

What do we know about Dark Energy?

Are the results observation dependent?

Are they parametrization (theoretical prior) dependent?

So, what next?

A bit of history

Universe is accelerating at present

Governing equations (aa

)2= 8πG

3 ρ− ka2

aa = −4πG

3 (ρ+ 3p)

Ordinary matter satisfying Strong Energy Condition (ρ+ 3p) ≥ 0cannot supply the effective “anti-gravity” required for acceleration

Who governs this expansion history ?

Cosmological Constant

(aa

)2= 8πG

3 ρ+ Λ3 −

ka2

aa = −4πG

3 (ρ+ 3p) + Λ3

M = 43πa

3(ρ+ 3p) =⇒ a = −GMa2 + Λ

3 a↓

Effective mass Attraction Repulsion

a(t) ∝[

sinh( 32

√Λ3 t)

]2/3

Fine-tuning problem

Cosmic coincidence problem

.....

Dynamical models: Dark Energy

(aa

)2= 8πG

3

∑i ρi −

ka2

aa = −4πG

3

∑i (ρi + 3pi )

pi < −ρi/3⇒ wi < −1/3 leads to acceleration ⇐⇒ violates SEC

Scalar field models

EM tensor components ρφ = 12 φ

2 + V (φ) ; pφ = 12 φ

2 − V (φ)

Quintessence: scalar field minimally coupled to gravity

K-essence: Kinetic energy driven

Tachyonic: Non-canonical

Phantom: Opposite sign in kinetic term

Dilatonic

(Generalized) Chaplygin gas

Too many candidates

Modified gravity models

∗ Einstein’s theory is not directly tested in cosmological scales.∗ Modify the gravity sector rather than the matter sector.

Brans-Dicke theory: G as a VEV of a geometric field, needvastly different values for solar system and cosmic scales

f(R) gravity: R in the action replaced by f(R), [e.g. R − µR ],

no unique choice

DGP model: Extra dimensional effects at large scale, fails toproduce small scale results

No unique candidate

Dark energy parametrizations

Observationally we probe two parameters of dark energy:

Density parameter (ΩDE) (by probing ΩM)

Equation of State (EOS) parameter (wDE)

Proposal

Instead of proposing individual models, can we parametrize darkenergy and constrain models from observations by constrainingthose parameters?

Parametrization of Hubble parameter

r(x) = H2(x)H2

0= Ω0

mx3 + A0 + A1x + A2x

2

with x = 1 + z ; Ω0m + A0 + A1 + A2 = 1 ; ρ0

c = 3H20

ρ = ρ0c

(A0 + A1x + A2x

2)

For A0 6= 0,A1 = 0 = A2 =⇒ ΛCDM

Either A1 6= 0 or A2 6= 0 =⇒ Dynamical dark energy

Parametrization of EOS

• CPL Parametrization

Fits a wide range of scalar field dark energy models including thesupergravity-inspired SUGRA dark energy models.

w(a) = w0 + wa(1− a)

= w0 + waz

1 + z

ρDE ∝ a−3(1+w0+wa)e−3wa(1−a)

Two parameter description: w0 = EOS at present , wa = itsvariation w.r.t. scale factor (or redshift).

For w0 ≥ −1,wa > 0 : dark energy is non-phantomthroughout

Otherwise, may show phantom behavior at some point

• SS Parametrization

Useful for slow-roll ‘thawing’ class of scalar field models having acanonical kinetic energy term.Motivation : to look for a unique dark energy evolution for scalarfield models that are constrained to evolve close to Λ.

w(a) = (1 + w0)×[√1 + (Ω−1

DE − 1)a−3 − (Ω−1DE − 1)a−3 tanh−1 1√

1+(Ω−1DE−1)a−3

]2

×[1√

ΩDE−(

1ΩDE− 1)

tanh−1√

ΩDE

]−2− 1

One model parameter: w0 = EOS at present

Rest is taken care of by the general cosmological parameterΩDE = dark energy density today.

• GCG Parametrization

p = − cρα

w(a) = − AA+(1−A)a−3(1+α) ; A = c

ρ1+αGCG

Two model parameters e.g A and α, with w(0) = −AFor (1 + α) > 0, w(a) behaves like a dust in the past andevolves towards negative values and becomes w = −1 in theasymptotic future. =⇒ ‘tracker/freezer’ behavior

For (1 + α) < 0, w(a) is frozen to w = −1 in the past and itslowly evolves towards higher values and eventually behaveslike a dust in the future. =⇒ ‘thawing’ behavior

Restricted to 0 < A < 1 only since for A > 1 singularityappears at finite past =⇒ non-phantom only

Dark Energy a la observations

Cosmic Microwave Background (CMB) dataShift parameter (position of peaks)Integrated Sachs-Wolfe effect (low-`)

Shift Parameter

DE ⇔ Shift in position of peaks by√

ΩmDD= Angular diameter distance (to LSS) ⇒ Shift Parameter

R =√

Ωmh2

|Ωk |h2χ(y)

χ(y) = sin y(k < 0) ; = y(k = 0) ; = sinh y(k > 0)

y =√|Ωk |

∫ zdeco

dz√Ωm(1+z)3+Ωk (1+z)2+ΩX (1+z)3(1+ωX )

χ2CMB(ωX ,Ωm,H0) =

[R(zdec,ωX ,Ωm,H0)−R

σR

]2

Integrated Sachs-Wolfe Effect

Some CMB anisotropies may be induced by passing through a timevarying gravitational potential

linear regime: integrated Sachs-Wolfe effect

non-linear regime: Rees-Sciama effect

Poisson equation : ∇2Φ = 4πGa2ρδ

Φ→ constant during matter domination→ time-varying when dark energy comes to dominate

(at large scales l ≤ 20)

Cl =∫

dkk PR(k)T 2

l (k)

T ISWl (k) = 2

∫dη exp−τ dΦ

dη jl(k(η − η0)

But cosmic variance!

Supernova Type Ia Data

Probe Luminosity distance: DL(z) = H0dL(z) via distance modulus

µ(z) = 5 log10(DL(z)) + µ0

χ2SN(w0

X ,Ω0m,H0) =

∑i

[µobs(zi )−µ(zi ;w

0X ,Ω

0m,H0)

σi

]2

Marginalizing over the nuisance parameter µ0,

χ2SN(w0

X ,Ω0m) = A− B2/C

A =∑

i


0X ,Ω

0m,µ0=0)

σi

]2

B =∑

i


0X ,Ω

0m,µ0=0)

σi

];C =

∑i

1σ2i

Union 2.1 compilation of 580 Supernovae at z = 0.015− 1.4,considered as standard candles

Baryon Acoustic Oscillation (BAO) data

Used to measure H(z) and angular diameter distance DA(z) via acombination

DV (z) =[(1 + z)2D2

A(z) czH(z)

]1/3

Confront models via a distance ratio

dz =rs(zdrag)DV (z)

rs(zdrag) = comoving sound horizon at a redshift wherebaryon-drag optical depth is unity

Give 6 data points:

WiggleZ : z = 0.44, 0.6, 0.73

SDSS DR7 : z = 0.35

SDSS DR9 : z = 0.57

6DF : z = 0.106

Hence calculate χ2BAO

Hubble Space Telescope Data (HST)

Use nearby Type-Ia Supernova data with Cepheid calibrations toconstrain the value of H0 directly.Combine and calculate χ2 for the analysis of HST data

χ2HST(w0

X ,Ω0m,H0) =

∑i

[Hobs(zi )−H(zi ;w

0X ,Ω

0m,H0)

σi

]2

Two methods of analysis

Riess et. al. (2011)

Efstathiou (2014)

Some more probes

Weak lensing

Galaxy clusters

Gamma ray bursts

X-ray observations

Growth factor

....

Dark Energy Perturbations

Can be important at horizon scales.Need to cross-correlate large scale CMB with large scale structures.But cosmic variance!Incorporated through sound speed squared c2

s .For canonical scalar, c2

s = 1

Dark energy from different datasets

Used all three parametrizations =⇒ Analysis is robust

Data ΛCDM CPL SS GCG

Planck (low-` + high-`) 7789.0 7787.4 7788.1 7789.0

WMAP-9 low-` polarization 2014.4 2014.436 2014.455 2014.383

BAO : SDSS DR7 0.410 0.073 0.265 0.451

BAO : SDSS DR9 0.826 0.793 0.677 0.777

BAO : 6DF 0.058 0.382 0.210 0.052

BAO : WiggleZ 0.020 0.069 0.033 0.019

SN : Union 2.1 545.127 546.1 545.675 545.131

HST 5.090 2.088 2.997 5.189

Total 10355.0 10351.4 10352.4 10355.0

Best fit χ2eff obtained in different model upon comparing against

CMB + non-CMB datasets using the Powell’s BOBYQA methodof iterative minimization.

Likelihood functions for CPL parametrization

0

0.5

1

-2 -1.5 -1 -0.5 0 0.5 1

Lik

elih

oo

d

w0

CPL model

Planck+WP

non-CMB

Planck+WP+non-CMB

0

0.5

1

-3 -2 -1 0 1 2 3

Lik

elih

oo

dwa

CPL model

Planck+WP

non-CMB

Planck+WP+non-CMB

Concordance with Planck 2015 paper: CPL

Likelihood functions for different parameters of EOS

0

0.5

1

-2 -1.5 -1 -0.5 0 0.5 1

Lik

elihood

w0

CPL model

Planck+WP

non-CMB

Planck+WP+non-CMB

0

0.5

1

-3 -2 -1 0 1 2 3

Lik

elihood

wa

CPL model

Planck+WP

non-CMB

Planck+WP+non-CMB

0

0.5

1

-2 -1.5 -1 -0.5 0

Lik

elihood

w0

SS model

Planck+WP

non-CMB

Planck+WP+non-CMB

0

0.5

1

-1 -0.9 -0.8 -0.7 -0.6

Lik

elih

ood

-A (w0)

GCG model

Planck+WP

non-CMB

Planck+WP+non-CMB

0

0.5

1

-3 -2 -1 0 1 2 3

Lik

elihood

α

GCG model

Planck+WP

non-CMB

Planck+WP+non-CMB

Mean value and 1σ range for CMB+non-CMB

Analysis: value of H0

CPL model

0.25 0.27 0.29 0.31 0.33Ωm

66

68

70

72

74

76

H0

GCG model

0.26 0.28 0.3 0.32Ωm

66

68

70

72

74

76

H0

SS model

0.26 0.28 0.3 0.32Ωm

66

68

70

72

74

H0

CPL model

-1.6 -1.4 -1.2 -1 -0.8 -0.6w0

66

68

70

72

74

76

H0

GCG model

-1 -0.95 -0.9 -0.85 -0.8 -0.75 -0.7w0

66

68

70

72

74

76

H0

SS model

-1.4 -1.3 -1.2 -1.1 -1 -0.9w0

66

68

70

72

74

H0

If phantom is forbidden by theoretical prior (GCG):

The parameters stay close to the values obtained in ΛCDMmodel analysis.H0 is not that degenerate with dark energy equation of statefor CMB.

If phantom is NOT forbidden by theoretical prior (CPL+SS):

Better fit to the CMB data comes with a large value of H0

⇒ agrees better with the HST data (better total χ2)But background cosmological parameter space (e.g., Ωm −H0)is dragged s.t. best-fit base model and that from Planckbecomes 2σ away.H0 becomes highly degenerate with dark energy EOS for CMBonly measurements.

Comparison with Planck 2015

Difference in analysis of HST data : Riess vs Efstathiou

Analysis: Equation of State

CPL model

-1.6 -1.4 -1.2 -1 -0.8 -0.6w0

-3

-2

-1

0

1

wa

Always non-phantom

GCG model

-1 -0.9 -0.8 -0.7-A (w0)

-3

-2

-1

0

α

Freezing/tracking

Thawing

0.0 0.5 1.0 1.5 2.0-1.6

-1.4

-1.2

-1.0

-0.8

z

w

0.0 0.5 1.0 1.5 2.0-1.6

-1.4

-1.2

-1.0

-0.8

z

w

0.0 0.5 1.0 1.5 2.0-1.00

-0.98

-0.96

-0.94

-0.92

-0.90

z

w

Top: CPL, Bottom left: SS, Bottom right: GCG

If phantom is forbidden by theoretical prior (GCG):

Show consistency between CMB and non-CMB dataBut they have marginally worse likelihood than otherparametrizations.CMB and non-CMB observations are separately sensitive tothe two model parameters but the joint constraint is consistentwith w = −1.

If phantom is NOT forbidden by theoretical prior (CPL+SS):

CMB data: the non-phantom equation of states stays at theedge of 2σ region.Non-CMB data: non-phantom behavior favored for everyparametrization considered.

Combined CMB + non-CMB data

Mean w and error bar depends on the parametrization.

SS and GCG parametrization: the nature of dark energy isbest constrained at high redshifts

CPL parametrization: the best constraints come in theredshift range of ≈ 0.2− 0.3

Just as aside...

Similar results by Novosyadlyj et.al. (JCAP): for datasetPlanck+HST+BAO+SNLS3 ΛCDM is outside 2σ confidenceregime, for dataset WMAP-9+HST+BAO+SNLS3 ΛCDM is1σ away from best fit.

PAN-STARRS1 shows tension with ΛCDM at 2.4σ with aconstant EOS (Rest et.al., 1310.3828)

Conclusion

Constraints on w and hence the nature of dark energy that weinfer from cosmological observation depends crucially on thechoice of the underlying parametrization of the EOS.

Can the apparent tension between CMB and non-CMB databe attributed to unknown systematics?

Unlikely!

Can it be due to different analysis of HST data (Riess vsEfsthathiou)?

Maybe

Can it be due to lack of a better theory/parametrization ofthe dark energy equation of state?

Most likely yes

Can a non-parametric reconstruction of w for the totaldataset help to infer about the correct nature of dark energy(or, Λ) without any priors on the form of w?

Conclusion

We don’t know :)

To Lambda or not to Lambda? - Indian Statistical Institutesupratik/talkaapcos15.pdf ·...

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