To Lambda or not to Lambda?

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

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 with Planck 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 Universe is accelerating at present ( ȧ ) 2 a = 8πG 3 ρ k a 2 ä a = 4πG 3 (ρ + 3p) Ordinary matter satisfying Strong Energy Condition (ρ + 3p) 0 cannot supply the effective anti-gravity required for acceleration Who governs this expansion history?

Cosmological Constant ( ȧ ) 2 a = 8πG 3 ρ + Λ 3 k a 2 ä a = 4πG 3 (ρ + 3p) + Λ 3 M = 4 3 πa3 (ρ + 3p) = ä = GM + Λ a 2 3 a Effective mass Attraction Repulsion a(t) [ ] 2/3 sinh( 3 Λ 2 3 t) Fine-tuning problem Cosmic coincidence problem...

Dynamical models: Dark Energy ( ȧ ) 2 a = 8πG 3 ä a = 4πG 3 i ρ i k a 2 i (ρ i + 3p i ) p i < ρ i /3 w i < 1/3 leads to acceleration violates SEC

Dynamical models: Dark Energy ( ȧ ) 2 a = 8πG 3 ä a = 4πG 3 i ρ i k a 2 i (ρ i + 3p i ) p i < ρ i /3 w i < 1/3 leads to acceleration violates SEC Scalar field models EM tensor components ρ φ = 1 2 φ 2 + V (φ) ; p φ = 1 2 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, need vastly different values for solar system and cosmic scales f(r) gravity: R in the action replaced by f(r), [e.g. R µ R ], no unique choice DGP model: Extra dimensional effects at large scale, fails to produce small scale results No unique candidate

Dark energy parametrizations Observationally we probe two parameters of dark energy: Density parameter (Ω DE ) (by probing Ω M ) Equation of State (EOS) parameter (w DE ) Proposal Instead of proposing individual models, can we parametrize dark energy and constrain models from observations by constraining those parameters?

Parametrization of Hubble parameter r(x) = H2 (x) H 2 0 = Ω 0 mx 3 + A 0 + A 1 x + A 2 x 2 with x = 1 + z ; Ω 0 m + A 0 + A 1 + A 2 = 1 ; ρ 0 c = 3H0 2 ρ = ρ 0 ( c A0 + A 1 x + A 2 x 2) For A 0 0, A 1 = 0 = A 2 = ΛCDM Either A 1 0 or A 2 0 = Dynamical dark energy

Parametrization of EOS CPL Parametrization Fits a wide range of scalar field dark energy models including the supergravity-inspired SUGRA dark energy models. w(a) = w 0 + w a (1 a) = w 0 + w a z 1 + z ρ DE a 3(1+w 0+w a) e 3wa(1 a) Two parameter description: w 0 = EOS at present, w a = its variation w.r.t. scale factor (or redshift). For w 0 1, w a > 0 : dark energy is non-phantom throughout Otherwise, may show phantom behavior at some point

SS Parametrization Useful for slow-roll thawing class of scalar field models having a canonical kinetic energy term. Motivation : to look for a unique dark energy evolution for scalar field models that are constrained to evolve close to Λ. w(a) = (1 + w 0 ) [ 1 + (Ω 1 DE 1)a 3 (Ω 1 DE 1)a 3 tanh 1 1 1+(Ω 1 [ ( ) 1 ΩDE 1 Ω DE 1 tanh 1 ] 2 Ω DE 1 DE 1)a 3 ] 2 One model parameter: w 0 = EOS at present Rest is taken care of by the general cosmological parameter Ω DE = dark energy density today.

GCG Parametrization p = c ρ α A w(a) = ; A = c A+(1 A)a 3(1+α) ρ 1+α GCG Two model parameters e.g A and α, with w(0) = A For (1 + α) > 0, w(a) behaves like a dust in the past and evolves towards negative values and becomes w = 1 in the asymptotic future. = tracker/freezer behavior For (1 + α) < 0, w(a) is frozen to w = 1 in the past and it slowly evolves towards higher values and eventually behaves like a dust in the future. = thawing behavior Restricted to 0 < A < 1 only since for A > 1 singularity appears at finite past = non-phantom only

Dark Energy a la observations Cosmic Microwave Background (CMB) data Shift parameter (position of peaks) Integrated Sachs-Wolfe effect (low-l)

Shift Parameter DE Shift in position of peaks by Ω m D D= Angular diameter distance (to LSS) Shift Parameter R = Ωmh 2 Ω k h 2 χ(y) χ(y) = sin y(k < 0) ; = y(k = 0) ; = sinh y(k > 0) y = Ω k z dec o dz Ω m(1+z) 3 +Ω k (1+z) 2 +Ω X (1+z) 3(1+ω X ) χ 2 CMB (ω X, Ω m, H 0 ) = [ ] 2 R(zdec,ω X,Ω m,h 0 ) R σ R

Integrated Sachs-Wolfe Effect Some CMB anisotropies may be induced by passing through a time varying gravitational potential linear regime: integrated Sachs-Wolfe effect non-linear regime: Rees-Sciama effect Poisson equation : 2 Φ = 4πGa 2 ρδ Φ constant during matter domination time-varying when dark energy comes to dominate (at large scales l 20) T ISW l C l = dk k P R(k)Tl 2 (k) (k) = 2 dη exp τ dφ dη j l(k(η η 0 ) But cosmic variance!

Supernova Type Ia Data Probe Luminosity distance: D L (z) = H 0 d L (z) via distance modulus µ(z) = 5 log 10 (D L (z)) + µ 0 χ 2 SN (w 0 X, Ω0 m, H 0 ) = i [ ] µobs (z i ) µ(z i ;wx 0,Ω0 m,h 0 ) 2 σ i Marginalizing over the nuisance parameter µ 0, B = i A = i χ 2 SN (w 0 X, Ω0 m) = A B 2 /C [ ] µobs (z i ) µ(z i ;wx 0,Ω0 m,µ 2 0=0) σ i [ ] µobs (z i ) µ(z i ;wx 0,Ω0 m,µ 0 =0) σ i ; C = i 1 σ 2 i 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 D A (z) via a combination [ ] 1/3 D V (z) = (1 + z) 2 DA 2 cz (z) H(z) Confront models via a distance ratio d z = rs(z drag) D V (z) r s (z drag ) = comoving sound horizon at a redshift where baryon-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 χ 2 BAO

Hubble Space Telescope Data (HST) Use nearby Type-Ia Supernova data with Cepheid calibrations to constrain the value of H 0 directly. Combine and calculate χ 2 for the analysis of HST data χ 2 HST (w 0 X, Ω0 m, H 0 ) = i [ ] Hobs (z i ) H(z i ;wx 0,Ω0 m,h 2 0) σ i 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 c 2 s. For canonical scalar, c 2 s = 1

Dark energy from different datasets Used all three parametrizations = Analysis is robust Data ΛCDM CPL SS GCG Planck (low-l + high-l) 7789.0 7787.4 7788.1 7789.0 WMAP-9 low-l 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 χ 2 eff obtained in different model upon comparing against CMB + non-cmb datasets using the Powell s BOBYQA method of iterative minimization.

Likelihood functions for CPL parametrization 1 CPL model 1 CPL model Planck+WP Planck+WP non-cmb non-cmb Planck+WP+non-CMB Planck+WP+non-CMB Likelihood 0.5 Likelihood 0.5 0-2 -1.5-1 -0.5 0 0.5 1 w 0 0-3 -2-1 0 1 2 3 w a

Concordance with Planck 2015 paper: CPL

Likelihood functions for different parameters of EOS CPL model CPL model SS model 1 1 1 Planck+WP Planck+WP Planck+WP non-cmb non-cmb non-cmb Planck+WP+non-CMB Planck+WP+non-CMB Planck+WP+non-CMB Likelihood 0.5 Likelihood 0.5 Likelihood 0.5 0-2 -1.5-1 -0.5 0 0.5 1 w 0 0-3 -2-1 0 1 2 3 w a 0-2 -1.5-1 w 0-0.5 0 1 GCG model Planck+WP 1 GCG model Planck+WP non-cmb non-cmb Planck+WP+non-CMB Planck+WP+non-CMB Likelihood 0.5 Likelihood 0.5 0-1 -0.9-0.8-0.7-0.6 -A (w 0 ) 0-3 -2-1 0 1 2 3 α

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

Analysis: value of H 0 76 CPL model 76 GCG model SS model 74 74 74 72 72 72 H 0 70 H 0 70 H 0 70 68 68 68 66 66 66 0.25 0.27 0.29 0.31 0.33 Ω m 0.26 0.28 0.3 0.32 Ω m 0.26 0.28 0.3 0.32 Ω m H 0 76 74 72 70 68 66 CPL model -1.6-1.4-1.2-1 -0.8-0.6 w 0 H 0 76 74 72 70 68 66 GCG model -1-0.95-0.9-0.85-0.8-0.75-0.7 w 0 H 0 74 72 70 68 66 SS model -1.4-1.3-1.2-1.1-1 -0.9 w 0

If phantom is forbidden by theoretical prior (GCG): The parameters stay close to the values obtained in ΛCDM model analysis. H 0 is not that degenerate with dark energy equation of state for CMB.

If phantom is forbidden by theoretical prior (GCG): The parameters stay close to the values obtained in ΛCDM model analysis. H 0 is not that degenerate with dark energy equation of state for CMB. If phantom is NOT forbidden by theoretical prior (CPL+SS): Better fit to the CMB data comes with a large value of H 0 agrees better with the HST data (better total χ 2 ) But background cosmological parameter space (e.g., Ω m H 0 ) is dragged s.t. best-fit base model and that from Planck becomes 2σ away. H 0 becomes highly degenerate with dark energy EOS for CMB only measurements.

Comparison with Planck 2015 Difference in analysis of HST data : Riess vs Efstathiou

Analysis: Equation of State w a CPL model 1 Always non-phantom 0-1 -2 0-1 α -2 GCG model Freezing/tracking Thawing -3-1.6-1.4-1.2-1 -0.8-0.6 w 0-3 -1-0.9-0.8-0.7 -A (w 0 )

0.8 1.0 w 1.2 1.4 1.6 0.0 0.5 1.0 1.5 2.0 z 0.8 0.90 1.0 0.92 0.94 w 1.2 w 0.96 1.4 0.98 1.6 0.0 0.5 1.0 1.5 2.0 z 1.00 0.0 0.5 1.0 1.5 2.0 z Top: CPL, Bottom left: SS, Bottom right: GCG

If phantom is forbidden by theoretical prior (GCG): Show consistency between CMB and non-cmb data But they have marginally worse likelihood than other parametrizations. CMB and non-cmb observations are separately sensitive to the two model parameters but the joint constraint is consistent with w = 1.

If phantom is forbidden by theoretical prior (GCG): Show consistency between CMB and non-cmb data But they have marginally worse likelihood than other parametrizations. CMB and non-cmb observations are separately sensitive to the two model parameters but the joint constraint is consistent with w = 1. If phantom is NOT forbidden by theoretical prior (CPL+SS): CMB data: the non-phantom equation of states stays at the edge of 2σ region. Non-CMB data: non-phantom behavior favored for every parametrization considered.

Combined CMB + non-cmb data Mean w and error bar depends on the parametrization. SS and GCG parametrization: the nature of dark energy is best constrained at high redshifts CPL parametrization: the best constraints come in the redshift range of 0.2 0.3 Just as aside... Similar results by Novosyadlyj et.al. (JCAP): for dataset Planck+HST+BAO+SNLS3 ΛCDM is outside 2σ confidence regime, for dataset WMAP-9+HST+BAO+SNLS3 ΛCDM is 1σ away from best fit. PAN-STARRS1 shows tension with ΛCDM at 2.4σ with a constant EOS (Rest et.al., 1310.3828)

Conclusion Constraints on w and hence the nature of dark energy that we infer from cosmological observation depends crucially on the choice of the underlying parametrization of the EOS.

Conclusion Constraints on w and hence the nature of dark energy that we infer from cosmological observation depends crucially on the choice of the underlying parametrization of the EOS. Can the apparent tension between CMB and non-cmb data be attributed to unknown systematics? Unlikely!

Conclusion Constraints on w and hence the nature of dark energy that we infer from cosmological observation depends crucially on the choice of the underlying parametrization of the EOS. Can the apparent tension between CMB and non-cmb data be attributed to unknown systematics? Unlikely! Can it be due to different analysis of HST data (Riess vs Efsthathiou)? Maybe

Conclusion Constraints on w and hence the nature of dark energy that we infer from cosmological observation depends crucially on the choice of the underlying parametrization of the EOS. Can the apparent tension between CMB and non-cmb data be attributed to unknown systematics? Unlikely! Can it be due to different analysis of HST data (Riess vs Efsthathiou)? Maybe Can it be due to lack of a better theory/parametrization of the dark energy equation of state? Most likely yes

Conclusion Constraints on w and hence the nature of dark energy that we infer from cosmological observation depends crucially on the choice of the underlying parametrization of the EOS. Can the apparent tension between CMB and non-cmb data be attributed to unknown systematics? Unlikely! Can it be due to different analysis of HST data (Riess vs Efsthathiou)? Maybe Can it be due to lack of a better theory/parametrization of the dark energy equation of state? Most likely yes Can a non-parametric reconstruction of w for the total dataset 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 :)