The Acceleration of the Universe Ruth A. Daly
Review of Data Indicating that the Universe is Accelerating 1). Observations of the Cosmic Microwave Background Radiation → Space curvature = 0 Numerous types of local measurements → Ωm ~ 0.3 (Clusters of Galaxies, LSS, Flows,….Age Problem ….) + Flat Universe → Dark Energy (CMB + Detailed Modeling → Dark Energy + an Accelerating Universe) 2). Type Ia Supernovae → Universe is Accelerating Radio Galaxies → Universe is Accelerating
CMBAngular Power Spectrum Observations of the Cosmic Microwave Background → Space curvature = 0
WMAP + Detailed Modeling
Supernovae Results from Knop et al. (2003)
Supernovae Results from Riess et al. (2004)
54 Type Ia SN: Constraints in a Λ model (Knop et al.2003)
157 Type Ia SN: Constraints in a Λ model (Riess et al. 2004)
SN Constraints in a quintessence model (Knop et al. 2003)
SN Constraints in a quintessence model (Riess et al. 2004)
FRIIb Radio Galaxies
Radio Galaxy Constraints in a quintessence model (Daly & Guerra 2002).
54 SN in a k=0 Quintessence model, w=P/ρ 54 SN in a k=0 quintessence model (from DG02)
20 RG + 54 SN in a Quintessence Model Results with 20 RG + 54 SN (from DG02)
The CMB + Any One of Many Types of Local Measurements (→ Ωm ~ 0.3) → Dark Energy and an Accelerating Universe Type Ia Supernovae alone → An Accelerating Universe FRIIb Radio Galaxies alone → An Accelerating Universe
Recent Results from DD03 & DD04The SN and RG methods → a set of luminosity or coordinate distances y(z)= Hoaor (z)
For k=0, (aor) = ∫ dt/a(t) = ∫ (á/a)-1 dz 1). Method 1: select GR & fE(z,w); use data to obtain the best fit values of the model. Einstein Equations (for k=0): (á/a)2=(8πG/3)∑ρi =Ho2[Ωom(1+z)3 + fE(z,w)] where ρE(z) = ρ0c fE(z,w) and w = PE /ρE ä/a= -(4πG/3)∑(ρi+3Pi) = -(4πG/3)[ρm+ρE+3PE] 2). Method 2: Differentiate the data aor(z) or y(z) + solve directly for q(z) = - äa/(á)2, E(z) = (á/a); can also obtain PE(z), ρE(z), and w(z) [DD03,04] q(z) and E(z) only depend upon the FRW metric!
Rather than assuming GR and integrating a model for fE(z,w) to obtain the best fit model parameters, we take y(z) to each source, do a robust numerical differentiation, solve for dy/dz and d2y/dz2→ obtainthe dimensionless expansion and acceleration rates: E(z) = (á/a)/Ho = (dy/dz)-1 q(z)=-äa/(á)2=-[1+(1+z)(dy/dz)-1(d2y/dz2)] Only assumes FRW (Valid for any homogeneous, isotropic space-time)!→ Independent of GR & Models
Derivation dτ2=dt2 – a2(t)[dr2/(1-kr2)+r2dθ2+r2sin2θ dφ2] for a light ray from source, dt=a(t)dr when k=0; dz/dt = -ao-1 (1+z)(dr/dz) since (1+z) = ao/a(t). Differentiating this, ά= -ao (1+z)-2 (dz/dt) = (1+z)-1(dr/dz)-1, or, with y = H0(a0r) H(z) ≡ (ά/a) = (d(aor)/dz)-1 = Ho (dy/dz)-1. → E(z) ≡ H(z)/Ho = (dy/dz)-1(Weinberg 1972) Differentiating ά,→ ä=-(1+z)-2 (dz/dt) (dr/dz)-1 [1+(1+z)(dr/dz)-1 (d2r/dz2)] or q(z)≡-(äa)/ά2=-[1+(1+z)(dy/dz)-1 d2y/dz2] (DD03) So, E(z) and q(z) can be obtained independent of GR & of specific models for the “dark energy!”
The Methodology y(z) Fit a parabola in a sliding window of ∆z around some z0 From the local fit coefficients, get y(z0), dy/dz and d2y/dz2 and their errors z0 z This is equivalent to a local Taylor expansion for y(z); a parabola is a minimum assumption local model for y(z). For noisy/sparse data, need a large ∆z: poor redshift resolution, but can determine trends …
Testing the Methodology Using Simulated (Pseudo-SNAP) Data From DD03; Assumes m = 0.3, = 0.7, and z = 0.4
Dimensionless Luminosity Distances to 78 SN and 20 RG
Initial results for the evolution of the expansion rate E(z) and the acceleration parameter q(z), obtained using 20 RG and 78 SNFound zT ≈ 0.45from DD03
Good agreement between RG and SN E(z) = (dy/dz)-1 From DD03; using 20 RG and 78 SN
New Results: Coordinate Distances y(z)157 Gold SN Sample + 20 RG
Comparison of SN and RG Distances In what follows, SN-only sample gives essentially the same results; RGs help tighten the error bars at the high-z end.
Evolution of the Expansion Rate
Valid for any theory of gravity, and any type of “DE”; only assumes k=0 and FRW metric Evolution of the Cosmic Acceleration - Transition redshift: zT ~ 0.4 q0= -.35 ±.15
Evolution of the Cosmic Acceleration Valid for any theory of gravity, and any type of “DE”; only assumes k=0 and FRW metric - Transition redshift: zT ~ 0.4 q0= -.35 ±.15
Can solve for the pressure PE, Energy Density fE, and equation of state w of the Dark Energy if a theory of gravity is assumed. Einstein Equations (for k=0): ä/a= - (4πG/3) [ρm + ρE + 3 PE] (á/a)2 = (8πG/3) [ρm + ρE] → PE = [E2(z)/3] [2 q(z) -1] or pE(z)=-(dy/dz)-2[1+(2/3)(1+z)(dy/dz)-1(d2y/dz2)], where pE(z)≡PE/ρoc With FRW + GR, we have PE(z)!
Pressure of the Dark Energy (Assumes GR) For models, p=- → direct measure of Λ We measure p0 = -0.60 ± 0.15 Since ρ0E= p0/w0, Ω0E = 0.60 ±0.15for w0 = -1
The Energy Density fE and equation of state w of the Dark Energy can be obtained if we assume a theory of Gravity and adopt a value of Ωm: Einstein Equations (for k=0): (á/a)2 = (8πG/3) [ρm + ρE] ä/a= - (4πG/3) [ρm + ρE + 3 PE] → fE(z) ≡ ρE(z)/ρoc = (dy/dz)-2 – Ωm(1+z)3 w(z) = -[1+(2/3)(1+z)(dy/dz)-1(d2y/dz2)/ [1-(dy/dz)2 Ωm(1+z)3], where w(z) = pE(z)/fE(z) = eq. of state
Relative Energy Density of the Dark Energy f0=0.62 ± 0.04
Evolution of the EOS Parameter w(z) We measure w0 = -0.9 0.1 Consistent with models, but possible evolution
Summary The Acceleration of the Universe is indicated by: CBM Obs. + Local Obs. → (k=0 + Ωm ~ 0.3); + Supernovae Type Ia (and Radio Galaxy) Obs. Though these determinations are model dependent, very similar results are obtained using different models (e.g. Λ, quintessence, ….); assumes GR. It is possible to use SN and RG observations to determine E(z), q(z), P(z), f(z), w(z), etc. in a model-independent fashion.
Assuming Only: Flatness + FRW → q(z) and E(z). Find that the universe is accelerating today, and was decelerating in the recent past (R.D. + Djorgovski 2003,2004). • + GR can solve for p(z) • For models, = -p → direct measure of Λ (independent of other methods) • + Ωom → ρE(z) and w(z) • Transition found by DD03 from accelerating to decelerating universe at zT ~ 0.4 is confirmed by DD04; Agrees with Riess et al. (2004) • Perhaps a hint that w is evolving with redshift • The current data provide some useful constraints for theoretical models; generally consistent with “concordance cosmology” (m ≈ 0.3, ≈ 0.7)
Evolution of the Cosmic Acceleration Valid for any theory of gravity, and any type of “DE”; only assumes k=0 and FRW metric - Transition redshift: zT ~ 0.4 q0= -.35 ±.15
Pressure of the Dark Energy (Assumes GR) For models, p=- → direct measure of Λ We measure p0 = -0.60 ± 0.15 Since ρ0E= p0/w0, Ω0E = 0.60 ±0.15for w0 = -1
