Lectures on Dark Energy Probes Eiichiro Komatsu (Max-Planck-Institut für Astrophysik) Challenges in Modern Cosmology, Natal, Brazil May 8 and 9, 2014 The lecture slides are available at http://www.mpa-garching.mpg.de/~komatsu/lectures--reviews.html
Topics • In this lecture, we will cover • Cosmic microwave background • Galaxy redshift surveys • Galaxy clusters • as “dark energy probes.” However, we do not have time to cover • Type Ia supernovae • Weak gravitational lensing
• Simple routines for computing various cosmological quantities [many of which are shown in this lecture] are available at • Cosmology Routine Library (CRL): • http://www.mpa-garching.mpg.de/~komatsu/crl/
Defining “Dark Energy” • It is often said that there are two approaches to explain the observed acceleration of the universe. • One is “ dark energy ,” and • The other is a “ modification to General Relativity .” • However, there is no clear distinction between them , unless we impose some constraints on what we mean by “dark energy.”
DE vs MG: Example #1 • Consider an action given by [with 8 π G=1] ✓ R + α R 2 ◆ Z Matter is minimally d 4 x √− g + L matter coupled to gravity via √ - g 2 • Perform a conformal transformation g µ ν → ˆ g µ ν = (1 + 2 α R ) g µ ν • Define a scalar field r 3 φ = 2 ln(1 + 2 α R ) • Then…
DE vs MG: Example #1 • The action becomes ˆ ! 2 − 1 Z R g µ ν ∂ µ φ∂ ν φ − V ( φ ) + e − 2 √ 2 p d 4 x 3 φ L matter − ˆ 2 ˆ g • with a potential V ( φ ) = 1 3 φ ⌘ 2 1 − e − √ ⇣ 2 8 α • Therefore, a modified GR model with R 2 is equivalent to a model with a dark energy field, φ , coupled to matter � α R 2 → f ( R ) • This is generic to models with
DE vs MG: Example #2 • Consider an action given by [with 8 π G=1] ✓ R + f ( R ) ◆ Z d 4 x √− g + L matter 2 • And a FLRW metric with scalar perturbations ds 2 = − (1 + 2 Ψ ) dt 2 + a 2 ( t )(1 + 2 Φ ) d x 2 • Then the relation between Φ and Ψ is given by d 2 f dR 2 r 2 ( Ψ + Φ ) = � r 2 ( δ R ) 6 = 0 1 + d f dR • (Here, “matter” does not have anisotropic stress)
DE vs MG: Example #2 • Consider an action given by [with 8 π G=1] ✓ R ◆ Z d 4 x √− g 2 + L dark energy + L matter • And anisotropic stress of dark energy j + P de ( r i r j � 1 T i j = P de δ i 3 δ i j r 2 ) π de • Then the relation between Φ and Ψ is given by r 2 ( Ψ + Φ ) = a 2 P de π de 6 = 0 • DE anisotropic stress can mimic f(R) gravity
Defining “Dark Energy” • Therefore, we shall use the following terminology: • By “dark energy”, we mean a fluid which • has an equation of state of P de < – ρ de /3, • does not couple to matter, and • does not have anisotropic stress • This “dark energy” fluid can be distinguished from modifications to General Relativity
Goals of Dark Energy Research • We wish to determine the nature of dark energy. But, where should we start ? • A breakthrough in science is often made when the standard model is ruled out. • “Standard model” in cosmology is the Λ CDM model. We wish to rule out dark energy being Λ , a cosmological constant • The most important goal of dark energy research is to find that the dark energy density, ρ de , depends on time
Measuring Dark Energy • We can measure the dark energy density only via its effect on the expansion of the universe. Namely, we wish to measure the Hubble expansion rate , H(z), as a function of redshifts H 2 ( z ) = 8 π G ρ matter (0)(1 + z ) 3 + ρ de ( z ) ⇥ ⇤ 3 • Energy conservation gives [with w(z)=P de (z)/ ρ de (z)] Z z ln ρ de ( z ) dz 0 1 + z 0 [1 + w ( z 0 )] ρ de (0) = 3 0
800 70.*sqrt(0.3*(1.+x)**3+0.7) 70.*sqrt(0.3*(1.+x)**3+0.7*(1.+x)**(3.*(1-0.9))) 70.*sqrt(0.3*(1.+x)**3+0.7*(1.+x)**(3.*(1-1.1))) 700 H(z): Small Effect! Hubble Expansion Rate, H(z) [km/s/Mpc] 600 500 400 300 w =–0.9 Ω m = 0 . 3 200 Ω de = 0 . 7 w =–1.1 100 H 0 = 70 km / s / Mpc 0 0 1 2 3 4 5 6 Redshift, z
140 70.*sqrt(0.3*(1.+x)**3+0.7) 70.*sqrt(0.3*(1.+x)**3+0.7*(1.+x)**(3.*(1-0.9))) 70.*sqrt(0.3*(1.+x)**3+0.7*(1.+x)**(3.*(1-1.1))) 120 Hubble Expansion Rate, H(z) [km/s/Mpc] 100 w =–0.9 Ω m = 0 . 3 80 Ω de = 0 . 7 w =–1.1 H 0 = 70 km / s / Mpc 60 40 • w >–1 : For a given value of H 0 at present, the expansion rate is greater in the past, as the dark 20 energy density increases toward high redshifts 0 0 0.2 0.4 0.6 0.8 1 Redshift, z
Z z ’redshift_da_w1.txt’u 1:($2*(1.+$1)) dz 0 ’redshift_da_w09.txt’u 1:($2*(1.+$1)) 6000 d A ( z ) = ’redshift_da_w11.txt’u 1:($2*(1.+$1)) H ( z 0 ) Comoving Angular Diameter Distance, d A (z) [Mpc/h] 0 Comoving Angular Diameter Distance 5000 Ω m = 0 . 3 w =–1.1 4000 Ω de = 0 . 7 w =–0.9 3000 2000 • w >–1 : For a given value of H 0 at present, d A is smaller, as the expansion rate is 1000 greater in the past 0 0 1 2 3 4 5 6 Redshift, z
Growth of Perturbation • The expansion of the universe also determines how fast perturbations grow. An intuitive argument is as follows. • The growth time scale of matter perturbations [free-fall time, t ff ] is given by d 2 r 1 dt 2 = − 4 π G ρ matter t ff ≈ r √ G ρ matter 3 • The matter perturbation growth is determined by competition between the free-fall time and the expansion time scale, t exp , t exp ≡ 1 1 H ≈ p G ( ρ matter + ρ de )
Growth of Perturbation • The matter perturbation cannot grow during the dark-energy-dominated era, ρ de >> ρ matter , because the expansion is too fast 1 1 t exp ⇡ ⇡ t ff p G ρ matter ⌧ p G ( ρ matter + ρ de ) • Therefore, measuring the [suppression of] growth rate of matter perturbations can also be used to measure the effect of dark energy on the expansion rate of the universe
Growth Equation • Writing the redshift dependence of matter density perturbations as δ matter ( z ) ∝ g ( z ) 1 + z • The evolution equation of g(z) is given by d 2 g 5 2 + 1 � dg 2( Ω k ( z ) − 3 w ( z ) Ω de ( z )) d ln(1 + z ) 2 − d ln(1 + z ) � 2 Ω k ( z ) + 3 + 2(1 − w ( z )) Ω de ( z ) g ( z ) = 0 *Strictly speaking, this formula is valid when the contribution of DE fluctuations to the gravitational potential is negligible compared to matter
’redshift_g_w1.txt’ ’redshift_g_w09.txt’ ’redshift_g_w11.txt’ 1 w =–1.1 Ω m = 0 . 3 Linear growth, g(z)=(1+z)D(z) 0.95 w =–0.9 Ω de = 0 . 7 0.9 • The growth is normalised to unity at high redshift, g(z) -> 1 for z >> 1 0.85 • w >–1 : For a given Ω de today, DE becomes dominant earlier for w >–1, 0.8 giving earlier/more suppression in the growth of matter perturbations 0.75 0 1 2 3 4 5 6 Redshift, z
Cosmic Microwave Background
DE vs CMB • Temperature anisotropy of the cosmic microwave background provides information on dark energy by • Providing the amplitude of fluctuations at z=1090 • Providing the angular diameter distance to z=1090 • Integrated Sachs-Wolfe (ISW) effect
Growth: Application #1 • Use the CMB data to fix the amplitude of WMAP5 fluctuations at z=1090 • Varying w then gives various values of the present-day matter fluctuation amplitude, σ 8 • Data on σ 8 [i.e., large- scale structure data at lower redshifts] can then determine the value of w [present]
Growth: Application #2 • Integrated Sachs-Wolfe effect [Sachs&Wolfe 1967] � • As CMB photons travel from z=1090 to the present epoch, their energies change due to time- dependent gravitational potentials p α p β dp µ dt + Γ µ [geodesic equation] = 0 αβ p 0 ds 2 = − (1 + 2 Ψ ) dt 2 + a 2 ( t )(1 + 2 Φ ) d x 2 with d [ln( ap ) + Ψ ] = ˙ Ψ − ˙ [ p 2 ≡ g ij p i p j ] Φ dt
Growth: Application #2 • Integrated Sachs-Wolfe effect Z t 0 δ T ISW dt ( ˙ Ψ − ˙ = Φ ) T t ∗ Z t 0 = 2 Ψ ( t MD ) dt ˙ g t MD • The right hand side vanishes during the matter-dominated (MD) era, while Ψ and Φ decay during the DE-dominated era • ISW is a direct probe of d g /d t
0.05 ’redshift_dgdlna_w1.txt’ ’redshift_dgdlna_w09.txt’ ’redshift_dgdlna_w11.txt’ 0 Linear growth derivative, dg/dlna=-dg/dln(1+z) w =–1.1 -0.05 Ω m = 0 . 3 -0.1 w =–0.9 Ω de = 0 . 7 -0.15 • The growth derivative vanishes at high redshifts where the universe is -0.2 dominated by matter -0.25 • w >–1 : For a given Ω de today, DE -0.3 becomes dominant earlier for w >–1, giving earlier suppression in the -0.35 growth of matter perturbations -0.4 0 1 2 3 4 5 6 Redshift, z
7000 ’lcdm_cl_lensed.dat’u 1:($2*2.726e6**2) ’wcdm_cl_lensed_w09.dat’u 1:($2*2.726e6**2) ’wcdm_cl_lensed_w11.dat’u 1:($2*2.726e6**2) CMB Temperature Power Spectrum, l(l+1)Cl/(2pi) [uK 2 ] 6000 Ω b = 0 . 05 Ω cdm = 0 . 25 5000 Ω de = 0 . 7 H 0 = 70 km / s / Mpc 4000 w =–1.1 w =–0.9 3000 2000 • The peak positions are given by l=k*d A , where d A 1000 is the angular diameter distance to z=1090. w>–1 shifts the peaks to the left because d A is smaller 0 100 200 300 400 500 600 700 800 900 1000 Multipole, l
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