Cosmology AS7009, 2008 Outline Lecture 5 Cosmological parameters Measuring distances Luminosity distance Angular -diameter distance Standard candles Magnitude system Supernova cosmology Dark energy Covers chapter 7 in Ryden
Cosmological parameters I Cosmological parameters II
Remember these ones? An endless number of subpopulations can Ω M: Matter be introduced if necessary… Ω R: Radiation Ω : Cold dark matter Ω Ω CDM Λor DE : Cosmological constant or dark energy Ω : Baryons Ω (or just Ω): Sum of the other Ωs bar tot Ω : Stars κ: Curvature (+1,0,-1) – related to Ω stars tot Ω CMBR : CMBR photons R0: Curvature radius of the Universe w : Equation of state of dark energy Ων: Neutrinos DE Ω H0: Hubble parameter at current time BH : Black holes -1 -1 Ω (often expressed as h: H 0=100 h km s Mpc ) Robots : Robots (see exercises) t0: Current age of the Universe
Deceleration parameter I A few others … Definition: q0: Deceleration parameter σ 8: Root -mean -square mass fluctuation aa a amplitude in spheres of size 8h -1 Mpc = − && = − && q0 2 2 τ: Electron -scattering optical depth a = aH = & t t0 t t0 η: Inhomogeneity parameter
ns: Slope of matter power spectrum > s q0 0 ⇒ Expansion slowing down (decelerat ion) zreion : Redshift of reionization < q0 0 ⇒ Expansion speeding up (accelerat ion) Neff : Effective number of neutrino species
Not really covered in this course…
1 Deceleration parameter II Cosmological distances I Acceleration equation → = 1 Ω + q0 ∑ w 0, 1( 3w) 2 w Radiation, matter & Λ → 1 q = Ω + Ω − ΩΛ 0 R 0, 2 M 0, 0, Benchmark model: → Ω ≈ Ω ≈ Ω ≈ D(z) depend on cosmological parameters R 0, ,0 M 0, ,3.0 Λ 0, 7.0 ⇒ Ω Ω H0, M, Λ etc. can be extracted from measurements of D(z) q ≈ − 55.0 Acceleration! Problem: In an expanding and/or curved Universe, there are 0 many ways to define D(z)
Cosmological distances II Luminosity distance I In a static, flat Universe, the brightness Proper distance of a light source is determined by Remember : Length of spatial geodesic at time t if scale the inverse-square law . factor is fixed at a(t). This is sometimes referred to as ”distance as measured by a rigid ruler ” However, in an expanding The proper distance is important for theoretical reasons, and/or curved Universe, this is not the case. but impossible to measure in practice, since you cannot halt the expansion of space! Other distance definitions: Inverse square law : Luminosity distance LS Angular size distance f = S 4πr 2 In a static Euclidian (flat) Universe, these would all be equivalent – but in our Universe, they’re not!
Luminosity distance II Luminosity distance III Why does the inverse square law not hold at cosmological distances? Geometry : Definition : Affects the area that photons are spread out 2/1 over = L dL 4πf
Expansion: Photons lose energy due to wavelength shift Time signals stretched by redshift
2 Luminosity distance IV Luminosity distance V
Radiation, matter and Λ: Flat, Λ only + z Benchmark = c 1( z) dz dL ∫ H Ω + 3 − Ω + Ω − + 2 +Ω 0 0 M 1( z) ( M Λ )(1 1 z) Λ Flat, matter only Approximation in a nearly flat Unive rse : dL − ≈ c +1 q0 dL z1 z H0 2 Note : z → 0 ⇒ c d ≈ z (Hubble's law) L H 0 3 6 0 z
Angular -diameter distance I Angular -diameter distance II
In a static, flat Universe, objects of a fixed length Definition : appear smaller if they are = l Problematic as a further away. dA In an expanding and/or θ cosmological probe… curved Universe, One can show that : No good standard this is not the case. rods/yardsticks have yet been discovered = dL dA 2 at cosmological distances Static, Euclidian space : (1 + z) l l θ l d = ≈ for small angles tan( θ ) θ
Angular diameter distance III How to use the luminosity distance Bizarre: After a certain redshift, distant objects start as a probe of cosmology appearing larger in the sky – not smaller! Remember : Flat, Λ only 2/1 = L dL 4πf
c 1( + z) z dz d = dA L ∫ 3 2 H 0 Ω 1( + z) − (Ω + Ω − )(1 1+ z) + Ω Benchmark 0 M M Λ Λ
Flat, matter only Observables : z and f If you know the intrinsic luminosity L of a light source , you can get information on H , Ω , ΩΛ… 0 3 6 0 M Standard candles: Light sources for which L can z be derived through some independent means
3 Standard candles I: Standard candles must be observable Cepheid Variables at high redshifts to be useful
Radially pulsating stars Period → Luminosity (Absolute Magnitude) → Distance Applicable out to ~ 30 Mpc (slightly beyond the Virgo galaxy cluster)
10000 Luminosity Luminosity (Lsolar ) 100
1 10 50 Time Period (days)
Standard candles II: Suggestion for Literature Exercise: Supernovae Type Ia Gamma-ray bursts as probes of cosmology Peak luminosity Useful at least out to almost constant! z~2 (~3000 Mpc ) -19
Probably formed in Absolute -17 binary system in magnitude which matter from a -15
red giant falls onto a -12 white dwarf 0 100 300 Days after maximum May be detectable up to z~10 But: Are they good standard candles?
The magnitude system I The magnitude system II In astronomy, one often measures the flux of light sourcs using photometry – i.e. the flux received within a well-defined filter
Filter profiles Apparent magnitude : = − f m 5.2 log Spectrum of freference light source
4 The magnitude system III Complications I: K -correction Luminosities are often given as absolute magnitudes, i.e. the apparent magnitude a light source of intrinsic luminosity L For two identical objects at different z, a given filter would have at a fixed distance of 10 pc probes different parts of the spectrum → Absolure magnitude : (and different physical processes) Low-z magnitudes cannot be directly = − L compared to high-z magnitudes M 5.2 log Lreference z=0 z=high Flux Flux B B dL m = M + 5log 10 pc Wavelength Wavelength dL m = M + 5log + 25 1 Mpc
Complications I: K -correction Complications II: Dust extinction
K-correction: An attempt to correct from observed (redshifted) to intrinsic (non-redshifted) spectrum What are these black stripes? = − mintrinsic mobs k(z)
Often a complicated function, based on assumptions about the shape of the source spectrum…
Wavelength dependence of Wavelength dependence of dust extinction II dust extinction dust extinction II Before contact with dust Radio After contact IR Flux with dust Optical UV Wavelength Star Radiation Dust = − λ Photons at infrared and radio wavelengths are less mintrinsic mobs A( ) affected by dust than optical or ultraviolet photons are Extinction correction
5 Supernova cosmology I
Supernova cosmology II Supernova cosmology III
ΩΛ > 0! Major breakthrough!
A few other probes of Combined constraints cosmological parameters
CMBR Lecture 7 Large scale structure Lecture 10 Benchmark model Ω =0.3, Ω =0.7 Galaxy clusters M Λ Suitable for -1 -1 H0=72 km s Mpc Weak gravitational lensing literature exercises Redshift shifts over time
6 Suggestion for Literature Exercise: Gravitational lensing Weak gravitational lensing
Distortion of background images by foreground matter
More on this also in the dark matter lecture… Unlensed Lensed
Dark energy and other alternatives The Big Rip I Phantom energy with equation of state w <-1 → → Alternatives to a cosmological constant : Dark energy grows over time Alternative fate of the Universe in which ≠ Dark energy with constant w -1 currently bound structures will get Dark energy with w(z ) disassembled in the future Modification of Friedman equation , for instance due to: Alternative theories of gravity Additional spatial dimensions Suitable for literature Breakdown of cosmological principle exercises Non -standard models of dark matter
The Big Rip II
Caldwell et al. (2003)
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