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1
Cosmology with Supernovae:Lecture 1
Josh Frieman
I Jayme Tiomno School of Cosmology, Rio de Janeiro, Brazil
July 2010
Hoje
• I. Cosmology Review• II. Observables: Age, Distances• III. Type Ia Supernovae as Standardizable
Candles• IV. Discovery Evidence for Cosmic
Acceleration• V. Current Constraints on Dark Energy
2
Coming Attractions• VI. Fitting SN Ia Light Curves & Cosmology
in detail (MLCS, SALT, rise vs. fall times)• VII. Systematic Errors in SN Ia Distances• VIII. Host-galaxy correlations• IX. SN Ia Theoretical Modeling• X. SN IIp Distances• XI. Models for Cosmic Acceleration• XII. Testing models with Future Surveys:
Photometric classification, SN Photo-z’s, & cosmology
3
References• Reviews:
Frieman, Turner, Huterer, Ann. Rev. of Astron. Astrophys., 46, 385 (2008)
Copeland, Sami, Tsujikawa, Int. Jour. Mod. Phys., D15, 1753 (2006)
Caldwell & Kamionkowski, Ann. Rev. Nucl. Part. Phys. (2009)
Silvestri & Trodden, Rep. Prog. Phys. 72:096901 (2009)
Kirshner, astro-ph/0910.0257
4
The only mode which preserves homogeneity and isotropy is overall expansion or contraction:
Cosmic scale factor
€
a(t)
6
On average, galaxies are at rest in these expanding(comoving) coordinates, and they are not expanding--they are gravitationally bound.
Wavelength of radiation scales with scale factor:
Redshift of light:
emitted at t1, observed at t2
€
a(t1)
€
a(t2)
€
λ ~ a(t)
€
1+ z =λ (t2)
λ (t1)=
a(t2)
a(t1)
7
Distance between galaxies:
where fixed comoving distance
Recession speed:
Hubble’s Law (1929)
€
a(t1)
€
a(t2)
€
υ =d(t2) − d(t1)
t2 − t1=
r[a(t2) − a(t1)]
t2 − t1
=d
a
da
dt≡ dH(t)
≈ dH0 for `small' t2 − t1
€
d(t) = a(t)r
€
r =
€
d(t2)
Recent Measurement of H0
9
HST Distances to 240 Cepheid variable stars in 6 SN Ia host galaxies
Riess, etal 2009
€
H0 = 74.2 ± 3.6 km/sec/Mpc
How does the expansion of the Universe change over time?
Gravity:
everything in the Universe attracts everything else
expect the expansion of the Universe should slow
down over time
Cosmological Dynamics
€
˙ ̇ a
a= −
4πG
3 i
∑ ρ i +3pi
c 2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
Equation of state parameter : wi = pi /ρ ic2
Non - relativistic matter : pm ~ ρ m v2, w ≈ 0
Relativistic particles : pr = ρ rc2 /3, w =1/3
€
H 2(t) =˙ a
a
⎛
⎝ ⎜
⎞
⎠ ⎟2
=8πG
3ρ i(t)
i
∑ −k
a2(t)FriedmannEquations
Density Pressure
Spatial curvature: k=0,+1,-1
Size of theUniverse
Cosmic Time
Empty
Today
In these cases, decreases with time, : ,expansion decelerates
€
˙ a
€
˙ ̇ a < 0
Cosmological Dynamics
€
˙ ̇ a
a= −
4πG
3 i
∑ ρ i +3pi
c 2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
Equation of state parameter : wi = pi /ρ ic2
Non - relativistic matter : pm ~ ρ m v2, w ≈ 0
Relativistic particles : pr = ρ rc2 /3, w =1/3
Dark Energy : component with negative pressure : wDE < −1/3
€
H 2(t) =˙ a
a
⎛
⎝ ⎜
⎞
⎠ ⎟2
=8πG
3ρ i(t)
i
∑ −k
a2(t)FriedmannEquations
16
Discovery of Cosmic Acceleration from High-redshiftSupernovae
Type Ia supernovae that exploded when the Universe was 2/3 its present size are ~25% fainter than expected
= 0.7 = 0.m = 1.
Log(distance)
redshift
Accelerating
Not accelerating
Cosmic Acceleration
This implies that increases with time: if we could watch the same galaxy over cosmic time, we would see its recession speed increase.
Exercise 1: A. Show that above statement is true. B. For a galaxy at d=100 Mpc, if H0=70 km/sec/Mpc =constant, what is the increase in its recession speed over a 10-year period? How feasible is it to measure that change?
€
˙ ̇ a > 0 →
˙ a = Ha increases with time
€
υ =Hd
Cosmic Acceleration
What can make the cosmic expansion speed up?
1. The Universe is filled with weird stuff that gives rise to `gravitational repulsion’. We call this Dark Energy
2. Einstein’s theory of General Relativity is wrong on cosmic distance scales.
3. We must drop the assumption of homogeneity/isotropy.
19
Cosmological Constant as Dark Energy
Einstein:
Zel’dovich and Lemaitre:
€
Gμν − Λgμν = 8πGTμν
Gμν = 8πGTμν + Λgμν
≡ 8πG Tμν (matter) + Tμν (vacuum)( )
€
Tμν (vac) =Λ
8πGgμν
ρ vac = T00 =Λ
8πG, pvac = Tii = −
Λ
8πG
wvac = −1 ⇒ H = constant ⇒ a(t)∝ exp(Ht)
Cosmological Constant as Dark Energy Quantum zero-point fluctuations: virtual particles continuously fluctuate into and out of the vacuum (via the Uncertainty principle).
Vacuum energy density in Quantum Field Theory:
Theory: Data:
Pauli
€
ρvac =Λ
8πG=
1
V
1
2h∑ ω = hc(k 2 + m2
0
M
∫ )1/ 2 d3k ~ M 4
wvac =pvac
ρ vac
= −1, ρ vac = const.
€
M ~ MPlanck = G−1/ 2 =1028 eV ⇒ ρ vac ~ 10112 eV4
ρ vac <10−10eV4
Cosmological Constant Problem
Components of the Universe
Dark Matter: clumps, holds galaxies and clusters togetherDark Energy: smoothly distributed, causes expansion of Universe to
speed up
€
ρm ~ a−3
€
ρr ~ a−4
€
ρDE ~ a−3(1+w )
€
wi(z) ≡pi
ρ i
˙ ρ i + 3Hρ i(1+ wi) = 0
=Log[a0/a(t)]
Equation of State parameter w determines Cosmic Evolution
Conservation of Energy-Momentum
23
• Depends on constituents of the Universe:
History of Cosmic Expansion
€
E 2(z) ≡H 2(z)
H02
= Ω i
i
∑ (1+ z)3(1+wi ) + Ωk (1+ z)2 for constant wi
= Ωm (1+ z)3 + ΩDE exp 3 (1+ w(z))d ln(1+ z)∫[ ] + 1− Ωm − ΩDE( ) 1+ z( )2
where
Ω i =ρ i
ρ crit
=ρ i
(3H02 /8πG)
24
Cosmological Observables
€
ds2 = c 2dt 2 − a2(t) dχ 2 + Sk2(χ ) dθ 2 + sin2 θ dφ2
{ }[ ]
= c 2dt 2 − a2(t)dr2
1− kr2+ r2 dθ 2 + sin2 θ dφ2
{ } ⎡
⎣ ⎢
⎤
⎦ ⎥
Friedmann-Robertson-WalkerMetric:
where
Comoving distance:€
r = Sk (χ ) = sinh(χ ), χ , sin(χ ) for k = −1,0,1
€
cdt = a dχ ⇒ χ =cdt
a∫ =
cdt
ada∫ da = c
da
a2H(a)∫
€
a =1
1+ z ⇒ da = −(1+ z)−2 dz = −a2dz
cdt
dada = adχ ⇒ −
c˙ a
a2dz = adχ ⇒ − cdz = H(z)dχ
Age of the Universe
25
€
cdt = adχ
t = adχ =da
aH(a)∫∫ =
dz
(1+ z)H(z)∫
t0 =1
H0
dz
(1+ z)E(z)0
∞
∫
where E(z) = H(z) /H0
26
Exercise 2:
A. For w=1(cosmological constant ) and k=0:
Derive an analytic expression for H0t0 in terms of
Plot
B. Do the same, but for
C. Suppose H0=70 km/sec/Mpc and t0=13.7 Gyr.
Determine in the 2 cases above.
D. Repeat part C but with H0=72.
€
E 2(z) =H 2(z)
H02
= Ωm (1+ z)3 + ΩDE exp 3 (1+ w(z))d ln(1+ z)∫[ ] + 1− Ωm − ΩDE( ) 1+ z( )2
€
E 2(a) =H 2(a)
H02
= Ωma−3 + ΩΛ
€
Ωm
€
H0t0 vs. Ωm
€
ΩΛ =0, Ωk ≠ 0
€
Ωm
Luminosity Distance• Source S at origin emits light at time t1 into solid angle d, received by observer O at coordinate distance r at time t0, with detector of area A:
S
A
r
Proper area of detector given by the metric:
Unit area detector at O subtends solid angle
at S.
Power emitted into d is
Energy flux received by O per unit area is
€
A = a0r dθ a0rsinθ dφ = a02r2dΩ
€
dΩ =1/a02r2
€
dP = L dΩ /4π
€
f =L dΩ
4π=
L
4πa02r2
Include Expansion
• Expansion reduces received flux due to 2 effects: 1. Photon energy redshifts:
2. Photons emitted at time intervals t1 arrive at time
intervals t0: €
Eγ (t0) = Eγ (t1) /(1+ z)
€
dt
a(t)t1
t0
∫ =dt
a(t)t1 +δ t1
t0 +δ t0
∫
dt
a(t)t1
t1 +δ t1
∫ +dt
a(t)t1 +δ t1
t0
∫ =dt
a(t)t1 +δ t1
t0
∫ +dt
a(t)t0
t0 +δ t0
∫
δt1a(t1)
=δt0
a(t0) ⇒
δt0
δt1=
a(t0)
a(t1)=1+ z
€
f =L dΩ
4π=
L
4πa02r2(1+ z)2
≡L
4πdL2
⇒ dL = a0r(1+ z) = (1+ z)2 dA
Luminosity DistanceConvention: choose a0=1
30
Worked Example I
For w=1(cosmological constant ):
Luminosity distance:
€
E 2(z) =H 2(z)
H02
= Ωm (1+ z)3 + ΩDE exp 3 (1+ w(z))d ln(1+ z)∫[ ] + 1− Ωm − ΩDE( ) 1+ z( )2
€
dL (z;Ωm,ΩΛ ) = r(1+ z) = c(1+ z)Sk
da
H0a2E(a)
∫ ⎛
⎝ ⎜
⎞
⎠ ⎟
= c(1+ z)Sk
da
H0a2[Ωma−3 + ΩΛ + (1− Ωm − ΩΛ )a−2]1/ 2∫
⎛
⎝ ⎜
⎞
⎠ ⎟
€
E 2(a) =H 2(a)
H02
= Ωma−3 + ΩΛ + 1− Ωm − ΩΛ( )a−2
31
Worked Example II
For a flat Universe (k=0) and constant Dark Energy equation of state w:
Luminosity distance:
€
E 2(z) =H 2(z)
H02
= Ωm (1+ z)3 + ΩDE exp 3 (1+ w(z))d ln(1+ z)∫[ ] + 1− Ωm − ΩDE( ) 1+ z( )2
€
E 2(z) =H 2(z)
H02
= (1− ΩDE )(1+ z)3 + ΩDE (1+ z)3(1+w )
€
dL (z;ΩDE ,w) = r(1+ z) = χ (1+ z) =c(1+ z)
H0
da
a2E(a)∫
=c(1+ z)
H0
1+ ΩDE[(1+ z)3w −1]−1/ 2
(1+ z)3 / 2∫ dz
Note: H0dL is independent of H0
Exercise 3
• Make the same plot for Worked Example I: plot curves of constant luminosity distance (for several choices of redshift between 0.1 and 1.0) in the plane of , choosing the distance for the model with as the fiducial.
• In the same plane, plot the boundary of the region between present acceleration and deceleration.
• Extra credit: in the same plane, plot the boundary of the region that expands forever vs. recollapses.
33
€
ΩΛvs. Ωm
€
ΩΛ = 0.7, Ωm = 0.3
34
Bolometric Distance Modulus• Logarithmic measures of luminosity and flux:
• Define distance modulus:
• For a population of standard candles (fixed M), measurements of vs. z, the Hubble diagram, constrain cosmological parameters.
€
M = −2.5log(L) + c1, m = −2.5log( f ) + c2
€
μ ≡m − M = 2.5log(L / f ) + c3 = 2.5log(4πdL2) + c3
= 5log[H0dL (z;Ωm,ΩDE ,w(z))]− 5log H0 + c4
flux measure redshift from spectra
Exercise 4
• Plot distance modulus vs redshift (z=0-1) for:• Flat model with• Flat model with• Open model with
– Plot first linear in z, then log z.
• Plot the residual of the first two models with respect to the third model
35
€
Ωm =1
€
ΩΛ =0.7, Ωm = 0.3
€
Ωm = 0.3
36
Discovery of Cosmic Acceleration from High-redshiftSupernovae
Type Ia supernovae that exploded when the Universe was 2/3 its present size are ~25% fainter than expected
= 0.7 = 0.m = 1.
Log(distance)
redshift
Accelerating
Not accelerating
37
Distance Modulus• Recall logarithmic measures of luminosity and flux:
• Define distance modulus:
• For a population of standard candles (fixed M) with known spectra (K) and known extinction (A), measurements of vs. z, the Hubble diagram, constrain cosmological parameters.
€
M i = −2.5log(Li) + c1, mi = −2.5log( f i) + c2
€
μ ≡mi − M j = 2.5log(L / f ) + K ij (z) + c3 = 2.5log(4πdL2) + K + c3
= 5log[H0dL (z;Ωm,ΩDE ,w(z))]− 5log H0 + K ij (z) + Ai + c4
denotes passband
38
K corrections due to redshiftSN spectrum
Rest-frame B band filter
Equivalent restframe i band filter at different redshifts
(iobs=7000-8500 A)
€
f i = Si(λ )Fobs(λ )dλ∫= (1+ z) Si∫ [λ rest (1+ z)]Frest (λ rest )dλ rest
39
Absolute vs. Relative Distances• Recall logarithmic measures of luminosity and flux:
• If Mi is known, from measurement of mi can infer absolute distance to
an object at redshift z, and thereby determine H0 (for z<<1, dL=cz/H0)
• If Mi (and H0) unknown but constant, from measurement of mi can
infer distance to object at redshift z1 relative to object at distance z2:
independent of H0
• Use low-redshift SNe to vertically `anchor’ the Hubble diagram, i.e., to determine
€
M i = −2.5log(Li) + c1, mi = −2.5log( f i) + c2
€
mi = 5log[H0dL ] − 5logH0 + M i + K(z) + c4
€
m1 − m2 = 5logd1
d2
⎛
⎝ ⎜
⎞
⎠ ⎟+ K1 − K2
€
M − 5logH0
Type Ia SupernovaeThermonuclear explosions of Carbon-Oxygen White Dwarfs
White Dwarf accretes mass from or merges with a companion star, growing to a critical mass~1.4Msun
(Chandrasekhar)
After ~1000 years of slow cooking, a violent explosion is triggered at or near the center, and the star is completely incinerated within seconds
In the core of the star, light elements are burned in fusion reactions to form Nickel. The radioactive decay of Nickel and Cobalt makes it shine for a couple of months
44
Type Ia SupernovaeGeneral properties:
• Homogeneous class* of events, only small (correlated) variations• Rise time: ~ 15 – 20 days• Decay time: many months• Bright: MB ~ – 19.5 at peak
No hydrogen in the spectra• Early spectra: Si, Ca, Mg, ...(absorption)• Late spectra: Fe, Ni,…(emission)• Very high velocities (~10,000 km/s)
SN Ia found in all types of galaxies, including ellipticals• Progenitor systems must have long lifetimes
*luminosity, color,spectra at max. light
How similar to one another?
Some real variations: absorption-line shapes at maximum
Connections to luminosity?
Matheson, etal, CfA sample
52
Model
SN Ia Light Curves in SDSS filters synthesized from composite template spectral sequence
SNe evolve in time from blue to red;
K-corrections are time-dependent
€
mi = −2.5logSi(λ )F(λ )dλ∫
Si(λ )dλ∫+ Ζ i
53
SN1998bu Type Ia
Multi-band Light curve
Extremely few light-curves are this well sampled
Suntzeff, etal
Jha, etal
Hernandez, etal
Lum
inos
ity
Time
m15
15 days
Empirical Correlation: Brighter SNe Ia decline more slowly and are bluerPhillips 1993
SN Ia Peak LuminosityEmpirically correlatedwith Light-Curve Decline Rate
Brighter Slower
Use to reduce Peak Luminosity Dispersion
Phillips 1993
Pea
k L
umin
osit
y
Rate of declineGarnavich, etal
56
Type Ia SNPeak Brightnessas calibratedStandard Candle
Peak brightnesscorrelates with decline rate
Variety of algorithms for modeling these correlations: corrected dist. modulus
After correction,~ 0.16 mag(~8% distance error)
Lum
inos
ity
Time
57
Published Light Curves for Nearby Supernovae
Low-z SNe:
Anchor Hubble diagram
Train Light-curve fitters
Need well-sampled, well-calibrated, multi-band light curves
59
Correction for Brightness-Decline relation reduces scatter in nearby SN Ia Hubble Diagram
Distance modulus for z<<1:
Corrected distance modulus is not a direct observable: estimated from a model for light-curve shape
€
m − M = 5logυ − 5log H0
Riess etal 1996
60
Acceleration Discovery Data:High-z SN Team
10 of 16 shown; transformed to SN rest-frame
Riess etal
Schmidt etal
V
B+1
61
Discovery of Cosmic Acceleration from High-redshiftSupernovae
Apply same brightness-decline relation at high z
Type Ia supernovae that exploded when the Universe was 2/3 its present size are ~25% fainter than expected
= 0.7 = 0.m = 1.
Log(distance)
redshift
Accelerating
Not accelerating
HZTSCP
Likelihood Analysis
This assumes errors in distance modulus estimates are Gaussian. More details on this next time.
62
€
−2ln L =(μ i − μ(zi;Ωm,ΩΛ ,H0)2
σ μ2
i
∑
Data Model
Exercise 5
• Carry out a likelihood analysis of using the High-Z Supernova Data of Riess, etal 1998: see following tables. Assume a fixed Hubble parameter for this exercise.
• Extra credit: marginalize over H0 with a flat prior.
64
€
ΩΛ , Ωm