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![Page 1: Science of the Dark Energy Survey Josh Frieman Fermilab and the University of Chicago Astronomy 41100 Lecture 2, Oct. 15, 2010.](https://reader038.fdocuments.net/reader038/viewer/2022110207/56649d565503460f94a343d6/html5/thumbnails/1.jpg)
Science of the Dark Energy Survey
Josh Frieman
Fermilab and the University of Chicago
Astronomy 41100Lecture 2, Oct. 15, 2010
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DES Collaboration Meeting
2
Go to:
http://astro.fnal.gov/desfall2010/Home.html
Science Working group meetings on Tuesday.Plenary sessions Wed-Fri.
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3
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)
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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
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Dark Energy: Alternatives to ΛThe smoothness of the Universe and the large-scalestructure of galaxies can be neatly explained if there was a much earlier epoch of cosmic acceleration that occurred a tiny fraction of a second after the Big Bang:
Primordial Inflation
Inflation ended, so it was not driven by the cosmological constant. This is a caution against theoretical prejudice for Λ as the cause of current acceleration (i.e., as the identity of dark energy).
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Light Scalar Fields as Dark Energy
Perhaps the Universe is not yet in its ground state. The `true’ vacuum energy (Λ) could be zero (for reasons yet unknown). Transient vacuum energy can exist if there is a field that takes a cosmologically long time to reach its ground state. This was the reasoning behind inflation. For this reasoning to apply now, we must postulate the existence of an extremely light scalar field, since the dynamical evolution of such a field is governed by
€
td ~1
m , td >1/H0 ⇒ m < H0 ~ 10−33eV
JF, Hill, Stebbins, Waga 1995
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7
Scalar Field as Dark Energy(inspired by inflation)
Dark Energy could be due to a very light scalar field j, slowly evolving in a potential, V(j):
Density & pressure:
Slow roll:
)(
)(2
21
221
ϕϕ
ϕϕρ
VP
V
−=
+=
&
&
V( )j
j
€
12
˙ ϕ 2 < V (ϕ )⇒ P < 0 ⇔ w < 0 and time - dependent
€
˙ ̇ ϕ + 3H ˙ ϕ +dV
dϕ= 0
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Scalar Field Dark Energy
Ultra-light particle: Dark Energy hardly clusters, nearly smoothEquation of state: usually, w > 1 and evolves in timeHierarchy problem: Why m/ ~ 1061?Weak coupling: Quartic self-coupling < 10122
General features:
meff < 3H0 ~ 10-33 eV (w < 0)(Potential > Kinetic Energy)
V ~ m22 ~ crit ~ 10-10 eV4
~ 1028 eV ~ MPlanck
aka quintessence
V( )j
j1028 eV
(10–3 eV)4
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The Coincidence Problem
Why do we live at the `special’ epoch when the dark energy density is comparable to the matter energy density?
matter ~ a-3
DE~ a-3(1+w)
a(t)Today
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Scalar Field Models & Coincidence
VV
Runaway potentialsDE/matter ratio constant(Tracker Solution)
Pseudo-Nambu Goldstone BosonLow mass protected by symmetry(Cf. axion) JF, Hill, Stebbins, Waga
V() = M4[1+cos(/f)]f ~ MPlanck M ~ 0.001 eV ~ m
e.g., e– or –n
MPl
Ratra & Peebles; Caldwell, etal
`Dynamics’ models
(Freezing models)
`Mass scale’ models
(Thawing models)
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PNGB Models
Tilted Mexican hat:
€
V (Φ) = λ ΦΦ* −f 2
2
⎛
⎝ ⎜
⎞
⎠ ⎟
2
+M 4 cos(Arg(Φ) −1( )
€
M 4 ~ 10−3eV << f ~ MPl
Frieman, Hill, Stebbins, Waga 2005
• Spontaneous symmetry breaking at scale f
• Explicit breaking at scale M
• Hierarchy protected by symmetry
f
M4
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12
Caldwell & Linder
Dynamical Evolution of Freezing vs. Thawing Models
Measuring w and its evolution can potentially distinguish between physical models for acceleration
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Runaway (Tracker) Potentials
Typically >> Mplanck today.
Must prevent terms of the form
V() ~ n+4 / Mplanckn
up to large n
What symmetry prevents them?
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14
Perturbations of Scalar Field Dark Energy
If it evolves in time, it must also vary in space.
h = synchronous gauge metric perturbation
Fluctuations distinguish this from a smooth “x-matter” or
Coble, Dodelson, Frieman 1997Caldwell et al, 1998
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15
Dark Energy Interactions
Couplings to visible particles must be small
Couplings cause long-range forces
Carroll 1998
Attractive force between lumps of
Frieman & Gradwohl 1990, 1992
Example: scalar field coupled to massive neutrino
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What about w < 1? The Big Rip• H(t) and a(t) increase with time and diverge in finite time e.g, for w=-1.1, tsing~100 Gyr• Scalar Field Models: need to violate null Energy condtion + p > 0: for example: L = ()2 V
Controlling instability requires cutoff at low mass scale
• Modified Gravity models apparently can achieve effective w < –1 without violating null Energy Condition
Caldwell, etal
Hoffman, etal
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Modified Gravity & Extra Dimensions
• 4-dimensional brane in 5-d Minkowski space• Matter lives on the brane• At large distances, gravity can leak off brane into the bulk, infinite 5th dimension Dvali, Gabadadze, Porrati
• Acceleration without vacuum energy on the brane, driven by brane curvature term• Action given by:
• Consistency problems: ghosts, strong coupling €
S = M53 d5X det g5∫ R5 + MPlanck
2 d4 x det g4∫ R4 + d4 x∫ det g4 Lm
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Modified Gravity
18
€
gAB = η AB + hAB | h |<<1 A,B = 0,1,2,3,4• Weak-field limit:
• Consider static source on the brane:
• Solution:
• In GR, 1/3 would be ½• Characteristic cross-over scale:
• For modes with p<<1/rc :• Gravity leaks off the brane: longer wavelength gravitons free to propagate into the bulk• Intermediate scales: scalar-tensor theory
€
Tμν ( p) μ,ν = 0,1,2,3
€
hμν ( p) =8πG
p2 + 2(G /G(5))pTμν ( p) −
1
3η μν Tα
α ( p) ⎛
⎝ ⎜
⎞
⎠ ⎟
where
G =1/ MPl2 and G(5) =1/ M5
3
€
rc =1
2
G(5)
G=
MPl2
2M53
€
hμν ~ p−1, corresponds to V (r) ~ r−2
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Cosmological Solutons• Modified Friedmann equation:
• Early times: H>>1/rc: ordinary behavior, decelerated expansion.• Late times: self-accelerating solution for (-)
• For we require
• At current epoch, deceleration parameter is
corresponds to weff=-0.8
€
H 2 ±H
rc
=ρ
3MPlanck2
€
H0 ~ rc−1 ~ 10−33eV
€
M5 ~ 1 GeV
€
H → H∞ = rc−1
€
q0 = 3Ωm (1+ Ωm )−1 −1
= −0.36 for Ωm = 0.27
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Growth of Perturbations
• Linear perturbations approximately satisfy:
• Can change growth factor by ~30% relative to GR• Motivates probing growth of structure in addition to expansion rate€
˙ ̇ δ + 2H ˙ δ = 4πρ m (t)δ(t)Geff (t)
where
Geff = G 1+3
β
⎛
⎝ ⎜
⎞
⎠ ⎟
and
β =1− 2rcH(t) 1+˙ H
3H 2
⎛
⎝ ⎜
⎞
⎠ ⎟
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Modified Gravity: f(R)
€
S = d4∫ x −g R ⇒ d4∫ x −g f (R)
€
e.g., f (R) =1
16πGR −
μ 2(n +1)
Rn
⎛
⎝ ⎜
⎞
⎠ ⎟
This model has self - accelerating vacuum solution, with
R =12H 2 = 3μ 2 .
This particular realization excluded by solar system tests, but variants evade them: Chameleon models hide deviations from GR on solar system scales
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22
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
= 5log[dL (z;Ωm,ΩDE ,w(z)) /10pc]
flux measure redshift from spectra
€
dL (z) = (1+ z)r = (1+ z)Sk (χ ) = (1+ z)Sk
dz
H(z)∫ ⎛
⎝ ⎜
⎞
⎠ ⎟
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23
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
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24
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
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25
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
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26
SN 1994D
Type Ia Supernovae as Standardizable Candles
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27
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SN Spectra
~1 week
after
maximum
light
Filippenko 1997
Ia
II
Ic
Ib
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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
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30
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
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SN Ia Spectral Homogeneity(to lowest order)
from SDSS Supernova Survey
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Spectral Homogeneity at fixed epoch
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SN2004ar z = 0.06 from SDSS galaxy spectrum
Galaxy-subtracted
Spectrum
SN Ia
template
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How similar to one another?
Some real variations: absorption-line shapes at maximum
Connections to luminosity?
Matheson, etal, CfA sample
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35Hsiao etal
Supernova Ia Spectral Evolution
Late times
Early times
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Layered
Chemical
Structure
provides
clues to
Explosion
physics
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37
SN1998bu Type Ia
Multi-band Light curve
Extremely few light-curves are this well sampled
Suntzeff, etal
Jha, etal
Hernandez, etal
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Lum
inos
ity
Time
m15
15 days
Empirical Correlation: Brighter SNe Ia decline more slowly and are bluerPhillips 1993
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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
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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
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41
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
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Low-z Data
42
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43
Carnegie
Supernova
Project
Nearby
Optical+
NIR LCs
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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
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Acceleration Discovery Data:High-z SN Team
10 of 16 shown; transformed to SN rest-frame
Riess etal
Schmidt etal
V
B+1
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Riess, etal High-z Data (1998)
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High-z Supernova Team data (1998)
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Likelihood Analysis
This assume
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−2ln L = χ 2 =(μ i − μmod (zi;Ωm ,ΩΛ,H0)2
σ μ ,i2
i
∑
Since μmod = 5log(H0dL ) − 5log(H0), let ˆ μ ≡ μmod (H0 = 70)
and define Δ i = μ i − ˆ μ . If we fix H0, then we are minimizing
ˆ χ 2 =Δ i
2
σ i2
i
∑
To marginalize over logH0 with flat prior, we instead minimize
χ mar2 = −2ln d(5logH0)exp −χ 2 /2( )∫
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥= ˆ χ 2 −
B2
C+ ln
C
2π
⎛
⎝ ⎜
⎞
⎠ ⎟,
where
B =Δ i
σ i2
i
∑ , C =1
σ i2
i
∑
σ μ ,i2 = σ μ , fit
2 + σ μ ,int2 + σ μ ,vel
2
Goliath etal 2001
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Exercise 5
• Carry out a likelihood analysis of using the High-Z Supernova Data of Riess, etal 1998: use Table 10 above for low-z data and the High-z table above for high-z SNe. Assume a fixed Hubble parameter for the first part of this exercise.
• 2nd part: repeat the exercise, but marginalizing over H0 with a flat prior, either numerically or using the analytic method of Goliath etal.
• Errors: assume intrinsic dispersion of
and fit dispersions from the tables and dispersion due to peculiar velocity from Kessler etal (0908.4274), Eqn. 28, with
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ΩΛ , Ωm
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σ μ,int = 0.15
€
σ z = 0.0012