宇宙微波背景辐射
郭宗宽
2014年理论物理大学生夏令营
2014.07.17
引言
•
•
•
•
什么是宇宙？
宇宙学的基本假设
宇宙学的发展历史
宇宙学观测
• 什么是宇宙？
 4.5 Gpc  15 Gly
太阳大，地球小，地球绕着太阳跑。地球大，月亮小，月亮绕着地球跑。
• 宇宙学的基本假设
–宇宙学原理（无边，无中心）
–爱因斯坦引力理论
–宇宙物质（暗能量+暗物质+重子+光子+中微
子+引力波）
𝐺𝜇𝜈 = 8𝜋𝐺𝑇𝜇𝜈
• 宇宙学的发展历史
1. 大爆炸宇宙学(1920s-1970s)
– 宇宙在膨胀（1929）
– BBN的预言与观测一致（1998）
– CMB的黑体谱（1994）
2. 标准模型(1980s-2000s)
– 暴胀+ΛCDM
3. 精确宇宙学(2000s-now)
– CMB, LSS(BAO, GC, WL), SNIa
– 宇宙学不再是个传说
• 宇宙学观测（电磁波）
 Radio: VLA, SKA, FAST, LOFAR
 Cosmic microwave background: COBE, WMAP, Planck
 Infrared: WISE, Spitzer, WFIRST (mid-2020s)
 Optical: Hubble (1990 ), 2dFGRS (1997  2002), SDSS
(2000  2014), CSTAR (2008  ), Euclid (2017/2018 ),
LSST (2022 )
 Ultraviolet: GALEX
 X-ray: Chandra, XMM-Newton
 Gamma-ray: Fermi LAT (0.02  300 GeV), H.E.S.S. (0.01 
10 TeV)
其他窗口
 Cosmic ray: PAMELA, Fermi, AMS-02
 Gravitational wave background: LIGO, Virgo, Advanced
LIGO, Advanced Virgo, AIGO, LISA
 Cosmic neutrino background
内容
1.
2.
3.
4.
5.
6.
宇宙微波背景（CMB）辐射的形成
CMB的发现和探测实验
CMB的数据分析
CMB各向异性的物理起源
CMB的宇宙学解释
现状与展望
1. CMB的形成
𝑝 + 𝑒− ↔ 𝐻 + 𝛾
𝛾 + 𝑒− ↔ 𝛾 + 𝑒−
decoupling during
recombination
400 cm−3 now
2. CMB的发现和探测实验
 The CMB was first predicted
by G. Gamow, R. Alpher and
R. Herman in 1948
T~5 K
 the first discovery of the CMB
radiation in 1964-1965 the Nobel
Prize in Physics 1978:
A.A. Penzias and R.W. Wilson
 It is interpreted by R. Wilson, B. Burke, R. Dicke and J. Peebles
in 1965.
 COBE (Cosmic Background Explorer) - the first generation CMB
experiment, launched on 18 Nov. 1989, 4 years
the Nobel Prize in Physics 2006: J.C. Mather and G.F. Smoot
Hot big bang
J.C. Mather
G.F. Smoot (DMR)
isotropy
 the COBE satellite experiments:
① the Far InfraRed Absolute
Spectrophotometer (FIRAS)
team
② the Differential Microwave
Radiometer (DMR) team
 advantages of satellite experiments:
• no atmospheric thermal emission
• full-sky map
 WMAP (Wilkinson Microwave Anisotropy Probe) - the second
generation CMB experiment, launched on 30 June 2001, 9 years
141°
23 GHz
33 GHz
41 GHz
61 GHz
• free-free emission: electron-ion scattering
• synchrotron emission: the acceleration of cosmic ray
electrons in magnetic fields
• thermal emission from dust
94 GHz
• foreground mask
• angular power spectrum of CMB
• WMAP science team publications
a)
b)
c)
d)
e)
2003, WMAP1, 14 papers, cited by 6873 records
2007, WMAP3, 5 papers, cited by 5289 records
2009, WMAP5, 8 papers, cited by 3527 records
2011, WMAP7, 6 papers, cited by 3803 records
2012, WMAP9, 2 papers, cited by 303 records
We have entered a new era of precision cosmology.
 Planck - the third generation CMB experiment, launched
on 14 May 2009, 30 months, 5 full-sky surveys
LFI: 30,44,70 GHz
HFI : 100,143,217,353,545,857 GHz
•
•
•
•
high sensitivity
wide frequency
full-sky coverage
high resolution ~7º,15′,5′
cosmological parameters
the temperature
angular power
spectrum
20 March 2013,
29 papers
1609 citations
 next generation space-based CMB experiment
• NASA: CMBPol
• ESA: COrE
 Other experiments
• ground-based experiments
ACBAR, CBI, VSA, QUaD, …
ACT, ACTPol (2013~)
SPT, SPTpol (95,150GHz, 2012~2015), SPT-3G (2016~2019)
BICEP1 (2006~2008), BICEP2 (150GHz, 2010~2012), BICEP3 (95GHz, 2016~)
POLARBEAR (150GHz, 2012~2013), POLARBEAR-2 (95,150GHz, 2016~)
QUBIC (r ~ 0.01, bolometer, interferometer)
• balloon-borne experiments
BOOMRANG, MAXIMA, …
EBEX (150,250,410GHz)
SPIDER (94,150GHz, Dec 2014)
PIPER (200,270,350,600GHz, 2015)
 South Pole Telescope (SPT)
10 meter telescope
3 frequencies (95, 150 and 220 GHz)
arXiv:1105.3182: SPT+WMAP7+BAO+H0
𝑁eff = 3.86 ± 0.42
arXiv:1212.6267: SPT+WMAP7+BAO+H0
∑𝑚𝜈 = 0.32 ± 0.11 eV
 Atacama Cosmology Telescope (ACT)
3 frequencies (148, 218, and 277 GHz)
6 meter telescope
3. CMB的数据分析
time-ordered data
 time-ordered data
full sky map
spectrum
parameter estimates
 the temperature anisotropies can
be expanded in spherical harmonics
∆𝑇(𝑛)
=
𝑇
𝑎𝑙𝑚 =
𝑎𝑙𝑚 𝑌𝑙𝑚 (𝑛)
𝑙𝑚
𝑑𝑛
∆𝑇(𝑛)
∗
𝑌𝑙𝑚 (𝑛)
𝑇
~ 10−5
𝑑𝑡 = 𝑃𝑡𝑖 𝑚𝑖 + 𝑛𝑡
 for Gaussian random fluctuations, the statistical properties of the
temperature field are determined by the angular power spectrum
∗
𝑎𝑙𝑚
𝑎𝑙′ 𝑚′ = 𝐶𝑙𝑇𝑇 𝛿𝑙𝑙′ 𝛿𝑚𝑚′
For a full sky, noiseless experiments,
𝐶𝑙𝑇𝑇
1
=
2𝑙 + 1
𝑎𝑙𝑚
2
𝑚
 cosmological parameter estimation
likelihood function for a full sky:
−2 ln ℒ =
𝑙
the sky-cut, MCMC
𝐶𝑙th + 𝒩𝑙
𝐶𝑙
(2𝑙 + 1) ln
+
−1
th
𝐶𝑙
𝐶𝑙 + 𝒩𝑙
4. CMB各向异性的物理起源
• primary CMB anisotropies (at recombination)
inflation model (Alan H. Guth in 1981)
𝛿𝜙 ⟺ 𝛿𝑔𝜇𝜈 ⇔ 𝛿𝑓 ⟺ 𝛿𝑇, 𝑈, 𝑄
• secondary CMB anisotropies (after recombination)
①
②
③
④
thermal/kinetic Sunyaev-Zel’dovich effect
integrated Sachs-Wolf effect
reionization
weak lensing effect
 inflation model
V (φ)
reheating
inflation
φ
for slow-roll inflation, the primordial power spectra of
scalar/tensor perturbations:
1 𝐻
𝒫𝑠 𝑘 =
2𝜖 2𝜋
𝐻
𝒫𝑇 𝑘 = 8
2𝜋
2
2
𝑘
𝑎𝐻
𝑘
𝑎𝐻
𝑛𝑠 −1
𝑛𝑇




Phase transition: Old inflation, New inflation
Kinetic term: K-inflation, Tachyon inflation, G-inflation
Modified gravity: R^2-inflation, Brane-world inflation, Extended inflation
Multiple fields: Hybrid inflation (waterfall field), Assisted inflation, N-flation, Matrixinflation, Double inflation, Curvaton-type inflation, Modulated inflation
 Slow-roll parameter: Large-field inflation, Small-field inflation
 Example: Chaotic inflation, Power-law inflation, Eternal inflation
 Interaction: Warm inflation, Trapped inflation
 Initial condition: Bounce inflation, Cyclic inflation
 Particle physics: Higgs inflation, Super-natural inflation
 SUSY: SUSY F-term inflation, SUSY D-term inflation, SUSY P-term inflation,
SUGRA inflation
 String theory:
open string inflationary models:
D3/𝐷3 brane inflation, Inflection point inflation, DBI inflation,Wilsonline inflation,
D3-D7 inflation
closed string inflationary models:
Racetrack inflation, N-flation, Axion monodromy, Kahler moduli inflation, Fibre
inflation, Poly-instanton inflaiton
 the coupled, linearized Boltzmann, Einstein and fluid equations
𝑓 𝑥, 𝑞, 𝑛, 𝜏 = 𝑓0 (𝑞) 1 + Ψ(𝑥, 𝑞, 𝑛, 𝜏)
𝑔𝑠
1
𝑓0 𝑞 = 3 𝜖(𝑞)/𝑘 𝑇
𝐵 ∓1
ℎ 𝑒
∆𝑇
𝑑 ln 𝑓0
∆(𝑥, 𝑛, 𝜏) ≡
=−
𝑇
𝑑 ln 𝑞
−1
Ψ
the metric in the synchronous gauge
𝑑𝑠 2 = 𝑎2 𝜏 [−𝑑𝜏 2 + (𝛿𝑖𝑗 + ℎ𝑖𝑗 )𝑑𝑥 𝑖 𝑑𝑥 𝑗 ]
ℎ𝑖𝑗 (𝑥, 𝜏) =
1
ℎ𝑖𝑗 = ℎ𝛿𝑖𝑗 + ℎ𝑖𝑗 ∥ + ℎ𝑖𝑗 ⊥ + ℎ𝑖𝑗 𝑇
3
1
3
𝑖𝑘∙𝑥
𝑑 𝑘𝑒 [𝑘𝑖 𝑘𝑗 ℎ 𝑘, 𝜏 + 𝑘𝑖 𝑘𝑗 − 𝛿𝑖𝑗 6𝜂(𝑘, 𝜏)]
3
the Einstein equations
ℋ
2
8𝜋 2
=
𝐺𝑎
3
the equations of state
𝜌𝑖 ,
𝑖
4𝜋 2
ℋ = − 𝐺𝑎
3
𝑤𝑖 = 𝑃𝑖 /𝜌𝑖
(𝜌𝑖 +3𝑃𝑖 )
𝑖
the linearized Einstein equations in k-space
𝑘2𝜂
1
− ℋ ℎ = −4𝜋𝐺𝑎2
2
𝑘 2 𝜂 = 4𝜋𝐺𝑎2
𝛿𝜌𝑖
𝑖
(𝜌𝑖 + 𝑃𝑖 )𝜃𝑖
𝑖
ℎ + 2ℋ ℎ − 2𝑘 2 𝜂 = −24𝜋𝐺𝑎2
𝛿𝑃𝑖
𝑖
ℎ + 6𝜂 + 2ℋ(ℎ + 6𝜂) − 2𝑘 2 𝜂 = −24𝜋𝐺𝑎2
(𝜌𝑖 + 𝑃𝑖 )𝜎𝑖
𝑖
the perturbed part of energy-momentum conservation equations
for the non-relativistic fluid (baryon, CDM, DE) in k-space
ℎ
𝛿𝑃
𝛿𝑖 = − 1 + 𝑤𝑖 𝜃𝑖 +
− 3ℋ
− 𝑤𝑖 𝛿𝑖
2
𝛿𝜌
𝑤𝑖
𝛿𝑃/𝛿𝜌 2
𝜃𝑖 = −ℋ 1 − 3𝑤𝑖 𝜃𝑖 −
𝜃𝑖 +
𝑘 𝛿 − 𝑘 2 𝜎𝑖
1 + 𝑤𝑖
1 + 𝑤𝑖
CDM:
1
𝛿𝑐 = − ℎ
2
baryon:
1
𝛿𝑏 = −𝜃𝑏 − ℎ
2
𝜃𝑏 = −ℋ𝜃𝑏 +
DE:
𝑐𝑠2 𝑘 2 𝛿𝑏
4𝜌𝛾
+
𝑎𝑛𝑒 𝜎𝑇 (𝜃𝛾 − 𝜃𝑏 )
3𝜌𝑏
the Boltzmann equation in the synchronous gauge for the photon
and neutrino components in k-space
𝜕Ψ
𝑞
𝑑 ln 𝑓0
ℎ + 6𝜂
1 𝜕𝑓
2
+𝑖 𝑘∙𝑛 Ψ+
𝜂−
(𝑘 ∙ 𝑛) =
𝜕𝜏
𝜖
𝑑 ln 𝑞
2
𝑓0 𝜕𝜏
for massless particles
𝐹 𝑘, 𝑛, 𝜏 ≡
𝐶
𝑞 2 𝑑𝑞𝑞𝑓0 (𝑞)Ψ
𝑞 2 𝑑𝑞𝑞𝑓0
F is expanded in a Legendre series
∞
F 𝑘, 𝑛, 𝜏 =
−𝑖
𝑙
2𝑙 + 1 𝐹𝑙 (𝑘, 𝜏)𝑃𝑙 (𝑘 ∙ 𝑛)
𝑙=0
using 𝑃0 𝜇 = 1, 𝑃1 𝜇 = 𝜇, 𝑃2 𝜇 = (3𝜇2 − 1)/2, the recursion
relation and the orthonormality of the Legendre polynomials we
obtain a hierarchy.
 CMB angular power spectrum
for the CMB photon
∆𝑇
𝑑 ln 𝑓0
∆ 𝑥, 𝑛, 𝜏 ≡
=−
𝑇
𝑑 ln 𝑞
∆ 𝑥, 𝑛, 𝜏 =
−1
𝐹𝛾
Ψ=
4
𝑑3 𝑘𝑒 𝑖𝑘∙𝑥 ∆ 𝑘, 𝑛, 𝜏
∞
∆ 𝑘, 𝑛, 𝜏 =
−𝑖
𝑙=0
𝑙
2𝑙 + 1 ∆𝑙 (𝑘, 𝜏)𝑃𝑙 (𝑘 ∙ 𝑛)
expanded in spherical harmonics
∆ 𝑥0 , 𝑛, 𝜏0 =
𝑑3 𝑘𝑒 𝑖𝑘∙𝑥0 ∆ 𝑘, 𝑛, 𝜏0
∆ 𝑘, 𝑛, 𝜏0 =
𝑎𝑙𝑚 (𝑘, 𝜏0 )𝑌𝑙𝑚 (𝑛)
𝑙,𝑚
𝑎𝑙𝑚 𝑘, 𝜏0 =
𝑑Ω ∆ 𝑘, 𝑛, 𝜏0 𝑌𝑙𝑚 𝑛 = 4𝜋∆𝑙 (𝑘, 𝜏0 )𝑌𝑙𝑚 𝑘
the angular power spectrum is
∗
𝑎𝑙𝑚
𝑎𝑙′ 𝑚′ = 𝐶𝑙 𝛿𝑙𝑙′ 𝛿𝑚𝑚′
initial conditions (radiation-dominated, outside the horizon,
adiabatic mode, isocurvature mode)
∆𝑙 𝑘, 𝜏 = 𝜓(𝑘)∆𝑙 𝑘, 𝜏
𝜓(𝑘1 )𝜓(𝑘2 ) = 𝒫 𝑘 𝛿(𝑘1 + 𝑘1 )
We obtain the result
𝐶𝑙 = 4𝜋
𝑑𝑘
𝒫 𝑘 |∆𝑙 (𝑘, 𝜏0 )|2
𝑘
 features of spectrum

large angular scales
integrated SZ effect (<10)
Sachs-Wolf effect (10~100)

intermediate scales
acoustic oscillations (100~1000)

small scales (>1000)
Silk damping: the dissipation of
small-scale perturbations caused
by photons' random walking out
of overdense regions.
For full accuracy, the Boltzmann equation must be solved to
follow the evolution of the photon distribution function.
5. CMB的宇宙学解释
 CMB temperature fluctuations
gravity
pressure
The stronger the contraction, the higher these peaks should be.
Acoustic oscillations are frozen in at recombination.
 CMB polarization
A monochromatic electromagnetic
wave propagating in the z direction has
an electric field vector
𝐸𝑥 = 𝑎𝑥 cos 𝜔𝑡 − 𝜉𝑥
𝐸𝑦 = 𝑎𝑦 cos 𝜔𝑡 − 𝜉𝑦
𝐼 = 𝑎𝑥 2 + 𝑎𝑦 2
𝑄 = 𝑎𝑥 2 − 𝑎𝑦 2
𝑈 = 2𝑎𝑥 𝑎𝑦 cos 𝜉𝑥 − 𝜉𝑦
𝑉 = 2𝑎𝑥 𝑎𝑦 sin 𝜉𝑥 − 𝜉𝑦
scalar mode
tensor mode with 𝑟 = 0.22
6. 现状与展望
 tension between Planck and astrophysical
measurements
𝐻0 , Ω𝑚 , 𝜎8 , 𝑟
 anomalies in the WMAP/Planck data
− the quadrupole-octopole alignment
− power deficit at low-l,
− hemispherical asymmetry,
− parity asymmetry
− the cold spot, …
 detection of the primordial tensor perturbations
 BICEP2 (Background Imaging of Cosmic Extragalactic Polarization)
experiment — evidence for primordial B-mode is first detected.
BICEP1 (2006-2008)
BICEP2 (2010-2012)
SPT
The Dark Sector Lab (DSL)
26 cm aperture
150 GHz
383.7 deg2
2010-2012
4 tiles
8×8 array of
detector pairs
antenna networks
band-defining filters
bolometers
scan strategy
 CMB极化数据分析
 raw timestreams (2010~2012)
Glitches and flux jumps are flagged.
 map making (T, Q, U)
𝑑 𝑡 = 𝑇 𝑛𝑡 + 𝑄 𝑛𝑡 cos 2𝜓𝑡 + 𝑈 𝑛𝑡 sin 2𝜓𝑡
𝑑± =
1 𝐴
𝑑 ± 𝑑𝐵 ,
2
1 𝛼2
𝑑− 𝛼
=
𝑑−𝛽
2 𝛼𝛽
𝛼𝛽
𝛽2
𝑑 + = 𝑇 𝑛𝑡
𝑄(𝑛𝑡 )
𝑈(𝑛𝑡 )
𝛼 = cos 2𝜓𝑡𝐴 − cos 2𝜓𝑡𝐵 ,
𝛽 = sin(2𝜓𝑡𝐴 ) − sin(2𝜓𝑡𝐵 )
detector transfer function, gain calibration, noise, beam function, polarization
leakage, …
 from maps to power spectra
(𝑄 ± 𝑖𝑈)′ 𝑛 = 𝑒 ∓2𝑖𝜓 (𝑄 ± 𝑖𝑈) 𝑛
𝑇 𝑛
=
𝑇
𝑇
𝑎𝑙𝑚
𝑇
𝑎𝑙𝑚
𝑌𝑙𝑚 𝑛
=
𝑑Ω
∗
𝑌𝑙𝑚
𝑇 𝑛
𝑛
𝑇
𝑙𝑚
𝑄 + 𝑖𝑈 𝑛 =
2
𝑎𝑙𝑚
2𝑌𝑙𝑚 𝑛
2
𝑎𝑙𝑚
=
∗
𝑑Ω 2𝑌𝑙𝑚
𝑛 𝑄 + 𝑖𝑈 𝑛
−2
𝑎𝑙𝑚
−2
𝑎𝑙𝑚
=
∗
𝑑Ω −2𝑌𝑙𝑚
𝑛 𝑄 − 𝑖𝑈 𝑛
𝑙𝑚
𝑄 − 𝑖𝑈 𝑛 =
−2𝑌𝑙𝑚
𝑛
𝑙𝑚
𝐸
𝑎𝑙𝑚
𝑌𝑙𝑚 𝑛
𝐸 𝑛 =
𝑙𝑚
𝐵
𝑎𝑙𝑚
𝑌𝑙𝑚 𝑛
𝐵 𝑛 =
𝐸
𝑎𝑙𝑚
𝐵
𝑎𝑙𝑚
𝑙𝑚
• rotationally invariant
• B has the opposite parity of T and E
• scalar modes contribute only to E
1 2
−2
= − 𝑎𝑙𝑚 + 𝑎𝑙𝑚
2
𝑖 2
−2
= 𝑎𝑙𝑚
− 𝑎𝑙𝑚
2
1
𝐶𝑙𝑇𝑇 =
2𝑙 + 1
1
𝐶𝑙𝐸𝐸 =
2𝑙 + 1
1
𝐶𝑙𝐵𝐵 =
2𝑙 + 1
1
𝐶𝑙𝑇𝐸 =
2𝑙 + 1
9 data bandpowers: ∆l=35, 20<l<340
𝑇∗ 𝑇
𝑎𝑙𝑚
𝑎𝑙𝑚
𝒟𝑏𝑇𝑇
𝓓𝑏 = 𝒟𝑏𝑇𝐸
𝒟𝑏𝑇𝐵
𝑚
𝐸∗ 𝐸
𝑎𝑙𝑚
𝑎𝑙𝑚
𝑚
𝐵∗ 𝐵
𝑎𝑙𝑚
𝑎𝑙𝑚
𝒟𝑏𝑇𝐸
𝒟𝑏𝐸𝐸
𝒟𝑏𝐸𝐵
𝒟𝑏𝑇𝐵
𝒟𝑏𝐸𝐵
𝒟𝑏𝐵𝐵
model bandpowers by band window functions
𝑚
𝒟𝑏𝑋𝑌 =
𝑇∗ 𝐸
𝑎𝑙𝑚
𝑎𝑙𝑚
𝑋𝑌 𝑋𝑌
𝑤𝑏,𝑙
𝒟𝑙
𝑙
𝑚
 cosmological parameter constraints
𝑋𝑏𝑇𝑇
𝑋𝑏𝐸𝐸
𝑋𝑏𝐵𝐵
𝑋𝑏𝑇𝐸
𝑋𝑏𝐸𝐵
𝑋𝑏𝑇𝐵
 direct likelihood
−1 𝑋
 bandpower likelihood: −2 log ℒ 𝓓𝑏 𝓓𝑏 = 𝑋𝑐 ℳ𝑐𝑐′
𝑐′
= vecp
𝑓 1/2
𝓓𝑏
𝑓 1/2
U𝑏 𝑔 D𝑏 U𝑏† 𝓓𝑏
−1/2
U𝑏 , D𝑏 = eig 𝓓𝑏
−1/2
𝓓𝑏 𝓓𝑏
𝑔 𝑥 = sign(𝑥 − 1) 2(𝑥 − ln𝑥 − 1)
 BICEP2数据
“BICEP2 2014 I: Detection of B-mode Polarization at Degree Angular
Scales”, arXiv:1403.3985, cited by 453 records
“BICEP2 2014 II: Experiment and Three-year Data Set”, arXiv:1403.4302
reionization bump
recombination bump
Step 1: is the data reliable?
① some unknown sources of systematic error
− foreground contribution, at least three frequencies (150 GHz)
− leakage
② data analysis pipeline (self-calibration)
③ likelihood method
“至于你信不信，我反正信了。”
353GHz, Q
Planck Collaboration, arXiv:1405.0871.
353GHz, U
Step 2: primordial gravitational wave?
①
②
③
④
⑤
to confirm that the signal is statistically isotropic
cosmic string
Faraday rotation
cosmic birefringence
reionization
Step 3: a tension with the Planck result 𝑟 < 0.11 2𝜎
①
②
③
④
⑤
⑥
a large negative running of scalar spectral index
step potential
fast-slow-roll inflation
non-Bunch-Davis vacuum, false vacuum, trans-Planck
anti-correlated tensor-curvature
anti-correlated iso-curvature
Step 4: what is the inflation field?
① Higgs field?
{𝑛𝑠 , 𝑟, 𝑁}
② for slow-roll inflation, the energy scale of inflation is
𝑉 1/4 ~2.25 × 1016 GeV 𝑟/0.2
1/4
③ challenge for slow-roll inflation with a large negative running
{𝑛𝑠 , 𝛼𝑠 , 𝑟, 𝑁}
Step 5: if confirmed by Planck, the tensor spectral index
① scale-invariant tensor spectrum (𝑛 𝑇 = 0)
② the standard consistency relation? (𝑛 𝑇 = −𝑟/8)
③ blue spectrum? (𝑛 𝑇 > 0)
string gas cosmology
bounce inflation
super-inflation before slow-roll inflation
𝑑 ln 𝒫(𝑘)
𝑑 ln 𝑘
method:
ln
𝑘𝑚𝑖𝑛
ln 𝒫(𝑘) =
𝑑 ln 𝒫(𝑘)
𝑑 ln 𝑘
𝑘
𝑘𝑚𝑖𝑛
+ ln 𝒫(𝑘𝑚𝑖𝑛 ) ,
𝑘 < 𝑘𝑚𝑖𝑛
𝑘 ∈ {𝑘𝑖 }
ln 𝒫 𝑘𝑖 ,
𝑘𝑖 < 𝑘 < 𝑘𝑖+1
cubic spline,
𝑘
ln
+ ln 𝒫(𝑘𝑚𝑎𝑥 ) ,
𝑘 > 𝑘𝑚𝑎𝑥
𝑘𝑚𝑎𝑥
𝑘𝑚𝑎𝑥
ZK Guo, D.J. Schwarz, YZ Zhang, JCAP 08 (2011) 031; ZK Guo, YZ Zhang, JCAP 11 (2011) 032;
ZK Guo, YZ Zhang, PRD 85 (2012) 103519.
Planck+WP+BICEP2
Planck+WP+BICEP2
B Hu, JW Hu, ZK Guo, RG Cai, arXiv:1404.3690.
谢谢！