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Analytical Spectra of RGWs and CMB Yang Zhang Astrophysics Center University of Science and Technology of China (USTC). Topics. Relic Gravitational Waves ( RGWs ) C l XX of CMB generated by RGWS and re-ionized C l XX generated by scalar perturbations - PowerPoint PPT PresentationTRANSCRIPT
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Analytical Spectra of
RGWs and CMB
Yang Zhang Astrophysics Center
University of Science and Technology of China (USTC)
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Topics
1. Relic Gravitational Waves ( RGWs )
2. ClXX of CMB generated by RGWS and re-ionized
3. ClXX generated by scalar perturbations
in synchronous gauge
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1. RGWsRobertson-Walker metric ds2=a2(t) [ -dt2 + (δij+hij) ]
perturbations hij = hδij/3+hij|| (scalar)
+hij┴ (vector)
+hijT (TT: RGWs)
after generated by inflation,
existing all the time,
existing everywhere,
broad distribution over (10 - 18 - 1010 ) Hz ,
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medium frequencies cavity: ν = 104 Hz MAGO, EXPLORER laser interferometer: ground, ν = 102 - 103 Hz LIGO, VIRGO, etcspace, ν = 10-3 - 100 Hz LISA , ASTROD, etc
high frequenciesGaussian laser beam ν = 109 - 1010 Hz waveguide ν = 108 Hz (Cruise & Ingley)
low frequencies CMB ν = (10-18 - 10-14) Hz WMAP, Planck, CMBPol, etc
pulsar timing ν = 10-9 Hz PPTA, etc
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Analytic calculations
equation :
solution :
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Initial condition: A, β, αT :
behavior during expansion : Wavelengths > horizon , |hij| = constant ; Wavelengths < horizon , |hij| = h/a(t);
→ lower |hij| at short wavelength λ
2 2 2( )( ( ) )i jij ijds a d h dx dx
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Other modification processes :
ν free-streaming ;
uud→p , QCD phase transition ;
e+e-→2γ, annihilation ;
accelerating expansion (dark energy ΩΛ) ;
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h(ν) and Ωg (ν) depend on inflation index β :
Class. Quant. Grav. 23, 3783 ( 2006 )
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h(ν) and Ωg (ν) depend on running index αT :
Phys.Rev.D80 084022 ( 2009 )
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neutrino free-streaming;
Phys.Rev.D75, 104009 (2007 )
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Phys.Rev.D77, 104016 (2008)
QCD phase transition, e-e+ annihilation,
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Dark energy reduces h(v) by a factor: Ωm /ΩΛ
Class. Quant. Grav. 22, 3405 ( 2005 )
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LIGO, Adv LIGO, LISA, DECIGO
Phys.Rev.D80 (2009) 084022
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PPTA
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BBN constraints:
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MAGO, EXPLORER:
Still short by ~7 orders of magnitudes in sensitivity;
PRD80, (2009) 084022
Gaussian beam:
Still short by ~5 orders in sensitivity;
PRD 78, 024041 (2008);
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RGWs might be directly detected via CMB Scalar : CTT, CEE, CTE
RGWs: CTT, CEE, CTE, CBB
WMAP5
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2 、 Analytic
ClTT, Cl
TE, ClEE, Cl
BB
generated by RGWs ,
and re-ionized
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anisotropies
polarization
Boltzmann eq for CMB photons :
Equivalent to :
with
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The formal integrations :
• where the visibility function for the decoupling process
fitted by two half Gaussian functions:
Carrying out time integration, one has
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Approximate, analytical solution :
where
with c ~ 0.6, b ~ 0.8
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Analytical CMB spectra :
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analytical, and numerical(CAMB)
Phys.Rev.D74 (2006) 083006; Phys.Rev. D78 ( 2008 ) 123005
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improvements :
1.effective range : l < 300 l < 600, covering the first 3 peaks;
2. errors only ~ 3%;
3. CTTl and CTE
l are also obtained ;
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ν- free-streaming has small modifications:
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ν- free-streaming :
1. amplitude reduced by 20 ~ 35 % for l > 100;
2. ClXX are shifted slightly to larger l with
Δl ∝ l. Δl =(1~5) ; (for the first two peaks)
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Zero multipole method:
to examine the value l ~ 50,where CTEl crosses 0.
Our results:Δl by NFS is the same order of magnitude as those caused by inflation index βinf and Ωb.
More works are needed before a conclusion can be made.
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inflation index β :
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WMAP5 constraint on CBBl :
Phys.Rev.D78, 123005 (2008)
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Reionized case
------- possibly by first generation of luminous stellar objects
------- likely occurred z = (6~ 20), uncertain yet; WMAP5 : ( sudden re-ionization ) z = 11 (95%CL).
------- a major process secondary only to the decoupling
V(t) consists of two parts : around z~1100 and around z~11
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• three models of reionization:• Sudden :
• η-linear :
• Z-linear :
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• Approximate, analytical solution :
with the coefficients
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a1 -- the probability of a polarized photon last scattered during decoupling,
a2 -- the probability of a polarized photon last scattered during reionization,
both depending upon the optical depth κr :
PRD79, 083002 (2009)
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Where hk(η) and dot hk(η) at decoupling and reionization
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Re-ionized CMB spectra
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Phys.Rev.D79, 083002 (2009)
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profile of CXXl determined by RGWs .
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Re-ionization bump• location is sensitive to time ηr.• height is sensitive to duration Δηr.
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κr - A degeneracy
Phys.Rev.D79, 083002 (2009)
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κr - β degeneracy
broken from the 2nd peak
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Re-ionization also shifts l0 around l=50
Therefore,Re-ionization has to be well studied before one can determine major cosmological parameters from CMB observational data.
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Contribution of baryon isocurvature is minor
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3 、 Analytic
ClTT, Cl
TE, ClEE
generated by scalar perturbations
(in synchronous gauge)
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scalar perturbations of metric
(synchronous gauge) :
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Boltzmann eq.:
formal sol. :
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several technique treatments :
1 。 Time integration2 。 Removing gauge modes 3 。 Joining at R=M4 。 Initial condition
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We get :
where
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Analytic result: (valid for l>400)
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Contributions by each term
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ClXX depend on inflation scalar index nS
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ClXX depend on baryon Ωb
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conclusion:
Still need to fully understand CMB,
The “precision cosmology” is yet to be reached.
Thank you!