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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!