Effects of a pre-inflation radiationdominated epoch to CMB anisotropy
I-Chin Wang (王怡青) NCKU(成功大學) & Kin-Wang Ng(吳建宏) ASIoP(中研院物理所)
Scalar field fluctuations in
Schwarzschild-de Sitter space-time
Hing-Tong Cho(曹慶堂) TKU(淡江大學) & Kin-Wang Ng(吳建宏) ASIoP(中研院物理所)
& I-Chin Wang (王怡青) NTNU(國立臺灣師範大學)
Dec 24 ,2009
Part one
Effects of a pre-inflation radiation-dominated epoch to CMB anisotropy
Outline
• Motivation: at l=2 WMAP data &ΛCDM model is not consistent
•
Our Model: we assume a pre-inflation radiation-dominated phase
•
Numerical results
•
Conclusion
before inflation
WMAP(Wilkinson Microwave Anisotropy Probe)
The key feature of inflation
a=eHt ,
a=t1/2 ,
a=t2/3
about 2.73k
ΛCDM model
I
s
Modes were out and then
entered our horizon again
K=aH
horizon crossing condition
deviation
Equation for inflation fluctuation
Our model
ΛCDM model
A ~ radiation component
B ~ vacuum energy
only inflation phase
1
inflation
1êH
0.5
0
-0.5
radiationdominated
Transition point for A=1
B=1 case
-1
-7.5
-5
-2.5
0
x
2.5
5
7.5
radiation
inflation
Initial condition
radiationdominated
when a is small
the horizon crossing k mode
The power spectrum is a smooth curve…
A ~ radiation component
B ~ vacuum energy
But…
A smooth power
spectrum is obtained
under two conditions
(1)
(2)
or
a(t) is a smooth transition function
initial a(t) can’t be too close to the phase transition point
it will lead to oscillations on power spectrum
HB=1 A=1L
1
HB=1000 A=1L
ai=0.0001
30
abrupt slope
0.9
0.85
k^H3ê2Lψ@aD
k^H3ê2Lψ@aD
0.95
ai=0.0001<<at
28
26
24
22
0.8
0
20
40
60
80
100
0
k
Ha=0.1 B=1 A=1L
1000
1500
k
2000
2500
3000
at=(A/B)1/4
Initial a is too close to phase transition point
1
500
Ha=0.5 B=1 A=1L
Ha=0.3 B=1 A=1L
1
1
0.9
0.85
0.95
k^H3ê2Lψ@aD
k^H3ê2Lψ@aD
k^H3ê2Lψ@aD
0.95
0.9
0.85
0.8
0.8
0
20
40
60
k
at=1
80
100
0.95
0.9
0.85
0.8
0
20
40
60
k
80
100
0
20
40
The choice of initial a is very important !
60
k
80
100
A ~ radiation component
Numerical results
B=1 A=7, 5, 1 and 0.1
B ~ vacuum energy
z ~ duration of inflation
B=1 and A=1
B=0.1 A=7, 5, 1 and 0.1
Using WMAP3 data to the chi-square fitting of the CMB anisotropy spectrum
Note the chi-square fitting for ΛCDM model is 47.09 in WMAP3 data
Using WMAP1 data to the chi-square fitting of the CMB anisotropy spectrum
Note the chi-square fitting for ΛCDM model is 64.99 in WMAP1 data
Using WMAP3 data to the chi-square fitting of the CMB anisotropy spectrum
Note the chi-square fitting for ΛCDM model is 47.09 in WMAP3 data
Using WMAP1 data to the chi-square fitting of the CMB anisotropy spectrum
Note the chi-square fitting for ΛCDM model is 64.99 in WMAP1 data
Conclusions :
(1) By adjusting the values of A( radiation component), B( vacuum energy ) and
z( duration of inflation) , it can lead to a low l=2 value. It also can determine how many efolds between phase transition point and today’s 60 e-folds .
1
0.5
1êH
7 e-folds 60 e-folds
0
(radiation-dominated)
-0.5
k^H3ê2Lψ@aD
(3) For choosing initial value a(t) , If the
initial value is too close to the transition
point , it will lead to an oscillating power
spectrum which is due to an improper
choice of initial a(t) . But if the initial a is in
the proper region (radiation-dominated
region) then the power spectrum is
almost the same and smooth. It’s initial
condition insensitive. Therefore, the
oscillation is from a improper choice of
initial a !
-1
-7.5
-5
-2.5
0
x
2.5
5
7.5
Phase transition
HB=1 A=1L
1
Ha=0.5 B=1 A=1L
1
0.95
k^H3ê2Lψ@aD
(2) One year WMAP data prefers our
model slightly and the pre-inflation
radiation-dominated phase was
taking place as 67 e-folds from the
end of inflation .
0.9
0.85
0.8
0.95
0.9
0.85
0.8
0
20
40
60
k
80
100
0
20
40
60
k
80
100
The value of A and B …
•
We choose B=1 and B=0.1 with A=7,5,1,0.1 to do the simulation . The value of B is
from the constraint
Where the slow-roll parameter is
The vacuum energy that drives inflation is given by
with
and
then
No observational constraint for the value of A , it can from the particle physics model
you choose .
Therefore , the following simulations are (1) B=1 & A=7,5,1,0.1 and (2) B=0.1 &
A=7,5,1,0.1
Quantitative comparison …
The chi-square fitting is the following
The measured ith band power
Theoretical value
The width of the error bar in measurement
We use this chi-square fitting method to do a comparison .
e-folding
The value of z correspond to an exponential expansion with the number of e-folding
from the start of inflation given by
The total e-folding is :
From observation that
Part two
Scalar field fluctuations in Schwarzschild-de Sitter space-time
arXiv:0905.2041
Outline
• Motivation:
• Our Model:
Still some deviations and “axis of evil” !?
we consider the inhomogenous background during
inflation, like a black hole in the space-time
• Numerical results
• Conclusion
ΛCDM model
I
s
still some deviation
Land & Magueijo 05
WMAP3
CMB sky map
Land & Magueijo 05
“Axis of Evil” ?
l=2, quadrupole
l=3, octopole
(1) Some deviations still exist
(2) Axis of evil ?
Our model : an inhomogenous background, a black-hole in it !
Equation for inflation fluctuation
Inflaton quantum fluctuation :
arXiv:0905.2041
ΛCDM model
homogenous
space-time
Our model
Schwarzschild-de Sitter (SdS) space-time
static coordinate
a black hole locates at the origin !
we have to use planar coordinate
conformal time
evaporation time for a Schwarzschild black hole with mass M :
and, evaporation time scale longer than that of inflation needs
which implies that
The process :
(1) Spherical symmetric
(2) quantization
(3) Using a perturbative approach to calculate the function
and assume the quantity
so that we can expand
also write
then solve the equation order by order.
Then we compute the two-point correlation function
the spectrum function
and
in powers of
as an expansion in powers of
A. Zeroth order
Bessel transform
then we have
and the solution is
To choose Bunch-Davies vacuum, we take
The solution is
and
scale-invariant power spectrum
of de-Sitter quantum fluctuation
B. First order
Use Green’s function
completeness property of spherical Bessel function
and taking
the equation becomes
The first order spectrum function is given by
Numerical results
A k-mode
oscillates when the mode is
still sub-horizon. Once the mode crosses out
the horizon,
stops oscillating and
gradually approaches a constant value.
3D plots
The asymptotic
value
is suppressed
in low l and low k regions.
Comparing to the zero-order
de Sitter power spectrum, it
is roughly decreased by the
expansion parameter .
Conclusion:
(1) We obtained
up to first-order perturbation results in the case that the black hole
located at the origin of the coordinates, for expansion parameter satisfied
.
(2) The suppressed power of
in low l and low k regions could give rise to a
suppression of the large –scale CMB anisotropy.
(3) The effect to the CMB, a detailed calculation is on going, including the black hole locates
somewhere else in the Universe.
example: the other model
CMBFast