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Non-linear evolution of matter power
spectrum in a closure theory
Takashi Hiramatsu
Institute for Cosmic Ray Research, University of Tokyo
Atsushi Taruya
Research center for the Early Universe, University of Tokyo
RESCEU international symposium @ University of Tokyo 11-14 Nov 2008
Closure Approxmation (CLA)
An alternative theoretical framework to measure the baryon
acoustic oscillation scale to probe the dark energy.
Based on fluid description of CDM+baryon components.
Diagramatic representation of naïve perturbation series.
Renormalised expressions of them in terms of three non-perturbative
quantities :
Power spectrum, Propagator, Vertex function
Truncation of the renormalised expressions at 1-loop order + tree-
level vertex function
Closed system coupled with power spectrum and propagator
Taruya, Hiramatsu, ApJ 674 (2008) 617
Contents
In this poster, we apply our closure formalism to 1-loop standard
perturbation theory (SPT) which can be realised by replacing all
quantities in the mode coupling term by linear solutions,
and show …
formalism consistent to 1-loop SPT, and how to solve
the closure equation derived in ApJ 674 (2008) 617,
numerical check to recover the 1-loop SPT results,
resultant power spectra in
time-varying dark energy model
DGP model where the Poisson equation has non-linear terms
coming from 2nd-order brane bending mode.
Actors
),,|,()();();(
);(212121
2211
kkkkk
kk
kabcD
cb
a
)(
)()(
k
kka
Ha
ki )v(
)',|()()';(
);(
kGkk
k
kabD
b
a
Non-linear power spectrum
Propagator
Non-linear vertex function
Irrotational
velocity field
Vector representation
of perturbative quantities
Replaced (CLA)
)',,()()2()',(),( 3 kRkkkk abDba
),()()2(),(),( 3 kPkkkk abDba
Script
Euler equation + conservation equation yields …
')',''|;(|)'',|;(|'')',|;(| kkk scasbcab GMdG
0
)',''|;(|)'',|;(|'')',|;(| kkkscasbcab RMdR
'
0
)'','|;(|)'',|;(|''
kk clal GNd
0
)'',|;(|)'',|;(|'')|;(| kkk asbscdabcd RMdP
0
)'',|;(|)'',|;(|'' kk albl GNd )( ba
)1log( z
The original closure equation is expressed by different forms. The present symmetric forms are
more suitable for numerical calculations than the original ones. In addition, while the original version
uses the growth rate as the time, we here use the scale factor in view of the application to modified
gravities.
Script
Integral kernels
),'()','()2(
'4)'',|;(|
3
3
kkkkkkk lrsapqas
kdM
)'',|;'(|)'',|,'(| kkk prql RG
)','()','()2(
'2)'',|;(|
3
3
kkkkkkk lrsapqal
kdN
)'',|;'(|)'',|,'(| kkk prqs RR
112 121222
In CLA, the vertex functions are replaced by ones with tree-level
approximation. Only have non-zero value.
Left-hand side operator
2224
10)(
H
H
HG mab
represents a correction of
the gravity constant appeared in the
linear Poisson equation. In the
standard theory, ,
while it is not generally true in
modified gravity theories.
),( k
1),( k
)(
ababab
Formal solution
)|;(|),'|;(|),|;(|)',|;(| 000 kkkk cdbdacab PGGR
0 0
),;(),'|;(|),|;(| 122121 kkk cdbdac NGGdd
'')''',|;(|''''')',|;(|)',|;(| kkk acabab gddgG
)',''|;(|)'','''|;(| kk sbcs GM
Complex combination of evolution equations yields…
Linearised closure equation = SPT
The solution of linearised closure equation coincides with
the prediction of 1-loop standardperturbation theory (SPT)
The closure equation mentioned above has all non-linear contributions
in 1-loop level except for the vertex function.
An approximation we can easily take here is to replace all quantities in
the right-hand by those obtained in the linear theory.
= ‘linearised closure equation’.
In our previous paper, we have proven an important fact :
Linear solution
Dropped all right-hand side terms
0)',|;(|0 kbcabG
0)',|;(|0 kbcabR
11
11)()'',|;(| 0
''0 kPeRsck
In the matter dominant Universe, the power spectrum takes the form :
where is the linearly extrapolated power spectrum at the present time. We use
this (neglecting the decaying solution) as the initial condition of the above equation, and
normalise the solution by a given present amplitude, .
)(0 kP
8
+ ( decaying solution )
As for the propagator, the initial condition is given by its definition :
ababG ),|;(| 00
0k
Linearised closure equation
Replacing all right-hand side quantities by linear ones
'
0 )',''()'',|;(|'')',|;(| sc
L
as
L
bcab GMdG kk
0
)',''|;(|)'',|;(|'')',|;(| 0kkk
sc
L
as
L
bcab RMdR
'
0
0
)'','|;(|)'',|;(|''
kk cl
L
al GNd
),'()','()2(
'4)'',|;(|
3
3
kkkkkkk lrsapq
L
as
kdM
)'',|;'(|)'',|,'(| 00 kkk prql RG
)','()','()2(
'2)'',|;(|
3
3
kkkkkkk lrsapq
L
al
kdN
)'',|;'(|)'',|,'(| 00 kkk prqs RR
Full closure Full closure
equation
linearised
equation
linearised
closure
equation
Standard Standard
perturbation
theory
linearised
)(H
),( kGeff
P(k) in general
cosmological models
Friedmann Eq.
Poisson Eq.
)'',',( kkkabcVertexes
(non-linear terms)
Born approx.Taruya, Hiramatsu, ApJ 674 (2008) 617
Hiramatsu, Taruya, in preparation
Koyama, Taruya, Hiramatsu,, in preparation
Strategy for numerics
;)()(),( ,m
mmabab kTGkG
n
nn ggdg )(2
)( )1()(
2
)1()1( nn ggg
n
abG abR k1. Expand and by a set of basis functions of
)(kTm
k
0
1
1mk mk 1mk
3. Replace the differential operator by the central difference
4. Apply the trapezoidal rule to the time-integrations
2. Integrate functions of appeared in and
5. Sequentially solve the recursive equations
abM abNk
cf. Valageas AA465 (2007) 725
Result 0 : demonstration of ‘Full’ closure Eq.
LCDM
1-loop SPT (blue) yields a
little large power because
of breakdown of SPT
Born approximation to the formal
solution of closure eq. suppresses
the small scale power.
Power spectrum in LCDM
Result 0 : demonstration of ‘Full’ closure Eq.
Leading to an anomalous
enhancement of power But, there is unphysical oscillation
due to truncation at 1-loop
Non-perturbative effect
acutually suppresses the
power on small scales
Propagator
Result I : Recovery of SPT results
d
df
f
ffab
EDSapprox 1
22
310
)( 2.
SPT frequently uses Einstein-de Sitter
approximation :
which is realised by a small modification of
left-hand side operator as
d
Ddf L )(log)(
Our numerical scheme can recover
the results of standard perturbation
theory with 1-loop corrections.
M 1
279.0M
817.08
961.0sn
)()(),( kDk n
n
L
)(LD : Linear
growth rate
Result II : Validity of EdS approx.
)1()( 0 awwaw a
Next we investigate a general dark energy
model in which the EOS parameter varies
like
),( 0 aww
The approximation does not work especially near the present time.
279.0M
817.08
961.0sn
Linder PRL 90 (2003) 091301
Non-linearity of Poisson equation
Extra time-dependent vertexes
0
)',''|;(|)'',|;(|'')',|;(| kkkscasbcab RMdR
'
0
)'','|;(|)'',|;(|''
kk clal GNd
)',|;(|)|;(|);()2(
'3
3
3
kk'k,k',k'k
rcpqapqr PP-d
);( 21211 k,k );,( 3212111 kk,k
)(42
2
FGa
km
Non-linearity of Poisson equation
Dvali-Gabadazde-Porratti braneworld model
0
)(1
6
1);(
2
2
2
1
2
21
3
2
21211
kk
HrC kkk,k
5
3
321211127
)1();,(
cc HrHr
kk,k
23121
H
HHrc
3
22
cr
HH
0|'|'
))'((
|'|'
))'((2
'
)'('22
2
22
2
22
222
kk
kkk
kk
kkkkk
kkkk
kk
2222
4
2
2)()(
3 ji
c
a
r
a
2
2
22
2
12
aa
)(3
11
2
1 22
2
2
O
a
Poisson eq.
Expanded up
to 2nd order
Non-linear
Poisson eq.
2
,
2222 )21(2)21()21( dydydxrdxdxAdtNds i
ic
ji
ij
Friedmann eq.
Brane bending mode
mode parameter
Brane is at y=0
Result III : Power spectrum in DGP model
Self-acceleration branch
138.0rc
36.10 crHgives a positive (dominant)
contribution to mode couplings.
In contrast, gives a
negative (sub-dominant) one. As a
result, the power on small scales
enhances about 3%.
211
2111
Neglecting
and 211
2111
Including the
extra vertexes
Summary
We checked the recovery of 1-loop SPT calculation.
As a demonstration, numerical solution of ‘full’ closure equation was presented. non-perturbative effects beyond 1-loop SPT.
We found the Einstein-de Sitter approximation woks well even beyond LCDM model at small z.
Finally we have demonstrated our method by applying to DGP model in which Poisson equation becomes to depend on wave number. This demonstration makes it clear that our unified method is capable of wide application.
Linearised closure equation = 1-loop standard perturbation theory
We have focused the following fact :
The confronting issue was …
Directly solve the (non-linear) simultaneous integro-partial-differential equation
This challenging task has provided ...
