arXiv:1705.00657v1 [astro-ph.CO] 1 May 2017
Prepared for submission to JCAP
Probing dark energy using
convergence power spectrum and
bi-spectrum
Bikash R. Dinda,a,1
a Centre
for Theoretical Physics, Jamia Millia Islamia,
New Delhi-110025, India
E-mail: [email protected]
Abstract. Weak lensing convergence statistics is a powerful tool to probe dark energy. Dark
energy plays an important role to the structure formation and the effects can be detected
through the convergence power spectrum, bi-spectrum etc. One of the most promising and
simplest dark energy model is the ΛCDM. However, it is worth investigating different dark
energy models with evolving equation of state of the dark energy. In this work, detectability
of different dark energy models from ΛCDM model has been explored through convergence
power spectrum and bi-spectrum.
1
Corresponding author.
Contents
1 Introduction
1
2 Background evolution with dark energy models
2
3 Evolution of perturbations
5
4 Linear solutions and linear matter power spectrum
6
5 Non-linear solutions and tree-level bi-spectrum
9
6 Convergence power spectrum and bi-spectrum
11
7 Conclusion
16
1
Introduction
A combination of different cosmological observations now points towards a concordance model
for our Universe where 1/3rd of the energy budget of the universe is in non-relativistic matter
comprising baryons and dark matter and the rest 2/3rd is cosmological constant Λ with a
constant equation of state w = −1 ([1]). This model, as popularly called ΛCDM model,
although is consistent with majority of cosmological meaurements, some recent observations
do predict important discrepencies in ΛCDM model ([2] - [5]) in addition to the theoretical
problems in ΛCDM, e.g, the fine tuning and cosmic coincidence problem ([6]).
That is why the major goal for upcoming high precision cosmological experiments is
to determine the evolution of the equation of state for the dark energy at percentage level
accuracy. Among these, experiments related to weak lensing measurements are particularly
promising in determining the nature of dark energy because of the high sensitivity of weak
lensing effect on both the background evolution of the Universe as well as on the growth of
structures ([7] - [20]).
Weak lensing is the distortions of galaxy imgaes due to grvitational bending of light by
the intervening large scale structures along the photon propagation. The measurment of weak
lensing around massive halos was first measured in nineties [21, 22]; but the first detection of
weak lensing by large scale structures was done independently by four groups in 2000 ([23] [26]). After that weak lensing has become one of the most accurate probe for our observable
Universe.
The main advantage of using weak lensing as cosmological probe compared to other
probes related to large scale structures is due to the fact that it solely depends on the underlying dark matter distribution and hence one can avoid the complicated bias modeling (visible
to dark matter). The other advantage of using weak lensing as a cosmological probe is due
to the relatively straightforward measurement of the galaxy shear which can be observed in
millions of galaxies in latest surveys. Correlation of galaxy shear across the sky together with
the redshift information of individual galaxies provides a 3-dimensional information of our
universe which is a powerful probe for dark energy.
Given the current and future surveys like Dark Energy Survey (DES [27]), Large Synoptic
Survey Telescope (LSST [28]), Euclid ([29]) and the Wide-Field Infrared Survey Telscope
–1–
(WFIRST [30]), the prospects of accurately measure the evolution of dark energy density and
its equation of state using weak lensing is extremely bright.
In this paper, we study the prospects of distinguishing any invidividual dark energy
model from ΛCDM model using weak lensing power specturm and bi-spectrum. We consider
the parameterization of dark energy equation of state e.g the CPL [31, 32] and GCG [33]
parameterization as well as thawing class of quintessence model for dark energy with powerlaw potentials ([34] - [38]) and compare their weak lensing signal with that from ΛCDM
model. This gives a broad picture about how far we can expect to distinguish different dark
energy models from ΛCDM model using weak lensing.
The paper is organised as: in section 2 background evolution has been discussed; in
section 3 perturbation in the matter has been studied with the above mentioned dark energy
models using Newtonian perturbations; in section 4 linear solutions to the perturbation and
linear matter power spectrum have been studied; In section 5 second order solutions to the
perturbation and tree-level matter bi-spectrum have been studied; in section 6 corresponding
convergence power spectrum and bi-spectrum have been discussed; and finally in section 7
conclusion has been given.
2
Background evolution with dark energy models
Considering flat Friedman-Robertson-Walker (FRW) background of the Universe with two
components, non-relativistic matter (baryons + dark matter) and dark energy, the Hubble
parameter, H can be expressed as
2
H =
H02
Z
h
(0) −3
(0)
Ωm a + (1 − Ωm ) exp[−3(
a
a0
0
1 + w(a ) i
da
)] ,
a0
0
(2.1)
where a is the scale factor, w(a) is the general time dependent equation of state of dark
(0)
energy, a0 , Ωm and H0 are the present day scale factor, matter density parameter and Hubble parameter respectively. Different dark energy models have different equations of state
p̄ (a)
(e.o.s), which is defined as w(a) ≡ ρ̄qq (a) , where ρ̄q and p̄q are background energy density
and pressure for dark energy respectively. We consider three types of dark energy models:
(I) models where the e.o.s for the dark energy is given by Chevallier-Polarski-Linder (CPL)
parametrization, (II) thawing class of minimally coupled canonical scalar field models (thawing quintessence) and (III) models where the e.o.s for the dark energy is given by the GCG
(generalized Chaplygin gas) parametrization. For CPL and GCG parametrizations, e.o.s w(a)
have analytical expressions whereas for quintessence models, background quantities have to
be evaluated numerically. Let us briefly discuss these models below.
(I) CPL parametrization:
In CPL parametrization, the e.o.s of dark energy is given by
w(a) = w0 + wa (1 − a),
(2.2)
where w0 and wa are two model parameters. w0 represents the present day e.o.s of dark
energy whereas wa gives its evolution with time [31, 32]. For any time, w < −1 and w > −1
correspond to phantom and non-phantom behaviours of dark energy respectively. For w0 <
−1 and wa < 0 (w0 > −1 and wa > 0), dark energy shows phantom (non-phantom) behaviour
for all time. The special case w0 = −1 and wa = 0 corresponds to the exact ΛCDM model.
–2–
(II) Quintessence:
The Lagrangian density for a minimally coupled scalar field can be written as ([34] - [38])
1
L = (∂ µ φ)(∂µ φ) − V (φ),
(2.3)
2
where φ is the field and V is the potential. The background energy density and pressure of
the quintessence become
1
ρ̄q = φ̇2 + V (φ),
2
1
p̄q = φ̇2 − V (φ),
2
(2.4)
respectively, where overdot represents derivative with respect to the cosmic time t. The
equation of motion for the scalar field is given by
φ̈ + 3H φ̇ + Vφ = 0,
(2.5)
where subscript φ is the derivative w.r.t the field φ. To study the background evolution
equations, it is a standard procedure to define few dimensionless quantities as given below:
dφ √
x = √ dN ,
6MP l
V φ
,
λ = −MP l
V
Ωq = x2 + y 2 ,
V
,
3HMP l
V φφ
Γ=V
,
Vφ2
y=√
w =γ−1=
x2 − y 2
,
x2 + y 2
(2.6)
where N = lna is the e-folding, MP l is the Planck mass and Ωq = 1 − Ωm is the energy
density parameter of the scalar field. Now, the background evolution can be studied through
an autonomous system of equations given by
p
dγ
= 3γ(γ − 2) + 3γΩq (2 − γ)λ,
dN
dΩq
= 3(1 − γ)Ωq (1 − Ωq ),
dN
p
dλ
= 3γΩq λ2 (1 − Γ).
dN
(2.7)
To solve the above coupled differential equations, we consider the thawing class of inital
conditions where the dark energy is initially frozen at the flat part of the potential (similar to
what happens in the early inflationary epoch). This gives γin ≈ 0 at the initial time (initial
redshift, zin = 1000 is considered in our calculations). In the subsequent calculations, we
assume γin = 10−10 and the results are not sensitive to the values of γin as long as γin 1.
The initial value for Ωq is chosen such that one gets its required value at present. The initial
value of λin controls the equation of state of the scalar field at present. For λin << 1, the
equation of state of the scalar field is always close to w = −1 (the cosmological constant);
–3–
0.8
CPL
w0 = - 0.9
1.0
wa = - 0.3
wa = -0.1
wa = 0.1
w(a)
w(a)
0.9
1.1
1.2
10 -3
10 -2
a
10 -1
0.0
10 0
GCG
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
10 -3
As = 0.9
w(a)
0.2
Quintessence
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
10 -2
a
10 -1
10 0
α = - 1.1
α = - 0.9
α = - 0.7
0.4
0.6
0.8
1.0
10 -3
10 -2
a
10 -1
10 0
Figure 1. w(a) vs. a plots for different dark energy models.
for higher values of λi , the equation of state starts deviating from the cosmological constant
behaviour. The quintessence models with this type of initial condition is called thawing class
of quintessence models where the e.o.s of the dark energy is −1 initially in matter dominated
era and with the expansion of the Universe the slope of e.o.s increses slowly and at late times
the e.o.s becomes greater than −1.
(III) GCG parametrization:
In GCG parametrization, the e.o.s of dark energy is given by [33]
w(a) = −
As
,
As + (1 − As )a−3(1+α)
(2.8)
where α and As are two model parameters. The special case As = 1 corresponds to ΛCDM
model. It is interesting to notice that the GCG parametrization incorporates both the thawing
and tracker behaviours of dark energy [33]. Thawing model corresponds to 1 + α < 0 whereas
the tracker model (where e.o.s of the dark energy initially in matter dominated era mimics
the background with a value nearly 0 and with the expansion of the Universe it decreases
towards −1 value at very late times and finally freezes to w ≈ −1 in future) corresponds to
1 + α > 0 ([34] - [36]).
The initial conditions of the background quantities for thawing quintessence (Ωin
q and λin ) and
the parameters for the CPL (w0 and wa ) and GCG (α and As ) parametrizations are chosen
(0)
such that for all the dark energy models at present w0 = −0.9 and Ωm = 0.3. For thawing
–4–
quintessence models linear, squared and inverse-squared potentials have been considered. For
CPL parametrization three models are chosen given by w0 = −0.9, wa = −0.3; w0 = −0.9,
wa = −0.1 and w0 = −0.9, wa = 0.1. The chosen models for GCG parametrization are
α = −1.1, As = 0.9; α = −0.9, As = 0.9 and α = −0.7, As = 0.9.
The equation of states of the above mentioned models for all three types of dark energy models
have been plotted in figure 1 to show how the equation of states evolve with expansion of
the Universe. The e.o.s of the model w0 = −0.9, wa = −0.3 in CPL parametrization is
−1.2 (phantom value) initially in matter dominated era (a << 1) and slowly increses with
expansion of the Universe and at redshift z = 0.5 the transition from phantom to non-phantom
regimes happens and finally at present time it becomes −0.9. At early matter dominated era
the e.o.s of the model w0 = −0.9, wa = −0.1 in CPL parametrization is nearly −1 which
slowly increases towards higher values at late times and goes to −0.9 at present. The e.o.s
of the model w0 = −0.9, wa = 0.1 has the negative slope unlike others two; it starts from
−0.8 (non-phantom value) initially and decreases towards −0.9 and remains always in nonphantom regime. As thawing class of quintessence models have been considered the equation
of states of all three quintessence models start from −1 initially and slowly slopes increase
towards −0.9 at present. The differences between three quintessence models are very small.
In GCG parametrization one thawing (α = −1.1) and two tracker models (α = −0.9 and
α = −0.7) have been considered with As = 0.9. So, the α = −1.1 has the similar behaviour
as the thawing quintesence models but the tracker models have different behaviour where e.o.s
has negative slopes. The e.o.s of the models α = −0.9 and α = −0.7 start from nearly 0 and
−0.5 initially at eraly matter dominated era (by which the dark energy tracks the background
initially) respectively and at late times decrese towards −0.9.
The above mentioned models have been considered throughout all the subsequent sections
(except for the last figure i.e. figure 8 where w0 = −0.85 has been considered instead of
w0 = −0.9 for certain purpose mentioned later).
3
Evolution of perturbations
On sub-horizon scales, the evolution of fluctuations in matter can be studied under Newtonian perturbation theory. On these scales, one can also ignore the fluctuations in the dark
energy component as dark energy only clusters on horizon/super-horizon scales. Hence on
sub-horizon scales, the dark energy affects the clustering of matter through the background
evolution only. Under these assumptions the evolution of fluctuations are governed by the
continuity, Euler and Poisson equations which are given below:
0
~
δm
+ ∇.[(1
+ δm )~vm ] = 0,
(3.1)
0
~
~ vm = −∇Φ,
~vm
+ H~vm + (~vm .∇)~
(3.2)
3
∇2 Φ = 4πGa2 (δρm ) = H2 Ωm δm ,
2
(3.3)
and
where prime ("0 ") denotes derivative w.r.t conformal time τ , δm is the matter energy density
m
contrast defined as δm ≡ ρmρ̄−ρ̄
with ρ̄m and ρm be the background and perturbed matter
m
energy densities respectively, ~vm is the velocity field of matter, H is the conformal Hubble
–5–
0
parameter defined as H ≡ aa , Φ is the Newtonian gravitational potential, G is the Newtonian
gravitational constant and Ωm is the matter energy density parameter ([39, 40]).
Assuming matter velocity field is irrotational, it can be completely described by its divergence
~ vm [39]. Using this, continuity equation (3.1) can be rewritten as
θm = ∇.~
0
~ m~vm ).
δm
+ θm = −∇.(δ
(3.4)
Now, taking divergence of the Euler equation (3.2) and using Poisson equation (3.3) into it,
the evolution equation of θm becomes
3
0
~ vm .∇)~
~ vm ].
θm
+ Hθm + H2 Ωm δm = −∇.[(~
2
In Fourier space, eqs. (3.4) and (3.5) can be written as ([39] - [47])
Z
Z
(3)
0
3
δ~k + θ~k = − d ~q1 d3 ~q2 δD (~k − q~1 − q~2 )α(q~1 , q~2 )θq~1 δq~2 ,
θ~k0
3
+ Hθ~k + H2 Ωm δ~k = −
2
Z
Z
3
d q~1
(3)
d3 q~2 δD (~k − q~1 − q~2 )β(q~1 , q~2 )θq~1 θq~2 ,
(3.5)
(3.6)
(3.7)
where
α(q~1 , q~2 ) = 1 +
β(q~1 , q~2 ) =
q~1 .q~2
,
q12
(q~1 + q~2 )2 (q~1 .q~2 )
,
2q12 q22
(3.8)
(3.9)
and ~k, q~1 & q~2 correspond to different wave modes in the Fourier space. Here, subscript
"m" has been omitted for the sake of simplified notation and in the subsequent sections this
notation will be used.
4
Linear solutions and linear matter power spectrum
In linear theory, the 2nd and higher order terms in perturbation equations can be neglected.
So, in the linear regime, eqs. (3.6) and (3.7) becomes
∂δ~lin
k
∂τ
+ θ~klin = 0,
(4.1)
∂θ~lin
3
+ Hθ~klin + H2 Ωm δ~klin = 0,
(4.2)
∂τ
2
where superscript ’lin’ stands for linear theory. Taking derivative of equation (4.1) and using
equation (4.2) into it, evolution equation for δ~lin becomes
k
k
∂ 2 δ~lin
k
∂τ 2
∂δ~lin
k
3
− H2 Ωm δ~klin = 0.
(4.3)
∂τ
2
This is the standard evolution equation for matter energy density contrast in linear regime in
the presence of smooth dark energy.
+H
–6–
In Fourier space, linear matter energy density contrast can be described through linear growth
function, D which is defined as
δ~klin (τ ) = D(τ )δ~kin ,
(4.4)
where δ~in is the initial density contrast at a sufficient initial time. Using above definition
k
of the linear growth function into equation (4.1), linear velocity field of the matter can be
obtained as
θ~klin = −H(τ )f (τ )D(τ )δ~kin .
(4.5)
where f is the linear growth rate which is defined as
f≡
d lnD
.
d lna
(4.6)
Putting equation (4.4) into equation (4.3) and using an identity
equation for the linear growth function becomes
H0
H2
= − 21 (1+3wΩq ), evolution
dD 3
d2 D 1 +
1
−
3wΩ
− Ωm D = 0,
(4.7)
q
dN 2 2
dN
2
where conformal time derivative has been transferred to the derivative w.r.t e-folding, N .
Being a 2nd order differential equation, it has two solutions; one solution is same as the
Hubble parameter which is the decaying mode and another one is the growing mode solution
which has to be computed numerically.
Let us denote growing and decaying mode growth functions as D+ and D− respectively and
dlnD−
dlnD+
the corresponding growth rate becomes f− =
and f+ =
respectively. In the
dlna
dlna
subsequent sections, growing mode solution has been considered.
Power spectrum:
Using statistical homogeneity and isotropy the matter power spectrum can be defined as
(3)
< δ~k (η)δ~k0 (η) >= δD (~k + ~k 0 )P (k, η).
(4.8)
2 and to have the proper
Putting equation (4.4) into equation (4.8), one can see that Plin ∝ D+
normalisation (here σ8 normalisation) the linear matter power spectrum is defined as
Plin (k, η) = Ak ns T 2 (k)
2 (η)
D+
2 (η ) ,
D+
0
(4.9)
where constant A is fixed by the σ8 normalisation with ns being the primordial scalar spectral
index and Eisenstein-Hu transfer function, T (k) has been considered. At present time we
(0)
(0)
fix all the power spectrum by σ8 normalisation by fixing Ωb = 0.05, Ωm = 0.3, H0 =
70km/s/M pc, ns = 0.96 and σ8 = 0.8.
ΛCDM
In all the figures the definition of %∆X is given by %∆X ≡ XMX−X
× 100, for any
ΛCDM
quantity X and for any model M .
In figure 2, percentage deviation in linear matter power spectrum from ΛCDM model
has been plotted for different dark energy models using equation (4.9). All the models are
chosen in such a way that for all of them, the present day equation of state is w0 = −0.9.
–7–
3.5
CPL
w0 = - 0.9
z=1
%∆P
2.5
wa = - 0.3
wa = -0.1
wa = 0.1
2.0
1.5
%∆P
3.0
1.0
0.5
10 -3
1.90
10 -1
10 -2
10 0
k [h Mpc ]
−1
Quintessence
9 CPL w0 = - 0.9
wa = - 0.3
8
wa = -0.1
wa = 0.1
7
6
5
4
3
2
1
00
2
4
z=1
%∆P
1.80
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
1.75
1.70
%∆P
1.85
1.65
1.60
10 -3
10 0
k [h Mpc −1 ]
GCG
z=1
As = 0.9
3.2
10
3.0
8
α = - 1.1
α = - 0.9
α = - 0.7
2.8
2.6
%∆P
%∆P
10 -1
10 -2
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.50
2.4
2.2
10 -3
k = 0.1 h Mpc −1
6
z
Quintessence
2
GCG
4
10 -1
k [h Mpc −1 ]
z
6
k = 0.1 h Mpc −1
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
8
10
As = 0.9
6
4
00
10 0
10
k = 0.1 h Mpc −1
α = - 1.1
α = - 0.9
α = - 0.7
2
10 -2
8
2
4
z
6
8
10
Figure 2. Percentage deviation in linear matter power spectrum from Λcdm model.
As expected, these deviations are scale independent as there is no dark energy clustering at
sub-horizon scales. For CPL parametrisation, models with wa > 0 has larger deviation from
ΛCDM than models with wa < 0. Note that with w0 > −1, models with wa > 0 is always
non-phantom in the past whereas models with wa < 0 become phantom at some point in the
past.
Both GCG and scalar field dark energy models are non-phantom models. GCG can act
as both thawing (α < −1) as well as tracking ( α > −1) models. The figure 2 shows that
tracker model has larger deviation from ΛCDM model than thawer models. In our analysis,
we consider thawer type of quintessence models and its power spectrum does not deviate
much from ΛCDM model for any of the potential considered (nearly 3%) and the differences
between three potentials are very small (sub-percentage level).
–8–
To summarise, models with always non-phantom behaviour is more probable to be distinguished from ΛCDM model; they can deviate 4 − 8% or more from ΛCDM behaviour.
Similarly tracker models deviates more from ΛCDM model than thawer models and differences can have 4 − 5% or more. Thawer models are less probable to be distinguished from
ΛCDM model and their deviations are less than 4 − 5% from ΛCDM model. The number we
quote are for w0 = −0.9; for larger values of w0 deviation from ΛCDM model will be larger.
5
Non-linear solutions and tree-level bi-spectrum
Having studied the dark energy effect on liner matter power specturm, we move on to study
its effect on the non linear regimes, more specifically on the three point correlation function,
the bi-spectrum.
Let us define a quantity η which is given by ([40],[41],[42])
η = lnD+ .
(5.1)
Using this definition eqs. (3.6) and (3.7) can be rewritten as [40]
Z
Z
∂δ~k
(3)
− Θ~k = d3 ~q1 d3 ~q2 δD (~k − q~1 − q~2 )α(q~1 , q~2 )Θq~1 δq~2 ,
∂η
∂Θ~k
3 Ωm
− Θ~k +
(Θ~k − δ~k ) =
∂η
2 f+2
Z
3
Z
d q~1
(3)
d3 q~2 δD (~k − q~1 − q~2 )β(q~1 , q~2 )Θq~1 Θq~2 ,
(5.2)
(5.3)
where a new quantity Θ is introduced which is related to θ given by
θ~k
.
(5.4)
Hf+
Defining this quantity has an advantage that in linear regime it is exactly same as the matter
energy density contrast which can be seen through eqs. (4.4), (4.5) and (5.4), and hence
Θ~k = −
Θ~lin
= δ~klin .
k
(5.5)
Solutions:
Equations (5.2) and (5.3) can be solved order by order using perturbative approach given by
δ~k =
∞
X
(n)
k
δ~
and
Θ~k =
n=1
(n)
k
where the nth order terms δ~
(n)
k
Z
δ~ (η) =
(n)
k
Θ~ (η) =
Z
d3 q~1 ...
Z
d3 q~1 ...
Z
∞
X
(n)
k
Θ~ ,
(5.6)
n=1
(n)
k
and Θ~
are given by
(3)
n
d3 q~n δD (~k − q~1 − ... − q~n )Fn (q~1 , ..., q~n , η)D+
(η)δq~in1 ...δq~inn ,
(5.7)
(3)
n
d3 q~n δD (~k − q~1 − ... − q~n )Gn (q~1 , ..., q~n , η)D+
(η)δq~in1 ...δq~inn ,
(5.8)
respectively.
–9–
Second order solutions:
Putting equation (5.6) into eqs. (5.2) and (5.3), the 2nd order perturbation equations become
(2)
k
∂δ~
∂η
(2)
k
∂Θ~
∂η
−
(2)
Θ~
k
−
(2)
Θ~
k
=
2
D+
(η)
Z
3
Z
d ~q1
(3)
in
d3 ~q2 δD (~k − q~1 − q~2 )αs (q~1 , q~2 )δqin
~1 δq~2 ,
3 Ωm (2)
(2)
2
+
(Θ~ − δ~ ) = D+
(η)
k
k
2 f+2
Z
3
d q~1
Z
(5.9)
(3)
in
d3 q~2 δD (~k − q~1 − q~2 )β(q~1 , q~2 )δqin
~1 δq~2 .
(5.10)
The right hand side of equation (5.9) is obtained after symmetrizing the right hand side
of equation (5.2). This is because, at 2nd order, the source terms Θq~1 and δq~2 in equation
(5.2) have to be linear and they are also equal (see equation (5.5)). Interchanging these two
linear source terms will not affect the evolution equation. Hence we need to symmetrize it
by introducing a quantity, αs (q~1 , q~2 ) = 21 [α(q~1 , q~2 ) + α(q~2 , q~1 )]. From eqs. (5.7) and (5.8), the
2nd order solutions are given by
Z
Z
(3)
(2)
2
(5.11)
δ~ (η) = D+
(η) d3 q~1 d3 q~2 δD (~k − q~1 − q~2 )F2 (q~1 , q~2 , η)δq~in1 δq~in2 ,
k
(2)
Θ~ (η)
k
=
2
(η)
D+
Z
3
d q~1
Z
(3)
d3 q~2 δD (~k − q~1 − q~2 )G2 (q~1 , q~2 , η)δq~in1 δq~in2 .
(5.12)
Now, using equations (5.11) and (5.12) in equations (5.9) and (5.10), the evolution equations
of F2 and G2 becomes
∂F2 (~q1 , ~q2 , η)
+ 2F2 (~q1 , ~q2 , η) − G2 (~q1 , ~q2 , η) = αs (~q1 , ~q2 ),
∂η
i
∂G2 (~q1 , ~q2 , η)
3 Ωm h
+ G2 (~q1 , ~q2 , η) +
G
(~
q
,
~
q
,
η)
−
F
(~
q
,
~
q
,
η)
= β(~q1 , ~q2 ),
2
1
2
2
1
2
∂η
2 f+2
(5.13)
(5.14)
In general these two coupled differential equations have to be solved numerically but fortunately approximate analytical solutions can be possible because
Ωm
≈ 1.
f+2
(5.15)
Except at very low redshifts the assumption (5.15) holds true [42]. Using this assumption,
equation (5.14) becomes
∂G2 (~q1 , ~q2 , η) 5
3
+ G2 (~q1 , ~q2 , η) − F2 (~q1 , ~q2 , η) ≈ β(~q1 , ~q2 ),
∂η
2
2
(5.16)
The coupled differential equations (5.13) and (5.16) can now be solved analytically and the
solutions for F2 and G2 are given by
F2 (~q1 , ~q2 , η) =
i 1h
i
i 7
1h
4h
5αs (~q1 , ~q2 )+2β(~q1 , ~q2 ) − 3αs (~q1 , ~q2 )+2β(~q1 , ~q2 ) e−η −
αs (~q1 , ~q2 )−β(~q1 , ~q2 ) e− 2 η ,
7
5
35
(5.17)
– 10 –
i 1h
i
i 7
1h
6h
3αs (~q1 , ~q2 )+4β(~q1 , ~q2 ) − 3αs (~q1 , ~q2 )+2β(~q1 , ~q2 ) e−η +
αs (~q1 , ~q2 )−β(~q1 , ~q2 ) e− 2 η ,
7
5
35
(5.18)
Here we have assumed the Gaussian initial condition at η = 0; at the initial time, both F2
and G2 vanishes which can be verified through equations (5.17) and (5.18) by putting η = 0.
One interesting point to notice that at sufficient late time the 2nd and 3rd terms in the r.h.s
of both the eqs. (5.17) and (5.18) becomes negligible compared to the 1st terms because of
the exponential factors. So, at sufficient late times, the solutions become
G2 (~q1 , ~q2 , η) =
F2 (~q1 , ~q2 ) '
i 5 1q
1h
q2 2
1
+
5αs (~q1 , ~q2 ) + 2β(~q1 , ~q2 ) = +
qˆ1 .qˆ2 + (qˆ1 .qˆ2 )2 ,
7
7 2 q2 q1
7
(5.19)
G2 (~q1 , ~q2 ) '
i 3 1q
1h
q2 4
1
+
qˆ1 .qˆ2 + (qˆ1 .qˆ2 )2 ,
3αs (~q1 , ~q2 ) + 4β(~q1 , ~q2 ) = +
7
7 2 q2 q1
7
(5.20)
which are standard results. To be precise, the numerical solution, F2 obtained from the
coupled differential eqs. (5.13) and (5.14) has been considered in the subsequent calculations.
Bi-spectrum:
The 3-point correlation functions of the matter energy density contrast i.e. the matter bispectrum is defined as
(3)
< δ~k1 (η)δ~k2 (η)δ~k3 (η) >= δD (~k1 + ~k2 + ~k3 )B(~k1 , ~k2 , ~k3 ; η),
(5.21)
with ~k1 + ~k2 + ~k3 = 0. Expanding δ~k (η) upto the second order the tree-level bi-spectrum
becomes [40]
B(~k1 , ~k2 , ~k3 ; η) = 2F2 (~k1 , ~k2 , η)Plin (k1 , η)Plin (k2 , η) + 2 − cycles.
(5.22)
In figure 3 percentage deviation in tree-level bi-spectrum for different dark energy models from
the ΛCDM model has been plotted using equation (5.22) for the equilateral configuration
(k1 = k2 = k3 = k). The behaviour of the deviations are similar as in the linear matter
power spectrum (see figure 2) but the magnitudes of the deviations become almost twice the
deviations in the linear matter power spectrum corresponding to the same models.
6
Convergence power spectrum and bi-spectrum
Weak lensing statistics is an important probe to the structure formation. The images of the
background galaxies are distorted by the gravitational lensing and this distortion effects are
quantified by a quantity called convergence. In Newtonian perturbation theory with the weak
lensing limit, the convergence (κ) in any particular direction n̂ in the sky can be related to
the weighted projection of the three dimensional matter energy density contrast integrated
along the line of sight, given by ([48],[49],[50],[51])
Z χ
κ(n̂, χ) =
W (χ0 )δ(n̂, χ0 )dχ0 ,
(6.1)
0
– 11 –
7 Equilateral CPL w0 = - 0.9 z = 1
−1
20 Equilateral CPL w0 = - 0.9 k = 0.1 h Mpc
6
%∆B
wa = - 0.3
wa = -0.1
wa = 0.1
4
3
%∆B
15
5
10
5
2
1 -3
10
k [h Mpc ]
Quintessence
00
10 0
−1
z=1
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
3.7
3.6
3.5
%∆B
%∆B
10 -1
10 -2
3.8 Equilateral
3.4
3.3
3.2
10 -3
10 -1
10 -2
k [h Mpc −1 ]
6
8
10
−1
25 Equilateral GCG k = 0.1 h Mpc As = 0.9
%∆B
5.5
z
−1
7 Equilateral Quintessence k = 0.1 h Mpc
6
5
4
3
2
V( φ ) ∝ φ
V( φ ) ∝ φ 2
1
V( φ ) ∝ φ 2
00
2
4
6
8
10
20
α = - 1.1
α = - 0.9
α = - 0.7
4
z
6.5
6.0
2
−
10 0
Equilateral GCG z = 1 As = 0.9
%∆B
wa = - 0.3
wa = -0.1
wa = 0.1
5.0
α = - 1.1
α = - 0.9
α = - 0.7
15
10
5
4.5
10 -3
10 -1
10 -2
00
10 0
k [h Mpc −1 ]
2
4
z
6
8
10
Figure 3. Percentage deviation in tree-level matter bi-spectrum from Λcdm model.
where χ is the comoving distance and the weight function W (χ) is given by
3
W (χ(z)) = Ω(0)
H 2 g(z)(1 + z),
2 m 0
(6.2)
where g/χ is the geometric lensing efficiency factor. g is given by
Z
χ∞
g(z) = χ(z)
χ
χ0 dχ n (χ ) 1 −
= χ(z)
χ
0 0
0
Z
z
∞
χ(z 0 ) dz 0 n(z 0 ) 1 −
,
χ(z)
(6.3)
where g(z) is weighted according to the source distribution n(z)
R ∞ (whose corresponding distri0
bution is n (χ) in χ space) with the normalization such that 0 n(z)dz = 1. Here the source
– 12 –
1.0
b1 = 2
b2 = 1.5
z0 = 0.9/1.412
n(z)
0.8
0.6
0.4
0.2
0.00.0
0.5
1.0
1.5
z
2.0
2.5
3.0
Figure 4. n(z) vs. z plot.
h b 2 i
which after normalisation becomes
distribution is considered as n(z) ∝ z b1 exp − zz0
n(z) =
h z b2 i
( 1+b1 ) z b1
z0
exp −
,
z0
z0
Γ 1+bb12+b2
(6.4)
with the three parameters b1 , b2 and z0 . Here b1 = 2, b2 = 1.5 and z0 = 0.9/1.412 are
considered which are similar to the Euclid Survey ([52],[53],[54],[55]).
In figure 4 the source distribution n(z) has been plotted with respect to the redshift z using
equation (6.4).
For the purpose to study the statistical correlations of the galaxy shears, the convergence can
be transformed in the multiple (l, m) space given by
Z
∗
,
(6.5)
κlm = dn̂κ(n̂, χ)Ylm
where Ylm are the spherical harmonics. Assuming statistical isotropy the convergence power
spectrum can be defined as
< κlm κl0 m0 >= δll0 δmm0 Pκ (l),
and using Limber approximation the convergence power spectrum becomes
Z χ∞
Z ∞
l
dz W 2 (z) l
W 2 (χ)
Pκ (l) =
dχ
P
(
,
χ)
=
P
,
z
.
χ2
χ
H(z) χ2 (z)
χ(z)
0
0
(6.6)
(6.7)
In figure 5 percentage deviation in convergence power spectrum for different dark energy
models from ΛCDM model has been plotted using equation (6.7). For all the dark energy
models the deviations increase with the increasing l. The models wa = −0.3, wa = −0.1
and wa = 0.1 in CPL parametrization with w0 = −0.9 have the deviations nearly 2.5%, 5%
and 7.5% respectively at l = 105 . Similar to the matter power spectrum the deviations in
convergence power spectrum are larger in the non-phantom models compared to the phantom
models. The deviations in the thawing quintessence models are nearly 4 − 4.5% at l = 105
and the differences are very small (sub-percentage) for three potentials. The inverse-squared
– 13 –
w0 = - 0.9
wa = - 0.3
wa = -0.1
wa = 0.1
%∆Pκ
CPL
10 2
10 3
l
%∆Pκ
%∆Pκ
0
1
2
3
4
5
6
7
81
10
10 4
10 5
GCG
1
2
3
4
5
6
7
81
10
10 2
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.010 1
As = 0.9
10 3
l
Quintessence
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
10 2
10 3
l
10 4
10 5
α = - 1.1
α = - 0.9
α = - 0.7
10 4
10 5
Figure 5. Percentage deviation in convergence power spectrum from Λcdm model.
potential has the maximum deviation whereas the linear potential has the minimum deviation.
The models α = −1.1, α = −0.9 and α = −0.7 in GCG parametrization with As = 0.9 have
the deviations nearly 6%, 6.5% and 7.5% respectively at l = 105 . Similar to the matter power
spectrum the deviations in convergence power spectrum are larger in the tracker models
compared to the thawing models.
The convergence bi-spectrum can be defined as
l
l2
l3
1
< κl1 m1 κl2 m2 κl3 m3 >=
Blκ1 l2 l3 ,
(6.8)
m1
m2
m3
where in the r.h.s Wigner 3-j symbol has been used and the above convergence bi-spectrum
(which is full-sky) is related to the flat-sky bi-spectrum given by
l
r
l2
l3
1
(2l1 + 1)(2l2 + 1)(2l3 + 1)
Blκ1 l2 l3 =
Bκ (~l1 , ~l2 , ~l3 ; z),
(6.9)
4π
0
0
0
and similar to the convergence power spectrum, the convergence bi-spectrum (using Limber
approximation) can be written as
Bκ (~l1 , ~l2 , ~l3 ; z) =
Z
0
∞
~l2
~l3
dz W 3 (z) ~l1
B
,
,
;
z
,
H(z) χ4 (z)
χ(z) χ(z) χ(z)
(6.10)
with ~l1 +~l2 +~l3 = 0. Since, in our subsequent plots, convergence bi-spectrum for different dark
energy models have been compared with the ΛCDM model, one need not to consider the full
– 14 –
0
CPL
Equilateral
w0 = - 0.9
wa = - 0.3
wa = -0.1
wa = 0.1
4
%∆Bκ
%∆Bκ
2
6
8
10 2
10 3
l
%∆Bκ
1010 1
10 4
1
2
3
4
5
6
7
8
9
1010 1
10 5
GCG
Equilateral
0
1
2
3
4
5
6
71
10
Quintessence
Equilateral
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
10 2
10 3
l
10 4
10 5
As = 0.9
α = - 1.1
α = - 0.9
α = - 0.7
10 2
10 3
l
10 4
10 5
Figure 6. Percentage deviation in convergence bi-spectrum from Λcdm model.
sky convergence bi-spectrum. Instead flat sky convergence bi-spectrum in equation (6.10) is
sufficient to compare as the ratio of convergence bi-specturm between two dark energy models
is the same for either full sky or flat sky case.
In figure 6 percentage deviation in convergence bi-spectrum for different dark energy models
from the ΛCDM model has been plotted using equation (6.10) for the equilateral configuration
(l1 = l2 = l3 = l). The behaviour of the deviations are similar as in the convergence power
spectrum (see figure 5) but the magnitudes of the deviations become nearly 1.5 times the
deviations in the convergence power spectrum corresponding to the same models. Similar to
the convergence power spectrum the deviations in convergence bi-spectrum for all the dark
energy models increase with the increasing l.
So far in all the bi-spectrum plots equilateral configuration has been considered. To see the
deviations in other configurations in figure 7 two different configurations have been considered
for the same parameter values and the configurations are folded shape (l1 = 2l2 = 2l3 = l)
and squeezed shape (l1 = l2 = 20l3 = l) respectively. Comparing figures 6 and 7, it is clear
that the deviations are almost same in the three different configurations and this is true for
the matter bi-spectrum also. So, it can be concluded that the deviations in the bi-spectrum
are highly insensitive to the shape of the bi-spectrum at least at the tree-level.
In figure 8 same graphs have been plotted but with present day e.o.s to be as −0.85 keeping
all the other normalisation intact. Naturally the deviations is larger in this case.
– 15 –
0
CPL
Folded
0
wa = - 0.3
wa = -0.1
wa = 0.1
2
4
6
8
0
1
2
3
4
5
6
71
10
0
10 3
10 4
Quintessence
Folded
10 2
l
wa = - 0.3
wa = -0.1
wa = 0.1
6
0
10 2
10 3
l
Quintessence
10 4
10 5
Squeezed
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
1
2
3
4
5
10 2
10 3
GCG
Folded
l
10 4
61
10
10 5
As = 0.9
0
α = - 1.1
α = - 0.9
α = - 0.7
4
6
10 2
GCG
10 3
l
Squeezed
10 4
8
10 5
As = 0.9
α = - 1.1
α = - 0.9
α = - 0.7
2
%∆Bκ
%∆Bκ
w0 = - 0.9
4
1010 1
10 5
V( φ ) ∝ φ
V( φ ) ∝ φ 2
V(φ) ∝ φ −2
2
1010 1
Squeezed
8
%∆Bκ
%∆Bκ
1010 1
CPL
2
%∆Bκ
%∆Bκ
w0 = - 0.9
4
6
8
10 2
10 3
l
10 4
1010 1
10 5
10 2
10 3
l
10 4
10 5
Figure 7. Percentage deviation in convergence bi-spectrum from Λcdm model.
7
Conclusion
The weak lensing statistics is a powerful tool to probe the dark energy and structure formation
in the Universe. The evolution of the background quantities as well as the perturbated
quantities in any dark energy model can be measured through weak lensing with high accuracy
by the current and future surveys like DES, LSST, Euclid, WFIRST etc.
Firstly, three types of dark energy models with evolving equation of state have been
considered to study their detectibility in the structure formation through weak lensing statistics. One is the most popular dark energy parametrization named CPL parametrization. The
next one is the minimally coupled canonical quintessence scalar field dark energy candidate
where thawing class (where initially equation of state of the dark energy is close to −1 due to
– 16 –
%∆Bκ
l
w0 = - 0.85
wa = - 0.3
wa = -0.1
wa = 0.1
10 4
1 Quintessence Equilateral w0 = - 0.85
V( φ ) ∝ φ
2
V( φ ) ∝ φ 2
3
V( φ ) ∝ φ 2
4
5
6
7
8
9
1010 1
10 4
10 5
10 2
10 3
−
%∆Bκ
%∆Bκ
0 CPL Equilateral
2
4
6
8
10
12
1410 1
10 2
10 3
10 5
l
GCG Equilateral
0
2
4
6
8
10
12
14
1610 1
10 2
10 3
l
As = 0.85
α = - 1.1
α = - 0.9
α = - 0.7
10 4
10 5
Figure 8. Same as figure 6 but changing the present day e.o.s to be −0.85.
the large Hubble damping and at late times hubble damping decreases due to the expansion
of the Universe and the equation of state of the scalar fields increases from −1 slowly) of
scalar field models with linear, squared and inverse-squared potentials have been considered
as because a broad class of potentials can give the proper thawing behaviour whearas to get
tracker behaviour (where initially scalar fields mimic the background matter density i.e. w
is close to 0 and at late times the equation of state of the scalar field freezes towards −1)
there are few potentials (e.g. cosine hyperbolic and double exponential potentials etc.) which
can give proper tracker behaviour. But to show the thawing versus tracker results next a
simple paramitrization named GCG parametrization has been considered where 1 + α < 0
gives thawing behaviour and 1 + α > 0 gives tracker behaviour (see figure 1).
Next, to compare all the dark energy models properly the normalisation has been taken
(0)
such that at present time the values of background quantities (H0 , w0 and Ωm ) as well as
(0)
the perturbed quantities (here power spectrum by σ8 normalisation with the same σ8 , ns
(0)
and Ωb values) are same for all the models.
Figure 2 shows that for all the dark energy models the deviations in linear matter power
spectrum from ΛCDM model are scale independent. The deviations increase with increasing
redshift and fix upto the maximum values after certain redshifts (5 to 10% or more). The
CPL parametrization shows the deviations for non-phantom models are larger compared to
the phantom models (upto 8 − 9% difference between wa = −0.3 and wa = 0.1 models with
w0 = −0.9). The quintessence models of thawing class show that the deviations are upto
2.5 − 3.5% and the differences between different potentials are very small (sub-percentage
– 17 –
level). It can also be noted that the deviation is the largest for inverse-squared potential
whereas it is the smallest for the linear potential. The deviations in the GCG parametrization
show that the deviations are larger for tracker models compared to the thawing models (nearly
upto 4%, 6% and 10% for α = −1.1, α = −0.9 and α = −0.7 models respectively with
As = 0.9).
Figure 3 shows that the deviations in the tree level matter bi-spectrum for all the dark
energy models from ΛCDM model have the similar behaviour as in the linear matter power
spectrum but the magnitude of the deviations are approximately twice the deviations in the
linear matter power spectrum which is quite obvious from eq. (5.22).
The deviations in convergence power spectrum for all the dark energy models from
ΛCDM model increase with increasing l (see figure 5). Similar to the deviations in linear
matter power spectrum and tree-level bi-spectrum, the deviations in convergence power spectrum are larger for non-phantom models (nearly 7.5% at l = 105 for the w0 = −0.9 & wa = 0.1
model) compared to phantom models (nearly 2.5% at l = 105 for the w0 = −0.9 & wa = −0.3
model) in CPL parametrization; the deviation is the largest for inverse-squared potential
(nearly 4.5% at l = 105 ) and the smallest for linear potential (nearly 4% at l = 105 ) in
thawing class of quintessence models; the deviations are larger in tracker models compared to
thawing models in GCG parametrization (nearly 6%, 6.5% and 7.5% for α = −1.1, α = −0.9
and α = −0.7 models respectively with As = 0.9 at l = 105 ).
The deviations in convergence bi-spectrum for all the dark energy models from ΛCDM
model have the similar behaviour as in convergence power spectrum but with larger deviations
(upto 2 − 3% more deviations for all the models accordingly. The deviations in convergence
bi-spectrum are almost same for equilateral (l1 = l2 = l3 = l), folded (here l1 = 2l2 = 2l3 = l)
and squeezed (here l1 = l2 = 20l3 = l) configurations for all the models accordingly (see figure
6 and 7).
Finally, the deviations from ΛCDM model become larger in all the corresponding quantities if the present day equation of state for all the dark energy models deviates more from
−1 accordingly (see figures 6 and 8 for the comparison between w0 = −0.9 and w0 = −0.85
normalised valued models in convergence bi-spectrum for equilateral configuration).
To summerise, the non-phantom models are more probable to be distinguished from
ΛCDM model compared to the phantom models and similarly the tracker models are more
probable to be distinguished from ΛCDM model compared to the thawing models.
In this paper linear power spectrum and tree level bi-spectrum have been studied. This
work can be extended to the non-linear matter power spectrum, the higher order correction to the tree level matter bi-spectrum and also to the higher oder correlation functions
(tri-spectrum etc.) by studying the higher order perturbation theory in more details and
corresponding to the convergence power spectrum, bi-spectrum etc.
Acknowledgments
The author would like to acknowledge CSIR, Govt. of India for financial support through SRF
scheme (No:09/466(0157)/2012-EMR-I). The author would also like to thank Anjan Ananda
Sen for exclusive discussions.
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