Lensing Effects on Gravitational Waves in a Clumpy Universe -Effects of Inhomogeneity on the Distance-Redshift Relation-

1.6 MB·

by  Chul-Moon Yoo

Department of Mathematics and Physics OCU-PHYS-244
Osaka City University AP-GR-32
Lensing Effects on Gravitational Waves in a Clumpy Universe
—Effects of Inhomogeneity on the Distance-Redshift Relation—
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Chul-Moon Yoo1, Ken-ichi Nakao2and Hiroshi Kozaki3
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Department of Mathematics and Physics, Graduate School of Science, Osaka City
n
a University, Osaka 558-8585, Japan
J

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Ryuichi Takahashi4

3
v Division of Theoretical Astronomy, National Astronomical Observatory of Japan, Mitaka,
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2 Tokyo 181-8588, Japan
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4
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ABSTRACT
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/
h
The distance-redshift relation determined by means of gravitational waves in
p
- the clumpy universe is simulated numerically by taking into account the effects
o
r of gravitational lensing. It is assumed that all of the matter in the universe
t
s
takes the form of randomly distributed point masses, each of which has the
a
:
v identical mass ML. Calculations are carried out in two extreme cases: λ
≫
Xi GML/c2 and λ GML/c2, where λ denotes the wavelength of gravitational
≪
r waves. In the first case, the distance-redshift relation for the fully homogeneous
a
and isotropic universe is reproduced with a small distance dispersion, whereas
in the second case, the distance dispersion is larger. This result suggests that
we might obtain information about the typical mass of lens objects through the
distance-redshift relation gleaned through observation of gravitational waves of
various wavelengths. In this paper, we show how to set limitations on the mass
M through the observation of gravitational waves in the clumpy universe model
L
described above.
Subject headings: gravitational lensing – gravitational waves – dark matter –
distance scale
– 2 –
1. Introduction
Owing to recent developments of observational techniques, it is expected that detection
of gravitational waves will be achieved in the near future. Laser interferometer gravitational
wave detectors, such as the Tokyo Advanced Medium-Scale Antenna (TAMA-300), the Laser
Interferometer Gravitational-Wave Observatory (LIGO), Variability of Irradiance and Grav-
ity Oscillations (VIRGO) and GEO-600, are in operation, and other projects, such as the
Large-scale Cryogenic Gravitational wave Telescope (LCGT), LIGO2, the Laser Interfer-
ometer Space Antenna (LISA), the Decihertz Interferometer Gravitational Wave Observa-
tory (DECIGO; Seto et al. (2001)), and the Big Bang Observer (BBO), are in the planning
stage. In preparation for future breakthroughs associated with these projects, it is useful
to discuss the information provided by gravitational wave data. Here we focus on inhomo-
geneities of our universe that can be investigated with gravitational wave observations.
Many cosmological observations suggest that our universe is globally homogeneous and
isotropic, in other words, that the time evolution of the global aspect is well approximated
by the Friedmann-Lemaˆıtre(FL) cosmological model. However, our universe is locally in-
homogeneous. The effects of these inhomogeneities on the evolution of our universe or on
cosmological observations draw our attention as one of the most important issues in cosmol-
ogy. However, it is difficult to directly observe the inhomogeneities with optical observations
because most of the matter in our universe is not luminous. One useful means of obtaining
information about aspects of inhomogeneities is to examine gravitational lensing. Gravita-
tional and electromagnetic waves are subject to gravitational lensing effects caused by inho-
mogeneously distributed matter around their trajectories. Therefore, we may find features
of inhomogeneities through both observational and theoretical studies of the gravitational
lensing effects in astronomically significant situations.
There are various candidates for the dark matter such as weakly interacting massive
particles (WIMPs) and massive compact halo objects(MACHOs). Several observations have
been undertaken to constrain the mass density of compact objects, Ω ; direct searches for
CO
MACHOs in the Milky Way have been performed by the MACHO and EROS collaborations
through microlensing surveys. The MACHO group (Alcock et al. 2000) group concluded
that the most likely halo fraction in form of compact objects with masses in the range 0.1-
1E-mail:c m [email protected]
2E-mail:[email protected]
3E-mail:[email protected]
4E-mail:[email protected]
– 3 –
1M is of about 20%. The EROS team concluded that the objects in the mass range from
⊙
2 10−7M to 1M cannot contribute more than 25% of the total halo (Afonso et al. 2003).
⊙ ⊙
×
However, the universal fraction of macroscopic dark matter could be significantly different
from these local estimates. Millilensing tests of gamma-ray bursts derive a limit on Ω of
CO
Ω < 0.1 in the mass range from 105 to 109M (Nemiroff et al. 2001). Multiple imaging
CO ⊙
searches in compact radio sources derive the limit on Ω as Ω . 0.013 in the mass range
CO CO
from 106 to 108M (Wilkinson et al. 2001). The mass range of these observational tests
⊙
are limited, and thus other methods to investigate the mass fraction of macroscopic compact
objects in wider mass range are needed.
In this paper, we show that it is possible to extract information about the properties of
macroscopic compact objects fromtheobservationaldataofgravitationalwaves byanalyzing
the gravitational lensing effects due to these compact objects. For this purpose, we consider
an idealized model of the inhomogeneous universe which has the following properties. First,
this model is a globally FL universe. Second, all of the matter takes the form of point
masses, each of which has the identical mass M . Finally, the point masses are uniformly
L
distributed.
One of the main targets of gravitational-wave astronomy is binary compact objects.
Their gravitational waves have much longer wavelengths (λ) than the optical, and further-
more, these are coherent. In typical optical observations, the wavelength (λ 1µm) might
∼
be much shorter than the Schwarzschild radii of lens objects.5 Thus, we usually analyze
the gravitational lensing effects on electromagnetic waves by using geometrical optics. In
contrast, we need wave optics for the gravitational lensing of gravitational waves, since the
wavelength of gravitational waves may be comparable to or longer than the Schwarzschild
radii of lens objects. For example, the wave-band of LISA is typically 1011 1014 cm; a point
−
mass of 105 108M has a Schwarzschild radius almost equal to the wavelengths within
⊙
−
that wave-band. When we consider gravitational lensing effects on the gravitational waves
with GM /c2 . λ, we have to take the wave effect (Nakamura 1998) into account. In
L
fact, remarkable differences between the extreme cases λ GM /c2 and λ GM /c2 have
L L
≫ ≪
already been reported (Takahashi & Nakamura 2003).
In order to obtain information about uniformly distributed compact objects by investi-
gatingthegravitationallensingeffectsinobservationaldata,wefocusontherelationbetween
the distance from the observer to the source of the gravitational waves and its redshift. The
5It does not conflict with any observationalresult that theremight be lens objects of Schwarzschildradii
muchsmallerthan 1µm. However,forexample, supernovaewhichmightbe the smallestand yetverybright
opticalsources,arelargerthantheEinsteinradiiofsuchsmall-masslensobjects,andthusthelensingeffects
due to those will be negligible.
– 4 –
distance is determined using the information contained in the amplitudes of the gravita-
tional waves from the so-called standard sirens (Holz & Hughes 2005). Although there are
several similar analyses using the optical observation of Type Ia supernovae (Holz & Linder
2005), gravitational lensing effects on gravitational waves can add new information about
the properties of the lensing compact objects to that obtained by optical observations, by
virtue of the wave effect, through which we can gain an understanding of the typical mass
and number density.
Hereweassumethattheredshiftofeachsourceisindependentlygivenbytheobservation
of the electromagnetic counterpart (Kocsis et al. 2006) or the waveform of the gravitational
waves (Markovic1993). Theluminositydistanced fromtheobserver tothesourceisgivenby
l
the waveform of the gravitational waves, as follows (Schutz 1986). Since the orbit of binary
will be quickly circularized by a gravitational radiation reaction (Peters & Mathews 1963;
Peters 1964), the effect of ellipticity is negligible when the emitted gravitational radiation
becomes so strong that we can observe it. Then, the two wave modes emitted from a binary
of two compact objects with masses m and m are given by
1 2
2 5/3 (πf)2/3cos(2πft)
h = Mchirp (cos2θ +1), (1)
+
d
l
2 5/3 (πf)2/3sin(2πft)
h = Mchirp 2cosθ, (2)
×
d
l
where f and θ are the frequency and the angle between the angular momentum vector of the
binary and the line-of-sight, respectively, and is the redshifted chirp mass, defined by
chirp
M
(m m )3/5
1 2
= (1+z)M = (1+z) . (3)
Mchirp chirp (m +m )1/5
1 2
Assuming a circular orbit, we have
df 96π8/3
= f11/3 5/3 . (4)
dt 5 Mchirp
Eliminating and θ from equations (1), (2), and (4), we can obtain d if there is no
chirp l
M
gravitational lensing effect on the gravitational waves. However, if the gravitational waves
are gravitationally lensed, their wave forms are changed. The gravitational lensing effects
may lead to an incorrect estimation of d , and thus we have to know how the gravitational
l
lensing changes the waveforms of the gravitational waves.
This paper is organized as follows. In 2, the basic theory of gravitational lensing is
§
introduced. We discuss the lensing probability in 3. Then we show the calculation method
§
and results for the distance-redshift relation in 4 and 5. Finally, 6 is devoted to the
§§ §
conclusions and summary. Throughout the paper, we adopt the unit of G = c = 1.
– 5 –
2. Review of gravitational lensing
In this section, we introduce basic equations for gravitational lensing.
2.1. Geometrical optics
In the case of a point-mass lens with mass M , the bending angle vector αˆ(ξ) is given
L
by (Schneider et al. 1992)
ξ
αˆ(ξ) = 4M , (5)
L ξ 2
| |
where ξ is the impact vector (see Fig. 1). For convenience, we consider the “straight” line A
from a source to the observer and define the intersection point of A with the lens plane as
the origin of the lens position ζ and the ray position γ. We note that, in general, the vector
ξ is not orthogonal to the line A. However, if the geometrically thin lens approximation is
valid, we can regard the vectors ξ, ζ, and γ as parallel to each other (Schneider et al. 1992).
We define D , D , and D as the angular diameter distances from the observer to the
S L LS
source, from the observer to the lens, and from the lens to the source, respectively. Using
simple trigonometry (see Fig. 1), we find the relation between the source position η and ξ,
D
η = Sξ D αˆ(ξ). (6)
LS
D −
L
Equation (6) becomes
D
0 = Sγ D αˆ(γ ζ) (7)
LS
D − −
L
in terms of ζ and γ.
Here we introduce the dimensionless vectors defined by
ξ η
x = and l = , (8)
ξ η
0 0
where ξ is the Einstein radius given by
0
D D
L LS
ξ = 4M (9)
0 L
D
r S
and
D
S
η = ξ . (10)
0 0
D
L
– 6 –
Fig. 1.— Geometry of gravitational lensing by a point-mass lens. The impact vector ξ
represents the relative position of a light ray on the lens plane to the lens position, and αˆ is
a vector whose norm is equal to the bending angle.
From Figure 1, we find the relation
ζ
l = . (11)
−ξ
0
The source will have two images at position (x ,x ). Substituting equation (5) into equa-
+ −
tion (6) we find
1
x = √l2 +4 l , (12)
±
2 ±
(cid:16) (cid:17)
where x = x and l = l . The total magnification factor µ is given by
± ±
| | | |
∂l ∂l l2 +2
µ = µ + µ = det + det = . (13)
| +| | −| ∂x ∂x l√l2 +4
(cid:12) (cid:18) +(cid:19)(cid:12) (cid:12) (cid:18) −(cid:19)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
2.2. Wave optics in gravitational lensing due to point mass lens
For simplicity, let us consider gravitational waves that propagate in asymptotically flat
spacetime with the metric
ds2 = (1+2U)dt2 +(1 2U)dr2, (14)
− −
where U is the gravitational potential and we assume U 1. If the wavelength of the
| | ≪
gravitational waves is much smaller than the typical curvature radius of the background
– 7 –
spacetime, the equation for the gravitational waves is equivalent to that of scalar waves.
Assuming monochromatic waves of an angular frequency ω, we have (Peters 1974)
2 +ω2 φ = 4ω2Uφ. (15)
∇
(cid:0) (cid:1)
For convenience, we introduce the amplification factor F defined by
φL(ω)
F = , (16)
φ(ω)
where φ is the plane wave with no lensing effects, and φL is the wave that is undergoing
lensing effects. Here note that φL is no longer a plane wave, although it was initially. In the
caseofthepoint-masslenswithmassM ,thegravitationalpotentialisgivenbyU = M /r.
L L
−
Then we have the solution of equation (15) in the form
πw w w i i i
F(ω;y) = exp +i ln Γ 1 w F w,1; wy2 , (17)
1 1
4 2 2 − 2 2 2
(cid:18) (cid:19) (cid:18) (cid:19)
h (cid:16) (cid:17)i
where Γ and F are the gamma function andthe confluent hypergeometric function, respec-
1 1
tively, and w := 4ωM (Peters 1974). Since U is much larger than the Hubble parameter
L
|∇ |
of the universe, by replacing ω with ω(1 + z ), we can use equation (17) in cosmological
L
situations.
2.3. The Dyer-Roeder distance
Intheclumpyuniversemodel,thelocalgeometryisinhomogeneous, butitsglobalaspect
might be well described by the FL universe whose metric is given by
dR2
ds2 = dT2 +a2(T) +R2dΩ2 , (18)
− 1+KR2
(cid:18) (cid:19)
where K = 1, 0, and 1, and dΩ2 is the round metric. However, the global aspect of the
−
clumpy universe is a vague notion since we do not have a definite mathematical prescription
to identify the global aspect of the locally inhomogeneous universe with the FL universe
model. Therefore, the above FL universe is a fictitious background universe; hereafter, we
will refer to this fictitious geometry (universe) as the “global geometry” (or universe). We
assume that the global geometry is well approximated by that of the FL universe filled with
dust and with a cosmological constant.
In the clumpy universe model, the distances D , D , and D in equation (6) or equa-
L S LS
tion (7) are replaced by the Dyer-Roeder (DR) distances (Kantowski 1969; Dyer & Roeder
– 8 –
1973), which are the observed angular diameter distances if the gravitational waves prop-
agate far from the point masses. The DR distance with smoothness parameter α˜ = 0
(Schneider et al. 1992) in the global universe defined above is given by
1+z z2 1 dz
1
D (z ,z ) = , (19)
DR 1 2
H (1+z)2 Ω (1+z)3 Ω (1+z)2 +Ω
0 Zz1 m0 − K0 Λ0
where z and z are the redshift of the obserpver and source, and Ω , H , and Ω are the
1 2 m0 0 Λ0
present valuesofthetotaldensity parameter, Hubbleparameterandnormalizedcosmological
constant in the global universe, respectively, and Ω = Ω + Ω 1. Hereafter, the
K0 m0 Λ0
−
subscript 0 means the present value. Some further discussions of the properties of the DR
distance are presented in Linder (1988), Seitz & Schneider (1994) , Kantowski (1998), and
Sereno et al. (2001).
3. The distribution of point masses
In order for the clumpy universe to be the same as a homogeneous andisotropic universe
from a global perspective, the point masses must be distributed uniformly. However, since
there are no isometries of homogeneity and isotropy in the clumpy universe, the notion of
“uniform distribution” cannot be introduced in a strict sense. Thus, we assume a uniform
distribution of the point masses with respect to the global geometry represented by equa-
tion (18) in a manner consistent with the mass density of this global universe. This universe
modeliscalled“theon-averageFriedmannuniverse.” Theassumptionsinthisuniverse model
are discussed in Seitz et al. (1994).
The comoving number density ρ of the point masses M is given by
n L
a3ρ 3Ω H2
ρ = = m0 0, (20)
n
M 8πM
L L
where ρ is the average mass density and we have set a = 1. We consider a past light cone in
0
the global universe. The vertex of this light cone is at the observer, at R = 0 and z = 0, and
is parametrized by the redshift z of its null geodesic generator. Then the comoving volume
of a spherical shell bounded by R(z) and R = R(z +∆z) on this light cone is given by
dR
∆V = 4πR2 ∆z. (21)
dz
Therefore, the number of point masses in this shell is given by
3Ω H2 dR
∆N = ρ ∆V = m0 0R2 ∆z, (22)
n
2M dz
L
– 9 –
where
dR 1 1+H2Ω (1+z)2D2(z)
= 0 K0 F , (23)
dz H0sΩm0(1+z)3 ΩK0(1+z)2 +ΩΛ0
−
whereD (z)istheangulardiameter distancefromthesourceoftheredshift z totheobserver
F
in the global universe, which is expressed as
1 sinX(z)
for Ω > 0
K0
H (1+z) √Ω
0 K0
 1
D (z) =  Y(z) for Ω = 0 , (24)
F  H (1+z) K0
 0
 1 sinhX(z)
for Ω < 0
K0
H (1+z) √ Ω
 0 − K0

where 
z dz′
X(z) = Ω , (25)
K0
| | Ω (1+z′)3 Ω (1+z′)2 +Ω
Z0 m0 − K0 Λ0
p
z dz′
p
Y(z) = . (26)
Ω (1+z′)3 Ω (1+z′)2 +Ω
Z0 m0 − K0 Λ0
The average number of point masses in the region y < ζ/ξ < y +∆y of this shell is given
p 0
by
2πξ2y∆y
p(y,z)∆y = 0 ∆N. (27)
4πa2R2
Substituting equations (22) and (23) into equation (27) and using equation (9), we have
1+H2Ω (1+z)2D2(z)
p(y,z)∆y = 3H Ω (1+z)2 0 K0 F
0 m0
sΩm0(1+z)3 ΩK0(1+z)2 +ΩΛ0
−
D (0,z)D (z,z )
DR DR S
y∆y∆z, (28)
× D (0,z )
DR S
where z is the redshift of the source and we have replaced each of distances in equation (9)
S
with the DR distance. We find that p(y,z) does not depend on M , and therefore the lensing
L
probability for a given source does not depend on M .
L
We note that the assumption of the distribution of point masses given in equation (28)
does not necessarily guarantee equality between the magnification in the global universe
defined by
2
D (0,z)
DR
µ := (29)
F
D (z)
(cid:18) F (cid:19)
and the average magnification in the clumpy universe. Sevral previous works have discussed
this issue (Weinberg 1976; Ellis et al. 1998; Claudel 2000; Rose 2001; Kibble & Lieu 2005).
– 10 –
4. The long-wavelength case, λ M
L
≫
In the case of λ M , the amplification factor F(ω(1 + z );y) is almost equal to
L L
≫
unity (see Fig. 2) because of diffraction effects (Nakamura 1998), and we have
φ φ
L
| − | = F(ω(1+z );y) 1 1. (30)
L
φ | − | ≪
| |
Thus we assume that a wave φ can be treated as a spherical wave whose center is at the
source during all stages of the propagation. Let us consider the situation in which N point
masses are located at (z ,y ), i = 1,...,N,, where i is the number assigned to each point
i i
mass in order of distance fromthe observer. If our assumption is valid, the wave φ is changed
whenever it propagates near each point mass as
N
φ F φ F F φ F φ, (31)
N N N−1 i
→ → → ··· →
i=1
Y
where
F = F (ω(1+z );y ). (32)
i i i
Thus the observed amplitude φ is given by
obs
| |
N
φ = F φ . (33)
obs i
| | | || |
i=1
Y
Then, the total amplification F is given by
total
| |
N
F = F . (34)
total i
| | | |
i=1
Y
We randomly distribute point masses so that the distribution of those is consistent with
equation (28). First, we divide the spherical region z < z into N concentric spherical
S
shells, each of which is bounded by two spheres, z = z ∆z/2 and z = z + ∆z/2. We
i i
−
take into account only the nearest lens in each shell. As described below, this procedure is
valid for large numbers N, i.e., small ∆z. We find from equation (28) that in the ith shell
z ∆z/2 < z < z + ∆z/2, there is one point mass within the region y Y on average,
i i i
− ≤
where Y is defined by
i
Yi
p(y,z )dy = 1. (35)
i
Z0
Then we randomly put a point mass within the region y Y in the ith shell. Here it is
i
≤
worth noting that Y is an increasing function of N, so by setting N large, we can take into
i