An analytical approximation of the luminosity distance in flat cosmologies with a cosmological constant

Abstract

We present an analytical approximation formula for the luminosity distance in
spatially flat cosmologies with dust and a cosmological constant. Apart from
the overall factor, the effect of non-zero cosmological constant in our formula
is written simply in terms of a rational function. We also show the approximate
formulae for the Dyer-Roeder distance (empty beam case) and the generalized
angular diameter distance from redshift $z_1$ to $z_2$, which are particularly
useful in analyzing the gravitational lens effects. Our formulae are widely
applicable over the range of the density parameter and the redshift with
sufficiently small uncertainties. In particular, in the range of density
parameter $0.3 \leq \Omega_{\rm m} \leq 1$ and redshift $0.03 \leq z \leq
1000$, the relative error for the luminosity distance by our formula is always
smaller than that of the recent work by Wickramasinghe and Ukwatta (2010).
Hence, we hope that our formulae will be an efficient and useful tool for
exploring various problems in observational cosmology.

Full text

Page 1

arXiv:1111.6396v1 [astro-ph.CO] 28 Nov 2011
1
An analytical approximation of the luminosity distance
in flat cosmologies with a cosmological constant
Masaru Adachi and Masumi Kasai∗)
Graduate School of Science and Technology, Hirosaki University,
Hirosaki 036-8561, Japan
We present an analytical approximation formula for the luminosity distance in spatially
flat cosmologies with dust and a cosmological constant.
the effect of non-zero cosmological constant in our formula is written simply in terms of
a rational function. We also show the approximate formulae for the Dyer-Roeder distance
(empty beam case) and the generalized angular diameter distance from redshift z1 to z2,
which are particularly useful in analyzing the gravitational lens effects. Our formulae are
widely applicable over the range of the density parameter and the redshift with sufficiently
small uncertainties. In particular, in the range of density parameter 0.3 ≤ Ωm ≤ 1 and
redshift 0.03 ≤ z ≤ 1000, the relative error for the luminosity distance by our formula is
always smaller than that of the recent work by Wickramasinghe and Ukwatta (2010). Hence,
we hope that our formulae will be an efficient and useful tool for exploring various problems
in observational cosmology.
Apart from the overall factor,
§1. Introduction
Current cosmological observations indicate that the universe is spatially flat
with a cosmological constant. In such a universe, calculations of the luminosity
distance and the angular diameter distance require numerical integrations and elliptic
functions.1)
In order to simplify the repeated numerical integrations, Pen2)has
developed an efficient fitting formula. Recently, Wickramasinghe and Ukwatta3)
have shown another analytical method, which runs faster than that of Pen2)and has
smaller error variations with respect to redshift z. (See also Ref. 4).)
In this paper, we present yet another analytical approximation to calculate the
luminosity distance as follows:
dL(z,Ωm) =2c
H0
1 + z
√Ωm
?
Φ?x(0,Ωm)?−
1 + 1.392x + 0.5121x2+ 0.03944x3,
1
√1 + zΦ(x(z,Ωm))
?
,(1.1)
Φ(x) =1 + 1.320x + 0.4415x2+ 0.02656x3
(1.2)
where c is the speed of light, H0is the Hubble constant, Ωmis the density parameter
of dust matter, related to the density parameter of vacuum energy ΩΛby Ωm+ΩΛ=
1, and
x(z,Ωm) =1 − Ωm
Ωm
Apart from the overall factor 1/√Ωm, the effect of non-zero cosmological constant
in our distance formula is written simply in terms of a rational function Φ(x).
The function Φ(x) has the following properties:
1
(1 + z)3.(1.3)
∗)E-mail: kasai@phys.hirosaki-u.ac.jp
typeset using PTPTEX.cls ?Ver.0.9?

Page 2

2
M. Adachi and M. Kasai
1. Φ = 1 for x = 0.
2. Φ is a monotonically decreasing function of x, dΦ/dx < 0, and Φ → 0.6735 for
x → ∞.
3. Φ is a monotonically increasing function of Ωm, ∂Φ/∂Ωm> 0, and Φ → 1 for
Ωm→ 1.
4. Φ is a monotonically increasing function of z, ∂Φ/∂z > 0, and Φ → 1 for z → ∞.
5. 0.6735 < Φ(x(0,Ωm)) < Φ(x(z,Ωm)) < 1 for 0 < z, 0 < Ωm< 1.
Note that our approximate formula is explicitly shown to be exact when Ωm= 1:
dL(z,1) =2c
H0(1 + z)
?
1 −
1
√1 + z
?
.(1.4)
§2.Approximation
The luminosity distance in flat cosmologies with a cosmological constant is given
by
dL(z,Ωm) =
c
H0(1 + z)
?1
1+z
√Ωmda′
1
da
?Ωma + (1 − Ωm)a4.(2.1)
We define
F =
?a
0
?Ωma′+ (1 − Ωm)a′4.(2.2)
The power series expansion of F with respect to a around a = 0 yields
F =√a
?
2 −1
7x +3
52x2−
5
152x3+ ···
?
, (2.3)
where
x =1 − Ωm
Ωm
a3. (2.4)
After expanding F up to O(x6), we can obtain the Pad´ e approximant to the following
order:
F =√a2 + b1x + b2x2+ b3x3
1 + c1x + c2x2+ c3x3,
where the numerical constants are determined as follows:
(2.5)
b1=4222975319
1599088274,(2.6)
b2=1138125153117
1288865148844,
7433983569773
139933930445920,
c1=635916643
456882364,
c2=14505955555
28326706568,
(2.7)
b3=
(2.8)
(2.9)
(2.10)

Page 3

An analytical approximation of the luminosity distance
3
c3=
44686179629
1133068262720. (2.11)
Setting a = 1/(1 + z), we finally obtain Eqs. (1.1)–(1.3).
In order to compare our method with that of Ref. 3), we calculate the following
relative error
∆E =|dappr
L
− dnum
dnum
L
L
|
× 100 (per cent), (2.12)
where dappr
approximate formula and numerical method, respectively.
L
and dnum
L
represent the values of luminosity distances calculated by using
0
0.1
0.2
0.3
0.1 1
redshift
10 100
relative error ∆E (%)
Wickramasinghe & Ukwatta 2010
This paper
Fig. 1. A comparison of the percentage rela-
tive error ∆E for two analytical methods
as a function of the redshift for Ωm = 0.3.
Fig. 1 shows a comparison of ∆E
for both analytical methods for Ωm =
0.3. It is shown that our method has a
smaller relative error for redshift range
0.03 ≤ z ≤ 100. Although the relative
error for our method is slightly worse in
the range 0.01 ≤ z < 0.03, it is still less
than 0.3 per cent.
Since our method is based on the
Taylor expansion and the Pad´ e approx-
imant with respect to x = a3(1 −
Ωm)/Ωm, it is evident that the error
in our method decreases monotonically
with increasing redshift z (or increasing
Ωm), i.e., with decreasing x.
Table I. The percentage relative error ∆E(%) for the luminosity distance by our formula Eq. (1.1)
Ωm
0.2
0.3
0.4
1
z = 0.03
1.87%
0.26%
0.03%
z = 0.1
1.38%
0.18%
0.01%
z = 1
0.25%
0.02%
< 0.01%
z = 10
0.08%
0.01%
<0.01%
z = 1000
0.06%
< 0.01%
< 0.01%
00000
Table I shows the relative percentage error ∆E in our method. It is apparent
that our approximate formula has sufficiently small uncertainties in the wide range
of parameters. Only one exception is the nearby region (say, z < 0.1) in the low
density (say, Ωm< 0.2) universe, where the relative error ∆E exceeds 1 per cent.
In such a nearby region (z ≪ 1), however, we may alternatively use the power series
expansion around z = 0 with sufficient accuracy.
In particular, in the range of density parameter 0.3 ≤ Ωm ≤ 1 and redshift
0.03 ≤ z ≤ 1000, the relative error for the luminosity distance in our formula is
always smaller than that of the recent work by Wickramasinghe and Ukwatta.3)

Page 4

4
M. Adachi and M. Kasai
§3. The empty beam case
The distance formula which takes the effect of clumpy distribution of matter
into account has proposed in Refs. 5),6), and later by Dyer and Roeder7)in more
general form, which is now known as the Dyer-Roeder distance. The Dyer-Roeder
luminosity distance for empty beam case in flat cosmologies is
DL(z,Ωm) =
c
H0(1 + z)2
?1
1+z
1
a2da
?Ωma + (1 − Ωm)a4. (3.1)
Our analytic formula for the empty beam is
DL(z,Ωm) =2
5
c
H0
(1 + z)2
√Ωm
?
Ψ(x(0,Ωm)) −
1
(1 + z)
5
2Ψ(x(z,Ωm))
?
,(3.2)
Ψ(x) =1 + 1.256x + 0.3804x2+ 0.0164x3
1 + 1.483x + 0.6072x2+ 0.0587x3,
where x is defined in Eq. (1.3). The formula can be obtained in the same way as that
in the previous section. The rational function Ψ(x) has the following properties:
1. Ψ = 1 for x = 0.
2. Ψ is a monotonically decreasing function of x, dΨ/dx < 0, and Ψ → 0.2792 for
x → ∞.
3. Ψ is a monotonically increasing function of Ωm, ∂Ψ/∂Ωm> 0, and Ψ → 1 for
Ωm→ 1.
4. Ψ is a monotonically increasing function of z, ∂Ψ/∂z > 0, and Ψ → 1 for
z → ∞.
5. 0.2792 < Ψ(x(0,Ωm)) < Ψ(x(z,Ωm)) < 1 for 0 < z, 0 < Ωm< 1.
Again, our approximate formula Eq. (3.2) is exact when Ωm= 1:
(3.3)
DL(z,1) =2
5
c
H0(1 + z)2
?
1 −
1
(1 + z)
5
2
?
. (3.4)
Note that no consideration on the Dyer-Roeder distance has been taken in the pre-
vious papers, e.g., Refs. 2),3).
Table II. The percentage relative error ∆E(%)
for the Dyer-Roeder distance (empty beam
case) by our formula Eq. (3.2)
Ωm
0.2
0.3
0.4
1
z = 0.03
0.90%
0.14%
0.02%
z = 0.1
0.68%
0.10%
0.01%
z = 1
0.21%
0.03%
< 0.01%
z = 10
0.15%
0.02%
< 0.01%
0000
Table II shows the relative error of
the our formula Eq. (3.2). The relative
error for 0.3 ≤ Ωm ≤ 1 is always less
than 0.15 per cent in the range 0.03 ≤
z ≤ 10. For Ωm = 0.2, the accuracy
gets slightly worse, but the error is still
less than 1 per cent in the same redshift
range.
We omitted the error calculations
for z > 10 for the following reasons.
First, the errors are sufficiently small in
those regions, and second, the Dyer-Roeder description is relevant only in the regions
where the clumpy distribution of matter becomes important.

Page 5

An analytical approximation of the luminosity distance
5
§4.The generalized angular diameter distance
Here we present the analytic formulae for the generalized angular diameter dis-
tance from redshift z = z1to z = z2, which is frequently required in analyzing the
gravitational lens effects. Here we only consider the case of flat cosmologies. For the
distance formulae in non-flat cases, see, e.g., Ref. 8). For the standard (filled beam)
case, the angular diameter distance from redshift z = z1to z = z2is
dA(z1,z2) =
c
H0
1
1 + z2
?
1
1+z1
1
1+z2
da
?Ωma + (1 − Ωm)a4. (4.1)
Our analytical approximation is simply
dA(z1,z2) =2c
H0
1
√Ωm(1 + z2)
?
1
√1 + z1Φ(x(z1,Ωm)) −
1
√1 + z2Φ(x(z2,Ωm))
?
(4.2)
,
where Φ(x) is defined in Eq. (1.2). The relative error of our formula Eq. (4.2) for
Ωm= 0.3 is less than 0.02 per cent in the range 0 ≤ z1< 1 for z2= 1, and less than
0.01 per cent in the range 0 ≤ z1< 3 for z2= 3.
For the empty beam case,
DA(z1,z2) =
c
H0(1 + z1)
?
1
1+z1
1
1+z2
a2da
?Ωma + (1 − Ωm)a4, (4.3)
and our approximate formula is
DA(z1,z2) =2
5
c
H0
(1 + z1)
√Ωm
?
1
(1 + z1)
5
2Ψ(x(z1,Ωm)) −
1
(1 + z2)
5
2Ψ(x(z2,Ωm))
?
(4.4)
,
where Ψ(x) is already defined in Eq. (3.3).
The reciprocity theorem holds between the angular diameter and luminosity
distances as follows: dL(z) = (1 + z)2dA(0,z), and DL(z) = (1 + z)2DA(0,z).
§5. Summary
We have presented a simple analytical approximation formula for the luminosity
distance in flat cosmologies with a cosmological constant. We have also shown the
approximate formulae for the Dyer-Roeder distance and the generalized angular
diameter distance from redshift z = z1 to z = z2, which are particularly useful
in analyzing the gravitational lens effects. Apart from the overall factor 1/√Ωm,
the effects of non-zero cosmological constant in our distance formulae are written
simply in terms of the rational functions Φ(x) for “filled beam case” and Ψ(x) for
“empty beam case”. Both are monotonically decreasing functions with respect to x,
and increasing ones with respect to redshift z and the density parameter Ωm.
Our formulae are widely applicable over the range of the density parameter
and the redshift with sufficiently small uncertainties. In particular, in the range