Effect of the cosmological constant on the bending of light and the cosmological lens equation

0.49 MB·English

by  Hideyoshi Arakida

Effect of the cosmological constant on the bending of light
and the cosmological lens equation
Hideyoshi ARAKIDA∗
Graduate School of Education, Iwate University, Morioka, Iwate 020-8550,Japan
Masumi KASAI†
Graduate School of Science and Technology, Hirosaki University, Hirosaki, Aomori 036-8561 Japan
(Dated: January 20, 2012)
WerevisittheeffectofcosmologicalconstantΛonthelightdeflectionanditsroleinthecosmolog-
ical lens equation. First, we re-examine the motion of photon in the Schwarzschild spacetime, and
explicitlydescribethetrajectoryofphotonanddeflectionangleαuptothesecond-orderinG. Then
thediscussionisextendedtothecontributionofthecosmologicalconstantΛintheSchwarzschild-de
2 SitterorKottlerspacetime. Contrarytothepreviousarguments,weemphasizethefollowingpoints:
1 (a) the cosmological constant Λ does appear in the orbital equation of light, (b) nevertheless the
0 bendingangle of light α does not change its form even if Λ6=0 since thecontribution of Λ is thor-
2 oughly absorbed into the definition of the impact parameter, and (c) the effect of Λ is completely
n involvedin theangular diameter distance DA.
a
J PACSnumbers: 95.30.Sf,98.62.Sb,98.80.Es

9
1
I. INTRODUCTION M or not and so on. In order to clear up the confusion,

we will revisit the effect of the cosmological constant on
]
c the light deflection and its role in the cosmological lens
Nowadays, it is widely regarded that the cosmological
q
equation.
- constant Λ or more generally dark energy is the most
r responsiblecandidatewhichexplainsacceleratingexpan-
g
[ sion of Universe. Nonetheless the details of cosmological
II. PHOTON TRAJECTORY IN
4 constantΛ or darkenergyarestillfar fromclear,then it SCHWARZSCHILD SPACETIME
ispreferableandworthytoclarifythe validityofthishy-
v
pothesis by means of not only cosmologicalobservations
5 Before discussing the influence of Λ on bending angle
but also another astronomical/astrophysicalones.
3
α,weshallbeginwithre-consideringthesolutionofpho-
7 Among such attempts, it would be the most natural
ton trajectory in the Schwarzschildspacetime. From the
6 idea to investigate the role ofcosmologicalconstantΛ in
Schwarzschildmetric in the Schwarzschildcoordinates,
0. theclassicaltestsofgeneralrelativity,e.g. theperihelion
1 advanceofplanetaryorbitandthebendingoflightpath. 2 rg 2 2 rg −1 2
ds =− 1− c dt + 1− dr
1 So far, it was shown that the cosmological constant Λ r r
1 causestheperihelionshiftofplanetsatleastinprinciple, (cid:16) (cid:17) (cid:16) (cid:17)2GM
2 2 2 2
: +r (dθ +sin θdφ ), r = , (1)
v even though its contribution is too small to detect in g c2
i the current measurement technique (see [1–3] and the
X references therein). and the condition for null geodesic ds2 =0, we have the
r geodesic equation for equatorial plane (θ =π/2),
While it has been believedfor a long time that Λ does
a
not contribute to the light deflection because there is du 2 1 1 1 E2
2 3
no Λ in the second-order ordinary differential equation = −u +r u , u≡ , ≡ , (2)
dφ b2 g r b2 c2L2
(ODE) of photon. However recently, Rindler and Ishak (cid:18) (cid:19)
[4] pointed out that Λ does affect the bending angle by in which the two constants of motion, E and L are total
using the Schwarzschild-de Sitter or Kottler metric and energy and angular momentum, respectively. Alterna-
the invariant formula of cosine. Subsequently many au- tively, Eq. (2) can be expressed in the form of second-
thors arguedits appearancein diversewaysandgeneral- order ODE as,
ityassistedthefactthatthereappearsΛinthedeflection
d2u 3
angle α, see [5] for review and the references therein and 2
=−u+ r u , (3)
also [6–12]. However, it seems that the conclusion has dφ2 2 g
not converged yet; for instance whether the leading or-
nevertheless,hereinafterweuseEq.(2)insteadofEq.(3).
der effect of Λ is coupled with the mass of central body
In order to obtain the photon trajectory up to the
second-order in G, let us put the solution of Eq. (2) u
as,
∗E-mail:[email protected] 1 2
†E-mail:[email protected] u= b sinφ+rgu1+rgu2 , (4)
(cid:0) (cid:1)
2
where u1 and u2 are, respectively, first O(G) and sec- In the same way as the Schwarzschild case, the differen-
ondO(G2)ordercorrectiontozeroth-ordersolutionu0 = tial equation of light is given by
sinφ/b (straight line). Hence u1 and u2 satisfy the fol-
2
lowing differential equations, du 1 2 3 Λ
= −u +r u + . (10)
dφ b2 g 3
3 (cid:18) (cid:19)
du1 sinφ 1 sin φ
= − u1+ , (5) It should be emphasized here that the geodesic equation
dφ cosφ 2b cosφ
of light Eq. (10) does obviously include Λ. Therefore,
du2 = −sinφu2 (6) previous arguments, such as “Λ does not appear in the
dφ cosφ geodesic equation of light”, would be overstated. Actu-
1 du1 2 2 3 2 ally,thesecond-orderODEderivedfromEq.(10)reduces
− +u1− u1sin φ . to Eq. (3), nevertheless its solution of light trajectory
2cosφ dφ b
"(cid:18) (cid:19) # should be obtained in such a way that the integration
constants satisfy Eq. (10).
Noting thatthe integrationconstantsofEqs.(5) and(6)
Furthermore, the impact parameter is the minimum
are chosen such that maximizing u (or minimizing r) for
valueofthecoordinaterifthelightraywereundeflected,
φ = π/2, then we obtain the trajectory of photon up to
i.e., r = 0. It is obvious from Eq. (10) that the impact
g
the second-order in G as,
parameter B in this case is defined by
1 1 rg 1 1 Λ
r = b sinφ+ 4b2 (3+cos2φ) (7) B2 ≡ b2 + 3. (11)
r2
+ g 37sinφ+30(π−2φ)cosφ−3sin3φ , Then, the form of Eq. (10) completely coincides with
64b3
Eq. (2), except that the impact parameter b is replaced
(cid:0) (cid:1)
by B. Therefore, the solution of Eq. (10) becomes,
where b is the impact parameter which represents the
minimum value of r-coordinate for the undeflected light
1 1 r
g
ray, i.e., rg = 0. The bending angle α is shortly derived r = B sinφ+ 4B2 (3+cos2φ) (12)
fromEq.(7)anditcoincideswiththefamousformulaby r2
[13], + g 37sinφ+30(π−2φ)cosφ−3sin3φ ,
64B3
r 15π r 2 (cid:0) (cid:1)
α = 2 g + g and deflection angle is,
b 16 b
4GM 15(cid:16)π(G(cid:17)M)2 3 α = 2rg + 15π rg 2
= + +O(G ). (8)
c2b 4 c4b2 B 16 B
4GM 15(cid:16)π(G(cid:17)M)2
3
= + +O(G ). (13)
AsimplederivationofαisgiveninAppendixB.Itshould c2B 4 c4B2
be mentioned about the validity of the solution for light
It is worthy to note that the contribution of Λ is incor-
trajectory. The appropriateness of our solution, Eq. (7)
porated in Eqs. (12) and (13) through Eq. (11). As a
can be verified readily by the direct substitution into
consequence, it is found that the cosmological constant
Eq. (2) and it is found that the residual terms are or-
der O(G3), then it is perfectly valid up to the order of Λ does appears in both the geodesic equationand its so-
G2. However, the photon trajectory given in previous lution, that is the trajectory of photon. However, the
effect of Λ is completely absorbed into the definition of
works such as Eq. (18) of [14] and Eq. (16) of [5] are in-
correct;infact, thereappearsO(G2)orderresidualterm the the impact parameter (see Eq.(11)). Hence it is dif-
ficult to distinguish the influence of Λ fromthe observed
in the solution of photon trajectory in [5, 14].
deflection angle.
When we expand Eq. (13) by using 1/B =
(1/b) 1+Λb2/3 ≃ (1/b)(1 + Λb2/6) and remain
III. CONTRIBUTION OF THE
O(M,MΛ) terms, it follows that
COSMOLOGICAL CONSTANT p
4GM 2GMbΛ
α≃ + , (14)
Now, let us investigate the contribution of Λ on light c2b 3c2
ray. For this purpose, we adopt the Schwarzschild-de
inwhichsecondtermcoincideswiththepreviousresults,
Sitter or Kottler metric [15],
e.g. Eq. (5) and below in [16] and the third term of
Eq. (15) in [10]. Hence it is found that these results are
r Λ
2 g 2 2 2
ds = − 1− − r c dt (9) included in Eq. (13) as a limiting case.
r 3
(cid:18) (cid:19) ItisclearthatthetrajectoryofphotonEq.(12)strictly
−1
rg Λ 2 2 2 2 2 2 satisfies Eq. (10) up to the second-order in G based on
+ 1− − r dr +r (dθ +sin θdφ ).
r 3 the result in previous section.
(cid:18) (cid:19)
3
IV. COSMOLOGICAL LENS EQUATION absorbedintothedefinitionoftheimpactparameter,and
(c) the effect of Λ is completely involved in the angular
Finally, we consider the contribution of Λ in the cos- diameter distance DA. Then, no modifications due to Λ
mological lens equation. Under the assumption that the appear even if the second-orderterms in G are included.
thin lens approximation is valid, the lens equation re- It should be mentioned that the similar conclusions
latestheobservedimagepositionangleθ totheunlensed are reached by other authors [6–9]. We hope that this
position angle β of the source as paper provides a simpler and clearer explanation of the
roleofΛinthebendingoflightandthecosmologicallens
DA(zL,zS) equation than the above-mentioned studies.
β =θ− α, (15)
D (0,z )
A S
where DA(z1,z2) denotes the angular diameter distance Appendix A: The angle between the radial direction
from the redshift z1 to z2, and the arguments zL and and the light trajectory
z are the redshift of the lens and the source, respec-
S
tively. For the distance formula D in the unper-
A
turbed Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) Rindler and Ishak [4] have derived the angle between
universe with Λ, see, e.g., [17]. the radial direction and the light trajectory from the in-
Up to the first order in G, the bending angle α in the variantcosine formula. Justfor a reference,here we give
present case is a more intuitive derivation. Consider a spherically sym-
metricspace. Themetricontheequatorialplaneθ =π/2
4GM is given by
α= . (16)
c2B
dr2
2 2 2
TheimpactparameterB isrelatedto theimageposition dℓ = +r dφ . (A1)
f(r)
angle θ by
B =D (0,z )θ. (17) In this space, the infinitesimal proper lengths along the
A L
radial direction and that perpendicular to the radial di-
Then, from Eqs. (15), (16), and (17), the lens equation rection are given by f(r)−1/2|dr| and r|dφ|, respectively.
is finally Then the tangent of the angle ψ between the radial di-
rection and the light trajectory is
4GMD (z ,z ) 1
A L S
β =θ− . (18)
c2DA(0,zL)DA(0,zS)θ r|dφ| dr −1
tanψ = = f(r)r , (A2)
f(r)−1/2|dr| dφ
Therefore, the contribution of Λ is completely involved (cid:12) (cid:12)
p (cid:12) (cid:12)
in the form of the angular diameter distance DA. No (cid:12) (cid:12)
modifications due to Λ appearevenif the term ofO(G2) whichisthesameformasEq.(16)in[4(cid:12)]. O(cid:12)nceweobtain
the solution r(φ) for the light trajectory, Eq. (A2) gives
in α is included. It should also be noted that Eq. (18)
the angle at any point (r(φ),φ)) on the trajectory.
is exactly the same form appeared in [17, 18], where the
authorshave shownthat the gravitationallensing effects
are strongly dependent on the value of the cosmological
constantandhencethey provideus withusefulmeansto
test the cosmologicalconstant.
Here,wementionthatinpaper[16],Serenointroduced
therelation,b0 =Ddϑ(seeEq.(8)belowof[16])whereb0
is the impact factor of “Schwarzschild lens” (see Eq. (8)
of[16]),D istheangular-diameterdistance,andϑisthe
d
angularseparation. Sinceinthecaseofcosmologicallens,
the cosmological distance, such as the angular diameter
distance, is defined with Λ, then b0 should be replaced
by B, instead.
V. SUMMARY
Webrieflysummarizeourconclusions. Contrarytothe FIG.1: Light trajectories on Schwarzschild geometry and on
previousarguments,(a)thecosmologicalconstantΛdoes the flat space which is tangent to the asymptotically flat re-
gion of the Schwarzschild geometry. The coordinates of the
appear in the orbital equation of light, (b) nevertheless
point P is (r(φ),φ))=(r(0),0).
the bending angle of light α does not change its form
even if Λ 6= 0 since the contribution of Λ is thoroughly
4
Appendix B: The bending angle in Schwarzschild Appendix C: The angle on Schwarzschild-de Sitter
geometry geometry
Using Eqs. (1), (7) and (A2), the angle ψ at the point Using Eqs. (9), (12), and (A2), the angle ψ between
P(r(0),0) is the radial direction and the light trajectory at the point
P′(r(0),0) on the Schwarzschild-de Sitter geometry is
dr −1 r 15π r 2 (see Fig. 3)
g g
ψ ≃ f(r(0))r(0) ≃ + . (B1)
dφ b 32 b
p (cid:12)(cid:12) (cid:12)(cid:12)φ=0 (cid:16) (cid:17) Λ b2 2 r 15π r 2
(cid:12) (cid:12) ψ ≃ 1− g + g . (C1)
Ctootnicsaidlleyrflaatflraetgisopnaco(cid:12)ef wth(cid:12)heicShchiswatraznsgcehniltdtgoeotmheetarsyym(sepe- s 3 (cid:18)rg(cid:19) (cid:18) b 32 (cid:16) b (cid:17) (cid:19)
Fig. 1). It is apparent from the property of an isosceles When linearized with respect to Λ,
triangle in the Euclidean geometry that the angle ψ at
the point P is a half of the bending angle α (see Fig. 2). r 15π r 2 Λb3 15π
g g 2
ψ ≃ + − − Λb , (C2)
Then, b 32 b 6 r 192
g
(cid:16) (cid:17)
r 15π r 2
α=2ψ ≃2 g + g . (B2) which is essentially a half of Eq. (18) in [5], although
b 16 b theydropthelastterminEq.(C2). Itshouldbeempha-
(cid:16) (cid:17)
sized since the Euclidean property does not hold on the
Schwarzschild-deSittergeometry,theangleψinEq.(C2)
is not a half of the bending angle α.
FIG. 2: Light trajectory on the flat space with the metric
dℓ2=dr2+r2dφ2. Theimpact parameterb and thebending
angle αare definedon this space.
FIG. 3: The angle ψ between the radial direction and the
light trajectory at P′ on Schwarzschild-deSitter geometry.
5
[1] A. W. Kerr, J. C. Hauck, B. Mashhoon, Class. Quant. 063003 (2010).
Grav., 20, 2727 (2003). [11] H.Miraghaei,M.Nouri-Zonoz,Gen.Rel.Grav.42,2947
[2] W.Rindler,Relativity: Special,General,andCosmolog- (2010)
ical (2nd Ed.), Oxford Univ.Press, New York (2006). [12] T.Biressa,J.A.deFreitasPacheco,Gen.Rel.Grav.,43,
[3] L. Iorio, Adv.Astron., id. 268647 (2008). 2649 (2011).
[4] W. Rindler, M. Ishak, Phys. Rev. D, 76, id. 043006 [13] R.Epstein,I.I.Shapiro, Phys.Rev.D,22,2947 (1980).
(2007). [14] J. Bodenner, C. M. Will, Am.J. Phys., 71, 770 (2003).
[5] M. Ishak,W.Rindler, Gen.Rel. Grav., 42, 2247 (2010). [15] F. Kottler, Annalen.Phys., 361, 401 (1918).
[6] K.Lake, Phys. Rev.D.65, id. 087301 (2002). [16] M. Sereno, Phys.Rev.Lett., 102, id. 021301 (2009).
[7] M. Park, Phys. Rev.D,78, id. 023014 (2008). [17] M. Fukugita, T. Futamase, M. Kasai, E. L. Turner, As-
[8] I.B. Khriplovich,A. A. Pomeransky,Int.J. Mod. Phys. trophys. J., 393, 3 (1992).
D,17, 2255 (2008). [18] M.Fukugita,T.Futamase,M.Kasai,Mon.Not.R.Astr.
[9] F.Simpson,J.A.Peacock,A.F.Heavens,MNRAS,402, Soc., 246, 24 (1990).
2009 (2010).
[10] A. Bhadra, S. Biswas, K. Sarkar, Phys. Rev. D, 82, id.