Quasi normal modes in Schwarzschild-DeSitter spacetime: A simple derivation of the level spacing of the frequencies

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by  T. Roy Choudhury

Quasi normal modes in Schwarzschild-DeSitter spacetime: A simple derivation of the
level spacing of the frequencies
T. Roy Choudhury
∗
SISSA/ISAS, via Beirut 2-4, 34014 Trieste, Italy
T. Padmanabhan
IUCAA, Ganeshkhind, Pune 411 007, India
(Dated: February 7, 2008)
It is known that the imaginary parts of the quasi normal mode (QNM) frequencies for the
Schwarzschildblackholeareevenlyspacedwithaspacingthatdependsonlyonthesurfacegravity.
On the other hand, for massless minimally coupled scalar fields, there exist no QNMs in the pure
DeSitterspacetime. ItisnotclearwhatthestructureoftheQNMswouldbefortheSchwarzschild-
DeSitter (SDS) spacetime, which is characterized by two different surface gravities. We provide a
4 simple derivation of the imaginary parts of the QNM frequencies for the SDS spacetime by calcu-
0 lating the scattering amplitudein the first Born approximation and determining its poles. We find
0 that, for the usual set of boundary conditions in which the incident wave is scattered off the black
2 holehorizon,theimaginary partsoftheQNMfrequencieshaveaequallyspaced structurewith the
n levelspacingdependingonthesurfacegravityoftheblackhole. Severalconceptualissuesrelatedto
a theQNMare discussed in thelight of this result and comparison with previouswork is presented.
J
PACSnumbers: 04.70.-s,04.30.-w,04.62.+v
9
1

I. INTRODUCTION The numerical studies were followed by a series of an-
2
v alyticalcalculationswhichweremainlybaseduponsome
4 It is well knownthat the quasi normalmodes (QNMs, approximationscheme, like for example, (i) approximat-
6 hereafter) are crucial in studying the gravitational and ing the scattering potential with some simple functions
0 electromagnetic perturbations around black hole space- sothattheproblembecomesexactlysolvable[24,25,26],
1
times (for comprehensive reviews and exhaustive list of (ii)usingWKB-liketechniques[27,28,29,30,31,32,33,
1
3 references see [1, 2]). The QNMs also seem to have an 34, 35, 36, 37, 38], (iii) Born approximation [39, 40] and
0 observationalsignificanceasthegravitationalwavespro- (iv) others [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
/ duced by the perturbations can, in principle, be used for 53, 54]. Nearly all of these schemes came with their own
c
q unambiguous detection of black holes. The early stud- limitations and rarely reproduced the exact form of the
- ies of the QNMs concentrated on numerical computa- QNM spectrum as given in equation (1) – in particular,
r tions (see, for example, [3, 4]) as analytical solutions of therewashardlyanyexplanationforthe ln3term. Ana-
g
: the perturbation equations were difficult to obtain. Es- lyticalproofsofequation(1)wereprovidedonlyrecently,
v sentially, one solved the perturbation equations numeri- using the methods of continued fractions [5, 12, 55, 56]
i
X cally for different initial conditions in a given black hole andalsobystudyingthemonodromyoftheperturbation
r spacetime. It was found that the results were, in gen- continued to the complex plane [57, 58].
a eral, independent of the initial conditions and depended Whilemostoftheanalyticalworkconcentratedonthe
mostly on the parameters characterizing the black hole realpart of QNM (due to the interest in the ln3 factor),
horizon. For example, for the Schwarzschild black hole, the structure of the imaginary part is equally intrigu-
the spectrum of the QNMs was found to have the struc- ing and requires physical understanding. In particular,
ture [5, 6, 7, 8, 9, 10] we need to understand why the imaginary part of k
n
in equation (1) has a simple, equally spaced structure,
1 ln3
k =iκ n+ + κ+O[n−1/2] (1) withκdeterminingthelevelspacing. Physicalquantities
n
2 2π with a quantized spectrum are always of interest when
(cid:18) (cid:19)
they have constant spacing. In the case of horizon area,
which depends only on the surface gravity of the black
forexample,onecanattempttorelateanequallyspaced
hole κ. Similarly, for charged (Reissner-Nordstr¨om) and
spectrum(obtainedinseveralinvestigations,e.g.,see[59]
rotating (Kerr) black holes, the spectrum was found to
andreferencestherein)totheintrinsiclimitationsinmea-
depend only on the parameters characterizing the hori-
suringlengthscalessmallerthanPlancklength[60]. But
zon,i.e.,the mass,chargeandangularmomentumofthe
theuniformspacingofQNMfrequenciesisapurelyclas-
black hole [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
sical resultandhence isharderto understandphysically.
23].
Itturnsout,however,thatonecanmakesomeprogress
in this direction. There are some simple derivations,
based on the Born approximation, which reproduce this
∗[email protected];http://www.sissa.it/~chou structure for the imaginary part of the QNM frequen-
2
cies for a large class of spherically symmetric spacetimes as kn = i(n−κ− +n+κ+) , where n± are two integers.
like the Schwarzschild black hole [39, 40]. The relation [This arXiv submission has been revised subsequently,
obtained,validforlargek (i.e.,forlargen),hasthe form after the appearance of the current work, removing this
claim]. Analytical calculation based on approximating
kn =inκ (for n≫1) (2) thescatteringpotentialbysomesimpleformgivesQNMs
proportional to both the surface gravities, depending on
(In the imaginary part of k , one obtains a factor n in-
n the time scale one is interested in [75]. Currently, we do
stead of n+(1/2) since the Born approximation is ex-
notseemtohaveanygeneralconsensusonthisparticular
pected to be valid only for n≫1). The important point
issue!
to note is that — though it is not possible to obtain the
There are a few more conceptual issues because of
realpartoftheQNMfrequenciesusingtheBornapprox-
which such a study (using a prototype metric with two
imation — this simple derivation does give the correct
horizons) is important. First is the possible connection
levelspacingforimaginaryvaluesofk . Hencethepossi-
n between the QNMs and thermodynamics of horizons. It
bilityofextendingtheformalismtospacetimeswithmore
was pointed out in [40] that the Born approximation re-
complicatedstructureisworthexamining. Followingthis
sult arises from integrals which are very similar to those
approach,weusetheBornapproximationtoexaminethe
which arise in the case of horizon thermodynamics (see
level spacing for spacetimes with two horizons.
e.g., equation (23) of [77]). In the case of spacetimes
The simplest spacetime with two horizons is that of a
withmultiplehorizons(likeSDS),thereisnouniquetem-
black hole in a spacetime with a cosmological constant,
perature except when the ratio of surface gravities is a
described by the Schwarzschild-DeSitter (SDS) metric.
rational number [78]. If the QNMs are related to the
The metric is characterized by the presence of a black
horizon thermodynamics (in some manner which is not
hole event horizon and a cosmological horizon. In re-
yet clearly understood), then it would be interesting to
cent times, studying such a spacetime has acquired fur-
see whether the level spacing in the SDS spacetime can
thersignificancebecauseofthecosmologicalobservations
give any idea about the temperature of spacetimes with
suggestingthe existenceofanon-zeropositive cosmolog-
multiple horizons. Second, it is wellknown that thermo-
ical constant ([61]; for reviews, see [62, 63]). While the
dynamics of gravitating systems depend crucially on the
observations can be explained by a wide class of models
ensemble which is used (see e.g., [79]) which translates
(see, e.g [64]), including those in which the cosmic equa-
into the boundary conditions on the horizon (see e.g.,
tionofstatecandependonspatialscale[65,66],virtually
[80]). If QNMs are related to horizon thermodynamics,
allthesemodelsapproachtheDeSitter(DS)spacetimeat
we will expect some similar kind of dependence on the
latetimesandatlargescales. Likeintheothercases,the
boundary condition for the wave modes in case of SDS
QNMsfortheSDSspacetimehavebeenstudiedbothnu-
spacetime. We shall see that this expectation is indeed
merically (see, for example, [67, 68, 69, 70, 71, 72]) and
borneout. Finally,therewerealsosomediscussioninthe
analytically (see, for example, [73, 74, 75]). However,
literature asto whether the QNMs depend on the region
since the SDS spacetime is characterized by two differ-
beyondthe horizons. Wewillseethatwecanmakesome
entsurfacegravities(correspondingtothetwohorizons),
comments regarding this issue.
thedependenceofthelevelspacingoftheimaginarypart
The structure of the paper is asfollows. InSec. II, we
of the QNM frequencies on the surface gravities is not
briefly review the results for the Schwarzschild metric.
obvious. The numerical studies fail to give a “clean”
The main problem of interest, the SDS metric, is taken
result like equation (1) as they seem to vary depend-
upinSec. III.We derivethe explicitformofthe scatter-
ing on the relative values of the two surface gravities.
ingamplitude using Bornapproximationanddiscussthe
A calculation based on the monodromy of the pertur-
structure of the QNM spectrum. The main conclusions
bation continued to the complex plane [76], similar to
are summarized and comparedwith other results in Sec.
what is done for the Schwarzschild case, gives a result
IV.
of the form kn = iκ− n+ 21 +real part, where κ− is
the surface gravity of the black hole horizon. The real
(cid:0) (cid:1)
part has a term analogous to ln3 whose actual form de-
II. WARM UP: QNMS FOR THE
pends on the values of the surface gravities. The above SCHWARZSCHILD METRIC
result is found to be in qualitative agreement with nu-
merical results for the near-extremalcase (i.e., when the
In this section we review the derivation of the QNMs
surface gravities are very close to each other), although
forthe SchwarzschildmetricusingthefirstBornapprox-
the exact numerical coefficients seem to differ [72]. It
imation. Let us start with a general class of spherically
is not clear whether the problem is with the numerics,
symmetric metrics of the form
or the analytical derivation misses out on some issues.
The abovecalculationindicates thatthe QNMspectrum ds2 =f(r)dt2−[f(r)]−1dr2−r2dΩ2 (3)
is independent of the surface gravity of the cosmological
horizon κ . On the other hand the first version of [39], with f(r) having the simple zero at r = r , i.e., f(r) ≃
+ 0
′
based on Born approximation, claimed that the spec- f (r )(r − r ). It was shown in [81] that spacetimes
0 0
trum of the imaginary part of the QNMs should behave described by the above class of metrics have a fairly
3
straightforward thermodynamic interpretation and – in purelyoutgoingwaveatspatialinfinity,i.e.,whichsatisfy
fact – Einstein’s equations can be expressed in the form the boundary conditions
of a thermodynamic relation TdS =dE−PdV for such
sthpeacseutrimfaecse,gwraitvhitythoef ttehmepheorraiztounre: being determined by F(r)∼ ee−ikirk∗r∗ ((aatt rr∗∗ →→−∞∞) ), (11)
(cid:26)
1 It is clear from the equation(9) that the QNMs actually
′
κ= 2|f (r0)| (4) correspond to case where Ain = 0 and are obtained by
calculatingthepolesofthescatteringamplitudeS(k)[see
Let us consider a massless scalar field φ satisfying the equation (10)]. Obtaining the exact form of the scatter-
wave equation (cid:3)φ = 0 in this spacetime. We look for ing amplitude for the potential (7) is non-trivial – hence
solutions to the wave equation in the form onehastoobtainitthroughsomeapproximationscheme.
It turns out that one canobtain the explicit form of this
1
φ= F(r)Y (Ω)eikt; Re(k)>0 (5) amplitude using the first Born approximation.
lm
r In general, the scattering amplitude in the Born ap-
proximation is given by the Fourier transform of the
Straightforward algebra now leads to a “Schrodinger
potential V(x) with respect to the momentum transfer
equation” for F given by:
q=k −k :
f i
d2
−dr2 +V(r) F(r)=k2F(r) (6) S(q)= dx V(x) e−iq·x (12)
(cid:20) ∗ (cid:21)
Z
where the potential is given by In one dimension, k and k should be parallel or anti-
i f
parallel; further we can take their magnitudes to be the
l(l+1) f′(r)
sameforscatteringinafixedpotential. Thennon-trivial
V(r)=f(r) + (7)
r2 r momentum transfer occurs only for k = −k so that
(cid:20) (cid:21) f i
q = −2k . From equation (9), the “incident” wave is
i
and the tortoise coordinate is defined as seen to be of the form eikr∗, giving the the scattering
dr amplitude as
r∗ ≡ f(r) (8) ∞
Z S(k) = dr∗V[r(r∗)]e2ikr∗
One can, in principle, solve the differential equation (6) Z−∞
given a particular set of boundary conditions. However, = ∞dr l(l+1) + f′(r) e2ikr∗(r) (13)
the equation, in general,does not havean exactsolution r2 r
Zr0 (cid:20) (cid:21)
and one has to solve it either numerically or by using
where we have omitted irrelevant constant factors. (Our
some approximation scheme. In this paper, we shall be
original problem was three-dimensional and we are not
using one such approximation scheme, namely, the first
working out the three-dimensional scattering amplitude
Born approximation.
in,say,s-wavelimit. Rather,wefirstmaptheproblemto
For a pure Schwarzschild black hole, we have f(r) =
an one-dimensional Schrodinger equation and study the
1−2M/r and the horizon is at r =2M. The potential
0
scattering amplitude in one dimension). This integral
V(r) in equation (7) vanishes at the horizon(r∗ →−∞)
picks up significant contribution only near the horizon
awnadveafutnscptaiotinalcainnfibnietyta(kre∗n→to∞be),pwlahniechwmaveeasnisntthhaetttwhoe where r∗ ≈ (1/2)κ−1ln(r/r0 −1), and it can be shown
thatthe approximateformofthe scatteringamplitude is
regions. Aclassofphysicallyacceptablesolutionsforthis
given by [40]
system has the asymptotic form
k
eikr∗ (at r∗ →−∞), S(k)≈constant factors×Γ 1+i (14)
F(r)∼ Aineikr∗ +Aoute−ikr∗ (at r∗ →∞). (9) (cid:18) κ(cid:19)
(cid:26)
Itisclearfromtheaboveexpressionthatthepolesofthe
The “incident” wave (Aineikr∗) for this class of solutions amplitude is given by the poles of the Gamma function,
is propagatingtowards the black hole, and hence the so- which occur at
lutions of the above form are appropriate for studying
scattering off the black hole. The scattering amplitude kn =inκ (for n≫1) (15)
for the above solution is simply given by
It also turns out that the integral (13) can be solved ex-
A actlyforapureSchwarzschildmetric[40],andtheimagi-
out
S(k)∝ (10)
narypartoftheQNMfrequenciesarefoundtobeexactly
A
in
identicaltowhatis obtainedabove. As discussedin Sec-
Now, the QNMs are defined to be those for which one tion1,theBornapproximationfailstoreproducethereal
has a purely in-going plane wave at the horizon and a part of the QNM spectrum.
4
As an aside, we would like to comment on the issue of the origin, i.e.,
whether QNMs depend on the form of the metric inside
the horizon and in particular on the singularity of the 0 (at r →0),
Schwarzschild metric at r =0. (These comments do not F(r)∼ e−ikr∗ (at r →H−1) (17)
(cid:26)
depend on the Born approximation but it is easy to see
Itturnsoutthatthewaveequationcanbesolvedexactly
theresultinthislimit). Theanswerisessentially“no”in
for the DS spacetime [82, 83], and for scalar fields sat-
the sense that if the Schwarzschild metric is modified in
isfying the wave equation (cid:3)φ = 0, there exist no modes
a small region around r =0, making it nonsingular, but
forwhichtheaboveboundaryconditionissatisfied. This
leaving the form of the metric unchanged for r ≥ 2M,
result is implicit in [68] and we include the details of the
the QNMsdonotchange;butthereis subtletyinthis is-
calculation in Appendix A for completeness.
sue. Itiseasytoseethat,intheBornapproximation,we
aredealingwithscatteringprobleminther∗ coordinates
with boundary conditions at r∗ = ±∞. This scattering
III. QNMS FOR THE
problem only depends on V[r(r∗)] which – in turn – de-
SCHWARZSCHILD-DESITTER METRIC
pends onlyonf(r)forr ≥2M. Soifwemodify thef(r)
for r < 2M, it does not change the Born approximation
results. What happens when we go beyond the Born We next consider the Schwarzschild-DeSitter (SDS)
approximation and consider the real part of QNMs, for spacetime,whichisdescribedbyasphericallysymmetric
example? Here we need to analytically continue to the metric of the form (3), with
complex values of r and r∗. Now the original definition
2M
of the problem, posed as a Schrodinger equation in (6), f(r)=1− −H2r2 (18)
r
againcaresonlyfor ther∗ coordinateandis well-defined
if boundary conditions are specified at r∗ = ±∞. But Let us denote the black hole event horizon and the cos-
the relationbetween r andr∗ [which againdepends only mological horizon by r− and r+ respectively. The cor-
on f(r) at r ≥ 2M] can be analytically continued for responding surface gravities are denoted as κ− and κ+
all r including near r = 0. This leads to a unique ana- respectively. Note that, by definition, both κ− and κ+
lyticstructureinthecomplexplaneandevenforr <2M are positive definite. The tortoise coordinate is given by
throughanalyticcontinuation,asthoughtheformoff(r)
isvalidallthewaytor =0! Moreexplicitly,thisanalyti- 1 r 1 r
calstructureisobtainedby: (i)definingf(r)forr ≥2M; r∗ = 2κ− ln(cid:12)r− −1(cid:12)− 2κ+ ln(cid:12)1− r+(cid:12)
(ii) defining r∗ using it; (iii) analytically continuing this 1 1 (cid:12) 1 (cid:12) r (cid:12) (cid:12)
relationto define r∗ and r in the complex plane and (iv) − (cid:12)(cid:12)− (cid:12)(cid:12)ln (cid:12)(cid:12) +1 (cid:12)(cid:12)
defining f(r) for all r including near r = 0 by analytic 2(cid:18)κ− κ+(cid:19) (cid:12)r−+r+ (cid:12)
continuation. This is independent of the actual form of (cid:12)(cid:12) (cid:12)(cid:12) (19)
(cid:12) (cid:12)
f(r)forr <2M andpreviousresultslikethosein[57,58]
dependonlyonthisstructure. Thisisgratifyingsincewe With this definition, the regions r ≤ r− and r ≥ r+
do not know the effects of quantum gravity,which could are mapped to r∗ ≤ −∞ and r∗ ≥ ∞ respectively, and
we will not require the regions beyond the two horizons.
modify the spacetime structure near the singularity.
(In particular, the form of f(r) inside the Schwarzschild
Itmightseemthattheaboveformalismcanbetrivially
horizon and the singularity at r = 0 are irrelevant in
applied to the case of a spacetime with a cosmological
what follows.) The potential in equation (7) reduces to
horizon,describedbythepureDSmetricwithf(r)=1−
H2r2. The potential term for calculating the scattering l(l+1) 2M
amplitude, given by equation (7), becomes V(r)=f(r) + −2H2 (20)
r2 r3
(cid:20) (cid:21)
which, because of the f(r) factor, vanishes at both the
l(l+1)
V(r)=(1−H2r2) −2H2 (16) horizons. Thus one can take the boundary conditions to
r2
(cid:20) (cid:21) besimpleplanewavesatthetwohorizons. Theusualset
of solutions used in a scattering problem is identical to
which, like in the pure Schwarzschild case, vanishes at equation (9) which, as mentioned earlier, is appropriate
the horizon r = H−1. However, in contrast to the for studying scattering off the black hole. Let us assume
Schwarzschildcase,itdoesnotvanishattheotherbound- thattheboundaryconditionswhichdefinetheQNMsare
ary,i.e.,atr =0. Itiseasytocheckthatneartheorigin, still given by (11), which imply that one has purely in-
V(r) ≃ −2H2 when l = 0, while it blows up as r−2 for going plane waves at the black hole horizon (r = r−)
l > 0. This implies that one cannot take the bound- and purely outgoing waves at the cosmological horizon
aryconditionsassimpleplanewaveslikeinequation(9). (r = r ). One should realize that this set of boundary
+
In fact, the set of boundary conditions used for study- conditionsimplies thatthe wavesarepropagating“into”
ing QNMsin the pure DSspaceis thatthe wavefunction the horizons at both the boundaries and is probably the
should be outgoing at the horizon, but should vanish at most reasonable set of conditions to be used.
5
We now apply the first Born approximation, with the with the scattering amplitude being given by the sum
“incident”wavebeingtakenaseikr∗ asbefore. Thescat-
tering amplitude is then given by S(k)=2MI3+l(l+1)I2−2H2I0 (26)
∞
The expression for I turns out to be [84] an integral
S(k)= dr∗V(r)e2ikr∗ (21) representationof thenAppell hypergeometric function F
−∞ 1
Z
which can be simplified to
r+ l(l+1) 2M In = (r+−r−)1−ik(1/κ+−1/κ−)(r++r−)ik(1/κ−−1/κ+)
S(k) = dr r2 + r3 −2H2 × r−−ik/κ−r+ik/κ+(r++2r−)ik(1/κ+−1/κ−)
Zr− (cid:20) (cid:21)
× r −1 ik/κ− 1− r −ik/κ+ × Γ 1+ κik− Γ 1− κik+
(cid:18)r− (cid:19)r i(cid:18)k(1/κ+r−+1(cid:19)/κ−) Γ(cid:16)(cid:16)2+ik(cid:17)hκ1i−k(cid:16)− κ1+i(cid:17)(cid:17)1 1
× 1+ (22) × r−nF 1+ ,n,ik − ,
(cid:18) r++r−(cid:19) − 1(cid:18) κ− (cid:20)κ− κ+(cid:21)
Like in the pure Schwarzschild case, this integral will 1 1 r+−r− r+−r−
2+ik − ;− ,− (27)
pick up contribution only near the horizons. How- (cid:20)κ− κ+(cid:21) r− r++2r−(cid:19)
ever, there are some crucial differences which need to
be taken care of. Near the black hole horizon, we have This expression can be further simplified and written
the usual relation r∗ ≈ (1/2)κ−−1ln(r/r− −1), and the in terms of the usual hypergeometric function 2F1 for
contribution is exactly similar to equation (14). On n=0,2,3–wegivetherelevantexpressionsinAppendix
the other hand, near the cosmological horizon, we have B for completeness. The pole structure of In can, how-
r∗ ≈−(1/2)κ−+1ln(1−r/r+),whichdiffersfromtheother ever, be determined from equation (27) itself. The com-
case in the signof the surface gravityterm. The scatter- binations of the form
ing amplitude will then be a sum of two contributions
F α,n,β,2+ik 1 − 1 ;r ,r
given by 1 κ− κ+ 1 2 (28)
(cid:16) h i (cid:17)
k Γ 2+ik 1 − 1
S(k) ≈ constant factors×Γ 1+i κ− κ+
κ− (cid:16) h i(cid:17)
(cid:18) (cid:19) which occur in the expression for I , do not have any
k n
+ constant factors×Γ 1−i (23) poles. (Even though both the denominator and numer-
κ
(cid:18) +(cid:19) ator have poles, the ratio does not.) The only poles
of I occur at the poles of the two Gamma functions
In this case, the poles of the amplitude is given by the n
poles of both the Gamma functions: Γ 1+ ik and Γ 1− ik . It is then straightforward
κ− κ+
to(cid:16)see tha(cid:17)t the po(cid:16)les of th(cid:17)e scattering amplitude S(k)
kn =inκ−, kn =−inκ+ (n≫1) (24) aregivenbyequation(24)–exactlyidenticaltowhatwe
The factthatthe amplitude wouldpickupcontributions obtained by evaluating the integral near the horizons.
from both the horizons was pointed out earlier in [39], Let us now discuss the QNM spectrum as obtained in
and — in the first version of [39] that appeared in the equation (24). Note that the QNMs are identified as the
arXiv — it was suggested that the poles would occur positive imaginary values of k, which implies that the
at kn = i(n−κ− +n+κ+). However, we have shown by QNMs in this case are given by kn = inκ−; the modes
the explicit calculation above that this is not correct; whicharedependentonκ+ correspondtonegativeimag-
summinguptwocontributionstogetvalueofanintegral inary values of k and hence do not represent QNMs. [In
is not the same as adding the arguments for poles. [We general,thepolesgivenbynegativeimaginaryvaluesofk
note that [39] has since been revised and this particular correspondtoboundstatesofthesystem. However,since
claim has been withdrawn]. Further, there is a crucial the potential, for l > 0, (20) is positive everywhere and
sign difference in the surface gravity of the cosmological vanishes at the boundaries, it can be shown that there
horizon. cannotexist any bound states for the system [85]. Thus,
This conclusion can be explicitly verified since, for- the poles given by kn = −inκ+ are physically irrelevant
tunately, one can evaluate the integral in (22) exactly. as far as this problem is concerned.]
Essentially we have to solve integrals of the form The above analysis indicates that the QNMs obtained
through the first Born approximation are independent
r+ r ik/κ− r −ik/κ+ of the cosmological horizon. This conclusion agrees, in
I = dr r−n −1 1−
n the large n limit, with the imaginary part of the QNM
Zr− (cid:18)r− (cid:19) (cid:18) r+(cid:19) spectrum obtained through the monodromy of the per-
r ik(1/κ+−1/κ−) turbation continued to the complex plane [76]. In view
× 1+ (25)
(cid:18) r++r−(cid:19) of the fact that there exists no QNMs for the pure DS
6
spacetime, “adding” a black hole near the origin merely on both the surface gravities κ− and κ+ depending on
introduces the QNMs corresponding to the black hole. the time-scaleoneis interestedin. Since we arestudying
Interpreted in the above manner, this result should not a time-independent situation, it is difficult to comment
be surprising. It is also clear that this result gives the on time-dependence of the QNM spectrum – however,
two correct limits, i.e., when H → 0, the level spacing our analysis indicates that if one starts with an incident
reduces to that corresponding to a Schwarzschild black wavepacketwhich is composedof monochromaticwaves
hole, while for M → 0, we have κ− → ∞, and hence propagating in both directions (i.e., terms of the form
there exists no QNMs for the pure DS spacetime. eikr∗ and e−ikr∗), then one might obtain a QNM spec-
As an aside, one can also consider a different set of trumwhichdependsonboththesurfacegravities. There
boundary conditions, where the incident wave is scat- is one more crucial difference between our results and
teredoffthecosmologicalhorizon. Itturnsoutthatsuch those obtained by the Poschl-Teller potential [75]– the
conditions will give QNMs proportional to κ+ [the de- levelspacinginthelatercaseis2κ±ratherthanκ±. Such
tails of the calculation are given in Appendix C]. How- a difference was noted in the case of QNMs obtained by
ever, these boundary conditions may not be physically BornapproximationfortheSchwarzschildblackhole[40]
relevant and are considered here just as a mathematical while comparingwith resultsobtainedbyapproximating
possibility. the potential [43]; the difference is probably related to
the incorrect use of q = k rather than q = 2k for mo-
i i
mentum transfer in the Born approximation.
IV. DISCUSSION MostofthenumericalcomputationsregardingtheSDS
spacetime concentrate on the near extremal case (where
We have used the first Born approximation to obtain κ+ ≈ κ−), and it is found that the imaginary part of
the QNM frequencies have an equally spaced structure
the QNM spectrum for the SDS spacetime. The ap-
with the spacing given by either of the surface gravities
proximationgivesthe correctlevelspacingfortheimagi-
(which are anyway equal to the lowest order) [69, 70,
naryvaluesoftheQNMfrequenciesfortheSchwarzschild
72]. This means that, to the lowest order, we do not
blackhole,andthe spacingis relatedtothe temperature
findanydisagreementbetweenourresultsandnumerical
corresponding to the horizon. On the other hand, there
computations. In other numerical computations, where
exist no QNMs for a massless minimally coupled scalar
the values of the two surface gravities are taken to be
field in a pure DS spacetime. It turns out that the situ-
widely different [67, 68], one obtains two sets of QNM
ation is more complicated in the SDS spacetime and de-
spectra proportional to the two surface gravities, each
pends on the type of scattering one is interested in, i.e.,
validatdifferenttime-scales. Atthisstage,weareunable
onthetypeofboundaryconditionsoneimposes. Onecan
tomakeanydirectcomparisonwithsuchresultssincethe
start with the usual set of conditions where an incident
issue of time-scales is difficult to settle in our approach.
waveispropagatingtowardstheblackholeandcalculate
In future studies for the SDS spacetime, it would be
the scattering amplitude. The poles of the amplitude
interesting to calculate the scattering amplitude using
will then represent boundary conditions appropriate for
somemorerigoroustechnique (like, say,whatis done for
QNMs. It turns out that for this case the QNM level
the pure Schwarzschild case [58]) and see how the real
spacing depends only on the surface gravity of the black
parts of the QNM frequencies depend on the different
hole κ−, as expected. Thus the introduction of a black
surface gravities and on the type of scattering.
hole in the DS spacetime brings along the appropriate
QNMs.
However, there exists another set of boundary condi-
APPENDIX A: QNMS FOR THE PURE
tions in which one starts with an incident wave propa-
DESITTER METRIC
gating towards the cosmological horizon. As shown in
Appendix C,itispossibletochoosethe boundarycondi- In this appendix, we give the details of the calcula-
tions and the definition of the scattering amplitude such tions for calculating the QNMs for the DS spacetime.
that the QNMlevelspacing inthis case depends only on Although most of the mathematical apparatus already
the surface gravity of the cosmological horizon κ . We exists in literature[68, 82, 83, 86, 87, 88], we include the
+
do not believe these boundary conditions are physically details for completeness and for emphasizing the conclu-
relevant. sion.
It was found earlier, based on the monodromy of the The radialwave equation for the DS metric [see equa-
perturbationcontinuedtothecomplexplane[76]thatthe tions (6) and (7)]
imaginary part of the QNM frequencies have an equally
2
spaced structure, with the spacing dependent only on d
− (1−H2r2)
κ−. It is not clear whether there exists any extensions " (cid:26) dr(cid:27)
of the above procedure for obtaining the other set of
l(l+1)
QNMs which are dependent on κ+. Analytical calcula- +(1−H2r2) −2H2 F(r)=k2F(r)
tions,basedonapproximatingthe potentialbyaPoschl- r2
(cid:26) (cid:27)(cid:21)
Teller form [75], gave a QNM spectrum which depends (A1)
7
can be reduced to the hypergeometric form by introduc- Then we have a much simpler expression:
ing a new variable z = r2H2. The solution, which is
regular at the origin, can be written as Γ(1+iak)Γ(1−ibk)
I = A
n k
Γ(2+ic )
k
F(r) = rl+1(1−H2r2)ik/2H × r−nF (1+ia ,n,ic ,2+ic ;
− 1 k k k
l ik l 3 ik 3
× 2F1 + , + + ,l+ ;H2r2 r+−r− r+−r−
2 2H 2 2 2H 2 − ,− (B3)
(cid:18) ((cid:19)A2) r− r++2r−(cid:19)
For calculating the scattering amplitude, we are only in-
where the normalization is arbitrary. The behaviour of
terested in the three quantities I ,I ,I . For n = 0, use
this solution near the horizon r =H−1 is given by 0 2 3
the relation
3
F(r) ∝ Γ l+ F (a,0,b,c,x,y)= F (a,b,c;y) (B4)
1 2 1
2
(cid:18) (cid:19)
Γ ik to obtain
× (1−H2r2)−ik/2H H
" Γ 2l + 2iHk Γ(cid:0) 2l(cid:1)+ 23 + 2iHk I0 = AkΓ(1+iak)Γ(1−ibk)
+ (1−H2r2)ik/2HΓ 2l(cid:0)− 2iHkΓ(cid:0)Γ−(cid:1) iH2lk(cid:0)+(cid:1) 23 − 2iHk #(cid:1) × 2F1(cid:16)1+iak,Γic(k2,+2+icic)k;−rr+++−2rr−−(cid:17) (B5)
k
(cid:0) (cid:1) (cid:0) ((cid:1)A3)
Similarly, for n=2, use the relation
According to the boundary conditions (17), the solu-
tion should be purely outgoing near the horizon, i.e., F (a,c−b,b,c,x,y)=(1−x)−a F a,b,c;y−x
F(r)∼(1−H2r2)ik/2H. [This followsfromthe factthat 1 2 1 1−x
r∗ =H−1tanh−1(Hr) for the DS metric which, near the (cid:18) ((cid:19)B6)
horizon, gives 1−H2r2 = sech2(Hr∗) ∼ e−2Hr∗.] This to write
implies that the QNMs are given by the poles of the ex-
pression F 1+ia ,2,ic ,2+ic ;−r+−r−,− r+−r− =
1 k k k
(cid:18) r− r++2r−(cid:19)
Γ(cid:0)2l + 2iHk (cid:1)ΓΓ(cid:0)iHk2l + 23 + 2iHk (cid:1) (A4) (cid:18)rr+−(cid:19)−1−iak2F1(cid:18)1+iak,ick,2+ick;r+(rr+2+−+r2−2r−)(cid:19)
The numeratorhastwosets(cid:0)of(cid:1)poles atk =iH(2n+l) (B7)
n,l
and k = iH(2n+l+3) for n = 0,1,2,.... However,
n,l
and hence
each of these poles is canceled by a similar pole of the
Gamma function in the denominator [68]. Hence, there r −1−iak
exist no QNMs for the pure DS spacetime which obey I = A Γ(1+ia )Γ(1−ib )r−2 +
2 k k k −
the boundary condition (17). (cid:18)r−(cid:19)
Note that this conclusion is only true for a massless, F 1+ia ,ic ,2+ic ; r+2−r−2
minimally coupled scalar field with wave equation (cid:3)φ = × 2 1 k k k r+(r++2r−)
0. Theredoexistwell-definedQNMsforamassivescalar (cid:16) Γ(2+ick) (cid:17)
field, or for a scalar field coupled to the Ricci scalar. (B8)
Finally, use
APPENDIX B: SIMPLIFIED EXPRESSIONS FOR
THE SCATTERING AMPLITUDE F (a,c−b+1,b,c,x,y)=(1−x)−a F a,b,c;y−x
1 2 1
1−x
In this appendix, we shall write the scattering ampli- (cid:20) (cid:18) (cid:19)
a x y−x
tude given by equations (26) and (27) in terms of the
+ F a+1,b,c+1;
2 1
more familiar hypergeometric functions. For notational c1−x 1−x
(cid:18) (cid:19)(cid:21)
convenience, let us define (B9)
k k 1 1
to obtain
a ≡ ,b ≡ ,c ≡a −b =k − (B1)
k k k k k
κ− κ+ (cid:18)κ− κ+(cid:19) r+−r− r+−r−
F 1+ia ,3,ic ,2+ic ;− ,− =
Also define 1(cid:18) k k k r− r++2r−(cid:19)
Ak =(r+−r−)1+ick(r++r−)ickr−−iakr+ibk(r++2r−)−ick r+ −1−iak F 1+ia ,ic ,2+ic ; r+2 −r−2
(B2) (cid:18)r−(cid:19) (cid:20)2 1(cid:18) k k k r+(r++2r−)(cid:19)
8
−(1+ia )r+−r− Note that in this case, the “incident” wave is propagat-
k
r+ ingtowardsthecosmologicalhorizon. Forthepureblack
F 1+ia ,ic ,3+ic ; r+2−r−2 hole case, these solutions are irrelevant as they corre-
×2 1 k k k r+(r++2r−) spond to the physically unacceptable case where the in-
(cid:16) 2+ic (cid:17) cident waves propagate towards spatial infinity. These
k
solutions are not relevant for the pure DS case too; this

 (B10) isbecausethe potentialisnon-zeroatr =0andhence it
is never possible to have an in-going plane wave at this
and hence region. However, in the case of the SDS spacetime, one
might consider the above case for studying scattering off
r −1−iak
I = A Γ(1+ia )Γ(1−ib )r−3 + the cosmological horizon at r∗ → ∞. Clearly, the scat-
3 k k k −
r− teringamplitudeinthiscasewillbeexactlylikeequation
(cid:18) (cid:19)
F 1+ia ,ic ,2+ic ; r+2−r−2 (10), i.e., S(k) ∝ A¯out/A¯in, and the QNMs, still defined
× 2 1(cid:16) k Γk(2+ick)k r+(r++2r−)(cid:17) bofytthhee sbcoautntedrainrygcaomndpiltitioundse.(11S)e,ttairneggAi¯vienn=by0thine p(oCl1es)
leads to the same boundary condition (11) as obtained

− (1+ia )r+−r− by setting Ain = 0 in (9). In that case, we will obtain
k r the same poles for the scattering matrix as before. If,
+
on the other hand, we also change the sign of k in the
×2F1(cid:16)1+iak,iΓc(k3,3++icic)k;r+(rr+2+−+r2−2r−)(cid:17) d“ienficniditeionnt”owfasvceatttrearvinelglinagmipnlitthuedeopwphoesnitewdeirceocntisoiidne,rththene
k
poles flip sign. In this case, the scattering amplitude in

(B11) the first Born approximationis given by
Todeterminethepolestructuresofthethreequantities
I0,I2,I3, note that all of them contain combinations of ∞
the form S(k)= dr∗V(r)e−2ikr∗ (C2)
−∞
Z
F (1+ia ,ic ,n+ic ;r )
2 1 k k k 1
(B12)
Γ(n+ic )
k
One might notice that both the quantities Note that because of the different form of the “incident”
F (1+ia ,ic ,n+ic ;r ) and Γ(n+ic ) have wave,thesignofkintheaboveequationisdifferentfrom
2 1 k k k 1 k
poles at negative integral values of n + ic – however, the corresponding equation (21) in the previous case.
k
their ratio turns out to be regular everywhere [84]. This The analysis of Sec. III goes through identically, except
implies thatthe poles ofI ,I ,I occur only atthe poles that there is a crucial difference in the sign of k. The
0 2 3
oftheGammafunctionsΓ(1+ia )andΓ(1−ib ),which explicit formofS(k) canbe calculatedexactly as in Sec.
k k
is what we have been using in the main text. III,andbecauseofthechangeinthesignofk,onewould
find that the poles occur at
APPENDIX C: QNMS FOR A DIFFERENT SET
OF BOUNDARY CONDITIONS
Oneshouldnotethatthe solutions(9)usedareappro- kn =−inκ−, kn =inκ+ (n≫1) (C3)
priate for describing scattering off the black hole. As a
mathematical possibility, it might be interesting to see
what happens if one uses a different boundary condition
This shows that the QNMs, given by positive imaginary
intheformofthesolutionswhichrepresentthescattering
k, are now determined by the surface gravity of the cos-
off the cosmological horizon, i.e.,
mological horizon, as expected. While this is certainly
F(r)∼ A¯ine−ikr∗ +A¯outeikr∗ (at r∗ →−∞), (C1) a(Cm1)atsheeemmsataicrtailficpioaslsfirboimlityt,htehpehbysoiucnaldaproyinctoonfdvitiieown.s in
e−ikr∗ (at r∗ →∞)
(cid:26)
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