Table Of ContentThe blackness of the cosmic microwave background spectrum as a probe of the
distance-duality relation
George F. R. Ellis1,∗ Robert Poltis1,† Jean-Philippe Uzan2,‡ and Amanda Weltman1§
1 Astrophysics, Cosmology and Gravitation Centre,
Department of Mathematics and Applied Mathematics,
University of Cape Town, Rondebosch 7701, South Africa
2 Institut d’Astrophysique de Paris, Universit´e Pierre & Marie Curie - Paris VI,
CNRS-UMR 7095, 98 bis, Bd Arago, 75014 Paris, France.
(Dated: 4 January 2013)
A violation of the reciprocity relation, which induces a violation of the distance duality relation,
reflectsitselfinachangeinthenormalisationofthecosmicmicrowavespectruminsuchawaythat
3
its spectrum is grey. We show that existing observational constraints imply that the reciprocity
1
relation cannot be violated by more than 0.01% between decoupling and today. We compare this
0
effecttoothersourcesofviolationofthedistancedualityrelationswhichinducespectraldistortion
2
of the cosmic microwave background spectrum.
n
a PACSnumbers: 98.80.Cq,04.80.Cc
J
9
2 In the standard cosmological model [1, 2], in which regardless of the metric and matter content of the
theuniverseisdescribedbyaspatiallyhomogeneousand spacetime. While r is related to the angular distance,
o
] isotropicgeometryoftheFriedmann-Lemaˆıtrefamily,the the solid angle dΩ2 cannot be measured so that r
O s s
luminosity distance D and angular diameter distance is not an observable quantity. If one further assumes
L
C
D are related by the distance duality relation, that the number of photons is conserved, the source
A
. angular distance is related to the luminosity distance,
h D (z)=(1+z)2D (z), (1)
p L A by DL = (1+z)rs, which leads to the distance duality
- wherez istheredshift. Thisrelationisactuallyfarmore relation (1), where DA(z)=ro.
o
general [3, 4]. It can be shown (see Ref. [5] for a demon-
r
t stration) that it holds in any spacetime in which (1) the Varying the distance-duality relation: Violations
s
a reciprocity relation holds and (2) the number of photons of the distance duality relation (1) can arise from a vio-
[ is conserved. lation of the reciprocity relation, which can occur in the
2 Thereciprocity relationconnectstheareadistancesup case where photons do not follow (unique) null geodesics
v and down the past lightcone and is a relation between (e.g. in theories involving torsion and/or non-metricity
2 the source angular distance, rs, and the observer area or birefringence [6]) or (2) from the non-conservation of
1 distance,ro. Theformerisdefinedbyconsideringabun- photons, which occur e.g. when photons are coupled to
3 dleofnullgeodesicsdivergingfromthesourceandwhich axions[7]ortogravitonsinanexternalmagneticfield[8],
1 subtends a solid angle dΩ2 at the source. This bundle toKaluza-Kleinmodesassociatedwithextra-dimensions
. s
1 has a cross section d2So at the observer and the source [9], or to a chameleon field [10–12]. The fact that such a
0 angular distance is defined by the relation violation of the distance duality relation can account for
3
a dimming of the supernovae luminosity [13, 14], since
1 d2S =r2dΩ2. (2)
o s s e.g. in the case of photon-axion mixing the luminosity
:
v Similarly, the observer area distance is defined by con- distance has to be rescaled as D /(cid:112)1−P (r) with
L γ−a
i
X sidering a reciprocal null geodesic bundle converging at Pγ−a being the probability for a photon to oscillate in
the observer by an axion after having propagated over a distance r, has
r
a motivated the design of many tests of this relation [15–
d2S =r2dΩ2. (3)
s o o 17, 22, 23] using independent measurements of D and
L
As long as photons propagate along null geodesics and DA [15, 17] based on the SZ effect and X-ray measure-
thegeodesicdeviationequationholdsthenthesetwodis- ments [16, 22, 23].
tances are related by the reciprocity relation [5] These tests play an important role in understanding
the physics behind the acceleration of cosmic expan-
r =(1+z)r , (4)
s o sion [24]. Defining the deviation from Eq. (1) as
D (z)
η(z)=(1+z)2 A , (5)
∗Electronicaddress: [email protected] D (z)
L
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected] so the distance-duality relation holds iff η(z)=1, it was
2
concluded [16] that η = 0.91±0.04 up to z (cid:39) 0.8. 1 In is the amount of radiation received by the detector per
particular, this sets strong constraints on photon-axion unit area per unit time, and is given by
oscillation models [15, 16]. It also allows us to prove [15]
that the dimming of the supernovae did not result from L 1
F = (9)
absorption by a grey-dust model [28]. 4πr2(1+z)2
s
It has also been pointed out [29–31] in the partic-
ular case of photon-axion mixing that the oscillation where the factor 4π arises from the integral of dΩ2 over
s
probability depends on the frequency of the photon [7], the whole sky [5]. The specific flux Fν received from the
so that a spectral distortion of the cosmic microwave source, i.e. the flux per unit frequency range, is given by
background spectrum was expected (see below).
L I[ν(1+z)]dν
F dν = (10)
Thecosmicmicrowavebackground(CMB)radiationis ν 4π r2(1+z)
s
considered to enjoy one of the most precise black body
spectra ever produced, that is, it has the Planck form where I(ν) is the source spectrum. Note that here ν
is the frequency measured by the observer, which corre-
2h 1 sponds to a frequency (1+z)ν at emission.
I (ν,T)=ν3f(ν/T), f(ν/T):= (6)
BB c2 ehν/kT −1 Whatisactuallymeasuredfromanextendedsourceby
adetectoristhespecificintensityI inasolidangledΩ2
ν o
whereIBB(ν,T)istheenergyperunittime(orthepower) in each direction of observation,
radiated per unit area of emitting surface in the normal
direction per unit solid angle per unit frequency by a F dν
I dν ≡ ν (11)
blackbodyattemperatureT atemission(histhePlanck ν dΩ2
o
constant,cthespeedoflightinavacuumandktheBoltz-
mann constant.) We note that the normalisation factor and reduces, using Eqs. (10) and (2-3), to
2h incorporated in the definition of f(ν/T) is crucial to
c2
the black body nature of the spectrum, deriving directly (cid:18)r (cid:19)2 I[ν(1+z)]dν
I dν =I o (12)
from quantum mechanics. ν s r (1+z)
s
The accuracy of the CMB black body spectrum
sets [32–34] constraints on various spectral distortions where I :=L/4πd2S is the source surface brightness in
s s
parametrized as the effective chemical potential µ, the that direction. It follows from Eq. (8) that
Compton parameter y and the free-free distortion pa-
rameter Yff of the order I[ν(1+z)]dν
I dν =I . (13)
ν s (1+z)3β(z)
|µ|<19×10−6, |y|<9×10−5, |Y |<15×10−6. (7)
ff
This expression is completely general (and does not as-
This sets stringent constraints on violations of the dis-
sume any specific geometry for the spacetime) and thus
tance duality relation induced by the non-conservation
holds in any curved or flat spacetime. In the laboratory,
of photons, as e.g. in the case of photon-axion mixing.
it simply means that the intensity of the radiation is in-
dependent of the distance from the source, as long as
Changes in the CMB Spectrum: Our goal is to
thesourcehasnorelativemotioncomparedtothedetec-
investigatetheeffectofaviolationofthedistanceduality
tor (z = 0). In cosmology, the relation depends only on
related to a violation of the reciprocity relation on the
redshift and is thus achromatic (so that the spectrum is
spectrum of CMB photons. We assume that Eq. (4) is
redshifted but not distorted) and is independent of area
modified to
distance.
In the case of the cosmic microwave background, the
r2 =(1+z)2r2β(z), (8)
s o photons are coupled to the electrons and baryons by
Thomson scattering up to recombination [1, 2] (see [18]
whereβ(0)=1; β(z)isafunctiondependingonthepar-
for a description of the physical origins of the different
ticular physical mechanism responsible for the violation
spectraldistortions). Becausethecollisiontermentering
of the reciprocity relation; that relation holds precisely
the Boltzmann equation has a very weak dependence on
iff β(z)=1.
the energy of the photon, the CMB spectrum enjoys a
First, the integrated flux of radiation F received from
Planck spectrum (6). Deviations from the Planck spec-
an isotropically emitting source of intrinsic luminosity L
trum induced by non-linear dynamics [35, 36] are negli-
gible as recalled in Eq. (7); see also [19–21].
Then, denoting the emission temperature by T , we
e
have that
1 Adeviationofη(z)awayfromunityinopticalwavelengthswas
studied in [25, 27] where [25] put strong constraints on any de- (cid:20) (cid:21)
ν(1+z)
viation from cosmic transparency in optical wavelengths, with I I[ν(1+z)]=ν3(1+z)3f . (14)
η(z)=1/(1+z)(cid:15) and(cid:15)=−0.04+0.08 at2σ. s T
−0.07 e
3
Using Eq. (13) and the relation ν ∝ (1+z)−1, which possible origins of such a violation are a violation of the
follows from the definition of redshift (which is purely a reciprocity relation, or non-conservation of the number
time dilation effect, so this relation is quite independent of photons.
of area distances), we finally get Generically,theevolutionofthedistributionfunctionf
oftheCMBisdiscussedintermsoftheBoltzmannequa-
ν3 (cid:20)ν(1+z)(cid:21) tion, L[f] = C[f]. In a Friedmann-Lemaˆıtre universe,
I = f (15)
ν β(z) T the distribution function is, by symmetry, a function of
e
the energy E and cosmic time t so that the Liouville
after simplifying by a factor (1+z)3. term reduces to L[f]=E∂ f−HE2∂ f. If the collision
t E
As long as β(z) = 1, the redshift prefactor combines term does not depend on energy, the integration of the
with the factor ν3 in Eq. (14) so that the initial Planck Boltzmann equation over energy gives the following evo-
spectrum with temperature Te remains a Planck spec- lution equation for the photon number density (see e.g.
trum Ref. [1, 42])
(cid:20) (cid:21)
ν
I =ν3f , (16) n˙ +3Hn=C˜. (23)
ν T(z)
In general, the collision term depends on energy, as for
with a redshifted temperature
exampleinthecaseofaxion-photonmixing,sotheeffect
T of photon non-conservation is expected to be chromatic.
T(z)= e . (17) Itfollowsthatthegeneralformoftheobservedspectrum
1+z
can be parameterized as
If β(z)(cid:54)=1, then the spectrum has the form
I(ν,T) = Φ (ν,z)I [ν,T(z)], (24)
0 BB
I =β−1(z)I [ν,T(z)]. (18)
ν BB
where
We conclude that if the reciprocity relation is violated,
theblackbodyspectrumisobservedasagreybodyspec- Φ0(ν,z): = β−1(z)η(ν,z). (25)
trum. Howeverthereisnospectraldistortion,sosuchan
The factor β is related to the violation of the reciprocity
achromaticeffectcanbeconfusedwithcalibrationerrors.
relation as shown above and the factor η(ν,z) to the
NotethatthegreybodyfactordoesnotimpactWien’s
non-conservation of photons. In Fig 1 we compare the
displacementlaw,thatthewavelengthλ atwhichthe
max factor Φ (ν,z) for photon-axion mixing, a violation of
intensity of the radiation is maximum obeys 0
thereciprocityrelation,andaµ-typespectraldistortion.
λ =b/T (19) It demonstrates that each physical phenomenon impacts
max
a different part of the CMB spectrum.
where Wien’s displacement constant is b = 2.8977721×
103 K.m. This corresponds to a frequency ν = Concerning the possibility of a violation of the reci-
max
58.8(T/1K)GHz. TheStefan-Boltzmannlawstatesthat procity relation, the COBE-FIRAS experiment [32, 33]
the power emitted per unit area of the surface of a black showed that the CMB photons have a black body
body is proportional to the fourth power of its absolute spectrum within 3.4×10−8ergs.cm−2.s−1.sr−1.cm over
temperature; j =σT4, where j is the total power radi- the frequency range from 2 to 20 cm−1. More important
∗ ∗
atedperunitareaandσ =5.67×108m2K4 istheStefan- concerning our work, it showed that the deviations are
Boltzmann constant. For a grey body, this is modified less than 0.03% of the peak brightness with an rms
to value of 0.01%. This means that the normalisation
of the spectrum can be considered accurate at this
j∗ =β−1(z)σT4 (20) level so that it indicates a constraint of the order
(cid:12) (cid:12)
(cid:12)β−1(zLSS)−1(cid:12) < 10−4 with zLSS ∼ 1100 for the
because all intensities get changed by this same factor.
redshift of the last scattering surface.
Hence, the Wien temperature
To check the order of magnitude of this effect, we fo-
T :=b/λ (21)
W max cus on the CMB monopole spectrum [33] and compare
and the Stefan-Boltzmann temperature to a 2.725K grey body spectrum for several values of
β−1(z ). TheratiooftheexpectedCMBmonopolesig-
LSS
T :={j β(z)/σ}1/4 (22) nal(18)tothemeasuredCMBmonopole(publiclyavail-
SB ∗
able at [38]) is depicted in Fig. 2. Error bars designate
are different iff β(z) (cid:54)= 1. The first is independent of z, the 1σ uncertainty of the FIRAS data. From Fig. 2 we
the second is not. see that any violation of the reciprocity relation must be
lessthanonepartin104 atthesurfaceoflastscattering:
In summary, a violation of the distance duality rela-
tion (1) has an imprint on the CMB spectrum. Two (cid:12)(cid:12)β−1(zLSS)−1(cid:12)(cid:12)<10−4. (26)
4
previous constraints [16], we use the fact that they were
1.5
basedonbolometricobservationssothattheyconcerned
the parameter η(z) defined in Eq. (5), that is related to
parameters introduced in this article by
(cid:45)body 1.0 η(z)=β−1(z)(cid:82) η((cid:82)ν,z)IBB[ν,T(z)]dν. (27)
black IBB[ν,T(z)]dν
I
(cid:144) We recall that η = 0.91±0.04 up to z (cid:39) 0.8 [16]. Note
I
thatEq.(26)givesmuchtighterlimitsoveramuchlonger
range of redshifts.
For completeness, we shall also mention that bounds
havebeensetontherelationbetweenCMBtemperature
10(cid:45)5 10(cid:45)4 0.001 0.01 0.1 1 10 and redshift, assuming a form T = T0(1+z)1−γ. The
Ν eV index γ may be constrained through measurements of
the Sunyaev-Zeldovich effect for redshifts z <1 and fine
FIG. 1: Comparison of the function Φ (ν,z) defined in
0 structureexcitationsinquasarspectroscopyforredshifts
Eq.(25)atredshiftz=0foraµ(cid:64)-ty(cid:68)pespectraldistortion(red
z > 1, specifically γ = −0.004±0.016 up to a redshift
dot-dashed),aviolationofthereciprocitytheoremleadingto
of z ∼ 3. [41]. A non-vanishing γ has been argued to
a grey body (green thick dashed) and photon-axion mixing
appear in models with decaying dark energy [39, 43, 44],
(black solid, blue dotted, and purple dashed) with different
physical parameters for the distribution of the intergalactic but it has been argued to have an unphysical ansatz
magnetic field, calculated with the results of Ref. [14]. Only [18]. Such constraints on the temperature are not easily
the solid black curve corresponds to realistic parameters. related to ours since these analyses assume a Planck
spectrum. However, when the spectrum is no longer a
Planck spectrum the different notions of temperatures
1.0010
differ. We have already noted the difference between
the Wien and Stefan-Boltzmann temperatures. The
latter is also related to the bolometric temperature in
1.0005
the case of spectral distortions. Note also that while the
S brightness and the bolometric temperatures agree at the
A
R background and first order level, it has been shown [40]
FI1.0000
I that the non-linear dynamics sources a y-type spectral
I (cid:144)
distortion and this would affect the brightness and thus
both the brightness and the bolometric temperatures.
0.9995
In such a case [45] it was proposed to use occupation
number temperature defined as the temperature of a
black body which would have the same number density
0.9990
100 200 300 400 500 600
Ν GHz of photons as the actual distribution (to be contrasted
with the bolometric temperature which is the temper-
FIG. 2: The ratio of the spectral radiance from a 2.725 K ature of the black body which carries the same energy
grey body spectrum (Eq. (18))(cid:64)to th(cid:68)at measured by FIRAS density as the actual distribution).
for β−1(z )−1=10−4 (black dotted), 10−4.2 (blue solid),
LSS
and 0 (cyan thick). Error bars are the 1σ uncertainty from Conclusion: We have shown that CMB spectral ob-
the FIRAS data. servations allow one to test the distance duality rela-
tion. We have emphasized the difference between the
imprintinducedbythenon-conservationofphotons,usu-
IncludingthespectraldistortionsoftheCMBmonopole allychromatic,andaviolationofthereciprocityrelation,
using the constraints listed in Eq. (7) (also see [33, 37]) which is achromatic. In the latter case we have shown
increases the deviation from unity of the ratio I/IFIRAS that the CMB spectrum is a grey spectrum, with the
plotted in Fig. 2, but does not significantly change any same shape as the CMB power spectrum, up to a nor-
constraint on β−1(zLSS). malisation factor. The FIRAS/COBE data allowed us
to set constraints of the order of 0.01% on the relative
In this article we have shown that a violation of the deviation from the reciprocity relation for the CMB.
reciprocity relation is associated with a deviation from As a final remark, we note that the limits above are
blackness and that the COBE/FIRAS data sets the con- for radiation coming from the surface of last scatter-
(cid:12) (cid:12)
straints (cid:12)β−1(zLSS)−1(cid:12) < 10−4. The second factor ing at z = 1100. However, it is likely that any effect
η(ν,z ) induces a spectral distortion and can be con- violating the DDR relation would be cumulative, and
LSS
strained independently; see e.g. Refs. [30, 31] for an ex- hence proportional to distance. While it probably im-
ampleofthecaseofphoton-axionmixing. Tocompareto pliesstrongerconstraintsforclosersources,norobustand
5
model-independent bound can be derived. Note however supported financially by the National Research Founda-
that our constraint improves those at low redshift by at tion. Any opinion, findings and conclusions or recom-
least two orders of magnitude. mendations expressed in this material are those of the
authors and therefore the NRF does not accept any lia-
bility in regard thereto. G.F.R. Ellis thanks the Institut
Acknowledgements d’Astrophysique de Paris for hospitality during the early
stage of this work and J.-P. Uzan thanks the Yukawa In-
WethankNabilaAghanim,Franc¸ois-XavierDesertfor stituteforTheoreticalPhysicsatKyotoUniversity,where
their insight on the CMB constraints and Cyril Pitrou thisworkwascompletedduringtheLong-termWorkshop
for his comments. This material is based upon work YITP-T-12-03 on“Gravity and Cosmology 2012”.
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