Table Of ContentA simple Hubble-like law in lieu of dark energy
Yves-Henri Sanejouand∗
UMR 6286 of CNRS,
5
1 Facult´e des Sciences et des Techniques,
0 Nantes, France.
2
v November 12, 2015
o
N
Abstract
1 studyistoprovidesafegroundsonwhichsuchcos-
1 mologies could be established.
Within the frame of the Λ cold dark matter
]
O paradigm, a dark energy component of unknown
2 Main hypothesis
originisexpectedtorepresentnearly70%oftheen-
C
ergy of the Universe. Herein, a non-standard form
h. oftheHubblelawisadvocated,withtheaimofpro- In the late 1920s, Edwin Hubble discovered a pro-
p viding safe grounds on which alternative cosmolo- portionality between zλ, the redshift of nearby
- galaxies,andD ,theirdistanceestimates[9]. He
o gies could be developed. Noteworthy, it yields an mes
wrote his law as follows:
r age-redshift relationship which is consistent with
t
as aavnaaillyasbisleofdgaatam.maT-orgaeythbeurrswtcitohunatss,tirtafiugrhtthfoerrwsaurgd- zλ = H0Dc mes (1)
[ 0
gests that the observable Universe has been eu-
where H is the Hubble constant, c , the speed of
6 clideanandstaticoverthelast12Gyr. Althougha 0 0
v light, with zλ being:
non-standard distance-duality relation is then re-
19 quired for interpreting luminosity distances, the z = λobs−λ0
λ
9 magnitude-redshift relationship obtained is com- λ0
2 patible with type Ia supernovae data.
where λ is the wavelength of the light received
. obs
1 from the galaxy, while λ is the wavelength mea-
0
0
1 Introduction sured for the same kind of source sitting on Earth.
4
In the late 1990s, using type Ia supernovae as
1
standard candles, it was shown that, for large val-
: Over the last twenty years, as a consequence of
v
ues of the distance, Hubble’s law is not linear any
i its numerous successes, the Λ cold dark matter
X more [10, 11]. Within the frame of ΛCDM, this
(ΛCDM) paradigm has reached the status of a
r ”concordance” cosmology [1, 2]. However, sev- deviation from linearity is in particular due to a
a non-zero, although very small [12], value of Λ, the
eral clouds are still obscuring the brilliance of this
cosmologicalconstant.
paradigm, one of the most notable being that it
Hereafter,itisinsteadassumedthat,asproposed
relies on a new kind of so-called ”dark energy”, of
by Hubble, the law he discovered is indeed linear.
unknownoriginbutaccountingforatleast68%[3],
However, it is also posited that, when Hubble de-
and up to 75% of the energy of the Universe [2, 4].
fined the redshift, in the nowadays standard way,
As long as this dominant component remains
he made the wrong choice. Specifically, herein, the
mysterious [5], alternative cosmologies need to be
physicallyrelevantformofHubble’slawisassumed
developed [6, 7, 8]. The purpose of the present
to be:
∗[email protected] zν =H0∆t (2)
1
Table 1: The two oldest objects presently known. 16
HD140283, an extremely metal-deficient subgiant,
14
is the oldest star known in our neighbourhood.
APM 08279+5255 is the oldest quasar known at 12
z ≈ 4. The age of APM 08279+5255 was ob-
taλinedthroughthemeasureoftheFe/Oabundance yr) 10
G
ratio [13]. e ( 8
g
A 6
Object z z Age Ref.
λ ν 4
(Gyr)
HD140283 0 0 14.5 [14] 2
APM 08279+5255 3.9 0.8 2.1 [15]
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Redshift
where ∆t is the photon time-of-flight between the
Figure1: Age-redshift(z ) relationship. Filledcir-
source and the observer, H being an actual con- λ
0
cles: ages of nine early-type stars, galaxies and
stant, and where z , the frequency-redshift, is:
ν
quasars (see text). Plain line: upper bound ex-
z = ν0−νobs pected with Told=14.5 and TH=15.6 Gyr, accord-
ν
ν ing to the Hubble-like law advocated herein.
0
ν beingthefrequencyofthelightreceivedfroma
obs
remotesource. Note thatwithD =c ∆t, when Being in our neighbourhood, HD140283 can
mes 0
z ≪ 1, eqn (2) can indeed be approximated by provide a fair estimate for T while, accord-
λ old
eqn (1) since, by definition: ing to eqn (4) and under the hypothesis that
APM 08279+5255 could be, nowadays, as old as
z
zν = λ (3) HD140283, the apparent age (Tobs = 2.1 Gyr) and
1+z
λ frequency-redshift(z =0.8)oftheformerallowto
ν
determine the Hubble time, namely:
3 Age-redshift relationship
T −T
T = old obs =15.6 Gyr
H
3.1 The age of the oldest stars zν
Consideringthe case of early-typestars or galaxies that is, a value consistent with recent estimates [3,
bornT Gyrago,T ,theirapparentageaccord- 16, 17]. Note that the above analysis is expected
old obs
ing to an Earth-based observer is: to yield an accurate value for TH if HD140283 and
APM 08279+5255 have, nowadays, the same age.
T =T −∆t Indeed, eqn (5) provides an upper-bound for the
obs old
age of early-type stars at any redshift.
that is, with eqn (2): Forinstance,ithasbeenclaimedthattwogalax-
ies found at z = 6 and z = 9.6 could be 0.8 [18]
λ λ
Tobs =Told−THzν (4) and 0.2 [19] Gyr old, respectively. According to
eqn(5),andasillustratedinFig.1,thecorrespond-
or, with eqn (3):
ing upper-bounds at these redshifts are indeed
z higher,namely,1.2and0.3Gyr,respectively. Fig.1
T =T −T λ (5)
obs old H1+z also shows that eqn (5) yields upper bounds that
λ
areovercurrentestimatesfortheagesof3C65[20],
where T = 1 is the Hubble time. Table 1 shows LBDS 53W069 [21], LBDS 53W091 [22], QSO
H H0
theageestimatesofwhatmaybethetwooldestob- B1422+231 [23] and GNS-zD1 [24], at z =1.175,
λ
jects presently known, at their respective redshift. 1.43, 1.55, 3.62 and 7.2, respectively.
2
3.2 A corollary
As a consequence of eqn (4), T >0 if: 100
obs
T
z < old (6)
ν T 80
H
z)
that is, with eqn (3) and the above values for Told 1+
and TH: z)/( 60
z <13 H(
λ
So, according to eqn (5), and under the assump-
40
tion that no object older than HD140283 or APM
08279+5255canbeobservednowadaysfromEarth,
itshouldnotbepossibletoobserveanygalaxyata
0 0.5 1 1.5 2
redshiftlargerthan13. Thisisconsistentwithcur-
rent knowledge, since the highest redshifts known Redshift
so far are around 10 [19, 25]. Note that this upper
limitwoulddriftifobjectsolderthanHD140283or Figure2: H1+(zzλλ) asafunctionofredshift(zλ). The
APM 08279+5255are discovered. dashedlinecorrespondsto H =63km.s−1.Mpc−1
0
(T = 15.6 Gyr). H(z ) data come from [27].
H λ
3.3 Another prediction
4 Distance versus redshift
On the other hand, as a consequence of eqn (2):
∂z
ν =H 4.1 Gamma-ray bursts
∂(∆t) 0
Without any explicit cosmology, going further re-
With∆t=t −t,takingeqn(3)intoaccountyields:
0
quires additional hypotheses.
∂z
λ =−H (1+z )2 (7) So,inordertogetinsightsabouttherelationship
∂t 0 λ between photon time-of-flight and distance mea-
where t and t are the observer and cosmic times, surements, let us turn to cumulative object counts
0
respectively. Measures of ∂zλ, obtained through and assume that:
∂t
studiesoftheageofpassivelyevolvinggalaxies,are
usually provided through H(zλ) [26, 27], which is n(z)=βDcd (9)
defined as follows [28]:
wheren(z)is thenumberofobjectswitharedshift
1 ∂z
H(z )=− λ lower than zλ, β, a constant, d being the effective
λ
1+zλ ∂t dimension of space when Dc is the light-traveldis-
that is, with eqn (7): tance, namely:
D =c ∆t (10)
H(z ) c 0
λ =H (8)
1+z 0 With eqn (2), (3) and (10), eqn (9) becomes:
λ
Thecorrespondingrelationshipexpectedwithinthe z
n(z)=βDd ( λ )d (11)
frame of ΛCDM is not that simple [28, 29]. As a H 1+z
λ
matter of fact, it has been claimed that eqn (8),
which is also a prediction of the R = c t cosmol- where D =c T is the Hubble length.
h 0 H 0 H
ogy[30,31],isruledoutbythedata[32]. However, In order to measure d, it is necessary to con-
backed by standard statistical analysis (χ2 = 13.4, sider a categoryof objects whose redshift is known
p-value = 0.71), and in spite of large error bars over a wide range, with few selection biases. In
(see Fig.2), a recent in-depth study shows that, on this respect, noteworthy because they are highly
the contrary,when compared to ΛCDM, eqn (8) is energetic, sources of gamma-ray bursts (GRB) are
favoured by model selection criteria [17]. attractive candidates.
3
lessfrequentmorethan12Gyrago(withT =15.6
H
Gyr). Itcouldalsomeanthat,duetothesensitivity
limits of Swift and follow-up telescopes that were
usedtomeasuretheredshifts,asignificantfraction
of the GRB with z >4 were missed.
λ
Oscillations aroundvalues predicted by eqn (11)
may also prove meaningful (see the inset of Fig. 3)
since, near z ≈ 0.7 (z ≈ 2.3), the residuals ex-
ν λ
hibit a large-scale fluctuation of the GRB density,
whose size (≈ 400 Mpc; δz ≈ 0.1) is of the or-
ν
der of the size of the largest voids known in our
neighbourhood [36, 37].
4.2 Angular diameter distance
If, as suggested by the above results, the Universe
is both euclidean and static, D , the angular di-
A
Figure 3: Cumulative count of long gamma-ray
ameter distance, is so that:
bursts (t > 0.8 s), as a function of redshift (z ).
90 λ
Plain line: least-square fit of the data, performed DA =Dc (12)
for GRB sources with z < 3.5. Inset: residuals
λ
(observed−expected), as a function of frequency- Taking into account eqn (2), (3) and (10) yields:
redshift (z ). z
ν D =D λ (13)
A H
1+z
λ
Thanks to the Swift mission launched ten years As a consequence, θ, the angular size:
ago [33, 34], the redshifts of 265 GRB sources s
θ =
have nowadays1 been determined with fair accu- D
A
racy. Moreover, in the Swift sample, most GRB
s being the actual size of the considered standard
with a duration over 0.8 s are expected to have
rod, becomes:
same physical origin [35]. Fig.3 shows the cumula-
tivecountforthecorrespondingsubsetof254GRB s 1
θ = (1+ ) (14)
sources, which is expected to be rather homoge-
D z
H λ
neous.
Indeed,althoughthismaynotbethecaseforultra-
With eqn (11), a least-square fit of these data
compact [38] or double-lobed radio sources [39],
yields:
it has been claimed that, over a wide range of
d=2.96±0.03
redshifts (up to z = 3.2), the average angular
λ
Such a result strongly suggests that GRB sources size of galaxies is approximately proportional to
arerandomlydistributedinanobservableUniverse z−1 [40, 41]. Note that, assuming the standard
λ
that is both euclidean and static. Indeed, in such cosmological model as correct, this fact can be ex-
a case, d = 3 and β = 4πρ , where ρ is the plained only if the average linear size of galaxies
3 grb grb
average density of observable GRB sources. Note with same luminosity is six times smaller at zλ =
that this result relies on the hypothesis that Swift 3.2 than at zλ = 0 [40].
data represent a fair sample of the GRB sources,
up to zλ ≈4. 4.3 Luminosity distance
Indeed, abovethis value, eqn(11) starts to over-
Theluminositydistance,D ,isrelatedtotheangu-
estimate the observed GRB counts. Such an over- L
lar distance through the distance-duality relation,
prediction could mean that GRB happened to be
that is:
1http://swift.gsfc.nasa.gov,2014, July5th. DL =DA(1+zλ)n (15)
4
46 55
44
50
s 42 s
u u
ul ul
d d 45
o 40 o
m m
e e
c 38 c
n n 40
a a
st st
Di 36 Di
35
34
32 30
0 0.5 1 1.5 0 2 4 6 8 10 12
Redshift Redshift
Figure 4: The distance modulus of supernovae Ia, Figure 5: Distance modulus, as a function of red-
asafunctionofredshift(z ). Filledcircles: the580 shift (z ). Plain line: ΛCDM, with Ω = 0.3 and
λ λ m
cases of the Union 2.1 compilation (error bars not Ω =0.7;dashedlines: n=1.65(above)andn= 3
Λ 2
shown). Plain-line: least square fit of these data (below).
(n=1.65±0.02,µ =18.21±0.02). Dashedline:
0
the n= 3 case (µ = 18.30 ± 0.01).
2 0 theory of gravity, n = 2 [45, 46]. Indeed, in this
context, n can not be lower than two while, in the
Together with eqn (2) and (3), (15) allows to context of a static Universe, the most likely values
write µ, the distance modulus: are either n = 0.5, as a consequence of the en-
ergyloss of the photons during their flight towards
µ=5log10(DL)+25 the observer, or n = 1, if time dilation of SNIa
lightcurves is also taken into account [47, 48, 49].
as follows:
However, if for instance the number of photons is
µ=5log (z (1+z )n−1)+µ +25 (16) not conserved during their travel, n can be larger
10 λ λ 0 than that [50].
where µ =5log (D ).
0 10 H
Nowadays, distance moduli have been measured 4.4 The distance-duality relation
for hundreds of supernovae of type Ia (SNIa) [42].
Deviationfromthe Etheringtonrelation(eqn(15))
AsshowninFig.4forthe580casesoftheUnion2.1
has been quantified through the η parameter,
compilation[43], anaccurateleast-squarefitofthe 0
which can be defined as follows:
data (χ2 = 571, p-value = 0.57)2 can be obtained
with eqn (16), which yields: D =D (1+z )2+η0 (17)
L A λ
n=1.65±0.02
that is, for small values of z :
λ
Note that when another type Ia supernova dataset D ≈D (1+z )2(1+η z )
L A λ 0 λ
is considered, namely, the 397 cases of the Con-
stitution compilation [44], the value found for n is By combining the Sunyaev-Zeldovicheffect and X-
similar (n=1.63±0.03). ray surface brightness for two samples of galaxy
Let us emphasize that, within the frame of clusters [51, 52], together with type Ia supernovae
ΛCDM,like inmostcosmologiesbasedona metric data so as to end with a model-independent cos-
mological test, it was shown that η = −0.28 ±
2The error estimates on the distance modulus measure- 0
ments were used for the χ2 calculation. This amounts to 0.22[46],whenasampleof25galaxyclusters[51]is
assumethatSNIaareperfect standardcandles. analysed,andη0 =−0.42±0.11[46],whenalarger
5
sample of 38 galaxy clusters [52] is considered. In And since, according to eqn (18), ν ∝t :
0 0
thelatercase,whenaredshiftbiasisaccountedfor,
η0 =−0.23±0.11 or η0 =−0.43±0.10,depending νobs =1− ∆t (19)
uponwhichtypeIasupernovaedatasetistakeninto ν t
0 0
account [53]. More recently, using a sample of 91
galaxy clusters and four different methods, η val-
0 In other words, if t = T , that is, if T is as-
0 h H
ueswerefoundtorangebetweenη =−0.08±0.10,
0 sumedtobethetimeelapsedsincephotonsstarted
and η =−0.17±0.17 [54].
0 to be emitted with non-vanishing frequencies then
So, while metric theories of gravity like ΛCDM
eqn (2), the Hubble-like law advocated in the
require η = 0, observed values have been found
0 present study, is recovered.
to favor the negative side, up to 4σ away from the
Of course, such a drift of atomic spectra should
ΛCDM prediction. On the other hand, all of them
show up in various physical domains, notewor-
but one are within 2σ of η =−0.35, the value ex-
0 thy as a consequence of a corresponding drift of
pected within the frame of the presentstudy (with
atomic clocks. In particular, an apparent increase
n=1.65).
of lengths measured through the time it takes for
electromagneticwavestogofromaplacetoanother
4.5 Another difference with ΛCDM
should be observed [60].
As shown in Fig.5, as far as distance moduli are
concerned, the difference between values predicted
with ΛCDM or eqn (16) becomes obvious for zλ > 6 Conclusion
2, when n = 1.65, or for z > 4, when n = 3. Al-
λ 2
though the fit of the supernovae data of the Union
2.1 compilation looks poor when n= 3 (χ2 = 652, A Hubble-like law, where the frequency-redshift
2
is proportional to the photon time-of-flight, yields
p-value = 0.02), it follows the values predicted by
an age-redshift relationship which is consistent
ΛCDM over a wider range of redshifts (Fig.5).
Note that n= 3 is expected within the frame of with available data. A straightforward analysis of
2
gamma-ray burst counts further suggests that the
a couple of alternative cosmologies [8, 55].
observable Universe has been euclidean and static
over the last 12 Gyr.
5 Possible meanings
Through a non-standard distance-duality rela-
tion, which is consistent with current knowledge,
Eqn (2) is so simple that, like the original Hubble
it also yields an alternative explanation for the lu-
law itself [8], it can be derived in many different
minosity distance data, alleviating the need for a
ways, based on a variety of physical ground [56].
dark energy component of unknown origin.
Noteworthy, it is a straightforward consequence of
Overall, the present study provides a frame,
the R =c t Universe [30, 31].
h 0
namely, a background that is euclidean and static,
Asanotherexample,letusassumethat,forsome
as previously advocated by others [40, 41, 61], as
yet unknown reason, a steady drift of atomic and
well as a set of relationships between redshifts and
molecularspectratakesplace[57,58,59]suchthat,
distanceswhichcouldbecomeusefulanchorsforthe
for any frequency:
ν ∝t (18) development of new cosmologies.
Furthermore, let us also assume that, during its
flight between the source and the observer,the en-
ergy of the photon is conserved, i.e., that its fre- Acknowledgements
quency does not change. As a consequence, ν ,
obs
thefrequencyofthephotonreceivedfromaremote
I thank Georges Paturel for fruitful discussions,
sourceisthefrequencythephotonhadwhenitwas
Maciej Bilicki for his useful comment, and referee
emitted at t=t −∆t:
0 BofEPL forhis carefulreadingofthe manuscript,
ν ∝t −∆t as well as for his constructive suggestions.
obs 0
6
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