*wol [email protected] eibu g.de 1
G owing en opy as an al e na i e o da k ene gy
Wol gang Salm*
Abs ac
To a oid he da k ene gy hypo hesis, we look a a ame o e e ence ha is a es ela i e o he dipole-co ec ed cosmic
mic owa e backg ound adia ion, and in i a me ic in which each obse e assigns a ime dila a ion o all o he bodies depending
on hei dis ance. Based on ou me ic we es ablish he Eins ein ield equa ions o weak ields. The adial equa ion can be
in e p e ed as Fokke -Planck equa ion o en opy; he eby, we link en opy o he leng h o he ime in e al a ailable o a
ansmi ed uni o in o ma ion. The g ow h o en opy leads o a dissipa ing o ce ela ed o he change in Helmhol z ee
ene gy, which d i es he accele a ed expansion o he uni e se. In ou app oach, he ligh o ecen ly obse ed massi e galaxies
wi h is emi ed no 500 – 700 My a e he big bang, bu la e han hal he age o he uni e se.
1. In oduc ion
The cu en ly p e e ed cosmological models a e based on
he RW–me ic, wich leads o he F iedmann ield equa ions.
The p e alen model in cosmology, he ΛCDM model, is
based on hese equa ions. Ne e heless, i uns in o se e e
di icul ies: Taking in o accoun he accele a ed expansion o
he uni e se, one has o in oduce a epulsi e componen ,
called da k ene gy, whose physical na u e is enigma ic.
A numbe o ideas ha e been p oposed o a oid he da k
ene gy hypo hesis. As in ou app oach, Easson e al.
p oposed en opy accele a ing he uni e se; bu he e,
unde lying he RW–me ic, en opy is associa ed wi h he
holog aphic in o ma ion s o age on he su ace sc een placed
a he ho izon o he uni e se. He e, we wo k wi hou his
concep . Fa nes p oposed a modi ied ΛCDM model,
whe e con inuously c ea ed nega i e masses can explain
bo h, da k ene gy as well as da k ma e phenomena.
In ensi e wo k o e 12 yea s, based on he RW–me ic, has
been done in he TRR 33 colabo a ion wi hou a
conclusi e b eak h ough. Hau e and Magain p oposed a
modi ica ion o he RW–me ic by in oducing a cosmological
ime in o de no o ha e o ely on he hypo hesis o da k
ene gy. We eel ha hei idea is pionee ing. Howe e , he
cosmological ime i sel is linked o he cu a u e o
space ime hus exceeding he scope o he unde lying RW–
me ic.
Using he RW–me ic, he edshi o he adia ion emi ed
om dis an sou ces co ela es wi h he g ow h o he cosmic
scale ac o , while he ime measu ed in an locally
ine ial mo ing e e ence ame lows uni o mly in he
uni e se. In ou app oach howe e , we a e d opping he
hypo hesis o a uni o m ime low he eby lea ing he
amewo k o como ing coo dina e sys ems. We assume,
ha any obse e in space possesses only i s own, local ime
scale. Due o he p inciple o cosmology, he ime scales o
obse e s should equal each o he s. Iso opy o space
demands ha he ime low does no depend on he
di ec ion in space; so, he edshi o he adia ion which is
emi ed by dis an sou ces and abso bed a he posi ion o an
obse e has o be iso op. This iso opy is gua an eed i we
choose a ame o e e ence which is a es ela i e o he
dipole co ec ed cosmic mic owa e backg ound (cmb)
adia ion; he o igin as posi ion o he obse e is eely
selec able. Independen o cosmological models, he
ba yonic acous ic oscilla ions o he cmb show ha he
spa ial cu a u e o space- ime is indis inguisible om ze o;
we ake his in o accoun by pu ing he adial scale ac o o
ou me ic o be one.
Whe eas usually, ´mass g ips space ime, elling i how o
cu e´ , he e, we will no s a wi h a gi en s ess-ene gy
enso , bu p opose ou me ic in sec. 2 and examine he
p ope ies o he co esponding s ess-ene gy enso in sec.
7. Unde lying ou me ic, we es ablish he Eins ein ield
equa ions o slowly a ying a iables as expec ed o he
p esen and u u e uni e se in sec. 3. In sec. 4, we p opose a
solu ion o he ield equa ions, which includes a dis ance-
dependen ime dila ion which is asc ibed o an objec a he
dis ance by an obse e a his posi ion . We may
conside his solu ion as an ex ension o he well known
e ec in special ela i i y, ha any obse e in his local
ine ial ame o e e ence asc ibes a ime dila a ion o all
o he d i eless mo ing objec s, e en ecip ocally. The
equa ions o he geodesic lines a e e alua ed in sec. 5 and
illus a ed in sec. 6, showing, ha any obse e no ices a
escape d i o all bodies away om his own posi ion and a
empo al g owing edshi o he cmb. A g ea e dis ances,
he imescale depends weake on he dis ance han in he
immedia e icini y o he obse e ; so, one does no expec
he edshi o ollow Hubble's law.
In sec. 8, we no ice, ha he adial equa ion has he shape o
he lux equa ion o en opy , he Fokke –Planck equa ion, i
2
is in e p e ed as he pe iod o ime which is disponible o
one uni o in o ma ion o be ansmi ed. In sec. 9, we
discuss ee Helmhol z en opy de e mining he expansion
a e o he uni e se.
The empi ical edshi -dis ance law o he SNIa adia ion
is quali a i ely in e p e ed in sec. 10. In ou app oach, he
ligh o ecen ly obse ed massi e galaxies wi h
is emi ed no only 500 – 700 My a e he big bang, bu
la e han hal o he age o he uni e se.
2. The me ic
In o de o mo i a e ou me ic, we s a wi h a gaussian
geodesic me ic wi h line elemen
,
hus omi ing angula coo dina es. In he case o he spa ially
la RW–me ic, we can iden i y wi h he cosmic adial scale
ac o , wi h he ime coo dina e and wi h
he dis ance . In ou me ic howe e , we swap he oles o
and ; we a e en i led o do so because an ex ension o he
wa eleng h o an obse ed adia ion canno be dis inguished
om a educ ion in equency. We use he line elemen
. (1)
He e, is he numbe o ime uni s passing in he
dis ance om an obse e a , i he measu es one
ime uni o himsel . In his p esen a ime he may se
. As shown in sec. 5, can be selec ed eely
in ou app oach. Whe eas, due o he p inciple o cosmology,
depends only on ime , may a y wi h ime
und dis ance .
Be o e he de ec ion o he cosmic mic owa e backg ound
(cmb) adia ion, he RW–me ic was he only concei able
ame o e e ence. In ou app oach, he ame o e e ence
has o be a es ela i e o he dipole co ec ed cmb; hus,
he iso opy o space is gua an eed.
Ou me ic has o desc ibe wo opposi e e olu ional
endencies o he uni e se: he con ac ion due o he
g a i a ion and he accele a ed expansion which we will
asc ibe o g owing en opy in sec. 8. So,
(2)
is ac o ised in wo pa s. Rega ding he u u e o he
uni e se, whe e g a i a ion is expec ed o loose impo ance,
we assume, ha he i s ac o
, wi h a ac ing
g a i a ional po en ial ha dly de ia es om one. Un il
sec ion 9, we will conside only he second pa , .
3. The Eins ein ield equa ions
The me ic (1) de e mines he o mulas o he Eins ein ield
equa ions. He e, we conside only e ms a ying weakly wi h
and , as expec ed in he u u e uni e se; so, he ea ly
uni e se is no conside ed. Thus, we s a wi h he Ricci–
enso in linea app oxima ion
. (3)
The enso elemen s in he ield equa ions a e gi en in Tab. 1.
While can be ime-dependen , he enso elemen s
con ain only de i a i es wi h espec o he dis ance ; we
ma k hem wi h dashes.
The sphe ical s ess-ene gy enso wi h ene gy densi y and
p essu e using me ic (1) is
. (4)
I will be conside ed in sec. 7.
Ricci enso elemen
Ricci enso elemen
Ricci enso elemen
0
Riemannscala
Eins ein enso elemen
Eins ein enso elemen
Eins ein enso elemen
Tab. 1: Tenso elemen s o he Eins ein ield equa ions o
weak ields, based on he me ic eq. (1) wi h .
He ewi h, he ield equa ions a e calcula ed:
, = cons an o g a i a ion)
o
(5)
o
(6)
o
(7)
4. Solu ions o he ield equa ions
Assuming ha he uni e se is homogenous, we seek
solu ions wi h spa ially cons an p und ρ o e y la ge
dis ances
. In his limi ,
in eqs. (5) and (6)
should no depend on . The me ic wi h scale ac o
mus be ime-dependen ; so, he solu ion o he
eqs. (5) ,(6), (7) is
(8)
3
wi h cons an . So, a all ixed loca ions , he
uni o ime inc eases and hus he equency o adia ion
emi ing sou ces dec eases wi h g owing dis ance . A he
posi ion o an obse e a , ,
independen o he ime .
Wi h
, we can w i e eq. (8) in he di usion o m
.
The Riemannscala
is independen o he
eely selec able posi ion o an obse e a ; hus, he
p inciple o cosmology is ull illed. In he ollowing, we
choose .
In he exponen ial unc ion eq. (8), we choose he nega i e
sign. We jus i y his conside ing Minkowski space ime as a
limi o cu ed space ime o weak ields: he e, any obse e
in his local ine ial sys em asc ibes a ime dila a ion wi h
o any o he mo ed ine ial sys em, e en
ecip ocally. This also applies o di e encies o eloci ies, i.e.
o d i eless mo ions in g a i a ional ields. Geodesic lines in
space ime glide o e in Minkowskian woldlines wi hou
up u e o he local me ic. Due o he nega i e sign, o
, we ep esen an expanding uni e se wi h densi y
and p essu e in eqs. (5) and (6) diminishing wi h ime
acco ding o
.
5. The equa ions o mo ion
We no e he equa ions o geodesic lines wi h Ch is o el
symbols
( :
. (9)
He e, we look o he equa ions o a body mo ing in adial
di ec ion ( ) in ou ime- a ying g a i a ional ield:
(10)
We in eg a e he i s eq. (10) ( ); wi h an app op ia e
cons an o in eg a ion, we ge
. (11)
So, we in e p e e as he p ope ime o he mo ing
body, .
The wo di e en ial equa ions eq. (10) can be sol ed
i e a i ely o calcula e eloci ies and posi ions o a body
mo ing along a geodesic line. By his way, he geodesics ha e
been d awn in Fig. 1. The e, he cu a u e o he su ace is
held cons an . He ewi h, he Fig. 2 shows he dis ance
a eled by a ecessing body as i is measu ed in he local
coo dina es o he pseudosphe e. The body s a s a 0 = 0 a
he ime wi h s a ing eloci y . The plo
coincides in good app oxima ion wi h he s aigh dashed line
which co espondes o a uni o m mo ion o he body.
6. Illus a ion o he escape mo ion o
bodies
In Fig. 1, we p esen he adial dependence o based
on he me ic eq. (8) as a snapsho o cons an ime .
Di e en o he usual p esen a ion, he ime axis is di ec ed
o he igh side and he dis ance upwa ds on he cu ed
su ace – he pseudosphe e – acco ding o he p esen a ion
in g aphical ime ables. The posi ion o an obse e is below
a . Wi h inc easing , he g owing ime in e als in he
g id co espond o a g owing ime dila a ion, , as
seen om he posi ion . The h ee cu es d awn in Fig.
1 a e geodesics s a ing a he ime wi h di e en
eloci ies , he lowe geodesic wi h ; du ing hei
un, he g id is ixed. Due o he locally dependen ime
dila a ion an obse e a an a bi a y posi ion
will obse e a adially symme ic escape mo emen o all
o he bodies, e en mu ually. Seen om ou side he cu ed
coo dina e g id, his mo ion appea s accele a ed in he
di ec ion o inc easing dis ance .
Fig. 1: Gaussian geodesic g id o he he su ace wi h a
cons an nega i e Riemann scala (pseudosphe e). In he
physical con ex o he RW–me ic, he e ical geodesic lines
a e he ime axes when posi ions a e ixed and he o hogonal
ho izon al lines desc ibe he possible posi ions o bodies a
ixed imes. Thus, an exponen ially expanding uni e se is
illus a ed by a adius o he pseudosphe e g owing
exponen ially wi h ime. Howe e , in ou con ex , he e ical
geodesics show he adial dis ances o bodies o an obse e
below a posi ion , he ho izon al cu ed lines ep esen
ime axes a he a ious posi ions . Th ee geodesics all
s a ing a he posi ion wi h di e en ini ial eloci ies
a e calcula ed i e a i ely and d awn in he pseudosphe e.
Due o he ime dila a ion g owing exponen ially wi h dis ance
, geodesic lines all exhibi an escape d i away om he
posi ion a . The lowes s a s wi h . Based on
he local coo dina es o he pseudosphe e, obse e s mo e
uni o mly along hei geodesic lines. On he o he hand, hei
mo ion appea s o be accele a ed, i conside ed om ou side
he su ace.
4
7. In e p e a ion o he s ess-ene gy
enso
Whe eas usually, ene gy densi y and p essu e p a e gi en
and he me ic is de i ed, he e, we ha e p oposed he me ic
in sec. 2; wha can we say abou he app op ia e s ess-
ene gy enso ? Since (see eqs. (5), (6)), i ollows
ha . To in e p e e his esul , we conside a hin
sphe ical mass bubble wi h adius : Inside he bubble, he
g a i a ional ield ene gy is anishing, ou side, i is nega i e.
I we enla ge he bubble by wi hou changing he mass
olume and mass ene gy o he bubble i sel , he ield
ou side o does no change; his is shown, simila o
he de i a ion o he Bi kho heo em, by w i ing down he
Ricci enso elemen o an iso ope, ime
dependen adially symme ic me ic . By inc easing he
ene gy densi y wi hin he sphe ical mass shell wi h hickness
om nega i e alues o ze o, he inc ease o he
g a i a ional ene gy o he sys em, , is comple ely
p o ided by he p essu e wo k .
Using eqs. (10), (11) o he accele a ion , we ge
(12)
He e, g is ela ed o he p ope ime
.
Due o eq. (12), he ield equa ion eq. (5) can be exp essed as
. (13)
Thus, coincides wi h he classical o mula o he ene gy
densi y o a g a i a ional ield,
, wi h he local
p ope ime unde lying he accele a ion in he
dis ance om an obse e . Whe eas in classical heo y, he
e e ence dis ance , whe e lies in in ini y, he e,
i is localized a he posi ion o any obse e ; hus, he sign is
e e sed.
In Gene al Rela i i y, se e e p oblems a ise when
g a i a ional ield ene gy has o be inco po a ed in he
Eins ein ield equa ions: he e, he ene gy–s ess enso
de e mina es he cu a u e o space ime, bu in u n,
cu a u e o space ime is connec ed wi h ene gy and mass.
Fo he case o weak ields, pos -New onian app oxima ions
a e equen ly used in which he non-linea , quad a ic
e ms in he Eins ein enso on he le side o he ield
equa ions a e assumed o be pa o he s ess-ene gy enso
on he igh side o he ield equa ions. He e, he Eins ein
enso elemen s and which we calcula e based on
he me ic eq. (8) consis exclusi ely o such e ms; his is
expec ed, since in eq. (8), we ha e no conside ed he
g a i a ional pa o he me ic in eq. (2) .
8. In o ma ion heo e ical analysis o eq. (5)
Elek omagne ic wa es anspo in o ma ion; he e, we
conside he connec ion be ween anspo ed in o ma ion
and ime dila a ion by means o an hough expe imen
using oy uni s: a wa e, which is emi ed wi h equency
Hz a a dis an posi ion A is de ec ed a posi ion B
wi h equency 0,5 Hz. An obse e a A may assume
ha he wa e anspo s an in o ma ion o 2 bi pe s (e.g. in
wo hal wa es); hen, B will ecei e in his second only 1 bi .
On he o he hand, he in o ma ion o 1 bi disposes o e a
ime in e all o 0,5 s a posi ion A, bu o 1 s in he ime
scale o posi ion B. This g ow h o empo ally eedom
co esponds o he g ow h o he a ailible olume e.g. in a
di usion p ocess wi h g owing en opy. Since is doubled
when going om A o B, we may connec he con en o
in o ma ion (en opy) pe second wi h ime dila a ion by
. (14)
I o example we ha e , he ime scale is sho ened
and bi .
We de ine
as he lux densi y o en opy.
Now, we wish o in es iga e he connec ion be ween he
en opy eq. (14) and he adial ield equa ion eq. (6). To do
so, we i s conside a gene al smoo h unc ion which
obeys he Fokke -Planck equa ion wi h anishing d i
coe izien :
. (15)
He e, he do means he pa ial de i a e wi h espec o he
ime and he dashes he de i a i es wi h espec o he
dis ance . I is cons an , eq. (15) educes o he di usion
equa ion. Eq. (15) can be ans o med in o a lux equa ion o
en opy : Di iding by , we ge
o
.
Wi h eq. (15) can be w i en as
. (16)
He e, ´ desc ibes he change o he lux o
en opy and is he sou ce e m o en opy.
-0,1
0
0,1
0,2
0,3
0,4
0,5
0,6
0 0,2 0,4 0,6 0,8 1
dis ance
p ope ime ds/c
Fig. 2. Dis ance a eled by a body; i is calcula ed
i e a i ely using eq. (10) based on he local coo dina es o
he pseudosphe e. The body s a s a a he ime
wi h s a ing eloci y . The uni s o he
dis ance and he eloci y in Fig. 2 a e no i ed o
expe imen al da a bu calcula ed in o de o display he
essen ial ea u es.
5
We compa e eq. (6) wi h eq. (16): o la ge , he adial ield
equa ion (6) can be in e p e ed as cons i uen o he Fokke -
Planck equa ion
, (17)
whe e is he lux densi y o
in o ma ion. The sou ce e m o en opy, , is ep esen ed
by he enso elemen G00 = G11 and he low change ´ is
gi en by he Ricci Tenso elemen R11. The empo al e olu ion
o en opy
(18)
is p opo ional o he Rieman scala ; i is posi e de ini e,
as equi ed by he p inciple o en opy.
In ou app oach, he accele a ion o he escape d i o he
uni e se,
(19)
is opposi e o he a ac ion p oduced by dis an g a i a ing
masses, desc ibed by he ac o
; i is connec ed
o he lux o en opy .
9. The modynamical aspec s
We now inco po a e a ac ing g a i a ion in ou model and
compa e i wi h a simple he modynamical desc ip ion. Due
o eq. (2), he g a i a ion is desc ibed by a imescale
g owing wi h dis ance om he posi ion o an
obse e a . Fo weak ields,
, (20)
whe e is he po en ial o he g a i a ional ield .
Combing eq. (20) wi h eq. (8), he en opy, desc ibing he
con en o ansmi ed in o ma ion pe ime uni is now
(21)
Fo weak ields,
; due o eq. (19), we ge
wo opposi e accele a ions
. (22)
They exp ess he g a i a ional comp ession and he en opic
expansion o .
In a simple model, we conside he p esen uni e se as a
sphe ical sys em wi h a cons an numbe o iden ical poin
pa icles, he galaxies, wi h g a i a ing mass , adius and
cons an spa ial densi y. The a e aged ene gy o one
pa icle due o he g a i a ional in e ac ion is
.
Since he a e aged dis ance o wo pa icles in he uni e se
is p opo ional o , we ha e
wi h an app op ia e
cons an . So, he g a i a ional accele a ion in eq. (22),
, is p opo ional o
. In he u u e, o la ge
alues o he coo dina e ime , ; so, he p ope ime
lows p opo ional o he coo dina e ime and he
a e aged dis ance o masses mo ing on geodesics is
g owing p opo ional o he p ope ime (see Fig. 2); so,
due o he en opic accele a ion, in he u u e, dissipa ion
domina es g a i a ion in eq. (22).
Compa ing ou geome ical app oach wi h he modynamics,
we equi e he maliza ion o he pa icles and in e a uni o m
empe a u e . The one-pa icle en opy is gi en by
. (23)
He e, is he inne ene gy o one pa icle, is he
Bol zmann cons an and is he ee Helmhol z ene gy. In an
ideal gas, desc ibes he numbe o
possible posi ions which can be andomly
´choosen´ i is he minimal equi ed olume pe pa icle.
As is well known, in an ideal gas, dissipa ion is d i en by he
g ow h o en opy wi hou any change o ene gy. In an eal
gas, en opy is esponsible, ha he gas dissipa es agains he
a ac ing molecula o ces he eby cooling down. Thus,
he e a e wo o ces :
(24)
a con ac ing po en ial o ce and a dissipa ing o ce ,
which is connec ed wi h he change o ee Helmhol z
ene gy. In ou app oach, we eplace he ´ ee choice´ o
posi ions by he ´ ee choice´ o he disponible ime in e al,
so
by . Then, using eq. (19)
. (25)
The ene gy a o ded o he expansion agains he con ac ing
o ce leads o he cooling o he sys em. On he o he
hand, he g ow h o en opy in i e e sible p ocesses is no
necessa ily accompanied by a change o ene gy;
ne e heless, i can be measu ed in e ms o ene gy, i an
al e na i e e e sible p ocess is conside ed wi h ini ial and
inal s a es equal o hose in he i e e sible p ocess. In ou
con ex , he ene ge ic ela ion eq. (13) gi es he ene gy
acco ding o an imagined e e sible expansion o he
uni e se, which is d i en by he en opic o ce . I is
posi i e, exp essing he ene gy a ailible in such a e e sible
p ocess due o he expansion.
10. E ec s leading o edshi o adia ion
We conside monoch oma ic ligh which is emi ed by a
sou ce mo ing along a geodesic line. The equency o ligh
emi ed om dis ance a he momen and de ec ed a
a he la e ime is shi ed due o
a) The al e a ion o he ime dila a ion du ing he un ime
o he ligh om he sou ce o he de ec o due
o he ime-dependence o he me ic.
b) The ime dila ion in he dis ance due o he loca ion
dependence o he me ic, see Fig. 1.
c) The Dopple shi caused by he escape eloci y o he
sou ce, see Fig. 2.
6
The shi o he wa eleng h
o ligh emi ed in he pas
om dis ance and de ec ed a is
. (26)
Unde lying ou me ic eq. (8), he ime de elopmen and he
dis ance dependence o a e balanced du ing he lying
ime o ligh a eling back om he dis ance in
he di ec ion o an obse e a ; so, we a e in i led o
neglec he e ec a) and b). By using local coo dina es
, we a e allowed o look a he p opaga ion o ligh as
in a Minkowski diag am (Fig. 3). Rega ding c), he obse e
no ices indi idual eloci ies o he bodies ushing away; a
his local ime , he obse e ecei es monoch oma ic ligh
which has been emi ed a di e en imes om he bodies.
Wi h g owing lying ime o he ligh and hus
g owing dis ance , he escape eloci ies
o he ligh emi ing bodies inc ease and hus he
Dopple shi s
(27)
o he emi ed adia ion. A alue co esponds o
; This escape eloci y is much highe han he ela i e
eloci ies o indi idual galaxies, in e sely as in he di usion o
molecules in gases.
We assume ha bodies can d i apa a di e en ini ial
speeds a e he big bang. Based on Fig. (2),
neglec ing he e a ding e ec o g a i y and ela i e
eloci ies, we assume ha he mo ion o di e en bodies is
uni o m. In linea app oxima ion, we ake in o accoun he
in luence o g a i y by assuming ha he escape eloci ies a
ime a e somewha educed compa ed o he uni o m
mo ion wi h . We exp ess his e ec by a co ec ion ac o
. So, in Fig. 3, ligh is emi ed om a body wi h eloci y
(28)
in dis ance a ime and egis a ed a ime . Since he
edshi da a co e dis ances wi h , we ha e
. (29)
The Hubble cons an is
.
Fig. 4 shows he g aph o eq. 29 wi h alue
.
The g aph is no i ed o expe imen al da a, bu shows he
ea u e, ha de ia es om he Hubble law o he
cosmological edshi . Unde lying he RW–me ic, his
ea u e is esponsible o he assump ion o da k ene gy o
exis . Since he Hubble esidual o is small, is only
sligh ly less han one.
Fo small dis ances , we may expand eq. (29) o ge
. (30)
This Hubble app oxima ion is ma ked in Fig. 4 as dashed line.
In ou app oach, he Dopple shi o he adia ion
co esponds o he emission ime . So, we can de ec
bodies emi ing adia ion only a imes
. Recen ly,
new obse a ions showed massi e galaxies wi h
only 500 – 700 My a e he big bang, when in e p e ed
in he amewo k o he s anda d model o cosmology .
In ou app oach, such high alues o need no co espond
o imes .
Fig. 3. Ske ch o he expanding uni e se as seen by an
obse e based on i s local coo dina e sys em.
Fig. 4. Redshi o he adia ion which is emi ed om
bodies in he dis ance . The s aigh line is d awn assuming
he Hubble law.
Re e ences
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Hau e C., Magain P. and Bi neaux J.: en opy (2017),
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Wheele , J. A.: A jou ney in o g a i y and space ime, p. xi,
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Fließbach T.: Allgemeine Rela i i ä s heo ie, Sp inge , 6.
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Risken H.: The Fokke -Planck equa ion, Sp inge -Ve lag
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Labbe, I.: na u e 616, 266-269 (2023)
0
0,2
0,4
0,6
0,8
1
1,2
1,4
0 5 10 15
z
/Gpc
dis ance
ligh escaping bodies
ligh
big bang ime
