$d_{\ell}(z)$ and BAO in the emergent gravity and the dark universe

0.39 MB·

by  Ding-fang Zeng

dℓ(z) and BAO in the emergent gravity and the dark universe
Ding-fang Zeng1,∗
1Theoretical Physics Division, College of Applied Sciences, Beijing University of Technology
We illustrate that ΛMOND cosmology following from E. Verlinde’s emergent gravity idea which
contains only constant dark energy and baryonic matters governed by linear inverse gravitation
forces at and beyond galaxy scales fit with the luminosity distance v.s. redshift relationship, i.e.
d (z) of type Ia supernovae equally well as the standard ΛCDM cosmology does. But in a rather
ℓ
broad and reasonable parameter space, ΛMOND gives too strong baryon acoustic oscillation, i.e.
BAO signals on the matter power spectrum contradicting with observations from various galaxy
surveyand countingexperiments.
7
1
PACSnumbers: 04.20.Cv,98.65Dx,95.35+d,95.36+x
0
2

n In a last month’s work [1], basing on insights from could its late time expansion features still be similar
a string theory, black hole physics and quantum informa- to that observed in the type Ia supernovae’s distance-
J
tion theory, Eric. Verlinde argues that the dark grav- redshiftrelationship? The secondis,could this ΛMOND

3 ity effects observed in galaxies and clusters convention- model reproduce the Λ plus Cold Dark Matter(ΛCDM

ally attributed to dark matters could be accounted for here after) model’s beautiful prediction of large scale
]
O by the modified newtonian dynamics (MOND hereafter) structure’s evolutionand growthor not? At first glance,
following from the emergence feature of gravitation and we may think that this two questions may not be an-
C
space-time itself. Although Verlinde does not quotient sweredproperlybeforeconcreteMONDformulationthus
.
h concrete method for modifying the newton gravitation well-established new gravitational field equation follow-
p theory,sothereisbigarbitrarinessesinitspredictionpos- ing from the emergent idea is spelled out. However, we
-
sibilities. His idea attracts much attention [2–9] as well will show in the following that this is not the case.
o
r ascriticisms[10]duetoitspotentialofkickingthelongly
st non-measureddarkmattercontentsoutofourknowledge
Xiv:1701.00690v1 [a wmrbtsgaieanoiirascemehveancHlysoenaelep.obrirxgulsnlgeessryesterrdeaoHaaruwvrrtvtvgso0iioaiactiaiottynaiatlalnasaiigtltontoixtrigoinfefioooeyonanrdsnerersesneais-tacscqyhsiarunbypu’aeclsvrserinenhvrtelihshconeeaerwuiswamssctamn.boenhicgpasccrbtVcegtoklheltireeaeucssdoearsqlo[n=pld1mliuafsalin1opnaiqrabd61t–nrpeaugiee1eHaogrmia-5orcnaifln0]-enparciatvtisgvnsrunaethiuevtrmadlotreuee.thisusrnceeeSegsla,eltdieyirhdtrfn,aeealocvaaecpttecwit.hhrtcgutiaoaasuh.nrr,pttrieaitsaetoihcffhrosontmelqfaeeyrntturmanghiajtaestueoe(trntwh1neosseidecrt)t---- utututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututMututututututututututututututututututututututututututututututututututut=utututututututututututututututututututututututututut4ututututututoutut3ututututπututututututututututututututautρututututututututututututututaututututututututututututututututututbut3utututmutututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututut
ar dominated universe a ∝ t23, a¨ ∝ −a−2, we expect that
the universe as a gravitation system should also experi- FIG. 1: o is arbitrary observer, while m an arbitrary co-
ence this square-inverseto linear inversetransitionas its movingtestparticleinanisotropicandhomogeneousexpand-
scale factor grows beyond ing universe. The energy conserving condition obeyed by m
and the mass/energy contents inside the spherical region M
as.f. a20 12 (2) centeredonosimply hastheform 21ma˙2r2−GMarm =k˜,with
magic ≈ 6 k˜ a constant determined byinitial conditions.
(cid:0) (cid:1)
This reasoning immediately brings us two questions ur-
AccordingtothestandardtextbookofS.Weinberg,in
gently. The first is, if the universe consists only of nor-
an isotropic and homogeneous universe
mal matters controlled by this modified gravitation the-
ory and constant dark energy(ΛMOND here after), then dr2
ds2 = dt2+a2(t) +r2dθ2+r2sin2θdφ2 (3)
− 1 kr2
(cid:2) − (cid:3)
simple newton mechanic and energy conservation laws
∗Electronicaddress: [email protected] are sufficient in determining dynamics of the scale fac-
2
tor a(t). Referring FIG.2 and captions there, according
to which the conventional Friedman equation could be
derived as follows
1 4πG(ρ +ρ )a3r3
ma˙2r2 m Λ =k˜ (4)
2 − 3 ar
a˙2 k 8πG
= (ρ +ρ ) (5)
⇒ a2 − a2 3 m Λ
while the energy conservation requires
ρ a3 =const.1, ρ =const.2 (6)
m Λ
Equations (5) and (6) constitute the full set of dynamic
constraints for a(t).
Now,letusapplytheabovemethodtotheMONDthe-
ory of E. Verlinde in which the matter-matter attractive
force has linearly inverse law at scales beyond galaxies
butthematter-darkenergyforcestillsatisfiesthe square
inverse law,
1GM m GM m FIG.2: Best fittingthe580datapointsoftheSCPUnion2.1
m m
Vmm =−ǫ (ar)ǫ ⇐F =−∇V =−(ar)1+ǫ (7) [16]with{Ωm,ΩΛ,H0}asfreefreeparametersintheΛMOND
cosmology by thesimple χ2-minimization method.
GM m GM m
mΛ mΛ
VmΛ =− ar ⇐F =−∇V =− a2r2 (8) Usingequation(10)andthestandarddefinitionincon-
ventionalsupernovaedataanalysis,wecanderiveoutthe
luminositydistance v.s. redshiftrelationinthe ΛMOND
1 1 a a
ǫ= + tanh 0 0 θ(as.f. a) (9) model,
2 2 a − as.f. a magic−
(cid:0) magic− (cid:1) a2c/(a H ) z dζ
We introduce a simple ǫ(a) function here to implement dℓ(z)= a0 Ωk0 120 sinn(cid:2)|Ωk|12 Z0 H(ζ)/H0(cid:3) (12)
the goalofchanging the early time square-inverselaw to | |
the latertime linearinverselawsmoothly, whereθ is the
usual heaviside step function featured by θ(x 6 0) = 0, H2(ζ)=H2 Ωma30 a 1−ǫ+Ω Ωka20 (13)
θ(0<x)=1. 0 a3 a Λ− a2
(cid:2) (cid:0) 0(cid:1) (cid:3)
Substituting the abovetwopotentialformulasintothe
conservation equation (4), what we get will become where we defined a0 ζ+1, sinn[x] = sin[x], x, sinh[x]
a ≡
as Ω > 0,= 0 and < 0 respectively. We are very lucky
k
a˙2 k 8πG ar/ǫ that the annoying factors r/ǫ appearing in (10) could be
a2 − a2 = 3 ρm(ar)ǫ +ρΛ (10) simplyabsorbedintothederfiǫnitionofΩ ,sothatwecan
(cid:2) (cid:3) m
now safely set ǫ=0 here. Now using observational data
In the case of k = 0 and the matter dominated era, the
complied in the SCP Union2.1 [16] and minimizations of
function a(t) could be explicitly solved out the following χ2-function
amond(t)= 8π3Grr/ǫǫ 2+1ǫ2+2 ǫt2+2ǫ|ǫ→0 (11) χ2 = [mth(zi,Ωm,H0,···)−mex(zi)]2 (14)
(cid:0) (cid:1) σ2
Xi i
Considering the fact that a˙ǫ→0 t0 const., while in
thestandardEinstein/Newtmononcdo∝smolo→giesa˙ t−13 0, mth(z,···)≡5log10[dℓ(z,···)/mpc]+25
∝ →
it is very surprising that the strengthened gravitation wefindthatthethreeparameter Ω ,Ω ,H ΛMOND
m Λ 0
{ }
force does not lead to strengthened deceleration and re- model fit with the observational data equally well with
collapsing evolutions of the universe. Of course, when the standard ΛCDM model. But with radically different
the constant dark energy is included, there existing re- best fitting parameters
collapses or not will be relevant with the relative weight
70.1km
ofρ ,ρ andk. Anyway,thisfactsuggestsusthatthis
m0 Λ0 ΛMOND: Ωm =0.81, ΩΛ =0.58,H0 = (15)
simple ΛMOND model may have radically different late s mpc
·
time expansion features relative to the standard ΛCDM 70.2km
ΛCDM: Ω =0.29, Ω =0.76,H = (16)
model. m Λ 0 s mpc
·
3
The former has χ2 = 562.313, while the latter has R a˙
v = 3iΘ + v˙ + v +ikΨ (28)
χ2 =562.40. Justaswe pointedoutunderequation(11) b − 1 τ˙ B a B
(cid:0) (cid:1)
that the strengthened gravitational force in this model
Under the so called tightly-coupling limit, Hu and
between matters does not make the acceleration of the
Sugiyama show [20] that this equation array has simple
universe difficult. Instead they make such accelerations
oscillation solution
more easier so that more less dark energy is need in ac-
complishing the observed acceleration!
Θ (η)+Φ(η)=[Θ (η)+Φ(η)]cos(kr ) (29)
0 0 s
The most big difficult a cosmological model without
k η
non-baryonic dark matter may encounter is that, it may + dy Φ(y) Ψ(y) sin k[r (η) r (y)]
lead to too strong baryonic acoustic oscillation (BAO) √3Z0 (cid:2) − (cid:3) { s − s }
signalonthepowerspectrumofmatterdistributionssuch
asthoseobservedtypicallyin2dFGRS[22]andSDSS[17, η 3ρ
18] galaxy survey and counting experiments. Recalling rs ≡Z dycs(y), cs ≡(3+3R)−12, R≡ 4ρB (30)
that in the standard cosmological perturbation theory 0 γ
[19] This is just the baryon acoustic oscillation. It originates
fromthesoundmodeoscillationoftherelativisticplasma
ds2 =a2(η) (1+2Ψ)dη2+(1+2Φ)δ dxidxj (17)
− ij in the early universe. At redshift z 1000, the recom-
(cid:2) (cid:3) ≈
bination occurs so that the big bang plasma becomes a
neutralgasandtheoscillationstopspropagatinganyfur-
a˙ a˙
k2Φ+3 φ˙ Ψ =4πGa2 ρ δ + (18) ther. Butperiodicspatialinhomogeneityfeatureitbrings
a − a D D
(cid:0) (cid:1) (cid:0) continues to exist and evolves to the present time. In
ρ δ +4ρ Θ +4ρ
B B γ 0 νN0 thestandardCDMmodel,thebaryonicandnon-baryonic
(cid:1)
darkmatterscoexistevenbeforetherecombination,with
the former to latter ratio equates about 1. Since dark
k2(Φ+Ψ)= 32πGa2 ρ Θ +ρ (19) 5
− γ 2 νN2 matters do not participatein the soundwaveoscillation,
(cid:0) (cid:1)
the strength of baryonic acoustic oscillation signals on
The last two equations follow from Fourier transforma-
the powerspectrumof totalmatters observedin the late
tions of the linearised Einstein equation. Quantities on
time universe is very small. In cosmologicalmodels such
their right hand side are just the 1st and 2nd multipole
asΛMOND where non-baryonicdarkmatters do not ex-
expansionsof the correspondingparticle’s statisticaldis-
ist at all, to explain the smallness of of this signal is the
tribution
main challenge.
p −1
photon: fΘ = eT[1+Θ(~x,pˆ,t)] 1 (20)
−
(cid:2) p (cid:3)−1
nutrino: fN = eT[1+N(~x,pˆ,t)] +1 (21)
dark matter: f =(cid:2) , (cid:3) (22)
δD
···
baryon matter: f = (23)
δB
···
iℓ 1
Θ (~x,t) dcosθΘ(~x,cosθ,t)P (cosθ) (24)
ℓ ≡−2Z−1 ℓ
, , δ δ , δ δ similarly defined
N0 N2 D ≡ D0 B ≡ D0
Unlike Ψ and Φ, all these multipole’s evolution is con-
trolled directly by the Boltzmann instead of Einstein
equation
FIG. 3: In standard ΛCDM model (green line), the BAO
df
=C[f(p~)] (25) signal manifests only as small wiggles on the matter power
dt spectrum. But in the ΛMOND model (red line), this signal
manifests as strong oscillations of the power spectrum line.
The concrete form C depends on the particle type and
In the left panel, we replace the dark matter of ΛCDM with
theirmutualinteractions. Atfirsttwolevels,the compo-
baryonic matters in ΛMOND, while in the right panel, we
nent equation relevant with the baryon acoustic oscilla-
replace it with dark energy. In both panels, H0 is set as
tion reads 70km/(s·Mpc).
Θ˙ +kΘ = Φ˙, τ(η) n σ a (26)
0 1 − ≡− e T We explore in the following if modifying the standard
Friedmannequationintotheform(9)-(10)andlettingall
k k iv linear perturbations evolve in this background the same
Θ˙1 Θ0 = Ψ+τ˙ Θ1 B (27) way as the usual Einstein-Botzmann formulae require
− 3 3 − 3
(cid:2) (cid:3)
4
brings us suppressions of the BAO signal as required To obtain a suppressed BAO signal, we try in FIG.4
by observations. Our logic is, although we do not know using Ω ,Ω parameters following from best fittings of
m Λ
whatthefullgravitationalfieldequationgrowslikeinthe the d (z) relation of type Ia supernovae in the previous
ℓ
framework of emergent idea, in an exactly isotropic and section, where spatial curvatures contribute remarkably
homogeneous universe its key features are captured by heavier to the energy contents of the universe. However,
(9)-(10), while its linear perturbation, as long as being even when we let the magic scale factor be a tuneable
second order partial differential equations, would then parameter, we do not obtain the required results. This
not deviate from the Einstein perturbations too much. means that, new mechanisms must be find to suppress
Under this logic, we integrate the whole system of the BAO signal in this ΛMOND cosmology to make it a
Einstein-Boltzmann differential equations by the stan- competing model of ΛCDM.
dard code of CAMB [21]. The results is displayed in
Conclusion: we derive out dynamic equations control-
FIG.3 and 4 explicitly. From FIG.3, we easily see that,
ling the evolution of scale factors in a simple ΛMOND
replacing the conventional cold dark matter with either
cosmologywhichcontainsonlyconstantdarkenergyand
baryonic matter of MOND or dark energy of cosmologi-
baryonic matters governed by linear inverse gravitation
calconstantbothbringustoostrongBAOsignalsonthe
forces at and beyond galaxy scales. We find that the
matter power spectrums measuredobservationally. Nev-
model fit with observational data type Ia supernovae’s
ertheless, in the former case, ΛMOND and ΛCDM has
luminositydistancev.s. redshiftrelationshipequallywell
approximatelythesamefirstpeakpositionsonthepower
withthestandardΛCDMmodeldoes. However,sinceno
spectrum. This is expectable because it could be proved
dark matter is assumed, the model predicts too strong
analytically basing on the tight-coupling approximation
baryonicacousticoscillationsignalsonthe matterpower
(29)whosevalidity hasnorelevancewithassumptionsof
spectrum than the standard ΛCDM does. Nevertheless,
the ΛMOND model.
ΛMOND has the same position of first BAO peak as
ΛCDMdoesifwereplacedarkmattersinthe latterwith
baryonic matters in the former. So a reasonable mecha-
nismtosuppressthestrengthofBAOsignalsmaybethe
mosturgentingredientofΛMONDinitsroadofgrowing
into competing models of ΛCDM.
Acknowledgements
FIG. 4: Similar as FIG.3, but we used Ωm,ΩΛ parameters This work is supported by Beijing Municipal Natu-
following from best fittings of the d (z) relation of type Ia ral Science Foundation, Grant No. Z2006015201001and
ℓ
supernovae. In the left panel we let as.f. = (1/6)1/2 as partly by the Open Project Program of State Key Lab-
magic
required by E. Verlinde’s argument. While in the right, we oratory of Theoretical Physics, Institute of Theoretical
tries to set it to more smaller values. Physics, Chinese Academy of Sciences, China.
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