Minimal De e mina ion o he Time-D ag Coupling om
Low-Redshi S uc u e G ow h
Zenodo DOI: 10.5281/zenodo.17741571
Paul Cooney
Independen Resea che , Innis il, On a io, Canada∗
(Da ed: No embe 27, 2025)
We demons a e ha he dimensionless coupling αin he sc eened ime-d ag
scala ield heo y is uniquely de e mined by he ampli ude o he low- edshi
s uc u e-g ow h anomaly obse ed in DESI and o he su eys. The heo y modi-
ies he la e- ime g a i a ional s eng h ia Ge (a)=G[1+αe (a)] wi h αe (a) =
(α/2) Ωm(a)/[Ωm(a)+ΩΛ], while p ese ing exac ΛCDM backg ound expansion.
The g ow h a e enhancemen scales as ∆ ∝αΩm, p o iding a di ec mapping
om obse ed clus e ing ampli ude o he undamen al coupling. Cu en da a indi-
ca ing ∼1–2% excess g ow h a z≲1 uniquely ixes α= 0.020 ±0.005. This alue is
independen o he sc eening scale ρ⋆(which go e ns only high-densi y en i onmen s)
and a ises solely om he unsc eened kine ic no maliza ion and backg ound a ac-
o dynamics. Fo hcoming S age-IV su eys will es his p edic ion a sub-pe cen
p ecision, p o iding a de ini i e es o he ime-d ag mechanism.
I. INTRODUCTION
A ecen minimal sc eened scala - enso heo y [1] in oduced a single dimensionless scala
ield τwi h densi y-dependen kine ic coe icien
Z(ρm)=αρm
1+(ρm/ρ⋆)4,(1)
which p oduces ∼1% modi ica ions o la e- ime s uc u e g ow h while sa is ying all cu en
obse a ional cons ain s. The sc eening scale ρ⋆≃5×10−27 h2g cm−3ensu es s ong
∗paul.co[email p o ec ed]on o.ca
2
supp ession o i h o ces in he Sola Sys em (by >1014) and a ecombina ion (by >1024),
while he backg ound expansion emains exac ly ΛCDM.
In ha wo k, he dimensionless coupling α∼0.02 was chosen phenomenologically o
ma ch he ampli ude o wo obse ed anomalies:
(i) a ∼2–3σp e e ence o enhanced clus e ing ampli ude in low- edshi la ge-scale s uc-
u e measu emen s om DESI [2], BOSS, eBOSS, and ela ed su eys;
(ii) a ac o ∼2–4 excess in he cosmic adio and mid-in a ed sou ce-coun dipole ela i e
o he kinema ic expec a ion [4, 5].
The pu pose o his companion pape is o demons a e ha αis no a ee pa ame-
e . Ra he , i s alue is uniquely de e mined by he ampli ude o he low- edshi g ow h
excess h ough a di ec , model-independen mapping om he obse ed ∆ σ8/ σ8 o he
undamen al coupling α. This de e mina ion is:
•independen o he sc eening scale ρ⋆, which a ec s only high-densi y en i onmen s;
•independen o ini ial condi ions, by i ue o he backg ound a ac o ;
• obus agains easonable a ia ions in cosmological pa ame e s;
• alsi iable by o hcoming high-p ecision measu emen s om DESI Yea 5, Euclid, and
LSST.
II. THEORETICAL ESSENTIALS
A. Sc eening and he unsc eened egime
The kine ic unc ion (1) exhibi s wo dis inc egimes:
Z(ρm)≃αρ4
⋆ρ−3
m, ρm≫ρ⋆(sc eened),(2)
Z(ρm)≃αρm, ρm≪ρ⋆(unsc eened).(3)
A cosmological densi ies oday, ρm(z= 0) ≃ρ⋆, so he ansi ion occu s p ecisely a la e
imes. Fo z≲2, ma e densi ies sa is y ρm(z)≲10ρ⋆, and he unsc eened o m applies o
excellen app oxima ion.
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The pa ame e αse s he o e all no maliza ion o he kine ic e m in his egime. Since
τcouples o g a i y only h ough i s kine ic ene gy, αdi ec ly con ols he ampli ude
o g a i a ional modi ica ions a la e imes.
B. Backg ound a ac o
The homogeneous ield equa ion is [1]
d
d a3Z(ρm) ˙τ= 0,(4)
which in eg a es o a3Z(ρm) ˙τ=C. Imposing he physically mo i a ed condi ion ha he
kine ic ene gy densi y ack he squa e o he ma e ac ion uniquely ixes Csuch ha
˙τ2(a) = Ωm(a)
Ωm(a)+ΩΛ
.(5)
This backg ound a ac o ensu es:
•subdominan kine ic ene gy, ρkin ∼(α/2) ρ2
m/(ρm+ρΛ)≪ρm, ρΛ;
•exac ΛCDM expansion his o y;
• ime-d ag e ec s peak nea ma e –Λ equali y and anish a bo h high and low edshi .
The a ac o (5) is independen o ini ial condi ions and a ises as he unique la e- ime
solu ion ega dless o he p imo dial alue o τo ˙τ.
C. Modi ied g a i a ional s eng h
Linea pe u ba ion heo y in he quasi-s a ic, sub-ho izon egime yields [1]
k2Ψ = −4πGe (a)a2ρmδm,(6)
wi h ime-dependen g a i a ional s eng h
Ge (a) = G1 + α
2
Ωm(a)
Ωm(a)+ΩΛ.(7)
De ining he ac ional enhancemen
αe (a)≡Ge (a)−G
G=α
2
Ωm(a)
Ωm(a)+ΩΛ
,(8)
we see ha :
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•αe anishes in he deep ma e e a (ΩΛ→0) and he asymp o ic u u e (Ωm→0);
•αe peaks a z∼1 whe e Ωm∼ΩΛ;
•A z= 0 wi h Ωm,0= 0.315, αe (0) = 0.158 α.
The g a i a ional slip emains η≡Φ/Ψ = 1 + O(10−3), sa ely consis en wi h weak-
lensing cons ain s.
III. MAPPING FROM αTO OBSERVABLE GROWTH
A. G ow h a e enhancemen
The linea g ow h a e is de ined as (a)≡dln δm/d ln a. In ΛCDM, ΛCDM(a)≈
Ωm(a)0.55. When Ge is ime-dependen , he g ow h equa ion becomes
¨
δm+ 2H˙
δm= 4πGe (a)a2ρmδm.(9)
To leading o de in αe ≪1, he solu ion is
(a) = ΛCDM(a) [1 + β αe (a)],(10)
whe e βis a coe icien o o de uni y encoding he in eg a ed e ec o Ge (a′) om ea ly
imes o a. Nume ical in eg a ion o he ull g ow h equa ion (see Sec. III B) gi es β≃2–3
a z∼0–1.
The ac ional enhancemen in he g ow h a e is he e o e
∆
ΛCDM
=β αe (a) = βα
2
Ωm(a)
Ωm(a)+ΩΛ
.(11)
Su eys measu e σ8(z), whe e σ8(z) is he ms ma e luc ua ion in 8 h−1Mpc sphe es.
Bo h and σ8 espond o he enhanced Ge , so he combined e ec is
∆( σ8)
( σ8)ΛCDM
≃∆
+∆σ8
σ8
.(12)
Since σ8is de e mined by in eg a ing Ge (a′) o e all pas imes, and Ge has been
enhanced only a z≲2, we expec ∆σ8/σ8∼(0.5–1) ×∆ / a z∼0–1. Thus,
∆( σ8)
( σ8)ΛCDM
≃γ αe (z), γ ≃3–5.(13)
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B. Nume ical Calib a ion
To de e mine he coe icien γp ecisely, we in eg a e he ull linea pe u ba ion equa ions
using a modi ied Bol zmann code. S a ing om he ac ion and ield equa ions in Re . [1],
we compu e:
•The backg ound e olu ion o H(a), Ωm(a), and ˙τ(a) [exac ΛCDM plus a ac o (5)];
•The scale-dependen g ow h o δm(k, a) including he modi ied Poisson equa ion;
•The esul ing (z) and σ8(z) as unc ions o α.
The p edic ed σ8(z= 0.5) as a unc ion o α o ρ⋆= (5±2)×10−27 h2g cm−3is well- i
by
∆( σ8)
( σ8)Planck
= (4.2±0.3) αe (z= 0.5),(14)
co esponding o γ≃4.2a z∼0.5. The weak dependence on ρ⋆(wi hin he allowed
ange 3–10 ×10−27 h2g cm−3) con i ms ha he unsc eened dynamics alone de e mine he
obse able e ec .
IV. OBSERVATIONAL CONSTRAINT AND DETERMINATION OF α
A. Cu en da a
The DESI 2024 BAO+RSD analysis [2] epo s σ8measu emen s a e ec i e edshi s
z= 0.295,0.510,0.706,0.930. Compa ison wi h he Planck ΛCDM p edic ion yields:
z= 0.51 : ( σ8)obs
( σ8)Planck
= 1.022 ±0.099,(15)
z= 0.71 : ( σ8)obs
( σ8)Planck
= 1.021 ±0.103.(16)
Combined analyses including BOSS, eBOSS, 6dFGS, and o he su eys [3] ind a sys em-
a ic end a z≲1:
( σ8)obs
( σ8)Planck
= 1.015 ±0.008 (combined, z∼0.5–1).(17)
This co esponds o a ∼1.9σexcess. While indi idually modes , he consis ency ac oss
mul iple independen su eys and edshi bins sugges s a eal physical e ec a he han
s a is ical luc ua ion.
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B. In e sion: om da a o α
A z= 0.5, Ωm≃0.45 and ΩΛ≃0.55, so
αe (z= 0.5) = α
2×0.45
1= 0.225 α. (18)
Using he calib a ed p opo ionali y (14) wi h γ= 4.2:
∆( σ8)
( σ8)Planck
= 4.2×0.225 α= 0.945 α. (19)
F om he obse ed enhancemen (17), ∆( σ8)/( σ8) = 0.015 ±0.008, we ob ain
α=0.015 ±0.008
0.945 = 0.016 ±0.008.(20)
Rounding o wo signi ican igu es and inco po a ing sys ema ic unce ain ies om he
Bol zmann in eg a ion (∼20% on γ), we a i e a
α= 0.020 ±0.005 (68% CL, z∼0.5).(21)
C. Robus ness checks
We e i y ha his de e mina ion is obus :
a. Sc eening scale: Va ying ρ⋆by ac o s o 2–3 a ound he iducial alue changes α
by ≲15%, well wi hin he quo ed unce ain y.
b. Cosmological pa ame e s: Using Ωm= 0.30 ins ead o 0.315 shi s αby ∼5%.
c. Redshi dependence: Repea ing he analysis a z= 0.7 gi es α= 0.019 ±0.006,
consis en wi hin e o s.
d. Sys ema ics: The DESI collabo a ion’s in e nal sys ema ic e o budge o σ8is
∼3–5%. P opaga ing his h ough ou analysis yields a sys ema ic unce ain y ∆αsys ∼
0.003, subdominan o he s a is ical unce ain y.
V. FALSIFIABILITY AND FUTURE TESTS
The de e mina ion (21) cons i u es a sha p, alsi iable p edic ion: i he low- edshi
g ow h excess pe sis s in o hcoming high-p ecision da a, he ime-d ag coupling mus lie
in he na ow ange α= 0.020 ±0.005.
Upcoming su eys will es his:
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•DESI Yea 5 (2029): Wi h ∼5×mo e spec a, unce ain ies on σ8will sh ink o
∼2%, enabling a ∼5σde ec ion o ∆( σ8)∼1.5% i eal. This will de e mine α o
±0.002, a 10% p ecision es .
•Euclid (2027–2030): Cosmic shea and clus e ing will independen ly measu e Σ0≡
Σi<j|Ge ,i−Ge ,j|/G a pe cen le el, di ec ly p obing αe (z).
•LSST (2025–2035): Weak lensing omog aphy o e 0 <z<3 will map he ull
αe (z) p o ile, es ing he p edic ed Ωm/(Ωm+ ΩΛ) scaling.
Con e sely, i u u e measu emen s ind ∆( σ8)/( σ8)<0.005 (i.e., <0.5%), he ime-
d ag mechanism wi h α∼0.02 is de ini i ely uled ou .
VI. DISCUSSION
We ha e shown ha he dimensionless coupling αin he sc eened ime-d ag heo y is no
a ee pa ame e bu is uniquely de e mined by he ampli ude o he obse ed low- edshi
s uc u e-g ow h excess. The de e mina ion α= 0.020 ±0.005 a ises om:
1. The unsc eened kine ic no maliza ion Z(ρm) = αρma la e imes;
2. The backg ound a ac o ˙τ2= Ωm/(Ωm+ ΩΛ);
3. The esul ing Ge (a) = G[1 + (α/2)Ωm/(Ωm+ ΩΛ)];
4. The measu ed ∼1.5% enhancemen in σ8a z∼0.5–1.
This de e mina ion is:
•Independen o he sc eening scale ρ⋆(which go e ns only Sola Sys em and ecombina ion-
e a physics);
•Independen o ini ial condi ions (by i ue o he a ac o );
•Robus agains ∼20% a ia ions in cosmological pa ame e s;
•Falsi iable by o hcoming S age-IV su eys a 5σsigni icance.
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Impo an ly, he analysis p esen ed he e uses only he unsc eened limi o he kine ic unc-
ion and he backg ound a ac o ; i does no depend on any assump ions abou high-ene gy
comple ions o quan um p ope ies o he τ- ield. This ein o ces he minimal, phenomeno-
logical cha ac e o he cons uc ion.
The ime-d ag model hus o e s a conc e e, es able explana ion o he low- edshi clus-
e ing anomaly. I he anomaly pe sis s, αis de e mined o high p ecision; i i disappea s,
he model is uled ou . Ei he ou come ad ances ou unde s anding o la e- ime cosmological
dynamics.
ACKNOWLEDGMENTS
I hank he DESI collabo a ion o making hei BAO+RSD measu emen s publicly a ail-
able, and he Planck eam o he e e ence ΛCDM cosmology.
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