A quasi-static Time-Field framework reproduces the low-redshift Hubble-diagram shape and yields falsifiable redshift-drift predictions

A quasi-s a ic Time-Field amewo k ep oduces he low- edshi
Hubble-diag am shape and yields alsi iable edshi -d i
p edic ions
E ic L. Le in1
1Independen Resea che - Kennebunkpo , Maine,
www. ime ield.ne (617) 283-4468
Oc obe 8, 2025
Abs ac
We es he quasi-s a ic A2 ealiza ion o he Time Field Hypo hesis (TFH) agains
Pan heon(+) Type Ia supe no ae a low edshi (
z≤
0
.
35). Using a scale- ee econs uc ion
o
µ
(
z
) wi h a mono one PCHIP spline and en o cing a mono onic p oxy
(
z
)
∝dL
(
z
)
/
(1+
z
)
2
,
we ep ojec o
µmodel
(
z
) and i only a cons an o se ∆
µ
, he eby abso bing he absolu e
dis ance scale. Adop ing a s anda d in insic dispe sion
σin
= 0
.
09 mag, we ob ain
χ2/ν
=
0
.
998 (
ν
= 703) wi h RMS esidual 0
.
104 mag on he binned low-
z
subse . In A2, ime
e olu ion en e s p ima ily h ough edshi d i a he han one-epoch dis ances; we p o ide
compac
ϵ
–
τ
o ecas s, whe e d
z/
d
∝
(1 +
z
)
ϵ
[
(
z
)]
/τ
. These esul s es ablish empi ical
low-zconsis ency o TFH while isola ing alsi iable ime-domain p edic ions.
1 In oduc ion
Fo nea ly a cen u y, he obse ed cosmological edshi has been in e p e ed as e idence o an
expanding uni e se wi hin he ΛCDM amewo k. While his model succeeds b oadly, ensions
such as he disag eemen be ween local and ea ly-uni e se de e mina ions o
H0
mo i a e
complemen a y app oaches. The Time Field Hypo hesis (TFH) explo es a lapse-s uc u ed
me ic as an al e na i e ou e o edshi phenomenology wi hou in oking an explici ime-
dependen scale ac o . P io concep ual d a s amed edshi as an in eg a ed g a i a ional
e ec in a s a ic, homogeneous uni e se and p oposed es able signa u es linking dis ance scales
and obse ed slopes o he Hubble diag am (ca ied o wa d he e as his o ical con ex and
mo i a ion).1
In his pape we: (i) pe o m a da a-d i en, scale- ee low-
z
econs uc ion o he Hubble-
diag am shape and es i s ep oduc ion in TFH-A2 wi hou using
H0
; (ii) p esen edshi -d i
p edic ions ha isola e he ime-domain consequences o A2 ia pa ame e s (
ϵ, τ, ⋆
); and (iii)
documen obus ness o easonable analysis choices.
Rela ion o Timescape and lapse ideas. F amewo ks ha dis inguish ope a ional clock
a es (e.g., Timescape) mo i a e conside ing lapse mappings. Ou Pa h B is conse a i e and
obse a ional: i al e s edshi in e ence wi hou changing EFEs, le ing da a decide whe he a
smoo h lapse can mimic low-zkinema ics while lea ing dynamics in ac .
1Concep ual aming adap ed om he au ho ’s p io d a on TFH and Hubble-law in e p e a ion.
1
2 Theo y: S a ic Lapse, EFEs, and Obse a ional Mapping
2.1 S a ic, sphe ically symme ic me ic
We wo k in a s a ic, sphe ically symme ic line elemen
ds2=−e2ψ( )c2d 2+e2Λ( )d 2+ 2dθ2+ sin2θdϕ2,(1)
wi h dimensionless po en ials
ψ
(
) and Λ(
). The mixed Eins ein equa ions o a pe ec luid
Tµν= diag(−ρc2, p, p, p) gi e he usual TOV s uc u e,
8πG
c2ρ=e−2Λ
22 Λ′−1 + e2Λ,(2)
8πG
c4p=e−2Λ
22 ψ′−1 + e2Λ,(3)
p′=−(ρc2+p)ψ′,(4)
whe e p imes deno e de i a i es d/d .
2.2 Null geodesics and g a i a ional edshi
In a s a ic space ime, he Killing ene gy along pho on geodesics implies a pu ely g a i a ional
(s a ic) edshi ,
1+z=νem
νobs
= expψ( )−ψ(0),dz
d = (1 + z)ψ′( ).(5)
Thus he local slope ψ′( ) con ols he z( ) ela ion in he baseline s a ic geome y.
2.3 An obse a ional lapse mapping (Pa h B)
We in oduce a phenomenological lapse mapping
γ
(
)
>
0 ha escales ope a ional clock a es
wi hou modi ying he EFEs,
ψe ( ) := ψ( ) + ln γ( ),1 + z= expψe ( )−ψe (0),(6)
leading o dz
d = (1 + z)ψ′
e ( ) = (1 + z)hψ′( ) + d
d ln γ( )i.(7)
We es whe he he kinema ics implied by
ψe
can ep oduce he obse ed low-
z
Hubble-diag am
shape; absolu e scale is no used.
2.4 Scale- ee dis ances and cosmog aphy link
Fo any µ(z), de ine a scale- ee p oxy
(z)∝dL(z)
(1+z)2=10µ(z)/5
(1+z)2.(8)
A small
z
, expanding Ψ(
)
≡ψe
(
) a ound
= 0 and using Eq.
(7)
one ob ains he s anda d
se ies H0dL(z)
c=z+1−q0
2z2−1−q0−3q2
0+j0
6z3+O(z4),(9)
wi h kinema ic pa ame e s (
H0, q0, j0
) encoded by de i a i es o Ψ. Because we i only a
cons an ∆µ(abso bing H0), ou low-zanalysis p obes shape.
2
Pa h A (comple eness). I one ins ead embeds he lapse in
g
(“Pa h A”), EFEs a e modi ied
ia ψ′and ψ′′, gene ally equi ing ex a s ess-ene gy; we do no pu sue ha he e.
3 Me hod: scale- ee econs uc ion and e alua ion
Da a and binning. Pan heon(+) low-
z
(
z≤
0
.
35). Exac -
z
in e se- a iance binning p oduces
s ic ly inc easing abscissa; binned esiduals a chi ed as ou /a2 lowz esiduals.cs .
Scale- ee econs uc ion. Fi mono one PCHIP o
µ
(
z
), e alua e on a dense g id, map
o
(
z
)
∝dL/
(1 +
z
)
2
wi h a cumula i e-maximum mono onici y en o cemen , hen ep ojec
µmodel = 5 log10[ (1+z)2]. Fi only ∆µ; e alua e χ2wi h σin added in quad a u e.
P ima y analysis choice. We adop
σin
= 0
.
09 mag o achie e
χ2/ν ≈
1 on he
z≤
0
.
35
subse .
4 Me hods: design choices and jus i ica ions
Why
z≤
0
.
35.Low edshi minimizes selec ion/popula ion d i ; he se ies in Eq.
(9)
con e ges
well; we es shape, no absolu e scale.
Exac -zbinning. Requi ed o s ic ly inc easing z(PCHIP) and s able weigh s.
Mono one PCHIP and
(
z
)mono onici y. P e en s oscilla ions; en o ces physical mono-
onici y in he e-p ojec ion.
Scale- ee i and ∆µ.Explici ly emo es H0/calib a ion dependence.
In insic dispe sion. σin uned wi hin s anda d SN p ac ice.
5 Resul s: low-zHubble-diag am shape
Why low-
z
i s . In A2, he s a ic lapse con ols one-epoch dis ances while ime e olu ion
appea s as edshi d i ,
˙z∝
(1 +
z
)
ϵ
[
(
z
)]
/τ
. Hence: es s a ic shape a low
z
, hen con on
d i .
zmax 0.35
σin [mag] 0.09
∆µ[mag] −0.023
χ2/ν (wi h ν= 703) 0.998
RMS [mag] 0.104
Table 1: A2 low-
z
baseline on Pan heon(+) binned SNe (
z≤
0
.
35). Values om
ou /a2 lowz summa y.json.
3
Figu e 1: Low-
z
Hubble diag am (binned Pan heon(+),
z≤
0
.
35) wi h A2 scale- ee e-p ojec ion
plus i ed o se ∆µ.
Figu e 2: Residuals µ−µmodel −∆µwi h σin = 0.09 mag in quad a u e.
6 Robus ness
We sweep
zmax ∈ {
0
.
30
,
0
.
35
,
0
.
40
}
and
σin ∈ {
0
.
08
,
0
.
09
,
0
.
10
}
; see
ou /a2 obus ness able.cs
.
Ac oss his ange χ2/ν ∼1 and RMS ∼0.10–0.11 mag.
4
Table 2: Robus ness o he edshi cu
zmax
and in insic dispe sion
σin
. Values a e compu ed
on he binned Pan heon(+) low-zsubse wi h he A2 scale- ee ep ojec ion.
zmax σin [mag] N χ2/ν RMS [mag] ∆µ[mag]
0.30 0.08 694 0.442 0.066 -0.013
0.30 0.09 694 0.375 0.066 -0.013
0.30 0.10 694 0.322 0.066 -0.013
0.35 0.08 704 1.187 0.105 -0.024
0.35 0.09 704 0.998 0.104 -0.023
0.35 0.10 704 0.847 0.104 -0.022
0.40 0.08 716 3.072 0.179 -0.043
0.40 0.09 716 2.625 0.178 -0.041
0.40 0.10 716 2.260 0.178 -0.039
7 Redshi -d i p edic ions (A2)
In he A2 pa ame e iza ion Φ = ψ+ϵ ( )h( ) wi h ˙
h= 1/τ, he ins an aneous d i is
dz
d ≈(1+z)ϵ [ (z)] 1
τ.(10)
Figu e 3shows a g id a
z
= 0
.
20; Figu e 4shows d i s.
z
. S a ic TFH (
ϵ
= 0) p edic s
dz/d = 0.
Figu e 3: Redshi d i a
z
= 0
.
20 o a small g id in (
ϵ, τ
). See
ou /a2 d i g id z0p20.cs
.
Figu e 4: D i s. z o a ep esen a i e (ϵ, τ, ⋆) choice.
5

8 Low-zcosmog aphy c oss-check
A cosmog aphic i o he same binned subse wi h a ee o se
M
e u ns smoo h (
q0, j0
) and
χ2/ν ∼1, consis en wi h he esidual sca e ; his is a sani y check, no a model compa ison.
9 Discussion
TFH-A2 ep oduces he Hubble-diag am shape a he
∼
0
.
10 mag le el wi hou absolu e calib a-
ion. One-epoch dis ances p obe he s a ic lapse; ime e olu ion appea s in edshi d i . Hence
TFH is es able ia (i) low-
z
shape ( alida ed he e) and (ii) ime-domain d i (p edic ed he e).
High-z/BAO will be ea ed sepa a ely.
On Pa h A (de e ed). An al e na i e is o embed he lapse di ec ly in
g
(Pa h A), which
al e s he EFEs ia
ψ′
and
ψ′′
and gene ally demands addi ional s ess–ene gy. A ull dynamical
ea men o Pa h A—including he sou cing equi ed o sa is y he ield equa ions—is de e ed
o a companion heo y pape .
10 Conclusions
A scale- ee econs uc ion o
µ
(
z
) o
z≤
0
.
35 shows TFH-A2 ep oduces he low-
z
Hubble-
diag am shape wi h
χ2/ν ≈
1 and RMS
≃
0
.
10 mag using only ∆
µ
. D i scales as d
z/
d
∝
(1+z)ϵ/τ (up o [ (z)]), in i ing decisi e cons ain s wi h long-baseline spec oscopy.
Da a and code a ailabili y. Figu es and machine- eadable ables a e in he supplemen ZIP;
Colab cells ep oduce all esul s om pan heon clean.cs .
Appendix A: Implemen a ion de ails
Binning and mono one spline. Exac -zin e se- a iance binning ensu es s ic ly inc easing
abscissa o PCHIP; mono onici y o (z) en o ced by cumula i e maximum.
E o model. σin = 0.09 mag (swep in [0.08, 0.10]).
A i ac s expo .
igs/a2 lowz hubble.[png|pd ]
,
igs/a2 lowz esiduals.[png|pd ]
,
ou /a2 lowz esiduals.cs
,
ou /a2 lowz summa y.json
, and (i gene a ed)
ou /a2 d i g id z0p20.cs
,
igs/a2 d i g id z0p20.[png|pd ], igs/a2 d i s z.[png|pd ].
Appendix B: Field-equa ion summa y, assump ions, and de i a-
ion map
B.1 Con en ions and scope
•Me ic signa u e (−,+,+,+); c e ained in o mulas; 8πG explici .
•S a ic, sphe ically symme ic line elemen ,
ds2=−e2ψ( )c2d 2+e2Λ( )d 2+ 2(dθ2+ sin2θdϕ2).
•S ess–ene gy (baseline) Tµν= diag(−ρc2, p, p, p).
•
Pa h B (obse a ional lapse) used in he main ex :
ψe
(
) =
ψ
(
) +
ln γ
(
) modi ies
edshi in e ence bu does no change he EFEs.
6
B.2 Iden i ies used in he main ex
8πG
c2ρ=e−2Λ
22 Λ′−1 + e2Λ,(11)
8πG
c4p=e−2Λ
22 ψ′−1 + e2Λ,(12)
p′=−(ρc2+p)ψ′,(13)
1+z= expψe ( )−ψe (0),dz
d = (1 + z)ψ′
e ( ).(14)
Scale- ee dis ance p oxy:
(z)∝dL(z)
(1+z)2=10µ(z)/5
(1+z)2.
B.3 Small-zse ies (cosmog aphy link, scale- ee)
Expanding Ψ( )≡ψe ( ) nea = 0 and using he edshi slope yields
H0dL(z)
c=z+1−q0
2z2−1−q0−3q2
0+j0
6z3+O(z4),
wi h (
H0, q0, j0
) unc ions o Ψ
1,
Ψ
2,
Ψ
3
. In his pape we i only a cons an ∆
µ
(abso bing
H0
)
and es shape h ough cu a u e e ms.
C oss-check ( educ ion o s anda d cosmog aphy). In he limi o a i ial obse a ional
lapse,
γ
(
)
→
1 so ha
ψe →ψ
, he small-
z
se ies in Eq.
(9)
educes o he usual cosmog aphic
expansion wi h (
H0, q0, j0
) de e mined by he de i a i es o
ψ
alone. This p o ides a consis ency
check ha he Pa h B mapping in oduces no spu ious e ms a low edshi beyond hose
encoded by ln γ.
B.4 De i a ion map (whe e o ind de ails in he supplemen )
Main- ex i em Supplemen sec ion
Me ic, Ch is o els, Ricci, Gµν§1–2
Mixed-componen EFEs & p′=−(ρc2+p)ψ′§3
S a ic edshi and dz/d §4
Obse a ional lapse map ψe =ψ+ ln γ§5
Small-zexpansion linking o (H0, q0, j0)§6
Pa h A ema k (embedding lapse in g )§7
B.5 Assump ions and limi a ions ( o his pape )
•Low-zscope (z≤0.35); absolu e scale abso bed in o ∆µ.
•No claim abou global ma e con en beyond s a ic EFEs used abo e.
•Pa h B is kinema ic/obse a ional; dynamical implica ions o Pa h A a e de e ed.
7
Appendix C: Symbols and no a ion
Symbol Meaning
ψ( ),Λ( ) S a ic me ic po en ials in Eq. (1)
γ( ) Obse a ional lapse mapping (Pa h B)
ψe ( )ψ( ) + ln γ( ), con ols in e ed edshi
ρ, p Ene gy densi y and p essu e (pe ec luid)
µ(z) Dis ance modulus; dL= 10µ/5(pc)
(z) Scale- ee p oxy ∝dL/(1+z)2
∆µFi ed cons an o se (abso bs absolu e scale)
σin In insic dispe sion added in quad a u e
ϵ, τ, ⋆A2 ime-pe u ba ion ampli ude, imescale, en elope adius
Supplemen a y Ma e ial
The ile EFE De i a ion 2 (Oc obe 2025) accompanies his manusc ip and con ains comple e
de i a ions: me ic connec ions and cu a u e, mixed-componen EFEs and hyd os a ic balance,
s a ic edshi and he obse a ional lapse mapping, and he small-
z
expansion linking
ψe
de i a i es o (H0, q0, j0). See Appendix B.4 o a de i a ion map.
The supplemen ZIP (Zenodo) con ains igu es (PNG+PDF), esiduals and summa y
(CSV/JSON), and he obus ness able. A Colab no ebook (no ed in he README) egene a es
all ou pu s om pan heon clean.cs . Ve sion and checksums a e lis ed in MANIFEST. x .
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