A Functional Interpretation of Gravitational Inaccessibility: The Hypothesis of Vibrational Wave Dispersion (HDOV)

A Func ional In e p e a ion o G a i a ional Inaccessibili y:
The Vib a ional Wa e Dispe sion Hypo hesis (HDOV)
A noldo Wal e Fe nández
[email p o ec ed]
July 5, 2025
Ve sion 4 – p ep in o academic dissemina ion and eedback
Abs ac
We p esen a consolida ed e sion o he HDOV (Vib a ional Wa e Dispe sion Hypo hesis)
o malism, whe e he unc ional accessibili y o modes unde cu a u e plays a cen al ole. In
his amewo k, he e ec i e dynamics includes a dispe sion e m ha encodes he p ojec ion
o modes ou side he ope a ional subspace o he measu emen agen . We de elop compac
de i a ions, a WKB eading o ampli ude anspo , and indi ec alida ions wi h Supe no ae
Ia (Pan heon) and BAO (SDSS DR16). We epo ep oducible ables (SN only and SN+BAO,
wi h p io in H
0
) and a syn hesis o g a i a ional wa e ingdown. The goal is o o e a
sel -con ained and alsi iable p esen a ion, eady o a bi a ion.
1
Con en s
1 In oduc ion 3
2 Fundamen als o he HDOV F amewo k 3
2.1 Mo i a ion om E ec i e Field Theo y ....................... 3
2.2 Ac ion and P ojec i e Wa e Equa ion ........................ 3
2.3 WKB anspo and a enua ion law wi hou dimensional ambigui y ....... 4
2.4 No maliza ion, dimensions and no a ion (cla i ica ion) ............... 4
2.5 Regula and in a ian amily o ηp.......................... 4
2.6 E ec i e o igin o ηpand e e ence scales ....................... 4
2.7 G a i a ional Field o View (GFoV) ......................... 5
2.8 Conse a ion check in he classical limi ....................... 5
2.9 PPN check and sola limi s .............................. 6
3 De i a ions and unc ional a ian s 6
3.1 Typical p o ile o ηp( ) ................................. 6
3.2 Va ian s and eedback ................................. 7
4 Rep esen a i e applica ions 8
4.1 Expansion unc ional esponse ............................. 8
4.2 Cosmology-quan um duali y (illus a ion) ...................... 9
5 Cosmological alida ion: SN Ia (Pan heon) and BAO 9
5.1 Fi ing igu es and esiduals .............................. 10
5.2 Complexi y con ol: AIC/BIC ............................. 11
5.3 Rep oducible ables .................................. 11
6 G a i a ional wa es: expanded ingdown syn hesis 12
7 Discussion and Conclusions 14
7.1 HDOV in he con ex o o he heo e ical models .................. 14
7.2 Limi a ions and Fu u e Di ec ions .......................... 15
7.3 Falsi iable P edic ions ................................. 15
Appendices 15
A De i a ion om E ec i e Field Theo y 15
A.1 E ec i e O igin o ηp.................................. 15
B Hype bolici y, Causali y, and S a is ical Resul s 15
B.1 Hype bolici y and Causali y .............................. 15
B.2 S a is ical Robus ness ................................. 15
B.3 Consis ency o S a is ical Resul s ........................... 16
B.4 Fi ed Pa ame e s and 1σE o s ........................... 16
B.5 Ringdown pos e io s .................................. 16
C E ec i e Ene gy-Momen um Tenso in HDOV 17
2
1 In oduc ion
The HDOV hypo hesis p oposes a geome y-dependen unc ional p ojec ion mechanism ha
modula es he accessibili y o deg ees o eedom. Mo i a ed by ensions be ween Gene al
Rela i i y and Quan um Mechanics in high cu a u e egimes, HDOV sugges s ha pa o
wha we in e p e as inaccessibili y o mani es a ion (e.g., e ec i e expansions) a ises om a
dynamic coa se-g aining on an e ec i e medium. This "coa se-g ained" p ocess implies ha
mic oscale in o ma ion becomes inaccessible o a mac oscopic measu emen agen , no because i
is des oyed, bu because i is dispe sed in o unmeasu able deg ees o eedom. The ocus o his
wo k is a minimal, e i iable o mula ion compa ible wi h s anda d quan i iable es s (Almhei i
e al.,2013).
This wo k ex ends p e ious o mula ions h ough: (i) a de i a ion om i s p inciples o
e ec i e ield heo y, (ii) a igo ous analysis o p edic abili y and alsi iabili y, and (iii) expanded
alida ion wi h cosmological and g a i a ional wa e da a. The model makes speci ic p edic ions
es able wi h LIGO/Vi go measu emen s and u u e p ecision cosmology expe imen s.
2 Fundamen als o he HDOV F amewo k
This sec ion in oduces he ma hema ical co e o he HDOV hypo hesis. The s a ing poin is an
e ec i e ac ion ha desc ibes a scala ield non-minimally coupled o g a i y, which gi es ise o
a modi ied wa e equa ion.
2.1 Mo i a ion om E ec i e Field Theo y
The choice o he p ojec ion pa ame e
ηp
is mo i a ed om e ec i e ield heo y in cu ed
space- imes (Bi ell and Da ies,1982;Pa ke and Toms,2009). The unc ional o m o
ηp
can be ob ained h ough unc ional eno maliza ion g oup echniques in cu ed space- imes,
whe e high-ene gy modes a e in eg a ed in a cu a u e-dependen manne . In he p esence o
cu a u e, he acuum de elops non-local co ela ions ha can induce dispe sion e ms dependen
on geome ic in a ian s. We s a om he mos gene al e ec i e ac ion compa ible wi h he
p inciples o co a iance and uni a i y:
Γe = ΓEH + Γma e + Γnon-local +. . . (1)
whe e Γnon-local cap u es quan um memo y e ec s dependen on he causal his o y.
2.2 Ac ion and P ojec i e Wa e Equa ion
The dynamics o he sys em is de i ed om he ollowing ac ion, which includes an explici
coupling e m be ween he scala ield Ψand he geome y, media ed by an auxilia y ield
χ
(
I
):
S=Zd4x√−gh1
21+2gcχ(I)gµν ∂µΨ∂νΨ−1
2m2Ψ2+1
2gµν ∂µχ(I)∂νχ(I)−Vχ(I)i.(2)
This ac ion can be ob ained as a low-ene gy limi o quan um g a i y heo ies ha inco po a e
g a i a ional decohe ence e ec s (Bassi and Ghi a di,2003a;Hu and Ve dague ,2004a). By
a ying his ac ion wi h espec o Ψ, and ea ing he coupling e m as a unc ional pa ame e
ηp≡χ(I), we ob ain he p ojec i e wa e equa ion:
∇µ
(1 + 2gcηp)∇µΨ+m2Ψ=0,(3)
3
2.3 WKB anspo and a enua ion law wi hou dimensional ambigui y
In he eikonal egime, Ψ =
A eiΘ
wi h
kµ
=
∇µ
Θand
kµ∇µ≡d
dλ
. S a ing om
∇µ

(1 +
2
gcηp
)
∇µ
Ψ

= 0 and sepa a ing o de s in he WKB expansion, he anspo e m o he
ampli ude is dln A
dλ =−1
2θ−1
2
d
dλ ln1+2gcηp, θ ≡ ∇µkµ.(4)
Fo |2gcηp|≪1,
dln A
dλ ≃ −1
2θ−gc
dηp
dλ .(5)
In eg a ing along he ay be ween λ0and λ,
ln A(λ)
A(λ0)=−1
2Zλ
λ0
θ dλ′−1
2ln 1+2gcηp(λ)
1+2gcηp(λ0)≃ −1
2Zλ
λ0
θ dλ′−gcηp(λ)−ηp(λ0).(6)
Law (6)isdimensionless and does no equi e in oducing a cons an wi h leng h dimension: he
geome ic e m (θ) and he unc ional a ia ion o ηpplay dis inc and compa ible oles.
2.4 No maliza ion, dimensions and no a ion (cla i ica ion)
To a oid no a ion collisions, we dis inguish: (i)
g≡de
(
gµν
)(only wi hin
√−g
) and (ii)
gc
as
he (dimensionless) coupling cons an ha mul iplies ηpin he e ec i e kine ics.
Dimensions and consis ency o he ac ion. We wo k in na u al uni s
ℏ
=
c
= 1, whe e
he ac ion is dimensionless and he Lag angian densi y has dimension o mass
4
. Fo a scala
Ψwi h dimension [Ψ] =
mass
, he s anda d kine ic e m
1
2gµν∂µ
Ψ
∂ν
Ψal eady has he co ec
dimension. The e o e, he mul iplica i e ac o (1 + 2
gcηp
) ha modula es he kine ics mus be
dimensionless. We hen equi e ha bo h
gc
and
ηp
be dimensionless. Wi h his con en ion, he
p ojec i e wa e equa ion we use in he ex
∇µ[(1 + 2 gcηp)∇µΨ]+m2Ψ=0
is dimensionally consis en (see he o m used in Sec ion 2.2).
2.5 Regula and in a ian amily o ηp
Fo
ηp
o be physically consis en , i is cons uc ed om geome ic in a ian s. A possible
unc ional amily is:
ηp=a1 s
n+a2
√K
Λ2
K
+a3
κ
Λκ
, n ∈[1,3],(7)
whe e each e m has speci ic physical mo i a ion: (
s
)
n
o sc eening,
√K
o idal cu a u e
(K e schmann), and κ o su ace g a i y.
2.6 E ec i e o igin o ηpand e e ence scales
Appendix A.1 illus a es how non-local e ms o he ype
Snonlocal ∼R
d
4x√−g1
M2R
(
□−1R
)
can induce an e ec i e unc ional coupling (Bi ell and Da ies,1982;Pa ke and Toms,2009).
Expanding (x)=αx +βx2+···, he i s ele an e m p oduces
ηp=αR
M2+O(R2/M4),
which is dimensionless because
R
has mass dimension
2
and
M
ixes he scale (e.g.,
M∼MPl
o
an in e media e scale o he e ec i e sec o ).
4
Analogously, when pa ame e izing
ηp
wi h geome ic in a ian s, scales a e explici ly in oduced
o make hem dimensionless:
ηp(z) = a1 s
n+a2
√K
Λ2
K
+a3
κ
Λκ
, n ∈[1,3],
whe e
K
is he K e schmann scala and
κ
he su ace g a i y. He e Λ
K
and Λ
κ
a e scale
pa ame e s (cons an s wi h mass dimension) ha gua an ee ha each quo ien is dimensionless.
This p esc ip ion makes he dimensionali y unambiguous and connec s wi h he unc ional amily
al eady explo ed in he ex .
2.7 G a i a ional Field o View (GFoV)
The GFoV is he bounda y whe e
ηp
induces a comple e a enua ion, implemen ing a decomposi ion
o he Hilbe space:
H
=
Hop ⊕ Hnoop
. In p ac ice, we ake “comple e a enua ion” as he
condi ion
ηp→η⋆
, whe e
η⋆
is he alue om which he modes a e p ojec ed ou o
Hop
. This
decomposi ion is analogous o he complemen a i y o ho izons (Almhei i e al.,2013) bu
implemen ed dynamically, as illus a ed in Figu e 1.
Unobse able S a es
(ou -o - ange modes)
Accessible S a es
P ojec ion
P is
Concep ual Scheme: P ojec ion in he Hilbe Space
HDOV
Figu e 1: Illus a ion o he concep o unc ional p ojec ion and he decomposi ion o he Hilbe
space in o ope a ional and non-ope a ional subspaces.
2.8 Conse a ion check in he classical limi
In ou e ec i e o mula ion,
Gµν
= 8
πG Tµν +THDOV
µν 
. By he Bianchi iden i y,
∇µGµν
= 0,
so
∇µTµν +THDOV µν
= 0. In he classical limi (
ηp→
0),
THDOV
µν →
0and s anda d
conse a ion is eco e ed. When he e a e luc ua ions, conse a ion holds on a e age, in he
sense o s ochas ic g a i y (Hu and Ve dague ,2004b;Bassi and Ghi a di,2003b).
5

Table 1: PPN bounds check (Cassini and Me cu y). ’Yes’ indica es ha i sa is ies he bound.
model gamma be a |gamma-1| |be a-1| gamma_pass be a_pass PPN_OK
HDOV 0.999980 1.000100 2e-05 0.0001 Yes Yes Yes
LCDM 1.000000 1.000000 0 0 Yes Yes Yes
2.9 PPN check and sola limi s
In he weak and quasi-s a ic egime, he s anda d PPN expansion is
g00
=
−
1+2
U−
2
βU2
+
. . .
and
gij
= (1 + 2
γU
)
δij
+
. . .
. Fo HDOV, he e ms o
ηp
co ec he coe icien s. Requi ing
compa ibili y wi h he Cassini bound (Be o i e al.,2003) and he p ecession o Me cu y
(Pi je a and Pi je ,2013), we ob ain:
|γ−1| ≃ 2×10−5,|β−1| ≃ 1×10−4,
alues ha a e wi hin he expe imen al bounds. The model is compa ible wi h he sola limi s
in he conside ed pa ame e ange. The esul s a e summa ized in Table 1.
3 De i a ions and unc ional a ian s
In he p e ious sec ions, a anspo law o he ampli ude
A
(
λ
)along he null ay beam was
ob ained om he p ojec i e wa e equa ion and he WKB eading. I we de ine he in e e ence
isibili y as
np(λ)≡|A(λ)|2
|A(λ0)|2,(8)
hen, abso bing he pu ely GR geome ic ocus in o a mul iplica i e ac o and keeping he
con ibu ion o HDOV i sel , we can e ec i ely w i e
np(λ) = exp −2gcηp(λ)−ηp(λ0)≡exp [−2u(λ)] ,(9)
whe e we ha e de ined
u(λ)≡gc[ηp(λ)−ηp(λ0)] .(10)
In cosmology we will simply w i e
np
(
z
)and
u
(
z
), so ha e e y unc ional choice o
ηp
(
z
)uniquely
induces a cu e
np
(
z
)o accessibili y o quan um isibili y. The igu es in his sec ion should
be unde s ood in his con ex : hey show how di e en pa ame e iza ions o
ηp
ansla e in o
di e en e ec i e laws o unc ional accessibili y.
3.1 Typical p o ile o ηp( )
The beha io o
ηp
(
)is key. Nea a compac objec , i decays apidly wi h dis ance. Figu e 2
shows an example o his unc ional p o ile.
6
Figu e 2: Func ional p o ile o ηp( )as a unc ion o he no malized adial dis ance / s.
3.2 Va ian s and eedback
In cosmology, di e en o ms o
ηp
(
z
)a e explo ed. Figu e 3p esen s a compa ison o he
unc ional accessibili y o di e en pa ame e iza ions. The HDOV amewo k allows o unc ional
eedback, whe e he dynamics and pa ame e s a e in e connec ed (Figu e 4).
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Redshi
z
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Quan um isibili y
np
(
z
)
In e e en ial isibili y o di e en unc ional o ms
np
(
z
)
Sigmoid ( = 1.03, = 10)
Gaussian ( = 1.55,
n
= 0.6, = 2.4)
Ra ional oo ( = 1.31)
Figu e 3: Compa ison o he unc ional accessibili y o di e en pa ame e iza ions o ηp(z).
7
0.6
0.7
0.8
0.9
p
(
z
)
0.1
0.2
0.3
0.4
a
(
z
) = 1
p
(
z
)
1000
2000
H
(
z
) [km/s/Mpc]
0
2
4
6
DL
(
z
) [Mpc]
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Redshi
z
20
25
(
z
)
HDOV unc ional chain lea ned by neu al ne wo k
Figu e 4: Diag am o he unc ional eedback chain in HDOV.
4 Rep esen a i e applica ions
4.1 Expansion unc ional esponse
The e ec i e scale ac o
ae
(
) =
a
(
)[1
−ηp
(
)] p oduces measu able de ia ions om ΛCDM,
especially in in e media e edshi s (1
< z <
3), o e ing a alsi iable p edic ion wi h su eys such
as DESI and Euclid (Figu e 5).
8
0.0 0.2 0.4 0.6 0.8 1.0
No malized cosmological ime
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Scale ac o
a
(
)
Expansion compa ison: HDOV s. CDM
HDOV:
a
(
) = 1
p
(
)
CDM: Accele a ed expansion
Figu e 5: Compa ison o he expansion his o y be ween HDOV and ΛCDM.
4.2 Cosmology-quan um duali y (illus a ion)
The connec ion wi h quan um ounda ions is illus a ed by he duali y be ween he cosmological
eco d (
ηp
(
z
) i ed o supe no a da a) and he quan um simula ion ( he same
ηp
p edic s loss o
cohe ence in analog sys ems (Ba celó e al.,2005)). This is illus a ed in Figu e 6.
0.0 0.5 1.0 1.5 2.0
Redshi
z
15
20
25
30
35
Dis ance modulus (
z
)
Sigmoidal HDOV:
= 1.030, = 10.00, z = 0.00
MSE HDOV = 256.91
HDOV i o Type Ia supe no ae
Sigmoidal HDOV
Pan heon+
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Redshi
z
0.0
0.2
0.4
0.6
0.8
1.0
(
z
)
Quan um cohe ence simula ion unde
np
(
z
)
Quan um isibili y
Dual mani es a ion o
np
(
z
)
: cosmology and quan um p ojec ion
Figu e 6: Illus a ion o he HDOV duali y: cosmological i (le ) and quan um simula ion
( igh ).
5 Cosmological alida ion: SN Ia (Pan heon) and BAO
We con as HDOV and ΛCDM wi h Supe no ae Ia (Pan heon) and BAO (SDSS DR16) da a,
applying he same ea men : co a iances, so Gaussian p io in H0, and in o ma ion c i e ia.
9
B.3 Consis ency o S a is ical Resul s
I migh be hough ha he e is a con adic ion be ween Table 2, which shows ∆BIC ≃10.48
( a o ing HDOV), and Tables 3 and 4, whe e he absolu e alues o AIC/BIC each di e en
magni udes and, aken li e ally, migh seem o a o ΛCDM. The di e ence is explained because
Table 2 uses a consis en de ini ion o BIC o e he no malized se SN+BAO+p io , while
Tables 3 and 4 epo absolu e alues a ec ed by addi i e no maliza ion cons an s associa ed
wi h he de ini ion o he likelihood. When compa ing models on an equal oo ing, ha is, using
he same de ini ion o BIC and he same da a se (Table 2), HDOV is a o ed wi h ∆
BIC ≃
10
.
5.
B.4 Fi ed Pa ame e s and 1σE o s
In Table 6we show he i ed pa ame e s o HDOV and ΛCDM, wi h 1
σ
unce ain ies de i ed
om a join i (SN+BAO+H0).
Table 6: Cosmological pa ame e s i ed wi h 1σe o s.
Pa ame e HDOV ΛCDM
H0[km/s/Mpc] 69.8±1.2 67.5±0.9
Ωm0.296 ±0.015 0.311 ±0.010
S(HDOV) 0.12 ±0.04 —
n(HDOV) 1.01 ±0.07 —
B.5 Ringdown pos e io s
Figu e 12 shows he join pos e io dis ibu ions o he (2,2,0) mode in he ( 220, τ220)plane,
compa ing i s unde Gene al Rela i i y (GR, blue poin s) and he HDOV amewo k (o ange
poin s). Each cloud co esponds o pos e io samples ob ained om he ingdown analysis, while
he c oss and s a ma ke s indica e he cen al alues a o ed by GR and HDOV, espec i ely.
The shi s a e small —o o de 1–2 Hz in 220 and ∼0.3 ms in τ220— ye la ge enough ha
u u e de ec o s wi h highe signal– o–noise in he ingdown phase could disc imina e be ween
bo h p edic ions. In his sense, he pos e io dis ibu ions o he 220 mode s eng hen he link
be ween he cosmological phenomenology o HDOV and s ong–g a i y es s.
16

Figu e 12: Pos e io dis ibu ions o he 220 mode in he (
220, τ220
)plane o GR (blue poin s)
and HDOV (o ange poin s). The c oss and s a ma ke s indica e he cen al alues a o ed by
GR and HDOV, espec i ely.
C E ec i e Ene gy-Momen um Tenso in HDOV
S a ing om he e ec i e ac ion S=Rd4x√−gh1
16πG R+ηpOnonlocal(gµν, ψ)i, we de ine he
ene gy-momen um enso associa ed wi h he HDOV pa as
THDOV
µν ≡ − 2
√−g
δSHDOV
δgµν .
In an FLRW backg ound, conside ing he non-local e m o i s o de
SHDOV ∼Zd4x√−gηp
M2R□−1R, (11)
whe e Mis an e ec i e mass scale ha con ols he con ibu ion o he non-local e m,
analogously o he ac o 1/M2in oduced in Sec ion 2.6. De ining he non-local quan i y
X≡□−1R, he e ec i e componen s ake he o m:
THDOV
00 ≃ −ηp3H˙
X−1
2R2, THDOV
ij ≃ηpa2( )δij ¨
X−1
2R2,(12)
whe e he do s deno e ime de i a i es wi h espec o cosmological ime , i.e. ˙
X≡∂ Xand
¨
X≡∂2
X. In he limi ηp→0(see Sec. 2.5), THDOV
µν →0, eco e ing GR.
Da a and code a ailabili y
All da a used in his wo k (Pan heon ca alogue Scolnic e al. (2018), BAO measu emen s om
SDSS DR16 Alam e al. (2021) and public g a i a ional-wa e ca alogues GWTC-3 e al. (LIGO
Scien i ic Collabo a ion e al., 2021(@)) a e eely a ailable.
17
All he sou ce code used in his wo k is dis ibu ed oge he wi h he
HDOV_ ep o_en_2025-10-06 ep oducibili y package, unde he s c/co e/ and
s c/co e_ ig/ di ec o ies. I con ains he sc ip s equi ed o ep oduce he cosmological i s,
he SN Ia and BAO igu es, he ingdown panels, and he associa ed me ics.
Decla a ions and con ibu ions
Con lic o in e es : The au ho decla es ha he e is no inancial o pe sonal con lic o
in e es ha could ha e in luenced he esul s p esen ed.
Au ho ship con ibu ions: A noldo Fe nández concei ed he HDOV hypo hesis, de eloped
he ma hema ical o malism, pe o med he nume ical analyses, and w o e he manusc ip .
ORCID: 0000-0003-3027-0450.
Re e ences
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