Ope a ional holog aphic app oach o acuum
ene gy and cosmic accele a ion
in he HDOV amewo k
Mas e equa ion, unc ional accessibili y and cosmological
consequences
A noldo Wal e Fe nández
[email p o ec ed]
PREPRINT — No embe 22, 2025
Abs ac
The acuum ca as ophe— he disc epancy o ∼10122 o de s o
magni ude be ween he ze o-poin ene gy densi y p edic ed by quan-
um ield heo y and he cosmologically in e ed alue o he e ec i e
cosmological cons an —is add essed he e by means o a s ic ly ope -
a ional and holog aphic app oach, consis en wi h Gene al Rela i i y
and wi h Bekens ein–Hawking ype in o ma ion bounds [1,2,3,4,5,
6].
The HDOV amewo k in oduces a scala ield o unc ional acces-
sibili y ηp, which pa ame izes he ac ion o he s a e space associ-
a ed wi h a gi en physical mode ha emains ope a ionally accessible
o a mac oscopic obse e in a speci ied en i onmen . Inco po a -
ing ηpin o a co a ian e ec i e ac ion gene a es an HDOV mas e
equa ion o he obse able ield Ψ; in he WKB egime, his equa-
ion induces an exponen ial a enua ion A∝exp[−gRχ(I)ηpdλ]o
he accessible ampli ude. The same no ion o accessibili y is applied
he e o high-ene gy modes o he quan um acuum, implemen ed as
a smoo h spec al weigh Wη(k)holog aphically egula ed in momen-
um space.
We show ha he accessibili y mechanism ha explains exponen-
ial a enua ions in p opaga ion (g a i a ional wa es, plasma signals,
em osecond TRXS, e c.) also egula es he e ec i e acuum ene gy
1
wi hou in oducing ad hoc cu -o s and in a manne compa ible in o -
de o magni ude wi h he obse ed cosmological cons an . This egu-
la ion is o mula ed as an HDOV Holog aphic P ojec ion Theo-
em, which s a es ha , unde accessibili y and holog aphic sa u a ion
hypo heses, he accessible acuum densi y is bounded om below by
Λc4/(8πG), independen ly o he de ailed o m o Wη(k). Th oughou
his wo k, he app oach mus be unde s ood in a s ic ly ope a ional
and holog aphic sense: he misma ch is a oided in he accessible sec-
o wi hou a emp ing o build a comple e mic ophysical heo y o
he quan um acuum. Finally, we p esen explici nume ical examples
o smoo h spec al weigh s compa ed wi h sha p cu -o s, calib a ed
wi h cu en cosmological pa ame e s.
2
1 Mo i a ion: he acuum ca as ophe and
cosmic accele a ion
The acuum ene gy densi y p edic ed by he nai e sum o ze o-poin modes in
quan um ield heo y (QFT) is, in na u al uni s, o he o de o he Planck
densi y. By con as , he e ec i e ene gy densi y associa ed wi h he cos-
mological cons an ha d i es he accele a ed expansion o he uni e se is
∼10−123 imes smalle [1,2]. This misma ch o ∼10122 is known as he
“ acuum ca as ophe” and cons i u es he mos se e e mani es a ion o he
cosmological cons an p oblem [1,2,3].
On he o he hand, p ecise cosmological obse a ions (CMB, BAO, ype
Ia supe no ae) indica e ha he uni e se is unde going accele a ed expansion
and ha a Λ- ype e m domina es he o al ene gy a la e imes [3]. The
ΛCDM model inco po a es his e m as an a p io i cosmological cons an
i ed o he da a. Howe e , om a heo e ical poin o iew, i emains
unclea why he quan um con ibu ion o he acuum akes exac ly ha
alue and no o he .
In his con ex , HDOV (Vib a ional Wa e Dispe sion Hypo hesis) p o-
poses a di e en app oach: ins ead o summing all possible modes wi hou
es ic ions, i is pos ula ed ha a mac oscopic obse e has access only o
an ope a ional ac ion o he ield deg ees o eedom, encoded in a scala
o unc ional accessibili y ηpand in an en i onmen al ga e χ(I) ha e lec s
he in e ac ion wi h he en i onmen [7,8,9].
P e ious sec ions o he HDOV p og am ha e shown ha he same ac-
cessibili y s uc u e explains: (i) exponen ial ampli ude a enua ions in he
p opaga ion o g a i a ional wa es; (ii) silencing o signals in in e plane a y
plasmas nea he heliopause; (iii) loss o cohe ence in em osecond TRXS ex-
pe imen s; (i ) cosmological causal disconnec ion in an accele a ing uni e se
[7,8]. He e we ex end ha mechanism o he acuum ene gy p oblem.
No e. In his wo k, pa icula ca e has been aken ega ding o mal ea -
men , e e ences and dimensional consis ency, bu he HDOV amewo k
applied o acuum ene gy and cosmic accele a ion emains an explo a o y
p oposal which, a he ime o his e sion, has no ye unde gone pee e-
iew in specialized jou nals. We p esen he a gumen s and esul s explici ly
so ha anyone can examine he assump ions, c i icize he me hodology, e-
p oduce he basic calcula ions and, i necessa y, e u e he conclusions. The
ela ed manusc ip s in which he HDOV p og am is de eloped in o he phys-
ical egimes [7,8,9] a e cu en ly in p epa a ion (2025) and/o a ailable as
p ep in s in open-access eposi o ies, so ha e e ees and in e es ed eade s
3
can consul he comple e amewo k.
2 HDOV mas e equa ion and unc ional ac-
cessibili y
In wha ollows we adop he (−,+,+,+) signa u e con en ion o he me ic
enso gµν and uni s wi h c= 1 unless explici ly s a ed o he wise.
2.1 E ec i e ac ion and ield equa ion
In HDOV we pos ula e ha he obse able ield Ψdoes no li e in isola ion:
i is imme sed in an en i onmen (geome y, plasma, quan um acuum, in-
s umen a ion, s ong g a i y, e c.) ha egula es which ac ion o he mode
is ac ually accessible. This accessibili y is encoded in a scala ηpand in an
en i onmen al ga e χ(I) ha indica es whe e he en i onmen couples e ec-
i ely [7,8]. In his no e we ea bo h ηpand χ(I)as e ec i e quan i ies
ha encode, a a mac oscopic le el, he a e age esponse o he en i onmen ;
we do no assign hem an independen ac ion no hei own equa ions o mo-
ion, so ha hei de ailed dynamics is le o u u e de elopmen s o he
HDOV amewo k. In pa icula , we do no a y he ac ion wi h espec o
ηpo χ(I); bo h ac as e ec i e backg ound ields, so ha he exac conse -
a ion o Tµν and he back- eac ion on he geome y a e only discussed in
he ixed-backg ound egime and lie beyond he scope o his no e.
A minimal co a ian e ec i e ac ion ha inco po a es his can be w i en
schema ically as
S=Zd4x√−gM2
Pl
2R+Lma + (1 + 2g χ(I)ηp)∇µΨ∇µΨ−m2Ψ2,(1)
whe e gis an accessibili y coupling, Ris he Ricci cu a u e scala and MPl
is he Planck scale. In (1) he accessibili y ac o mul iplies only he kine ic
e m o he ield Ψ, while he es mass mis kep ixed. This choice gua an ees
ha he e ec i e ield equa ion (2) is de i ed in he s anda d way om he
a ia ion o he ac ion and e lec s he ac ha , a his s age o he HDOV
p og am, unc ional accessibili y is in e p e ed as a modula ion o ampli ude
anspo a he han as a mic oscopic mass gene a ion mechanism.
Va ying he ac ion (1) wi h espec o Ψwe ob ain he HDOV mas e
equa ion:
∇µ(1 + 2g χ(I)ηp)∇µΨ+m2Ψ=0.(2)
4
In all egimes explo ed in his wo k we explici ly impose he condi ion
1 + 2g χ(I)ηp>0, so ha he e ec i e kine ic e m p ese es i s sign and
no ghos modes o linea -le el ene gy ins abili ies appea .
Physical in e p e a ion: he mul iplica i e ac o (1 + 2g χ(I)ηp)does
no emo e deg ees o eedom; i modula es which pa o Ψis ope a ionally
ealizable in ha egion o space- ime.
2.2 WKB decomposi ion: phase and ampli ude
We adop a WKB- ype ansa z,
Ψ(x)=A(x)eiΘ(x),(3)
whe e A(x)is a slowly a ying ampli ude and Θ(x)a apidly a ying phase.
We de ine he local wa e 4- ec o kµ=∇µΘ. Fo an obse e wi h 4- eloci y
uµ, he locally obse ed equency is ωobs =−kµuµ, i.e. he a e o inc ease
o he phase along hei p ope ime.
Inse ing (3) in o (2) and sepa a ing leading-o de e ms in he phase om
lowe -o de e ms in g adien s o A(x), one ob ains wo coupled equa ions:
an eikonal equa ion o Θ( ay p opaga ion) and a anspo equa ion o A.
In pa icula , he anspo equa ion akes he o m
dln A
dλ =−g χ(I)(λ)ηp(λ)−1
2θ(λ),(4)
whe e λis he a ine pa ame e along he ay and θ(λ)is he geome ic
expansion o he beam ( ocusing/de ocusing, g a i a ional lensing, geome ic
opening, e c.).
In eg a ing (4) we ob ain
A(λ)∝exp"−gZλχ(I)(λ′)ηp(λ′)dλ′#×exp"−1
2Zλθ(λ′)dλ′#.(5)
The i s ac o , in ol ing χ(I)ηp, is unc ional accessibili y: how
much o he mode emains physically suppo ed by he en i onmen . The
second ac o , in ol ing θ, is pu ely geome ic ( ocusing o a enua ion by
opening).
Key poin : he exponen ial a enua ion in (5) does no say ha he mode
is des oyed. I says ha i ceases o be accessible o he obse e in eg a ing
along λ. This is exac ly wha we see in: (a) signals ha “swi ch o ” as
hey app oach g a i a ional ho izons; (b) loss o cohe ence in in e plane a y
plasmas nea he heliopause; (c) loss o isibili y in em osecond TRXS; (d)
cosmological causal disconnec ion [7,8,9].
5
F om a echnical iewpoin , he WKB de i a ion o (4) and (5) mus be
unde s ood as a leading-o de app oxima ion in smoo h g adien s o Aand
o (1 + 2g χ(I)ηp)along he ay. In si ua ions wi h sha p a ia ions o ηpo
wi h addi ional nonlinea i ies, one expec s highe -o de co ec ions o en e ,
which a e no modeled in his no e.
3 Applica ion o he quan um acuum: holo-
g aphic spec al weigh
3.1 Vacuum ene gy egula ized by accessibili y
The ze o-poin ene gy densi y in QFT, in la space, can be w i en schema -
ically as
ρ ac ∼ℏc
4π2Z∞
0k3dk. (6)
This in eg al di e ges in he ul a iole : modes wi h a bi a ily la ge kcon-
ibu e unbounded ene gy [1,2].
Ins ead o imposing a ha d cu -o by hand, HDOV p oposes ha unc-
ional accessibili y ηpinduces a smoo h spec al weigh o e acuum modes,
analogous o he exponen ial ac o in (5). In ui i ely: he e is a limi o
how much ene gy can be ope a ionally concen a ed in a ini e egion be-
o e iola ing holog aphic bounds; beyond a ce ain scale, modes cease o
be accessible be o e being able o sus ain hemsel es wi hou g a i a ional
collapse.
We hus in oduce a spec al accessibili y weigh Wη(k):
ρe
ac =ℏc
4π2Z∞
0k3Wη(k)dk, (7)
wi h
Wη(k) = exp"− k
k0!α#, α > 0.(8)
The unc ion Wη(k)is no a ha d cu -o ; i is a smoo h exponen ial sup-
p ession ha says: “modes wi h k≫k0 o mally exis in he ield, bu hei
accessible pa decays exponen ially” [9].
F om he iewpoin o s anda d quan um ield heo y, he exp ession (7)
mus be in e p e ed agains he backg ound o he usual acuum eno maliza-
ion p ocedu e: we assume ha he e e ence s a e in la space has al eady
been sub ac ed, and ha he densi y ρe
ac co esponds only o he accessi-
ble/ope a ional pa o esidual acuum luc ua ions. The accessibili y weigh
6
Wη(k) hus ac s on he physically ope a i e sec o o he acuum ene gy. The
shape pa ame e αcon ols how as he supp ession in he ul a iole is, bu
as long as α=O(1) he o de o magni ude o ρe
ac is domina ed by k0, so
ha he esul ing phenomenology is obus agains easonable a ia ions o
α.
This is he spec al (in k) e sion o he same exponen ial mechanism
ha al eady appea ed in (5) in he con ex o space- ime p opaga ion. I is
he same physics: unc ional accessibili y ηp ha a enua es he obse able
pa o a mode [7].
3.2 Holog aphic condi ion and scale ixing
The me e in oduc ion o Wη(k)in (7) makes he in eg al con e gen , bu
lea es he alue o k0unde e mined. To ix i wi hou a bi a iness, we in oke
Bekens ein–Hawking ype holog aphic bounds [4,5,6].
The maximal en opy o a egion o adius Rdoes no g ow as he olume
R3bu as he a ea R2:
Smax ∼A
4L2
P∼πR2
L2
P
.(9)
T ansla ed in o an e ec i e ene gy densi y, his implies ha he accessible
ene gy in a olume canno exceed he limi imposed by ho izon o ma ion.
Schema ically:
ρe
ac L4
P≲1
4π,(10)
whe e in (10) we ha e ew i en he Bekens ein–Hawking bound in e ms o
he Planck densi y, assuming an e ec i e ho izon o o de H−1; his ans-
la ion is heu is ic and only in ended o ix an o de -o -magni ude bound o
ρe
ac in accele a ing cosmological con ex s.
Equi alen ly, we can in oduce he second spec al momen
C≡Z∞
0k2Wη(k)dk, (11)
which con ols he e ec i e numbe o accessible modes in a gi en olume.
The holog aphic condi ion hen imposes a maximum alue (o , in he sa u-
a ion scena io, a ixed alue) o C, de e mined by he a ea o he e ec i e
ho izon. Fo a la e- ime de Si e uni e se wi h ho izon adius RH=c/HΛ
and Λ=3H2
Λ/c2, one has SH∝R2
Hand, consequen ly, C∝R2
H∝1/Λ. In
he nex sec ion we show ha , o any smoo h and mono onically dec easing
weigh unc ion ha sa u a es his condi ion, he accessible ene gy densi y
is con olled by he cosmological e m in Eins ein’s equa ions.
7
4 HDOV holog aphic p ojec ion heo em
The abo e s uc u e can be condensed in o a simple a ia ional esul , which
makes explici he independence o he accessible acuum densi y om he
de ailed shape o he spec al weigh , p o ided ha accessibili y and holo-
g aphic in o ma ion cons ain s a e sa is ied.
Theo em 1 (HDOV holog aphic p ojec ion).Le Wη(k)∈[0,1] be a mono-
onically dec easing unc ion o he wa enumbe modulus k, su icien ly eg-
ula o he momen s
C=Z∞
0k2Wη(k)dk, I3=Z∞
0k3Wη(k)dk (12)
o exis . Le ρe
ac be he accessible acuum ene gy densi y de ined by (7). I
he momen Cis ixed by a holog aphic condi ion associa ed wi h a cosmolog-
ical ho izon o adius RH(la e- ime de Si e ) and he Bekens ein–Hawking
bounds a e sa u a ed, hen he accessible ene gy densi y sa is ies he holo-
g aphic lowe bound
ρe
ac ≥Λc4
8πG,(13)
independen ly o he de ailed o m o Wη(k), p o ided ha i espec s mono-
onici y, boundedness and holog aphic consis ency. The equali y (o an a -
bi a ily close nume ical app oxima ion) is ob ained when Wη(k)app oaches
he e ec i e cu -o W⋆
η(k) = Θ(k⋆−k).
P oo . The e ec i e ho izon o adius RHinduces a holog aphic bound on
he numbe o accessible modes in a olume Vcon ained wi hin ha ho izon.
This numbe is gi en, in e ms o Wη(k), by
Ne =V
2π2Z∞
0k2Wη(k)dk =V
2π2C. (14)
The holog aphic condi ion iden i ies Ne wi h he maximal en opy SHassoci-
a ed wi h he ho izon, measu ed in uni s o an elemen a y en opy s0=O(1):
Ne ≤SH
s0
, SH=AH
4L2
P
=πR2
H
L2
P
.(15)
In he sa u a ion egime his yields an e ec i e alue o C, de e mined
solely by he ho izon adius RHand, consequen ly, by Λin la e- ime de
Si e (RH∼c/HΛ,Λ=3H2
Λ/c2).
Once Cis ixed, he unc ional
I3[Wη] = Z∞
0k3Wη(k)dk, (16)
8
unde he cons ain s 0≤Wη≤1and Wηmono onically dec easing, is min-
imized by dec easing ea angemen (see, o ins ance, he so-called ba h ub
p inciple in ea angemen inequali ies) wi h he e ec i e cu -o
W⋆
η(k) = Θ(k⋆−k), k⋆= (3C)1/3,(17)
so ha
Imin
3=Zk⋆
0k3dk =k4
⋆
4=(3C)4/3
4.(18)
The minimum accessible ene gy densi y compa ible wi h Cis hen
ρe ,min
ac =ℏc
4π2Imin
3=ℏc
16π2(3C)4/3.(19)
Consis ency wi h Eins ein’s ield equa ions in la e- ime de Si e equi es
ha he e ec i e ene gy densi y sus aining he ho izon be p ecisely he cos-
mological e m,
ρe ,min
ac =Λc4
8πG ≡ρΛ.(20)
This condi ion ixes he alue o C(and hence o k⋆) uniquely. Any o he
smoo h weigh unc ion Wη(k) ha sa is ies he same holog aphic condi ion
sha es he same Cand p oduces an accessible densi y
ρe
ac =ℏc
4π2I3[Wη]≥ρe ,min
ac =Λc4
8πG,(21)
wi h equali y a ained by he e ec i e cu -o W⋆
ηand by sui ably uned
amilies o smoo h weigh s. Consequen ly, he accessible densi y consis en
wi h he cosmological ho izon is bounded om below by (13) and app oaches
equali y when he spec al weigh app oaches he e ec i e cu -o .
Rema k 1 (Independence o he shape o Wη(k)).Fo smoo h amilies such
as Wη(k) = exp[−(k/k0)α]o Wη(k) = (1 + (k/k0)α)−1, he scale pa ame e
k0is ixed by imposing ha he ene gy in eg al R∞
0k3Wη(k)dk ep oduces (o
sa u a es) he bound (13). Wi hin easonable anges o α=O(1), he esul -
ing densi y is obus and app oaches he ab up cu -o case, bu wi h a smoo h
ansi ion in he ul a iole , consis en wi h he ope a ional in e p e a ion o
unc ional accessibili y.
5 Nume ical implemen a ion and ep esen a-
i e igu es
In his sec ion we illus a e he abo e cons uc ion wi h explici nume ical
examples o smoo h spec al weigh s, compa ed wi h e ec i e sha p cu -o s.
9
•Labo a o y (TRXS): exponen ial decays o accessibili y, analogous
o (5), measu ed in ul a as dynamics whe e he signal decays ollowing
a selec i e accessibili y law and no only he mal dissipa ion [7].
•Plasmas/heliopause: accessibili y indica o s ηp(o ope a ional equi -
alen s such as κlocal) an icipa e egime ansi ions ( o example, he-
liopause c ossing) be o e s anda d magne ohyd odynamic diagnos ics
[7,8].
In all cases, he unde lying physics is he same: a ac ion o he mode
becomes non-ope a ional o he obse e and i s accessible ampli ude decays
acco ding o he exponen ial law al eady seen in (5).
F om a quan i a i e s andpoin , hese p edic ions mus s ill be conside ed
explo a o y: hey iden i y obse a ional con as windows o he HDOV
amewo k, bu hei alida ion will equi e de ailed calcula ions, nume i-
cal simula ions and sys ema ic compa isons wi h eal da a in each o hese
egimes.
7 Discussion
The concep ual esul is s ong and simple:
•No hing “ eally disappea s”.
•Unde ce ain geome ic, ene ge ic o holog aphic condi ions [4,5,6],
pa s o he ield cease o be accessible o a ini e obse e be o e io-
la ing physical bounds (ho izon o ma ion, in o ma ion capaci y, e c.).
•This loss o accessibili y is seen om he ou side as ampli ude a en-
ua ion, equency shi s o as he e ec ha in s anda d models is
pa ame ized as “da k ene gy” in he accele a ed expansion o he uni-
e se [3,8,9].
HDOV places all o his unde a single mas e equa ion (2) and a single
exponen ial accessibili y law (5). Wha in di e en sub ields is desc ibed as:
(a) decohe ence, (b) dissipa ion, (c) loss o con as , (d) an e ec i e da k-
ene gy-like e m in cosmology, appea s he e as he same phenomenon: a
ac ion o he s a e space becomes non-ope a ional o he obse e , egu-
la ed by ηpand holog aphic bounds.
I is impo an o emphasize ha his app oach does no aim o sol e
he cosmological cons an p oblem a a undamen al le el no o modi y he
16
unde lying QFT: i simply shows ha , once ope a ional limi a ions on in-
o ma ion access (encoded in ηpand he spec al weigh Wη(k)) a e aken
in o accoun oge he wi h holog aphic bounds, he e ec i e acuum ene gy
en e ing Eins ein’s equa ions is na u ally egula ed o he obse ed o de
o magni ude. In his sense, HDOV p o ides a consis en egula ion o he
accessible sec o , wi hou pos ula ing new da k luids o undamen al cosmo-
logical cons an s ixed by hand.
Al hough he amewo k p esen ed he e is in e nally consis en , se e al
impo an issues emain open o u u e wo k: (i) cons uc ing an explici
heo y o he dynamics o ηpand i s coupling o he geome y and ma e-
ial con en (in his no e we ea ηpand χ(I)as non-dynamical e ec i e
ields); (ii) de i ing in a mo e mic ophysical way he conc e e o m o Wη(k)
and he scale k0, which he e a e ixed phenomenologically by a global holo-
g aphic condi ion; and (iii) de eloping quan i a i e and sys ema ic analyses
o he cosmological and s ong-g a i y p edic ions o HDOV, including de-
ailed compa isons wi h ΛCDM using public da a and s anda d s a is ical
c i e ia (BIC, AIC, e c.). These limi a ions do no a ec he o mal cohe ence
o he cons uc ion, bu clea ly delimi he s ill p elimina y and ope a ional
cha ac e o he esul s p esen ed he e.
8 Conclusions
We ha e p esen ed a uni ied amewo k (HDOV) in which:
•A unc ional accessibili y ield ηp, coupled h ough he en i onmen al
ga e χ(I), modi ies he e ec i e equa ion o mo ion o he obse able
ield Ψ, equa ion (2).
•In he WKB egime, his equa ion induces an exponen ial accessibili y
law (5) ha explains a enua ions obse ed in con ex s as di e se as
g a i a ional wa es, in e plane a y plasmas and TRXS expe imen s.
•The same mechanism, applied o he spec um o acuum modes, gen-
e a es an e ec i e acuum densi y ρe
ac ha is ini e and compa ible in
o de o magni ude wi h he obse ed cosmological cons an , in a way
consis en wi h holog aphic bounds.
•Cosmic accele a ion can be in e p e ed as a mani es a ion o unc ional
inaccessibili y (loss o ope a ional causal connec i i y) ins ead o a mys-
e ious da k luid.
17
The p oposal is delibe a ely ope a ional: i does no aim o de i e he
en i e y o acuum physics om i s p inciples, bu a he o p o ide a co-
he en amewo k in which he accessible pa o acuum ene gy and cosmo-
logical dynamics is egula ed h ough unc ional accessibili y and holog aphic
bounds. Ins ead o modi ying undamen al QFT, we show ha ope a ional
limi s on access o in o ma ion (encoded in ηpand Wη(k)) na u ally egula e
he e ec i e acuum ene gy, econciling mic oscopic p edic ions wi h cos-
mological obse a ions wi hin a sel -consis en holog aphic amewo k. This
app oach opens a na u al a ena o con as HDOV wi h ΛCDM and o he
phenomenological da k ene gy models in p ecision cosmology.
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 con ibu ions. A noldo Fe nández concei ed he HDOV hy-
po hesis, de eloped he ma hema ical o malism, pe o med he nume ical
analyses and w o e he manusc ip .
Re e ences
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18
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