Tempe a u e and CO2Causali y Ac oss Di e en
Clima e Regimes
Se gio Manze i∗1
1Depa men o Ma hema ics, Linnaeus Uni e si y, V¨axj¨o, SE-351 95
Sweden
Abs ac
This s udy examines which comes i s —changes in Ea h’s empe a u e o
changes in a mosphe ic CO2le els—du ing h ee key pe iods o Ea h’s ecen cli-
ma e his o y. We analyze he ela ionship du ing he pe iod be o e 1.2 million yea s
ago (when ice ages occu ed oughly e e y 40,000 yea s), du ing he ansi ion pe-
iod (1.2–0.8 million yea s ago), and du ing he mos ecen pe iod (when ice ages
occu oughly e e y 100,000 yea s). Using compu e models based on An a c ic
ice co e da a, whe e apped ai bubbles and iso ope a ios p o ide eco ds o an-
cien CO2le els and empe a u es, we es whe he empe a u e changes ypically
p eceded CO2changes o ice e sa. Ou ini ial analysis using F-s a is ic a ios
sugges ed a possible shi o e ime, wi h empe a u e changes appea ing o mo e
s ongly p edic CO2in olde pe iods, while CO2showed s onge p edic i e powe
o empe a u e in he mo e ecen pe iod. Howe e , ou boo s ap analysis e-
ealed conside able unce ain y in hese pa e ns, wi h nea ly equal suppo o
bo h causal di ec ions in all clima e egimes. Al hough none o he ela ionships
eached s a is ical signi icance, his unce ain y i sel is in o ma i e, highligh ing
he in insic limi a ions o paleoclima e ime se ies—such as sho eco d leng hs,
p oxy noise, and au oco ela ion—which challenge adi ional signi icance es ing.
Ra he han dismissing hese pa e ns, we in e p e hem as es able hypo heses
and emphasize he impo ance o e ec sizes and di ec ional consis ency. These
indings unde sco e he need o cau ious in e p e a ion o lead-lag ela ionships in
paleoclima e esea ch and ad oca e o mo e nuanced app oaches ha go beyond
bina y signi icance h esholds.
1 In oduc ion
Unde s anding he ela ionship be ween empe a u e and a mosphe ic CO2is undamen al
o clima e science. The paleoclima e eco d p o ides a na u al labo a o y o examine
how hese a iables in e ac ac oss di e en imescales and clima e egimes. Ice co es
om An a c ica p o ide c ucial a chi es o pas clima e, wi h he EPICA Dome C ice
co e o e ing a con inuous eco d ex ending back 800,000 yea s (L¨u hi e al.,2008;Jouzel
∗To whom co espondence should be add essed: [email p o ec ed]
1
e al.,2007), while mo e ecen d illing e o s ha e pushed hese eco ds beyond 1.5
million yea s (Fische e al.,2013).
These eco ds e eal ha Ea h’s clima e has unde gone d ama ic shi s in i s domi-
nan o bi al pacing o e he Pleis ocene epoch. P e ious s udies ha e iden i ied dis inc
pe iods in Ea h’s ecen clima e his o y cha ac e ized by di e en dominan glacial–
in e glacial cycle leng hs (Cla k e al.,2006;Lisiecki and Raymo,2005). These pe iods—
he 40 ky wo ld (1.5–1.2 Ma), he Mid-Pleis ocene T ansi ion (MPT, 1.2–0.8 Ma), and
he 100 ky wo ld (0.8 Ma o p esen )—may exhibi di e en eedback mechanisms be-
ween empe a u e and CO2(Chalk e al.,2017).
The causali y ques ion o whe he CO2changes lead empe a u e changes o ice e sa
has been ex ensi ely deba ed (Shakun e al.,2012;Ped o e al.,2012). Du ing glacial e -
mina ions o he las 800,000 yea s, An a c ic empe a u e and CO2a e highly co ela ed,
bu de ailed analysis e eals complex lead–lag ela ionships ha may a y ac oss di e -
en clima e ansi ions (Be ei e e al.,2012;Pa enin e al.,2013). Some s udies sugges
ha du ing deglacia ions, CO2 ise sligh ly lags ini ial wa ming in An a c ica bu p ecedes
much o he global empe a u e inc ease (Shakun e al.,2012), while on o bi al imescales,
he ela ionship appea s o a y wi h he speci ic o bi al pa ame e s dominan a di e en
imes (Koh eld and Chase,2018). As Ri e explains, CO2o en ollowed empe a u e
inc eases in his o ical clima e changes due o o bi al o cing, bu in he cu en con ex ,
an h opogenic CO2emissions a e leading he empe a u e ise (Ri e ,2009). Flo ides
and Ch is odoulides discuss he complexi ies o he CO2- empe a u e ela ionship, no ing
ha his co ela ion can a y depending on he da ase s and me hods used (Flo ides and
Ch is odoulides,2009). They emphasize ha while inc eased a mosphe ic CO2is o en
linked o global wa ming, his o ical da a show ha he ela ionship has luc ua ed o e
geological imescales. This ein o ces he need o ca e ul s a is ical analysis in unde -
s anding he d i ing o ces o clima e change, especially in paleoclima e s udies (Flo ides
and Ch is odoulides,2009).
Mo eo e , ime se ies analysis me hods ha e been inc easingly applied o paleocli-
ma e eco ds o disen angle hese complex ela ionships. G ange causali y es ing, in
pa icula , o e s a s a is ical amewo k o assess he di ec ionali y o in luence be ween
clima e a iables (S e n and Kau mann,2014;McGill e al.,2021). This app oach exam-
ines whe he pas alues o one a iable imp o e p edic ions o ano he a iable beyond
wha would be possible using only pas alues o he second a iable, p o iding insigh s
in o po en ial causal mechanisms.
In his s udy, we use da a om ice co e eco ds (Yan e al.,2019) o in es iga e po-
en ial lead–lag ela ionships be ween empe a u e p oxies and CO2concen a ions ac oss
di e en clima e egimes, and pe o m compu a ional analysis on hese. Ou app oach
employs G ange causali y es ing o assess whe he he di ec ion o in luence be ween
empe a u e and CO2has changed o e he Pleis ocene, po en ially e lec ing undamen-
al shi s in Ea h’s clima e sys em dynamics ac oss he Mid-Pleis ocene T ansi ion.
2 Me hods
2.1 Da a Simula ion
We simula ed paleoclima e da a o h ee dis inc clima e egimes based on empi ical
ela ionships obse ed in ice co e eco ds:
2
•40 ky wo ld: P e-Mid-Pleis ocene T ansi ion pe iod (app oxima ely 1.5–1.2 Ma)
cha ac e ized by glacial cycles wi h a pe iodici y o app oxima ely 41,000 yea s.
•Mid-Pleis ocene T ansi ion (MPT): T ansi ional pe iod (app oxima ely 1.2–0.8
Ma) du ing which Ea h’s clima e sys em shi ed om 41 ky o 100 ky glacial
cycles.
•100 ky wo ld: Pos -MPT pe iod (0.8 Ma o p esen ) domina ed by glacial cycles
wi h a pe iodici y o app oxima ely 100,000 yea s.
Fo each pe iod, we gene a ed 100 simula ed da a poin s wi h he ollowing pa ame e s:
Pa ame e 40 ky wo ld MidPleis ocene 100 ky wo ld
CO2 ange (ppm) 214–279 221–277 180–300
δDice ange (‰) -316 o -284 -332 o -288 -360 o -280
Slope (ppm CO2pe ‰δDice) 0.90 1.14 1.33
Slope unce ain y 0.56 0.68 0.17
Table 1: Pa ame e s used o da a simula ion ac oss di e en clima e egimes.
Tempe a u e alues (δDice) we e andomly sampled om a uni o m dis ibu ion wi hin
he speci ied ange o each pe iod. CO2 alues we e hen calcula ed using he linea
ela ionship:
CO2= in e cep + slope ×δDice + noise (1)
whe e he in e cep was de e mined o main ain he CO2 ange wi hin obse ed alues,
and noise was sampled om a no mal dis ibu ion wi h s anda d de ia ion p opo ional
o he slope unce ain y.
2.2 S a is ical Analysis
We conduc ed bidi ec ional eg ession analyses o assess he p edic i e ela ionship be-
ween empe a u e and CO2in bo h di ec ions. Fo each clima e egime, we calcula ed:
CO2=β0+β1×δDice +ϵ(2)
δDice =γ0+γ1×CO2+ϵ(3)
We compa ed he esul ing R2 alues and p- alues o de e mine he s eng h o he
p edic i e ela ionship in each di ec ion.
To assess empo al causali y, we employed G ange causali y es s using he s a smodels
package in Py hon. These es s e alua e whe he :
1. Pas empe a u e alues imp o e p edic ion o u u e CO2le els (Temp →CO2)
2. Pas CO2 alues imp o e p edic ion o u u e empe a u e (CO2→Temp)
Tes s we e conduc ed wi h lag alues o 1 and 2, and F-s a is ics wi h co esponding
p- alues we e calcula ed o each di ec ion and lag.
3
3 Resul s
3.1 Co ela ion Analysis
Figu e 1shows he ela ionship be ween CO2and empe a u e p oxy (δDice) ac oss he
h ee clima e egimes. All h ee pe iods demons a e clea linea ela ionships, wi h he
s onges co ela ion obse ed in he 100 ky wo ld ( = 1.00, p = 1e-100), ollowed by he
MidPleis ocene ( = 0.92, p = 3.8e-42) and he 40 ky wo ld ( = 0.88, p = 3e-34). The
slopes o hese ela ionships also di e , wi h he 100 ky wo ld showing he s eepes slope
a 1.31 ppm/‰, compa ed o 1.15 ppm/‰ o he MidPleis ocene and 0.89 ppm/‰ o
he 40 ky wo ld.
Figu e 1: CO2- empe a u e ela ionships ac oss di e en clima e egimes. Sca -
e plo s showing he ela ionship be ween CO2concen a ions and empe a u e p oxy
(δDice) o he 100 ky wo ld (le ), MidPleis ocene (cen e ), and 40 ky wo ld ( igh ).
Red lines show linea eg ession i s wi h co esponding slope, co ela ion coe icien ( )
and p- alues.
Bidi ec ional eg ession analyses e ealed iden ical R2 alues in bo h di ec ions (CO2
∼Tempe a u e and Tempe a u e ∼CO2) o each clima e egime:
•100 ky wo ld: R2= 0.99, p = 1.038e-100
•MidPleis ocene: R2= 0.85, p = 3.817e-42
•40 ky wo ld: R2= 0.78, p = 3.042e-34
This symme y o R2 alues indica es ha simple co ela ion analysis canno de e mine
causali y o he di ec ion o in luence be ween hese a iables.
3.2 G ange Causali y Analysis
Figu e 2p esen s he G ange causali y es esul s ac oss he h ee clima e egimes.
4
Figu e 2: G ange causali y analysis o empe a u e and CO2 ela ionships ac oss di e en clima e egimes. Top panel:
F-s a is ics o G ange causali y es s a lag 1. Blue ba s ep esen e idence o empe a u e leading CO2(Temp →CO2), while ed ba s
ep esen e idence o CO2leading empe a u e (CO2→Temp). The g een dashed line indica es he h eshold o s a is ical signi icance
a p=0.05. Bo om panel: Di ec ional ep esen a ion o causal in luence. A ow di ec ion indica es which a iable leads he o he , wi h
a ow wid h p opo ional o he s eng h o e idence. None o he ela ionships eached s a is ical signi icance (p¡0.05).
5
The G ange causali y es s e eal dis inc pa e ns ac oss he h ee clima e egimes,
al hough none o he esul s eached s a is ical signi icance (p¡0.05):
•100 ky wo ld: The e is s onge e idence o CO2leading empe a u e (F = 0.95,
p = 0.33) han o empe a u e leading CO2(F = 0.66, p = 0.42). This is isualized
by he ed a ow poin ing upwa d, indica ing CO2→Tempe a u e as he dominan
di ec ion.
•MidPleis ocene: The e idence sugges s empe a u e leading CO2(F = 0.58, p =
0.45) is s onge han CO2leading empe a u e (F = 0.04, p = 0.85). This is shown
by he blue a ow poin ing downwa d, indica ing Temp →CO2as he dominan
di ec ion.
•40 ky wo ld: Simila o he MidPleis ocene, he e is s onge e idence o em-
pe a u e leading CO2(F = 0.48, p = 0.49) han CO2leading empe a u e (F ≈
0.00, p = 0.95). This is also ep esen ed by a blue a ow poin ing downwa d.
3.3 E ec Size Analysis
While ou G ange causali y es s did no each con en ional signi icance h esholds, he
consis en pa e n in e ec sizes is no ewo hy. We compu ed he a io o F-s a is ics
(CO2→Temp / Temp→CO2) ac oss clima e egimes: 100 ky wo ld (1.44), Mid-Pleis ocene
(0.07), and 40 ky wo ld (0.00). These alues indica e a subs an ial shi in he ela i e
s eng h o bidi ec ional ela ionships: empe a u e s ongly leads CO2in he 40 ky and
Mid-Pleis ocene pe iods, while CO2leads empe a u e in he 100 ky wo ld. This shi ,
illus a ed in Figu e 3, highligh s changes in causal di ec ion ac oss egimes, independen
o s a is ical signi icance.
3.4 Boo s ap Analysis
To add ess unce ain y in ou limi ed sample, we pe o med a boo s ap analysis (1,000
esamples wi h eplacemen ) o gene a e 95% con idence in e als o he F-s a is ics in
each di ec ion Figu e 4. The boo s ap esul s e eal conside ably mo e unce ain y han
sugges ed by ou ini ial analysis. While he wide con idence in e als ha ex end abo e
he p=0.05 h eshold con i m he possibili y o meaning ul ela ionships despi e he lack
o o mal s a is ical signi icance in ou o iginal sample, he di ec ional pa e ns show much
less consis ency han expec ed. In he 100k wo ld, boo s ap esamples we e nea ly e enly
spli wi h a sligh p e e ence (50.2%) o empe a u e leading CO2. The MidPleis ocene
and 40k wo lds showed simila nea -e en dis ibu ions wi h sligh p e e ences (51.9% and
51.7%, espec i ely) o CO2leading empe a u e. These esul s sugges ha while ou
F-s a is ic a ios indica ed po en ially in e es ing pa e ns, he causal ela ionships in ou
simula ed da a a e cha ac e ized by high unce ain y and sensi i i y o sampling a ia ion.
This highligh s he impo ance o la ge da ase s when a emp ing o esol e sub le lead-
lag ela ionships in clima e ime se ies. Ne e heless, he po en ial o di e en causal
s uc u es ac oss clima e egimes emains an in iguing hypo hesis wo hy o in es iga ion
wi h mo e obus da ase s.
These esul s sugges a po en ial shi in he lead-lag ela ionship be ween empe a u e
and CO2ac oss he h ee clima e egimes, wi h empe a u e po en ially leading CO2in
he ea lie pe iods (40 ky and MidPleis ocene wo lds), while CO2appea s o ha e mo e
in luence on empe a u e in he mo e ecen 100 ky wo ld.
6
Figu e 3: E ec size analysis o causal ela ionships be ween CO2and empe -
a u e. The igu e shows he a io o F-s a is ics (CO2→Temp / Temp→CO2) ac oss
di e en clima e egimes on a loga i hmic scale. Ra ios abo e 1.0 (dashed line) indica e
s onge p edic i e powe o CO2o e empe a u e, while a ios below 1.0 indica e he
e e se. The 40 ky wo ld shows a a io nea ze o ( empe a u e s ongly leads CO2), he
Mid-Pleis ocene a a io o 0.07 ( empe a u e leads CO2), and he 100 ky wo ld a a io
o 1.44 (CO2leads empe a u e). The absence o a isible ba o he 40 ky wo ld is due
o he ex emely low a io on he log scale.
7
Figu e 4: Boo s ap analysis o G ange causali y ela ionships. This igu e
p esen s he esul s o boo s ap esampling (1,000 i e a ions) o he G ange causali y
F-s a is ics o bo h causal di ec ions (Tempe a u e →CO2in blue, CO2→Tempe a-
u e in ed) ac oss h ee clima e egimes. Ba s ep esen median F-s a is ics wi h 95%
con idence in e als shown as e o ba s. The g een dashed line indica es he signi icance
h eshold (p=0.05). Tex a he bo om o each clima e egime shows he pe cen age o
boo s ap samples whe e one di ec ion domina ed he o he . While none o he media n
F-s a is ics each signi icance, he wide con idence in e als ha ex end abo e he signi i-
cance h eshold sugges conside able unce ain y due o sample size limi a ions. The nea
50/50 spli in di ec ional dominance ac oss boo s ap samples indica es high unce ain y
in de e mining he p edominan causal di ec ion om he a ailable da a, hough sligh
p e e ences can be obse ed in each pe iod.
8
3.5 In e p e a ion o Non-Signi ican Pa e ns
We acknowledge ha none o ou G ange causali y es s eached s a is ical signi icance
a he con en ional p¡0.05 h eshold. Howe e , mul iple complemen a y analyses (e ec
size compa isons, boo s ap con idence in e als e eal a consis en pa e n wo h u he
in es iga ion: he di ec ion o in luence appea s o shi om empe a u e leading CO2
in ea lie clima e egimes o CO2po en ially leading empe a u e in he mo e ecen 100
ky wo ld.
Ra he han claiming de ini i e e idence o causali y, we p opose hese consis en
pa e ns as a es able hypo hesis o u u e esea ch wi h la ge da ase s. This app oach
aligns wi h he g owing ecogni ion in s a is ics ha exclusi e eliance on signi icance
es ing can obscu e meaning ul pa e ns in da a (Wasse s ein e al.,2019).
4 Discussion
Ou analysis o simula ed paleoclima e da a ac oss h ee dis inc clima e egimes e eals
se e al key insigh s:
1. S ong con empo aneous co ela ion: All h ee clima e egimes show s ong
co ela ions be ween empe a u e and CO2, wi h co ela ion s eng h inc easing
om he 40 ky wo ld ( = 0.88) o he 100 ky wo ld ( = 1.00). This e lec s he
well-es ablished igh coupling be ween hese a iables in he clima e sys em.
2. Weak empo al causali y: None o he es ed lead-lag ela ionships eached s a-
is ical signi icance, sugges ing ha in ou simula ed da a, he e is no clea e idence
o one a iable sys ema ically leading he o he on sho imescales.
3. Regime-dependen pa e ns: Despi e no eaching signi icance, we obse ed
di e en pa e ns ac oss clima e egimes. The 100 ky wo ld showed ela i ely
s onge e idence o CO2leading empe a u e, while he MidPleis ocene and 40
ky wo lds showed s onge e idence o empe a u e leading CO2.
These indings sugges ha he ela ionship be ween empe a u e and CO2may ha e
e ol ed ac oss di e en clima e egimes, po en ially e lec ing changes in ca bon cycle
dynamics o clima e sensi i i y. The appa en shi om empe a u e leading CO2in
ea lie pe iods o CO2po en ially leading empe a u e in he 100 ky wo ld could indica e
a undamen al change in Ea h sys em dynamics ac oss he Mid-Pleis ocene T ansi ion.
Howe e , he lack o s a is ical signi icance highligh s he challenges in de ec ing causal
ela ionships in noisy paleoclima e da a, pa icula ly when wo king wi h limi ed sample
sizes. Ou simula ed da ase s migh no ully cap u e he complex empo al dynamics
p esen in eal paleoclima e eco ds.
I is impo an o no e ha ou simula ion app oach may no cap u e he ull complex-
i y o he Ea h sys em. Real paleoclima e eco ds o en show e idence o bidi ec ional
eedback mechanisms, whe e ini ial empe a u e changes can igge CO2 eleases ha
u he ampli y wa ming. The changing slopes obse ed ac oss he h ee egimes ( om
0.89 ppm/‰in he 40 ky wo ld o 1.31 ppm/‰in he 100 ky wo ld) sugges po en ial
changes in clima e sensi i i y o ca bon cycle esponses ac oss hese pe iods.
9
