Temperature and CO2 Causality Across Different Climate Regimes

Temperature and CO2Causality Across Different Climate Regimes Sergio Manzetti∗1 1Department of Mathematics, Linnaeus University, V¨axj¨o, SE-351 95 Sweden Abstract This study examines which comes first—changes in Earth’s temperature or changes in atmospheric CO2levels—during three key periods of Earth’s recent climate history. We analyze the relationship during the period before 1.2 million years ago (when ice ages occurred roughly every 40,000 years), during the transition period (1.2–0.8 million years ago), and during the most recent period (when ice ages occur roughly every 100,000 years). Using computer models based on Antarctic ice core data, where trapped air bubbles and isotope ratios provide records of ancient CO2levels and temperatures, we test whether temperature changes typically preceded CO2changes or vice versa. Our initial analysis using F-statistic ratios suggested a possible shift over time, with temperature changes appearing to more strongly predict CO2in older periods, while CO2showed stronger predictive power for temperature in the more recent period. However, our bootstrap analysis revealed considerable uncertainty in these patterns, with nearly equal support for both causal directions in all climate regimes. Although none of the relationships reached statistical significance, this uncertainty itself is informative, highlighting the intrinsic limitations of paleoclimate time series—such as short record lengths, proxy noise, and autocorrelation—which challenge traditional significance testing. Rather than dismissing these patterns, we interpret them as testable hypotheses and emphasize the importance of effect sizes and directional consistency. These findings underscore the need for cautious interpretation of lead-lag relationships in paleoclimate research and advocate for more nuanced approaches that go beyond binary significance thresholds. 1 Introduction Understanding the relationship between temperature and atmospheric CO2is fundamental to climate science. The paleoclimate record provides a natural laboratory to examine how these variables interact across different timescales and climate regimes. Ice cores from Antarctica provide crucial archives of past climate, with the EPICA Dome C ice core offering a continuous record extending back 800,000 years (L¨uthi et al.,2008;Jouzel ∗To whom correspondence should be addressed: [email protected] 1 et al.,2007), while more recent drilling efforts have pushed these records beyond 1.5 million years (Fischer et al.,2013). These records reveal that Earth’s climate has undergone dramatic shifts in its dominant orbital pacing over the Pleistocene epoch. Previous studies have identified distinct periods in Earth’s recent climate history characterized by different dominant glacial– interglacial cycle lengths (Clark et al.,2006;Lisiecki and Raymo,2005). These periods— the 40 kyr world (1.5–1.2 Ma), the Mid-Pleistocene Transition (MPT, 1.2–0.8 Ma), and the 100 kyr world (0.8 Ma to present)—may exhibit different feedback mechanisms between temperature and CO2(Chalk et al.,2017). The causality question of whether CO2changes lead temperature changes or vice versa has been extensively debated (Shakun et al.,2012;Pedro et al.,2012). During glacial terminations of the last 800,000 years, Antarctic temperature and CO2are highly correlated, but detailed analysis reveals complex lead–lag relationships that may vary across different climate transitions (Bereiter et al.,2012;Parrenin et al.,2013). Some studies suggest that during deglaciations, CO2rise slightly lags initial warming in Antarctica but precedes much of the global temperature increase (Shakun et al.,2012), while on orbital timescales, the relationship appears to vary with the specific orbital parameters dominant at different times (Kohfeld and Chase,2018). As Ritter explains, CO2often followed temperature increases in historical climate changes due to orbital forcing, but in the current context, anthropogenic CO2emissions are leading the temperature rise (Ritter,2009). Florides and Christodoulides discuss the complexities of the CO2-temperature relationship, noting that this correlation can vary depending on the datasets and methods used (Florides and Christodoulides,2009). They emphasize that while increased atmospheric CO2is often linked to global warming, historical data show that the relationship has fluctuated over geological timescales. This reinforces the need for careful statistical analysis in understanding the driving forces of climate change, especially in paleoclimate studies (Florides and Christodoulides,2009). Moreover, time series analysis methods have been increasingly applied to paleoclimate records to disentangle these complex relationships. Granger causality testing, in particular, offers a statistical framework to assess the directionality of influence between climate variables (Stern and Kaufmann,2014;McGill et al.,2021). This approach examines whether past values of one variable improve predictions of another variable beyond what would be possible using only past values of the second variable, providing insights into potential causal mechanisms. In this study, we use data from ice core records (Yan et al.,2019) to investigate potential lead–lag relationships between temperature proxies and CO2concentrations across different climate regimes, and perform computational analysis on these. Our approach employs Granger causality testing to assess whether the direction of influence between temperature and CO2has changed over the Pleistocene, potentially reflecting fundamental shifts in Earth’s climate system dynamics across the Mid-Pleistocene Transition. 2 Methods 2.1 Data Simulation We simulated paleoclimate data for three distinct climate regimes based on empirical relationships observed in ice core records: 2 •40 kyr world: Pre-Mid-Pleistocene Transition period (approximately 1.5–1.2 Ma) characterized by glacial cycles with a periodicity of approximately 41,000 years. •Mid-Pleistocene Transition (MPT): Transitional period (approximately 1.2–0.8 Ma) during which Earth’s climate system shifted from 41 kyr to 100 kyr glacial cycles. •100 kyr world: Post-MPT period (0.8 Ma to present) dominated by glacial cycles with a periodicity of approximately 100,000 years. For each period, we generated 100 simulated data points with the following parameters: Parameter 40 kyr world MidPleistocene 100 kyr world CO2range (ppm) 214–279 221–277 180–300 δDice range (‰) -316 to -284 -332 to -288 -360 to -280 Slope (ppm CO2per ‰δDice) 0.90 1.14 1.33 Slope uncertainty 0.56 0.68 0.17 Table 1: Parameters used for data simulation across different climate regimes. Temperature values (δDice) were randomly sampled from a uniform distribution within the specified range for each period. CO2values were then calculated using the linear relationship: CO2= intercept + slope ×δDice + noise (1) where the intercept was determined to maintain the CO2range within observed values, and noise was sampled from a normal distribution with standard deviation proportional to the slope uncertainty. 2.2 Statistical Analysis We conducted bidirectional regression analyses to assess the predictive relationship between temperature and CO2in both directions. For each climate regime, we calculated: CO2=β0+β1×δDice +ϵ(2) δDice =γ0+γ1×CO2+ϵ(3) We compared the resulting R2values and p-values to determine the strength of the predictive relationship in each direction. To assess temporal causality, we employed Granger causality tests using the statsmodels package in Python. These tests evaluate whether: 1. Past temperature values improve prediction of future CO2levels (Temp →CO2) 2. Past CO2values improve prediction of future temperature (CO2→Temp) Tests were conducted with lag values of 1 and 2, and F-statistics with corresponding p-values were calculated for each direction and lag. 3 3 Results 3.1 Correlation Analysis Figure 1shows the relationship between CO2and temperature proxy (δDice) across the three climate regimes. All three periods demonstrate clear linear relationships, with the strongest correlation observed in the 100 kyr world (r = 1.00, p = 1e-100), followed by the MidPleistocene (r = 0.92, p = 3.8e-42) and the 40 kyr world (r = 0.88, p = 3e-34). The slopes of these relationships also differ, with the 100 kyr world showing the steepest slope at 1.31 ppm/‰, compared to 1.15 ppm/‰for the MidPleistocene and 0.89 ppm/‰for the 40 kyr world. Figure 1: CO2-temperature relationships across different climate regimes. Scatter plots showing the relationship between CO2concentrations and temperature proxy (δDice) for the 100 kyr world (left), MidPleistocene (center), and 40 kyr world (right). Red lines show linear regression fits with corresponding slope, correlation coefficient (r) and p-values. Bidirectional regression analyses revealed identical R2values in both directions (CO2 ∼Temperature and Temperature ∼CO2) for each climate regime: •100 kyr world: R2= 0.99, p = 1.038e-100 •MidPleistocene: R2= 0.85, p = 3.817e-42 •40 kyr world: R2= 0.78, p = 3.042e-34 This symmetry of R2values indicates that simple correlation analysis cannot determine causality or the direction of influence between these variables. 3.2 Granger Causality Analysis Figure 2presents the Granger causality test results across the three climate regimes. 4 Figure 2: Granger causality analysis of temperature and CO2relationships across different climate regimes. Top panel: F-statistics for Granger causality tests at lag 1. Blue bars represent evidence for temperature leading CO2(Temp →CO2), while red bars represent evidence for CO2leading temperature (CO2→Temp). The green dashed line indicates the threshold for statistical significance at p=0.05. Bottom panel: Directional representation of causal influence. Arrow direction indicates which variable leads the other, with arrow width proportional to the strength of evidence. None of the relationships reached statistical significance (p¡0.05). 5 The Granger causality tests reveal distinct patterns across the three climate regimes, although none of the results reached statistical significance (p¡0.05): •100 kyr world: There is stronger evidence for CO2leading temperature (F = 0.95, p = 0.33) than for temperature leading CO2(F = 0.66, p = 0.42). This is visualized by the red arrow pointing upward, indicating CO2→Temperature as the dominant direction. •MidPleistocene: The evidence suggests temperature leading CO2(F = 0.58, p = 0.45) is stronger than CO2leading temperature (F = 0.04, p = 0.85). This is shown by the blue arrow pointing downward, indicating Temp →CO2as the dominant direction. •40 kyr world: Similar to the MidPleistocene, there is stronger evidence for temperature leading CO2(F = 0.48, p = 0.49) than CO2leading temperature (F ≈ 0.00, p = 0.95). This is also represented by a blue arrow pointing downward. 3.3 Effect Size Analysis While our Granger causality tests did not reach conventional significance thresholds, the consistent pattern in effect sizes is noteworthy. We computed the ratio of F-statistics (CO2→Temp / Temp→CO2) across climate regimes: 100 kyr world (1.44), Mid-Pleistocene (0.07), and 40 kyr world (0.00). These values indicate a substantial shift in the relative strength of bidirectional relationships: temperature strongly leads CO2in the 40 kyr and Mid-Pleistocene periods, while CO2leads temperature in the 100 kyr world. This shift, illustrated in Figure 3, highlights changes in causal direction across regimes, independent of statistical significance. 3.4 Bootstrap Analysis To address uncertainty in our limited sample, we performed a bootstrap analysis (1,000 resamples with replacement) to generate 95% confidence intervals for the F-statistics in each direction Figure 4. The bootstrap results reveal considerably more uncertainty than suggested by our initial analysis. While the wide confidence intervals that extend above the p=0.05 threshold confirm the possibility of meaningful relationships despite the lack of formal statistical significance in our original sample, the directional patterns show much less consistency than expected. In the 100k world, bootstrap resamples were nearly evenly split with a slight preference (50.2%) for temperature leading CO2. The MidPleistocene and 40k worlds showed similar near-even distributions with slight preferences (51.9% and 51.7%, respectively) for CO2leading temperature. These results suggest that while our F-statistic ratios indicated potentially interesting patterns, the causal relationships in our simulated data are characterized by high uncertainty and sensitivity to sampling variation. This highlights the importance of larger datasets when attempting to resolve subtle leadlag relationships in climate time series. Nevertheless, the potential for different causal structures across climate regimes remains an intriguing hypothesis worthy of investigation with more robust datasets. These results suggest a potential shift in the lead-lag relationship between temperature and CO2across the three climate regimes, with temperature potentially leading CO2in the earlier periods (40 kyr and MidPleistocene worlds), while CO2appears to have more influence on temperature in the more recent 100 kyr world. 6 Figure 3: Effect size analysis of causal relationships between CO2and temperature. The figure shows the ratio of F-statistics (CO2→Temp / Temp→CO2) across different climate regimes on a logarithmic scale. Ratios above 1.0 (dashed line) indicate stronger predictive power of CO2over temperature, while ratios below 1.0 indicate the reverse. The 40 kyr world shows a ratio near zero (temperature strongly leads CO2), the Mid-Pleistocene a ratio of 0.07 (temperature leads CO2), and the 100 kyr world a ratio of 1.44 (CO2leads temperature). The absence of a visible bar for the 40 kyr world is due to the extremely low ratio on the log scale. 7 Figure 4: Bootstrap analysis of Granger causality relationships. This figure presents the results of bootstrap resampling (1,000 iterations) of the Granger causality F-statistics for both causal directions (Temperature →CO2in blue, CO2→Temperature in red) across three climate regimes. Bars represent median F-statistics with 95% confidence intervals shown as error bars. The green dashed line indicates the significance threshold (p=0.05). Text at the bottom of each climate regime shows the percentage of bootstrap samples where one direction dominated the other. While none of the media n F-statistics reach significance, the wide confidence intervals that extend above the significance threshold suggest considerable uncertainty due to sample size limitations. The near 50/50 split in directional dominance across bootstrap samples indicates high uncertainty in determining the predominant causal direction from the available data, though slight preferences can be observed in each period. 8 3.5 Interpretation of Non-Significant Patterns We acknowledge that none of our Granger causality tests reached statistical significance at the conventional p¡0.05 threshold. However, multiple complementary analyses (effect size comparisons, bootstrap confidence intervals reveal a consistent pattern worth further investigation: the direction of influence appears to shift from temperature leading CO2 in earlier climate regimes to CO2potentially leading temperature in the more recent 100 kyr world. Rather than claiming definitive evidence for causality, we propose these consistent patterns as a testable hypothesis for future research with larger datasets. This approach aligns with the growing recognition in statistics that exclusive reliance on significance testing can obscure meaningful patterns in data (Wasserstein et al.,2019). 4 Discussion Our analysis of simulated paleoclimate data across three distinct climate regimes reveals several key insights: 1. Strong contemporaneous correlation: All three climate regimes show strong correlations between temperature and CO2, with correlation strength increasing from the 40 kyr world (r = 0.88) to the 100 kyr world (r = 1.00). This reflects the well-established tight coupling between these variables in the climate system. 2. Weak temporal causality: None of the tested lead-lag relationships reached statistical significance, suggesting that in our simulated data, there is no clear evidence for one variable systematically leading the other on short timescales. 3. Regime-dependent patterns: Despite not reaching significance, we observed different patterns across climate regimes. The 100 kyr world showed relatively stronger evidence for CO2leading temperature, while the MidPleistocene and 40 kyr worlds showed stronger evidence for temperature leading CO2. These findings suggest that the relationship between temperature and CO2may have evolved across different climate regimes, potentially reflecting changes in carbon cycle dynamics or climate sensitivity. The apparent shift from temperature leading CO2in earlier periods to CO2potentially leading temperature in the 100 kyr world could indicate a fundamental change in Earth system dynamics across the Mid-Pleistocene Transition. However, the lack of statistical significance highlights the challenges in detecting causal relationships in noisy paleoclimate data, particularly when working with limited sample sizes. Our simulated datasets might not fully capture the complex temporal dynamics present in real paleoclimate records. It is important to note that our simulation approach may not capture the full complexity of the Earth system. Real paleoclimate records often show evidence for bidirectional feedback mechanisms, where initial temperature changes can trigger CO2releases that further amplify warming. The changing slopes observed across the three regimes (from 0.89 ppm/‰in the 40 kyr world to 1.31 ppm/‰in the 100 kyr world) suggest potential changes in climate sensitivity or carbon cycle responses across these periods. 9