#### 3.2. Temperature versus Atmospheric Carbon Dioxide

Temperature and atmospheric CO

_{2} concentration proxies plotted in the same time series panel (

Figure 5) show an apparent dissociation and even an antiphasic relationship. For example, a CO

_{2} concentration peak near 415 My occurs near a temperature trough at 445 My. Similarly, CO

_{2} concentration peaks around 285 Mybp coincide with a temperature trough at about 280 My and also with the Permo-Carboniferous glacial period (labeled 2 in

Figure 5). In more recent time periods, where data sampling resolution is greater, the same trend is visually evident. The atmospheric CO

_{2} concentration peak near 200 My occurs during a cooling climate, as does another, smaller CO

_{2} concentration peak at approximately 37 My. The shorter cooling periods of the Phanerozoic, labeled 1–10 in

Figure 5, do not appear qualitatively, at least, to bear any definitive relationship with fluctuations in the atmospheric concentration of CO

_{2}.

Regression of linearly-detrended temperature proxies (

Figure 3b, lower red curve) against atmospheric CO

_{2} concentration proxy data reveals a weak but discernible

negative correlation between CO

_{2} concentration and T (

Figure 6). Contrary to the conventional expectation, therefore, as the concentration of atmospheric CO

_{2} increased during the Phanerozoic climate, T decreased. This finding is consistent with the apparent weak antiphasic relation between atmospheric CO

_{2} concentration proxies and T suggested by visual examination of empirical data (

Figure 5). The percent of variance in T that can be explained by variance in atmospheric CO

_{2} concentration, or conversely,

R^{2} × 100, is 3.6% (

Figure 6). Therefore, more than 95% of the variance in T is explained by unidentified variables other than the atmospheric concentration of CO

_{2}. Regression of non-detrended temperature (

Figure 3b, upper red curve) against atmospheric CO

_{2} concentration shows a weak but discernible positive correlation between CO

_{2} concentration and T. This weak positive association may result from the general decline in temperature accompanied by a weak overall decline in CO

_{2} concentration (trendline in

Figure 4).

The correlation coefficients between the concentration of CO

_{2} in the atmosphere and T were computed also across 15 shorter time segments of the Phanerozoic. These time periods were selected to include or bracket the three major glacial periods of the Phanerozoic, ten global cooling events identified by stratigraphic indicators, and major transitions between warming and cooling of the Earth designated by the bar across the top of

Figure 5. The analysis was done separately for the most recent time periods of the Phanerozoic, where the sampling resolution was highest (

Table 1), and for the older time periods of the Phanerozoic, where the sampling resolution was lower (

Table 2). In both cases all correlation coefficients between the atmospheric concentration of CO

_{2} and T were computed both for non-detrended and linearly-detrended temperature data (

Table 1 and

Table 2, column D1). The typical averaging resolution in

Table 1 is one My, although resolutions down to 59 Ky were obtained over some time periods of the recent Phanerozoic (see below).

Table 2 is also based on one-My interval averaging, although CO

_{2} values were interpolated in about half the cases and therefore inferences are correspondingly weaker as noted in the Methods.

For the most highly-resolved Phanerozoic data (

Table 1), 12/15 (80.0%) Pearson correlation coefficients computed between atmospheric CO

_{2} concentration proxies and T proxies are non-discernible (

p > 0.05). Of the three discernible correlation coefficients, all are negative, i.e., T and atmospheric CO

_{2} concentration are inversely related across the corresponding time periods. Use of the distribution-free Spearman Rho correlation coefficient yields similar conclusions: 10/13 Spearman correlation coefficients computed (76.9%) are non-discernible and all discernible correlation coefficients are negative (

Table 1). The similarity of results obtained using parametric and non-parametric statistics suggests that conclusions from the former were not affected by the underlying assumptions (normality, independence, equal variance).

The most recent Phanerozoic was sampled most frequently in both the temperature and atmospheric CO

_{2} concentration proxy databases used here (

Figure 2). Correlation coefficients over these “high-resolution” regions are designated by the superscript “c” in

Table 1, and include Entries # 1 and # 3. The average sample resolutions over these time periods are 105 Ky (Entry # 1) and 59 Ky (Entry # 3), a three order-of-magnitude improvement over the one-My resolution that characterizes most paleoclimate data evaluated here. The respective correlation coefficients between temperature and atmospheric CO

_{2} concentration proxies are nonetheless non-discernible, consistent with the majority of correlation analyses. Correlation analysis for the highest-resolution data, therefore, yield the same conclusions as from the broader dataset.

Sample datapoints are sufficiently frequent in the recent Phanerozoic that individual datapoints of temperature proxies can be closely matched (±<1%) with individual datapoints of atmospheric CO

_{2} concentration proxies, eliminating the need for interpolation or averaging in bins. These matched-pair data are the strongest available for correlation analysis and are designated by the superscript “d” (parametric) and “e” (non-parametric) in

Table 1, comprising Entries # 1, # 2, and # 4–6. The sampling resolution over these regions is computed by dividing the duration of the corresponding period by the sample size. To illustrate the period from 26 to 0 Mybp (Entry # 6 in

Table 1), the Pearson correlation coefficient is −0.03, the mean relative age difference is −0.22%, the number of sample datapoints is 154, and, therefore, the mean sampling interval is 169 Ky. The mean sampling interval for all of these high-resolution calculations is 199 Ky. Of the ten matched-pair correlation coefficients computed over the early Phanerozoic (five non-parametric), eight are non-discernible, two are discernible, and both discernible correlation coefficients are negative. Therefore, the most powerful and highly-resolved matched-pair regression analysis possible using these proxy databases yields the same conclusions as drawn from the entire dataset.

For the less highly-resolved older Phanerozoic data (

Table 2), 14/20 (70.0%) Pearson correlation coefficients computed between atmospheric CO

_{2} concentration and T are non-discernible. Of the six discernible correlation coefficients, two are negative. For the less-sampled older Phanerozoic (

Table 2), 17/20 (85.0%) Spearman correlation coefficients are non-discernible. Of the three discernible Spearman correlation coefficients, one is negative. Combining atmospheric CO

_{2} concentration vs. T correlation coefficients from both tables, 53/68 (77.9%) are non-discernible, and of the 15 discernible correlation coefficients, nine (60.0%) are negative. These data collectively support the conclusion that the atmospheric concentration of CO

_{2} was largely decoupled from T over the majority of the Phanerozoic climate.

Spectral analyses of time series of atmospheric CO

_{2} concentration and T over the Phanerozoic Eon (

Figure 7) reveal non-random distribution of spectral density peaks at My time scales. Close association of atmospheric CO

_{2} concentration and T cycles would be indicated by spectral density peaks at the same period in the respective periodograms. Instead, the periodogram for atmospheric CO

_{2} concentration shows spectral peaks at 2.6, 3.7, 5.3, 6.5 and 9.4 My, while the periodogram for T shows similar but smaller peaks at lower frequencies, including 2.6, 3.9 and 5.2 My, but lower frequency (higher period) peaks that are not matched in the atmospheric CO

_{2} concentration periodogram occur at 6.0, 6.8, 7.8, 8.9, 11.3 and 14.6 My. The finding that periodograms of atmospheric CO

_{2} concentration proxies and T proxies exhibit different frequency profiles implies that atmospheric CO

_{2} concentration and T oscillated at different frequencies during the Phanerozoic, consistent with disassociation between the respective cycles. This conclusion is corroborated by the auto- and cross-correlation analysis presented below.

#### 3.3. Marginal RF of Temperature by Atmospheric CO_{2}

The absence of a discernible correlation between atmospheric CO

_{2} concentration and T over most of the Phanerozoic, as demonstrated above, appears to contravene the widely-accepted view about the relationship between atmospheric CO

_{2} and temperature, by which increases in atmospheric CO

_{2} concentration cause corresponding increases in T owing to increased radiative forcing. Moreover, this finding from the ancient climate appears to be inconsistent with the well-established positive correlation between atmospheric CO

_{2} concentration and T across glacial cycles of the last 400–800 Ky [

60,

61]. I sought to resolve these apparent paradoxes by evaluating a more direct functional measure of the warming effect of atmospheric CO

_{2} concentration on T, radiative forcing (RF), quantified using the well-known logarithmic relationship between RF by atmospheric CO

_{2} (RF

_{CO2}) and its atmospheric concentration. The logarithmic RF

_{CO2} curve, established more than a century ago [

10], implies a saturation effect, or diminishing returns, in which the marginal forcing power of atmospheric CO

_{2} declines as CO

_{2} concentration in the atmosphere increases. I hypothesized that the consequent decline in absolute and marginal forcing at high atmospheric CO

_{2} concentrations over the Phanerozoic climate might explain the absence of discernible correlation between the atmospheric concentration of atmospheric CO

_{2} concentration and T simply because large swings in atmospheric CO

_{2} concentration are then expected to have little effect on marginal forcing.

To evaluate this possibility, the RF

_{CO2} forcing curve was first constructed using the MODTRAN atmospheric absorption/transmittance code (

Figure 8a). Six conditions were modeled, represented by successively lower curves in each part of

Figure 8a,b: tropical latitudes, mid-latitudes, and the sub-Arctic, each modeled assuming one of two cloud conditions, clear-sky or cumulus clouds. The respective altitudes of the tropopause are 17.0, 10.9 and 9.0 km. The global mean tropopause of the International Standard Atmosphere is 10.95 km in altitude, corresponding to the mid-latitude (green) curves in

Figure 8, which is therefore considered most representative of mean global forcing. Each latitude was modeled with no clouds or rain (upper curve of each color pair in

Figure 8) and with a cloud cover consisting of a cumulus cloud base 0.66 km above the surface and a top at 2.7 km above the surface (lower curve of each color pair). Additional MODTRAN settings used to construct

Figure 8 are the model default values, namely: CH

_{4} (ppm) = 1.7, Tropical Ozone (ppb) = 28, Stratospheric Ozone scale = 1, Water Vapor Scale = 1, Freon Scale = 1, and Temperature Offset, °C = 0. The RF curves in

Figure 8a demonstrate that the absolute value of forcing at the tropopause increases with atmospheric opacity (thickness) and is therefore greatest in tropical latitudes and least in the sub-Arctic, as already well-established [

48]. The shape of the logarithmic curve is similar at different latitudes, although the absolute value of forcing decreases at progressively higher latitudes, also as expected.

The ΔRF

_{CO2} curves (

Figure 8b) were constructed by difference analysis of each radiative forcing curve (

Figure 8a). Each datapoint in every RF curve was subtracted from the next higher datapoint in the same curve to compute the marginal change in forcing over the corresponding range of atmospheric CO

_{2} concentrations for the identified latitudes and cloud conditions. The resulting ΔRF

_{CO2} curves (

Figure 8b) are shown here only for the natural range of atmospheric CO

_{2} concentration, i.e., ~200–6000 ppmv CO

_{2}, because only these values are normally relevant to forcing of T. Tropical latitudes were used to compute ΔRF

_{CO2} for correlation analysis because the majority of empirical datapoints in the databases used are from paleotropical environments (Methods). Forcing is higher than the global mean in the tropics, however, owing to greater atmospheric opacity. Mid-latitudes are more representative of the global mean, and were therefore used to compute the decay rates of incremental forcing versus the atmospheric concentration of CO

_{2}.

The mid-latitude forcing curve in

Figure 8a corresponding to a clear sky (arrow in

Figure 8b) is best fit (

R^{2} = 0.9918) by the logarithmic function:

This function explains 99.18% of the variance in forcing associated with CO

_{2} concentration and conversely. The marginal forcing curves (

Figure 8b) can be fit by both exponential and power functions. The corresponding mid-latitude curve in

Figure 8b corresponding to a clear sky, for example, is best fit (

R^{2} = 0.9917) by the power function:

This power function explains 99.17% of the variance in ΔRF

_{CO2} associated with CO

_{2}. An exponential function, however, also provides a reasonable fit (

R^{2} = 0.8265) to the same ΔRF

_{CO2} data:

Given that both power and exponential functions provide acceptable fits to the marginal forcing curves, I used two corresponding measures to quantify the rate of decay of marginal forcing by CO_{2}: the half-life and the exponential decay constant, respectively. Half-life is the time required to decline to half the original value and is most appropriate for a power function. The exponential decay constant is the time required for marginal forcing to decline to 1/e or 36.79% of its original (maximum) value and is applicable only to an exponential function. Both the half-lives and the exponential decay constants were calculated here using the best-fit mid-latitude clear-sky marginal forcing curves and are expressed as the corresponding concentrations of atmospheric CO_{2} in units of ppmv.

For mid latitudes, MODTRAN calculations show that ΔRF

_{CO2} peaks at 3.7 W/m

^{2} at an atmospheric concentration of CO

_{2} of 200 ppmv, near the minimal CO

_{2} concentration encountered in nature during glacial cycling (~180 ppmv) [

60,

61]. From that peak ΔRF

_{CO2} declines continuously with increasing atmospheric CO

_{2} concentration (

Figure 8b) as RF increases continuously and logarithmically (

Figure 8a). Using the above Equations (3) and (4) respectively, ΔRF

_{CO2} declines to half its initial value at an atmospheric concentration CO

_{2} of 337.15 ppmv, and to 1/

e or 36.79% of its initial value at 366.66 ppmv (

Figure 8b). Both the half life and the exponential decay constant are similar across forcing curves computed here as expected (

Figure 8b). The half-life and exponential decay constant, therefore, are comparable across latitudes while absolute forcing varies by more than 300%. At the current atmospheric concentration of CO

_{2} (~407 ppmv) (

Figure 8b), CO

_{2} forcing computed using Equation (3) above has declined by nearly two-thirds, to 41.56% of its maximum natural forcing power.

If ΔRF

_{CO2} is a more direct indicator of the impact of CO

_{2} on temperature than atmospheric concentration as hypothesized, then the correlation between ΔRF

_{CO2} and T over the Phanerozoic Eon might be expected to be positive and statistically discernible. This hypothesis is confirmed (

Figure 9). This analysis entailed averaging atmospheric CO

_{2} concentration in one-My bins over the recent Phanerozoic and either averaging or interpolating CO

_{2} values over the older Phanerozoic (Methods). Owing to the relatively large sample size, the Pearson correlation coefficient is statistically discernible despite its small value (

R = 0.16,

n = 199), with the consequence that only a small fraction (2.56%) of the variance in T can be explained by variance in ΔRF

_{CO2} (

Figure 9). Even though the correlation coefficient between ΔRF

_{CO2} and T is positive and discernible as hypothesized, therefore, the correlation coefficient can be considered negligible and the maximum effect of ΔRF

_{CO2} on T is for practical purposes insignificant (<95%).

The correlation coefficients between ΔRF

_{CO2} and T were computed also across the same 15 smaller time periods of the Phanerozoic Eon bracketing all major Phanerozoic climate transitions as done above for atmospheric CO

_{2} concentration for both non-detrended and linearly-detrended temperature data. For the most highly-resolved Phanerozoic data (

Table 1), 12/15 (80.0%) Pearson correlation coefficients between δ

^{18}O*(−1) and ΔRF

_{CO2} are non-discernible (

p > 0.05). Of the three discernible correlation coefficients, all are positive. Use of the distribution-free Spearman Rho correlation coefficient yields similar conclusions: 10/13 Spearman correlation coefficients computed (76.9%) are non-discernible and all discernible correlation coefficients are positive (

Table 1). High-resolution and matched-pair correlation analyses gave similar or identical results. The most recent period of the Phanerozoic (

Table 1, Entry # 5), where the sampling resolution is highest, shows moderate-to-strong positive ΔRF

_{CO2}/T correlations for both detrended and non-detrended temperature data. Therefore, for this most recent cooling period, where sample resolution is greatest, about 24–28% of the variance in T is explained by variance in ΔRF

_{CO2} and conversely.

For the less-resolved data from older time periods of the Phanerozoic Eon (

Table 2), 14/20 (70.0%) Pearson correlation coefficients between δ

^{18}O*(−1) and ΔRF

_{CO2} are non-discernible. Of the six discernible correlation coefficients, four are negative. For the less-resolved older Phanerozoic time periods (

Table 2), 15/20 (75.0%) Spearman correlation coefficients are non-discernible. Of the five discernible Spearman correlation coefficients, three are negative. Therefore, although the data from this period of the Phanerozoic (

Table 2) are less resolved, conclusions drawn from their analysis are generally similar to those drawn using more highly-resolved data from the more recent Phanerozoic (

Table 1). The main exception is that discernible correlations computed using more highly-resolved data (

Table 1) are more positive, while those computed from less-resolved data (

Table 2) include more negative values. Combining RF

_{CO2} vs. T correlation coefficients from both tables, 51/68 (75.0%) are non-discernible, and of the 17 discernible correlation coefficients, seven (41.2%) are negative. These results collectively suggest a somewhat stronger effect of ΔRF

_{CO2} on T than observed above for the effects of atmospheric concentration of CO

_{2} on T, but the cumulative results nonetheless require the conclusion that ΔRF

_{CO2} was largely decoupled from T, or at most weakly coupled with T, over the majority of the Phanerozoic climate.

The spectral periodograms of ΔRF

_{CO2} and T show substantial similarities both in peak periods and relative energy (

Figure 10). Both profiles show clusters of peaks at 1–4, 5–9 and 14 My periods. These similarities signify comparable oscillations of ΔRF

_{CO2} and T across the Phanerozoic, unlike the relationship between atmospheric CO

_{2} concentration and T (cf.

Figure 10 with

Figure 7). Oscillation at similar periods in turn implies the possibility of an association between the corresponding cycles, although a mutual, reciprocal or common influence of a third variable cannot be excluded without further analysis. These conclusions are generally corroborated using auto- and cross-correlation analysis as described next.

#### 3.4. Auto- and Cross-Correlation

The typical approach to the detection of oscillations in time series is spectral analysis as illustrated above. Alternatively, qualitative inspection of time series followed by auto- and cross-correlation to confirm qualitative impressions quantitatively can be used to test non-random periodicity and phase relationships between oscillating variables that are not evident from spectral analysis. Qualitative inspection of time series of δ

^{18}O*(−1), atmospheric CO

_{2} concentration and ΔRF

_{CO2}, particularly during the high-resolution and relatively flat period from 175 to 80 Mybp (

Figure 11) shows apparent non-random oscillation of all three climate variables over time. The temperature-proxy oscillation (δ

^{18}O*(−1), red curve in

Figure 11) is coupled tightly with peaks in the oscillation of the strontium ratio (

^{87}Sr/

^{86}Sr, purple arrows in

Figure 11) as reported by Prokoph et al. [

28] and estimated here visually. Strontium ratios are typically interpreted as a proxy for riverine influx to the ocean attributable to climate change [

62] and are therefore expected to be coupled with temperature as confirmed in

Figure 11. The CO

_{2} oscillation does not appear tightly coupled with temperature-proxy oscillations, however, which is consistent with the correlation analysis presented above, but is clearly antiphasic with the oscillation of ΔRF

_{CO2} (yellow arrows in

Figure 11). This antiphasic relationship is expected given the derivation of ΔRF

_{CO2} from atmospheric CO

_{2} concentration, although detailed waveforms and phase relationships are difficult to predict a priori owing to the power and/or exponential relationship between atmospheric CO

_{2} concentration and forcing.

To evaluate these qualitative impressions quantitatively, auto- and cross-correlation coefficients between different time series were used. Autocorrelation can be used to detect non-random periodicity in time series by comparing a progressively lagged time series to itself. In this comparison the time series are shifted relative to each other by one datapoint at a time (successive lag orders) and the correlation coefficients between the variables are computed for each shifted dataset. If the autocorrelation coefficient oscillates as a function of order, and if individual autocorrelation coefficients are discernibly different from zero at peaks and/or troughs, then non-random periodicity is indicated. Cross-correlation between different time series is done similarly, except the time series of two different variables are compared to each other. Discernible correlation coefficients that oscillate with increasing shift order confirm non-random periodicity (autocorrelation) and at the same time establish phase relationships between oscillating variables (cross-correlation), which are beyond the capacities of spectral analysis.

Autocorrelation across the time period from 174 to 0 Mybp demonstrates non-random periodicity in all three major variables evaluated here (

Figure 12) as anticipated from the corresponding time series (

Figure 11). The correlation profile across increasing lag order for δ

^{18}O*(−1) (

Figure 12a) is qualitatively dissimilar from the profile of atmospheric CO

_{2} concentration (

Figure 12b) and ΔRF

_{CO2} (

Figure 12c). The profiles for atmospheric CO

_{2} concentration and ΔRF

_{CO2} are similar, as expected from derivation of the latter from the former. In both the atmospheric CO

_{2} concentration and ΔRF

_{CO2} profiles, both short and long cycles can be detected qualitatively by modulation of the corresponding correlation coefficients, demarcated by double-headed open arrows in

Figure 12,

Figure 13,

Figure 14,

Figure 15 and

Figure 16. For autocorrelation across the longer time period, the short and long cycles of atmospheric CO

_{2} concentration average 16.9 and 66.8 My, respectively, similar to the comparable values for ΔRF

_{CO2} (17.2 and 66.2 My). Cyclic variation of the autocorrelation coefficients from discernibly negative to positive as lag order increases demonstrates non-random periodicity in all three variables. Dominant cycles of forcing and temperature appear in both spectral periodograms in (

Figure 10) and autocorrelelograms (

Figure 12) at ~15 My.

Similar autocorrelation analysis restricted to the period of high sampling resolution and relatively flat baseline during the Phanerozoic (174 to 87 Mybp) gives similar conclusions (

Figure 13). All three climate variables show non-random periodicity, and the autocorrelation profile for δ

^{18}O*(−1) is qualitatively dissimilar from corresponding profiles of both atmospheric CO

_{2} concentration and ΔRF

_{CO2}, while the autocorrelation profiles of atmospheric CO

_{2} concentration and ΔRF

_{CO2} are qualitatively similar. Estimates of cycle duration from autocorrelelograms of all three variables over this shorter analytic time period are similar, namely, 16.8, 18.0 and 19.0 My for δ

^{18}O*(−1), atmospheric CO

_{2} concentration and ΔRF

_{CO2}, respectively.

The cross-correlation analyses of the same three time series for the same time periods of the Phanerozoic (

Figure 14,

Figure 15 and

Figure 16) enable assessment of the relationship between the three climate variables studied here. Cross-correlation between δ

^{18}O*(−1) and atmospheric CO

_{2} concentration (

Figure 14) shows periodicity on long (

Figure 14a) and short (

Figure 14b) time scales at respective cycle periods of 70.0 and 16.0 My. The cross-correlation between δ

^{18}O*(−1) and ΔRF

_{CO2} (

Figure 15) show similar periodicity with long and short cycles at about 55.0 and 17.0 My (

Figure 15a,b respectively). The cross-correlation between atmospheric CO

_{2} concentration and ΔRF

_{CO2}, in contrast, shows strongly negative correlation coefficients at low-order lags that maximize at zero order lag, signifying the precise antiphasic relationship between these variables that is qualitatively evident in

Figure 11. Both cross-correlation profiles show short-period cycles of about 16.0 My, and the broader time scale (

Figure 16a) again shows the long cycle of about 70.0 My. The short cycle is evident also in the spectral periodograms described above (

Figure 7 and

Figure 10).