Proxies for the atmospheric concentration of COacross the Phanerozoic Eon show a weak declining trend over the Phanerozoic Eon ( Figure 4 ). The baseline remains relatively constant near 1000 parts per million by volume (ppmv) with numerous smaller troughs and larger peaks as high as nearly 6000 ppmv at 200 Mybp ( Figure 4 ). Therefore, the steady and steep decline in mean global temperature during the Phanerozoic is not attributable to a corresponding much weaker decline in atmospheric COconcentration. As discussed in the Methods, the smaller number of proxy datapoints for atmospheric COconcentration yields a less complete record than for T, with the consequence that the averaged curve (red curve in Figure 4 ) exhibits intermittent gaps over the represented span of the Phanerozoic. The most complete continuous proxy record of atmospheric COconcentration is from ~174 to 0 Mybp. The most densely- and continuously-sampled record is the last 80 My, which encompasses the Paleocene-Eocene Thermal Maximum starting at approximately 56 Mybp and includes the steady global cooling that has taken place since that time.

Temperature proxies show a steady decline across the Phanerozoic Eon modulated by slow temperature cycles, both in raw data ( Figure 3 a) and in curves fitted to these data using diverse filtering algorithms ( Figure 3 a,b). Non-detrended temperature-proxy data provide the clearest and most accurate view of the temperature profile of the ancient climate, declining by an estimated 8–9 °C over 522 My [ 38 ] ( Figure 3 ). However, linearly-detrended temperature-proxy data provide a clearer indication of the periodicity of the climate during the Phanerozoic (lower curves in Figure 3 b). The original analysis of Veizer et al. [ 38 ] shows a temperature-proxy periodicity of 135–150 My and a cycle amplitude (trough-to-peak) convertible to ~4 °C ( Figure 3 b, purple curve). The expanded database of Prokoph et al. [ 28 ] shows approximately the same periodicity and a slightly reduced cycle temperature-proxy amplitude ( Figure 3 b, red curve). These findings collectively demonstrate that temperature oscillated during the Phanerozoic on a long time scale, with an average periodicity estimated visually as 135–150 My. Spectral analysis [ 28 ] placed the energy density peak at 120 My, but as noted by the authors, that estimate was based on a single cycle [ 28 ] (p. 127).

Spectral analyses of time series of atmospheric COconcentration and T over the Phanerozoic Eon ( Figure 7 ) reveal non-random distribution of spectral density peaks at My time scales. Close association of atmospheric COconcentration and T cycles would be indicated by spectral density peaks at the same period in the respective periodograms. Instead, the periodogram for atmospheric COconcentration 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 COconcentration periodogram occur at 6.0, 6.8, 7.8, 8.9, 11.3 and 14.6 My. The finding that periodograms of atmospheric COconcentration proxies and T proxies exhibit different frequency profiles implies that atmospheric COconcentration 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.

For the less highly-resolved older Phanerozoic data ( Table 2 ), 14/20 (70.0%) Pearson correlation coefficients computed between atmospheric COconcentration 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 COconcentration 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 COwas largely decoupled from T over the majority of the Phanerozoic climate.

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 COconcentration 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.

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

For the most highly-resolved Phanerozoic data ( Table 1 ), 12/15 (80.0%) Pearson correlation coefficients computed between atmospheric COconcentration proxies and T proxies are non-discernible (> 0.05). Of the three discernible correlation coefficients, all are negative, i.e., T and atmospheric COconcentration 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 correlation coefficients between the concentration of COin 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 COand 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 COvalues were interpolated in about half the cases and therefore inferences are correspondingly weaker as noted in the Methods.

Regression of linearly-detrended temperature proxies ( Figure 3 b, lower red curve) against atmospheric COconcentration proxy data reveals a weak but discerniblecorrelation between COconcentration and T ( Figure 6 ). Contrary to the conventional expectation, therefore, as the concentration of atmospheric COincreased during the Phanerozoic climate, T decreased. This finding is consistent with the apparent weak antiphasic relation between atmospheric COconcentration 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 COconcentration, or conversely,× 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. Regression of non-detrended temperature ( Figure 3 b, upper red curve) against atmospheric COconcentration shows a weak but discernible positive correlation between COconcentration and T. This weak positive association may result from the general decline in temperature accompanied by a weak overall decline in COconcentration (trendline in Figure 4 ).

Temperature and atmospheric COconcentration proxies plotted in the same time series panel ( Figure 5 ) show an apparent dissociation and even an antiphasic relationship. For example, a COconcentration peak near 415 My occurs near a temperature trough at 445 My. Similarly, COconcentration 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 COconcentration peak near 200 My occurs during a cooling climate, as does another, smaller COconcentration 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

3.3. Marginal RF of Temperature by Atmospheric CO 2

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 [ 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 [ 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. The absence of a discernible correlation between atmospheric COconcentration and T over most of the Phanerozoic, as demonstrated above, appears to contravene the widely-accepted view about the relationship between atmospheric COand temperature, by which increases in atmospheric COconcentration 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 COconcentration 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 COconcentration on T, radiative forcing (RF), quantified using the well-known logarithmic relationship between RF by atmospheric CO(RF) and its atmospheric concentration. The logarithmic RFcurve, established more than a century ago [ 10 ], implies a saturation effect, or diminishing returns, in which the marginal forcing power of atmospheric COdeclines as COconcentration in the atmosphere increases. I hypothesized that the consequent decline in absolute and marginal forcing at high atmospheric COconcentrations over the Phanerozoic climate might explain the absence of discernible correlation between the atmospheric concentration of atmospheric COconcentration and T simply because large swings in atmospheric COconcentration are then expected to have little effect on marginal forcing.

CO2 forcing curve was first constructed using the MODTRAN atmospheric absorption/transmittance code ( 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 To evaluate this possibility, the RFforcing curve was first constructed using the MODTRAN atmospheric absorption/transmittance code ( Figure 8 a). Six conditions were modeled, represented by successively lower curves in each part of Figure 8 a,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(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 8 a 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.

CO2 curves ( 2 concentrations for the identified latitudes and cloud conditions. The resulting ΔRF CO2 curves ( 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 ΔRFcurves ( Figure 8 b) were constructed by difference analysis of each radiative forcing curve ( Figure 8 a). 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 COconcentrations for the identified latitudes and cloud conditions. The resulting ΔRFcurves ( Figure 8 b) are shown here only for the natural range of atmospheric COconcentration, i.e., ~200–6000 ppmv CO, because only these values are normally relevant to forcing of T. Tropical latitudes were used to compute ΔRFfor 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

R 2 = 0.9918) by the logarithmic function: y = 3.4221ln(x) + 1.4926 (2) The mid-latitude forcing curve in Figure 8 a corresponding to a clear sky (arrow in Figure 8 b) is best fit (= 0.9918) by the logarithmic function:

2 concentration and conversely. The marginal forcing curves ( R 2 = 0.9917) by the power function: y = 562.43x−0.982 (3) This function explains 99.18% of the variance in forcing associated with COconcentration and conversely. The marginal forcing curves ( Figure 8 b) can be fit by both exponential and power functions. The corresponding mid-latitude curve in Figure 8 b corresponding to a clear sky, for example, is best fit (= 0.9917) by the power function:

CO2 associated with CO 2 . An exponential function, however, also provides a reasonable fit ( R 2 = 0.8265) to the same ΔRF CO2 data: y = 0.9745 e −4E−04x (4) This power function explains 99.17% of the variance in ΔRFassociated with CO. An exponential function, however, also provides a reasonable fit (= 0.8265) to the same ΔRFdata:

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.

CO2 peaks at 3.7 W/m2 at an atmospheric concentration of CO 2 of 200 ppmv, near the minimal CO 2 concentration encountered in nature during glacial cycling (~180 ppmv) [ CO2 declines continuously with increasing atmospheric CO 2 concentration ( 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 ( 2 (~407 ppmv) ( 2 forcing computed using Equation (3) above has declined by nearly two-thirds, to 41.56% of its maximum natural forcing power. For mid latitudes, MODTRAN calculations show that ΔRFpeaks at 3.7 W/mat an atmospheric concentration of COof 200 ppmv, near the minimal COconcentration encountered in nature during glacial cycling (~180 ppmv) [ 60 61 ]. From that peak ΔRFdeclines continuously with increasing atmospheric COconcentration ( Figure 8 b) as RF increases continuously and logarithmically ( Figure 8 a). Using the above Equations (3) and (4) respectively, ΔRFdeclines to half its initial value at an atmospheric concentration COof 337.15 ppmv, and to 1/or 36.79% of its initial value at 366.66 ppmv ( Figure 8 b). Both the half life and the exponential decay constant are similar across forcing curves computed here as expected ( Figure 8 b). 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(~407 ppmv) ( Figure 8 b), COforcing computed using Equation (3) above has declined by nearly two-thirds, to 41.56% of its maximum natural forcing power.

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 ( 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 ( 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%). If ΔRFis a more direct indicator of the impact of COon temperature than atmospheric concentration as hypothesized, then the correlation between ΔRFand 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 COconcentration in one-My bins over the recent Phanerozoic and either averaging or interpolating COvalues over the older Phanerozoic (Methods). Owing to the relatively large sample size, the Pearson correlation coefficient is statistically discernible despite its small value (= 0.16,= 199), with the consequence that only a small fraction (2.56%) of the variance in T can be explained by variance in ΔRF Figure 9 ). Even though the correlation coefficient between ΔRFand T is positive and discernible as hypothesized, therefore, the correlation coefficient can be considered negligible and the maximum effect of ΔRFon T is for practical purposes insignificant (<95%).

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 (18O*(−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 ( 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. The correlation coefficients between ΔRFand 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 COconcentration 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 δO*(−1) and ΔRFare non-discernible (> 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/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 ΔRFand conversely.

18O*(−1) and ΔRF CO2 are non-discernible. Of the six discernible correlation coefficients, four are negative. For the less-resolved older Phanerozoic time periods ( 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. For the less-resolved data from older time periods of the Phanerozoic Eon ( Table 2 ), 14/20 (70.0%) Pearson correlation coefficients between δO*(−1) and ΔRFare 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 RFvs. 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 ΔRFon T than observed above for the effects of atmospheric concentration of COon T, but the cumulative results nonetheless require the conclusion that ΔRFwas largely decoupled from T, or at most weakly coupled with T, over the majority of the Phanerozoic climate.