## Introduction

Remaining carbon budgets (RCBs) represent the future cumulative CO2 emissions that would be consistent with keeping global warming to a specified level, such as those mentioned in the Paris Agreement1,2,3,4, and play an important role in framing the objectives of national and international climate policy. Despite being conceptually simple, RCBs have been defined and estimated in various ways and with many different underlying assumptions, resulting in a wide range of “best estimates” across different studies2. Moreover, most of these estimates of remaining budgets account for only a subset of the relevant uncertain processes and often omit the contribution of key uncertain processes (such as permafrost thaw or future scenario uncertainty, among others)1,2,5,6,7,8,9,10,11,12,13. Given the relevance of carbon budgets to inform climate policy discussions and decisions14, it is essential that the key uncertainties associated with the RCB are not only understood but also quantified and integrated into its main estimate.

The IPCC Special Report on Global Warming of 1.5 °C (hereafter SR1.5; ref. 1) is a prominent recent assessment of the RCB. SR1.5 used a new approach of a segmented framework2 that allows for calculating the RCB directly from an estimate of TCRE15,16,17 (informed by both models and observational constraints), an estimate of the anthropogenic warming to date, and information on the temperature response to the future evolution of non-CO2 emissions (generated by running reduced complexity climate model emulators, such as FaIR18,19 or MAGICC20). SR1.5 assessed other sources of uncertainties of the RCB, such as those arising from historical temperature uncertainty, recent CO2 emissions uncertainty, non-CO2 forcing and response, non-CO2 scenario variation, uncertainty in the shape of the TCRE distribution, and consideration of under-represented Earth system feedbacks (ref. 1; Table 2.2 therein). However, these uncertainties were not incorporated into a single distribution of the RCB, but were rather assessed individually as additional, uncertain factors. The SR1.5 budget assessment reflects the best available information at the time, and highlights a key knowledge gap related to how the distribution of the RCB is affected by uncertainties arising from both geophysical and socioeconomic processes1,2. While the segmented framework2 used in SR1.5 allows for the assessment of different factors contributing to the RCB estimates, it remains unclear how uncertainties in each of these individual factors affect the overall distribution of the remaining budget estimate.

Here we quantify the distribution of TCRE and the remaining budgets. Our framework allows for both: (1) a clear separation of the effect of individual uncertain factors affecting TCRE and the RCB, as well as (2) an estimate of the combined effect of different uncertain factors on the overall distribution of TCRE and the RCB. We derive uncertainty estimates from a combination of observation-based and modelled quantities, as well as from the subset of future emission scenarios that reach net-zero CO2 emissions before 2100—i.e. those that include the rapid CO2 emissions reductions required to meet the goals of the Paris Agreement. Furthermore, we distinguish here between geophysical uncertainty (associated with uncertain physical or biogeochemical processes in the climate system) and socioeconomic uncertainty (associated with human socioeconomic systems and decision-making processes). We characterise these two types of uncertainties differently in our framework. The geophysical uncertainty can be represented quantitatively by probability distributions reflecting current scientific knowledge. However, the socioeconomic uncertainty is not well suited to be quantified within a formal probability framework owing to its dependence on collective and individual human decision-making processes, as well as subjective choices and perceptions of decision-makers.

In the framework presented here, we define the RCB for a given temperature limit as a function of five input parameters and their respective distributions. The first two input parameters (anthropogenic warming to date, and cumulative historical CO2 emissions) are derived from observation-based data, and the third (the current non-CO2 fraction of total anthropogenic forcing) can be estimated from observationally-constrained model simulations. Together, these three parameters integrate the effects of geophysical uncertainty on the TCRE and its distribution. The fourth geophysical parameter (the unrealised warming from past CO2 emissions) is included to allow us to estimate carbon budgets associated with ambitious mitigation scenarios leading to net-zero CO2 emissions. In such scenarios, the TCRE alone may not provide a robust estimate of the CO2-induced warming21 as it does not account for the effect of shifting patterns of warming associated with temperature stabilisation, nor feedbacks that manifest fully on longer time scales, such as permafrost carbon release. The final input parameter (non-CO2 fraction of total anthropogenic forcing at the time of net-zero CO2 emissions) captures future non-CO2 forcing uncertainty. This future forcing parameter varies with the uncertainty in the forcing response to a specified level of non-CO2 emissions as well as the effect of socioeconomic pathway uncertainty associated with mitigation choices that influence the level of future non-CO2 emissions.

We first express the TCRE as a function of the current anthropogenic contribution to observed warming (∆Tanth), the current non-CO2 fraction of total anthropogenic forcing (fnc) and historical cumulative CO2 emissions (E) (see ‘Methods’ section for details of the derivation, and Table 1 for description of parameters):

$$TCRE = \frac{{\Delta T_{anth}}}{E} \times \left( {1 - f_{nc}} \right).$$
(1)

Using this TCRE relationship (Eq. (1)), we then derive an expression for the total carbon budget (TCB) associated with a given temperature limit (∆Tlim), defined as the total CO2 emissions from the pre-industrial reference period until the time that CO2 emissions reach net-zero:

$$TCB = E \,\times \left( {\frac{{\Delta T_{lim} - \Delta T_{ZEC}}}{{\Delta T_{anth}}}} \right) \times \left( {\frac{{1 - f_{nc}^ \ast }}{{1 - f_{nc}}}} \right).$$
(2)

Additional parameters in Eq. (2) represent the potential unrealised warming from past CO2 emissions, also referred to as the Zero-Emission Commitment (∆TZEC), and the expected non-CO2 fraction of total anthropogenic forcing at the time that CO2 emissions reach net-zero (f*nc) (see ‘Methods’ section for details of derivation).

Conceptually, this equation (2) for the TCB can be understood as a function of three terms: (1) cumulative historical CO2 emissions (E); (2) the available future warming between present-day and the temperature target $$\left( {\frac{{\Delta T_{lim} - \Delta T_{ZEC}}}{{\Delta T_{anth}}}} \right)$$; and (3) the time-evolving non-CO2 contribution to temperature change, as represented by the ratio of future to present-day forcing fractions $$\left( {\frac{{1 - f_{nc}^ \ast }}{{1 - f_{nc}}}} \right)$$. Finally, from Eq. (2), the RCB (defined as the total CO2 emissions from present day until the time that CO2 emissions reach net-zero, consistent with global temperatures reaching a desired warming level) can be calculated by subtracting historical CO2 emissions from the total budget, to arrive at:

$$RCB = E \,\times \left( {\left( {\frac{{\Delta T_{lim} - \Delta T_{ZEC}}}{{\Delta T_{anth}}}} \right)\left( {\frac{{1 - f_{nc}^ \ast }}{{1 - f_{nc}}}} \right) - 1} \right).$$
(3)

We used Eqs. (1) and (3) to obtain the distribution of the TCRE and RCBs, respectively, by randomly sampling the input distributions (summarised in Table 1, and detailed in ‘Methods’ section and Supplementary Tables S1 and S2). We estimated the distribution of $$\Delta T_{anth}$$ for the year 2019 using the method of ref. 22, based on the average of the three observational temperature datasets with full spatial coverage. The distributions for E and $$\Delta T_{ZEC}$$ are based on Gaussian distributions fitted to cumulative CO2 emissions from 1870 to 2019 from the Global Carbon Project23, and model-simulated Zero Emissions Commitment (ZEC) values from the coordinated model intercomparison project ZECMIP24, respectively. We represent the current non-CO2 forcing fraction as a 30-year average from 1990 to 2019, and computed the uncertainty range for $$f_{nc}$$ using the FaIR emulator18 driven by scenarios from SR1.5 database25,26. Future non-CO2 forcing fraction ($$f_{nc}^ \ast$$) values are defined as a linear function of $$f_{nc}$$ to reflect the range across SR1.5 scenarios that is caused by geophysical forcing uncertainty, for the 30 years prior to the time of net-zero CO2 emissions in each scenario. Finally, we applied a constant offset to this function to capture the scenario-based variation (socioeconomic uncertainty) in the relationship between current and future non-CO2 forcing fractions (see ‘Methods’ section).

## Results and discussion

### Observation-based estimate of the TCRE distribution

Our median TCRE estimate is 0.44 °C per 1000 GtCO2, with a 5–95% range of 0.32–0.62 °C per 1000 GtCO2 (Fig. 1; ‘Main case’; Supplementary Table S3). This is a similar median estimate, though with a much narrower range as compared to the assessed AR5 range29 that was used in SR1.5 (median 0.44 °C per 1000 GtCO2; 1-sigma range 0.22–0.68 °C; Fig. 1a grey distribution). Our 5–95% TCRE range also agrees well with previously reported TCRE estimates (Refs. 11,15,18,30,31,32,33,34,35,36,37,38,39,40,41,42; Fig. 2c). We do however find a stronger constraint on the left-hand side of the distribution (low TCRE values, with sharply increasing probability above 0.25 °C/1000 GtCO2), while the right-hand side of this distribution has a wider tail. This right-skewed distribution shape of our observationally-constrained TCRE estimate is physically related to the possibility of a large negative aerosol forcing40.

We further assessed the sensitivity of our TCRE estimate to varying the three input parameters. The TCRE distribution is most sensitive to changes in the current non-CO2 forcing ($$f_{nc}$$) parameter (Eq. (1)). Assuming zero uncertainty in $$f_{nc}$$ results in a TCRE 5–95% range that is 33% narrower, whereas assuming zero uncertainty in E or ∆Tanth decreases the TCRE 5–95% range by only 13% and 10%, respectively. Given that the dominant contribution of $$f_{nc}$$ uncertainty is from aerosol forcing uncertainty, where a stronger aerosol forcing implies a smaller total non-CO2 forcing and therefore a larger TCRE, we can expect that better constraining observed aerosol forcing would have a large impact on our median TCRE estimate and its distribution.

### Remaining carbon budgets integrating key geophysical uncertainties

Our median RCB for 1.5 °C is 440 GtCO2 from 2020 onwards, representing a 50% chance of stabilising warming at or below 1.5 °C. (Note that we report remaining budgets rounded to the nearest 10 GtCO2, following SR1.5). The corresponding budget for a 67% chance of remaining below the target is 230 GtCO2 from the year 2020 onwards (Fig. 2a; see Supplementary Fig. S4 for 1.75 and 2 °C budgets). Our median and 67% budget estimates for 1.5 °C are smaller by 60 GtCO2 and 110 GtCO2, respectively, than those reported by SR1.5 (table 2.2 therein, adjusted to 2020 onwards using observed CO2 emissions for 2018–2019). Our lower median budget compared to SR1.5 can be explained by the explicit representation of a broader range of sources of geophysical uncertainty in our framework; indeed, our “no uncertainty” median estimate shown in Fig. 2b is almost equal to the SR1.5 median value.

In our estimate of the RCB and its distribution, we have therefore internalised the sources of geophysical uncertainty which were reported separately by SR1.5: (1) current non-CO2 forcing uncertainty is now explicitly represented by uncertainty in our parameter $$f_{nc}$$; (2) future non-CO2 forcing uncertainty is internally consistent with historical uncertainty such that higher and lower values of $$f_{nc}$$ are paired with correspondingly higher and lower values of future $$f_{nc}^ \ast$$; (3) recent emission uncertainty is folded into the uncertainty in E; (4) historical temperature uncertainty is captured by our ∆Tanth distribution; (5) our TCRE distribution reflects the distributions of the input parameters and is therefore implicit in the remaining budget distribution; (6) adjustments for under-represented feedbacks in ESMs are embedded in our method, given that our TCRE estimate is derived from observed quantities that include the effect of all relevant processes operating in the Earth system; and (7) we have additionally included uncertainty in ∆TZEC which was only recently quantified24 and thus not included as a quantified uncertainty in the SR1.5 (or any other) carbon budget analysis. On near-term decadal time scales relevant to achieving the 1.5 °C or 2.0 °C target, this ∆TZEC term also accounts for the additional changes in global mean temperature due to feedbacks, such as permafrost carbon release24 that are not captured by the TCRE but may contribute to warming on longer time scales.

The integration of uncertainties and use of observational constraints on our input parameters has allowed us to generate a considerably narrower spread in the estimate of the RCB compared to that provided by SR1.5 (see grey bars on Fig. 2b showing the SR1.5 budget distribution: lower grey bar for the likely estimate of the remaining budget based on spread in TCRE alone, and the upper grey bar for the sum of all additional geophysical uncertainties as reported by SR1.5). It is worth noting however, that the spread of our RCBs estimate does include negative values, with a 17% chance that the RCB for 1.5 °C is less than zero (i.e. is already exceeded). This outcome could arise due to current and/or unrealised future warming being at the higher end of their respective distributions, or in the case that the current non-CO2 forcing fraction is small or negative owing to very strong current aerosol forcing. In this case, we would expect 1.5 °C to be exceeded even in the absence of additional emissions, and any future emissions between now and the time of net-zero CO2 emissions would cause temperatures to rise further above this threshold.

Of the four uncertain geophysical parameters in Eq. (3), the current non-CO2 forcing fraction ($$f_{nc}$$), anthropogenic warming (∆Tanth) and unrealised warming (∆TZEC) all had a substantial effect on our RCB distribution (Fig. 2b). Setting the uncertainty of each parameter to zero increased the 67% RCB for 1.5 °C from 230 to 260, 290 and 300 GtCO2 for $$f_{nc}$$, ∆Tanth and ∆TZEC, respectively (Supplementary Table S4). Interestingly, decreasing the uncertainty in historical cumulative emissions (E) had almost no effect on the spread of the RCB distribution. This can be understood as a result of two opposing effects. First, higher E would decrease the estimated TCRE, leading to an increase in the estimate of the TCB from Eq. (2). However, higher E also means that a larger portion of the total budget has already been emitted, which consequently decreases the available future emissions. The result is a remaining budget estimate that is almost insensitive to the uncertainty associated with historical cumulative emissions.

### Effect of socioeconomic uncertainties on the remaining carbon budget

The results presented above (and shown in Fig. 2) integrate the effect of geophysical uncertainties on the RCB. However, earlier studies2,14 also highlighted the importance of socioeconomic uncertainties that affect future pathways of non-CO2 forcing. This source of uncertainty can be explored in our framework by adjusting the relationship between current ($$f_{nc}$$) and future ($$f_{nc}^ \ast$$) non-CO2 forcing fractions to reflect the variation in non-CO2 forcing across future scenarios that is caused by socioeconomic uncertainty in future emission pathways. This relationship between $$f_{nc}$$ and $$f_{nc}^ \ast$$ at the time of net-zero emissions is shown in Fig. 3a, where each grey dot represents an individual scenario available in the SR1.5 scenario database25,26, and the isolines indicate the RCB for each combination of $$f_{nc}$$ and $$f_{nc}^ \ast$$. Here, $$f_{nc}$$ represents geophysical uncertainty associated with the estimate of today’s non-CO2 forcing (shown by the horizontal spread of the grey points; Fig. 3a). In contrast, $$f_{nc}^ \ast$$ represents both geophysical uncertainty associated with future forcing, and socioeconomic uncertainty associated with future non-CO2 emissions scenarios (represented by the vertical spread of grey points; Fig. 3a).

To separate the geophysical and socioeconomic contributions to the spread in $$f_{nc}^ \ast$$, we first used the positive correlation between current $$f_{nc}$$ values and future $$f_{nc}^ \ast$$ values (thick solid line) to propagate current forcing uncertainty onto the future scenarios, such that each $$f_{nc}$$ value is mapped to a unique and consistent $$f_{nc}^ \ast$$ value (based on the scenarios from SR1.5 database25,26). This approach avoids using combinations of current and future forcing values that are outside the SR1.5 scenario range, and captures the effect of both historical and future non-CO2 forcing uncertainty associated with the “average” future scenario (this is the approach used for the “main case” results shown in Fig. 2). To assess the effect of socioeconomic scenario uncertainty, we then used the vertical spread of $$f_{nc}^ \ast$$ values around the best-fit line as a representation of the scenario-based variation in $$f_{nc}^ \ast$$ that is mostly independent of the geophysical variation in $$f_{nc}$$. We therefore adjusted the intercept of the regression line using offset values of ±0.05 which reflect the 5–95% spread across best-fit lines to each individual scenario in the ensemble (see ‘Methods' section). In doing so, we are able to assess the sensitivity of the RCB estimate to the portion of future non-CO2 forcing fraction variation that is caused by socioeconomic uncertainties affecting future non-CO2 emission pathways.

This socioeconomic scenario uncertainty results in a substantial change in the RCB (Fig. 3b). Future non-CO2 forcing fraction values that are 0.05 higher (vertical offset shown by dashed line, Fig. 3a) decrease the remaining budget distribution for 1.5 °C by 170 GtCO2, causing its median value to decrease to 270 GtCO2. Conversely, if $$f_{nc}^ \ast$$ values are 0.05 lower, the RCB increases by the same amount and its median becomes 610 GtCO2. These changes in the median remaining budget (based on plausible values of $$f_{nc}^ \ast$$ resulting from scenario variation) are smaller than the estimated uncertainty in the remaining SR1.5 budget for 1.5 °C due to non-CO2 scenario variation of ±250 GtCO2 (SR1.5, table 2.2. therein, represented by the two dark grey bars in Fig. 3b). We are therefore able to provide here a robust quantitative basis that both supports and constrains the SR1.5 assessment of how future non-CO2 emissions will affect the spread of the RCB.

## Implications of incorporating uncertainties into TCRE and remaining carbon budget estimates

Here we explicitly integrated of a full spectrum of geophysical uncertainties into our TCRE and RCB estimates. The resulting TCRE distribution is consistent with the TCRE range used by SR1.5, but narrower, and more constrained at lower values (which follows from the empirical distributions of anthropogenic warming and (1 − $$f_{nc}$$) that are also more constrained at lower values). Our RCBs have a similar range as those reported in SR1.5, despite including a considerably larger set of uncertain processes. SR1.5 reported only the uncertainty in TCRE as an integrated part of the RCB uncertainty (reporting additional sources of uncertainty separately). Our framework, on the contrary, integrates these sources of geophysical uncertainty into a single distribution of the RCB. The overall uncertainty in our estimate of the RCB for future temperature targets is more constrained than the sum of all sources of uncertainty reported by SR1.5.

We show further that estimates of the remaining budget can be affected by mitigation scenario choices that determine the difference between future ($$f_{nc}^ \ast$$) and current ($$f_{nc}$$) non-CO2 forcing, and provide an alternative quantification compared to that reported by SR1.5. Scenarios with higher $$f_{nc}^ \ast$$ values are associated with substantially lower RCBs. This effect would be particularly pronounced in the case of a small or negative current $$f_{nc}$$ (such as would be associated with a strongly negative current aerosol forcing) which could plausibly lead to a large future $$f_{nc}^ \ast$$ value as a result of decreased aerosol emissions from decreased fossil fuel use. Similarly, failure to mitigate non-CO2 greenhouse gas emissions, such as methane from sources other than fossil fuels would lead to an increase in the future non-CO2 forcing fraction. In all cases, the estimates of RCBs consistent with keeping warming well-below 2 or 1.5 °C as indicated in the UN Paris Agreement imply a strong limit on allowable future CO2 emissions, and highlight that immediate measures are required to bring down global CO2 emissions to net zero in the coming decades. Our results also illustrate that human choices regarding how stringently we reduce non-CO2 emissions can markedly decrease or increase the size of the RCB and affect the overall global decarbonisation challenge43.

The framework described here presents a tool for exploring how the RCB distribution is affected by changes in assumptions and uncertainties in each of its determining components. We note that this framework is subject to several assumptions. First, we assume that the TCRE (as estimated from present-day observations) is the same as the TCRE that would be estimated at the time 1.5 °C is reached in a scenario of continued increasing emissions. This assumption is supported by many previous studies which have shown that in Earth System Models (ESMs), the TCRE is a good predictor of CO2-induced temperature changes over this range of warming levels as well as across a range of different emission scenarios15,29,32,44,45.

Second, we assume that all relevant feedbacks (including those not well represented by current models, such as permafrost carbon loss or methane release from wetlands) are reflected in the observational data that we have used to constrain the TCRE. We assume further that contributions from these feedbacks will not accelerate sufficiently over the next few decades as to change the observation-based TCRE estimate. This assumption is supported by recent analyses which have shown that the magnitude of the permafrost carbon feedback scales with cumulative CO2 emissions, suggesting that an observationally-constrained TCRE will remain a reasonable estimate of the climate response to future CO2 emissions8. We recognise, however, the potential for non-linear climate responses to cumulative CO2 emissions to become larger at higher warming levels. In addition, for higher warming levels, the possibility of tipping elements in the climate system increases46. Our framework is therefore best suited to estimating the remaining budget for warming levels below 2 °C during this century.

Third, we have represented the CO2-induced temperature change at the time of net-zero CO2 emissions as a function of both the TCRE and the ZEC. Here, the TCRE represents the transient warming from a given amount of cumulative emissions, and the ZEC reflects the additional warming or cooling that could occur in a scenario with rapidly decreasing CO2 emissions as a result of the transition from transient to equilibrium CO2-induced warming. This formulation allows for the lagged response of permafrost and other longer timescale feedbacks to manifest during the time that global CO2 emissions decrease to net-zero, and for the uncertainty in this response to be captured by uncertainty in the ∆TZEC parameter. This formulation also accounts for the so-called “pattern effect”, whereby the strength of physical climate feedbacks is expected to increase over time as a result of changing warming patterns47,48,49. However, this pattern effect may also lead to deviations from a linear forcing-temperature relationship over time, which we have not accounted for in the representation of non-CO2 forcing in our framework. While we expect non-linearities associated with the pattern effect or with feedbacks, such as permafrost thaw to be relatively small for temperatures below 2 °C, further research is needed to quantify if such effects would introduce further adjustments of the RCBs in our framework.

Finally, we assume that the temperature response is approximately proportional to effective radiative forcing, and therefore, that the non-CO2 warming contribution can be approximated from the non-CO2 forcing fraction ($$f_{nc}$$ and $$f_{nc}^ \ast$$, for the present day and future, respectively)50. As shown in Supplementary Fig. S5, the 30-year (1990–2019) average of $$f_{nc}$$ that we have used here is a good proxy for the non-CO2 warming fraction in the year 2019. Given that the non-CO2 forcing fraction has increased substantially over recent decades in response to decreasing or stable aerosol emissions with increasing CO2 forcing (see Supplementary Fig. S5), this suggests that there is about a 15-year lag between changing forcing fractions and the realised warming fraction associated with this change. Consequently, we expect there to also be a similar lag between future non-CO2 forcing changes and the resulting non-CO2 contribution to future warming. By selecting the 30-year average preceding the year of net-zero CO2 emissions in each scenario, we expect that we are correctly estimating the non-CO2 contribution to observed warming at the net-zero year. However, our framework does not include any information or assumptions about what occurs after the net-zero year; clearly both the continuing trajectory of non-CO2 forcing, combined with whether CO2 emissions remain at net-zero or become net-negative, will determine whether global temperatures successfully stabilise at the target temperature in subsequent years.

The framework we have described here allows for an estimate of the TCRE and its distribution by incorporating input parameters that can be derived from (and themselves constrained by) observational data. It also allows for more comprehensive and integrated treatment of uncertainties associated with estimates of the RCB. The resulting overall uncertainty in our estimates of the RCB for future temperature targets is substantially narrower than previous assessments. Our median estimate of the 1.5 °C RCB is consistent with a scenario of global CO2 emissions that reach net-zero around the year 2040, emphasising the central requirement of rapid CO2 emission reductions to retain any reasonable chance of meeting this temperature target. Our framework also shows explicitly how uncertainty in each parameter contributes to the overall distribution of the RCB, and is able to quantify the effect of mitigation decisions that will determine the non-CO2 contribution to future temperature changes. We emphasise that carbon budget estimates will need to be continually updated as scientific knowledge, and on-going CO2 emissions and mitigation efforts progress over the coming years51. In particular, narrower constraints on the strength of current aerosol forcing and its potential to change as a result of decarbonisation or air pollution control efforts52,53 will be key to decreasing the overall uncertainty in estimates of the RCB. The new estimate and improved quantification of uncertainties provided here demonstrate that both CO2 and non-CO2 greenhouse gas emissions must be decreased as quickly as possible to maintain a possibility of not surpassing the global temperature goal of the Paris Agreement.

## Methods

### Transient climate response to cumulative CO2 emissions

The transient climate response to cumulative CO2 emissions (TCRE) is defined as the transient warming of the climate system per unit of CO2 emitted:

$$TCRE = \frac{{\Delta T_{CO_2}}}{E},$$
(4)

where $$\Delta T_{CO_2}$$ is the CO2-induced warming, and E is the total historical CO2 emissions. The TCRE is usually formally estimated at the point of doubled CO2 in a CO2-only climate model simulation. However, given its time and scenario independence, the TCRE can also be estimated using observed quantities to estimate the CO2-only contribution to historical temperature change. Here, we approximate the historical temperature change caused by total historical CO2 emissions as:

$$\Delta T_{CO2} = \Delta T_{anth} \times \frac{{F_{CO2}}}{{F_{anth}}} = \Delta T_{anth} \times \left( {1 - f_{nc}} \right).$$
(5)

Here, $$\Delta T_{anth}$$ is an estimate of the anthropogenic contribution to observed warming. $$\frac{{F_{CO2}}}{{F_{anth}}}$$ is the ratio of CO2 to total anthropogenic forcing, which we rewrite as $$(1 - f_{nc})$$, where $$f_{nc}$$ is the non-CO2 fraction of total anthropogenic effective radiative forcing (i.e. $$f_{nc}$$ is $$\frac{{F_{non - CO2}}}{{F_{anth}}}$$, and $$F_{anth}$$ is a sum of $$F_{CO_2}$$ and $$F_{non - CO_2}$$). We can therefore estimate the TCRE as:

$$TCRE = \frac{{\Delta T_{anth}}}{E} \times (1 - f_{nc}).$$
(6)

### Remaining carbon budgets

Rearranging Eq. (6), historical cumulative emissions can be expressed as:

$$E = \frac{{\Delta T_{anth}}}{{TCRE}} \times (1 - f_{nc}).$$
(7)

Similarly, the TCB associated with some future temperature target ($$\Delta T_{lim}$$, e.g. 1.5, 1.75 or 2.0 °C of anthropogenic warming since the pre-industrial level (1850–1900); refs. 1,2,4) can be expressed as:

$$TCB = \frac{{\Delta T_{lim}}}{{TCRE}} \times (1 -{\,} f_{nc}^ \ast ).$$
(8)

Here, $$f_{nc}^ \ast$$ is now the future non-CO2 forcing fraction that occurs at the time that the temperature target is reached.

This formulation assumes that the TCRE is a robust predictor of CO2-induced warming between present-day and the time that the temperature target is reached. This has been shown by many previous studies to be a reasonable assumption in the case that emissions continue to increase15,16,17,32,44. However, this assumption may not hold for ambitious mitigation scenarios as a result of the transition from transient to equilibrium CO2-induced warming that would occur as CO2 emissions decrease to zero. As a result, Eq. (8) holds only for scenarios with increasing emissions which reach and then exceed the temperature limit. To generalise this to the case of decreasing emission scenarios, we therefore introduce an additional term ($$\Delta T_{ZEC}$$) that represents the warming or cooling that would occur after CO2 emissions are set abruptly to zero from a scenario with increasing emissions24,54. This ZEC represents unrealised warming or cooling from past CO2 emissions only, and allows us to approximate the CO2-induced warming when emissions reach net-zero ($$\Delta T_{CO_2 - net - zero}$$) as:

$$\Delta T_{CO_2 - net - zero} = TCRE \times TCB + \Delta T_{ZEC}.$$
(9)

Incorporating this relationship into Eq. (8) results in the following equation for the TCB:

$$TCB = \frac{{\left( {\Delta T_{lim} - \Delta T_{ZEC}} \right)}}{{TCRE}} \times (1 - f_{nc}^ \ast ),$$
(10)

where $$\Delta T_{lim}$$ represents the temperature target (e.g. 1.5, 1.75 or 2.0 °C of anthropogenic warming since 1850–1900, as before), but now this temperature change can be associated with a given TCB at the time that CO2 emissions reach net-zero.

Substituting Eq. (6) into Eq. (10) results in the following equation to calculate the TCB (included as Eq. (2) in the main manuscript):

$$TCB = E \times \left( {\frac{{\Delta T_{lim} - \Delta T_{ZEC}}}{{\Delta T_{anth}}}} \right) \times \left( {\frac{{1 - f_{nc}^ \ast }}{{1 - f_{nc}}}} \right).$$
(11)

Finally, the RCB for a given temperature limit is the difference between the above TCB (Eq. (11)) and the total historical CO2 emissions E. We therefore arrive at Eq. (3) in the main manuscript:

$$RCB = E \times \left( {\left( {\frac{{\Delta T_{lim} - \Delta T_{ZEC}}}{{\Delta T_{anth}}}} \right)\left( {\frac{{1 - f_{nc}^ \ast }}{{1 - f_{nc}}}} \right) - 1} \right).$$
(12)

This framework therefore can be used to calculate the RCB as a function of the three parameters used to estimate the TCRE ($$\Delta T_{anth}$$, E and $$f_{nc}$$) and the two additional parameters $$f_{nc}^ \ast$$ and $$\Delta T_{ZEC}$$.

Note that we have not included an adjustment to the RCB in our framework to represent the effect of so-called under-represented Earth system feedbacks (such as permafrost carbon cycle feedbacks). This adjustment was assessed in SR1.5 on the grounds that the TCRE values used in their analysis were derived largely from ESMs which did not include permafrost and other potentially important Earth system feedbacks. However, in our framework, we are estimating the TCRE from observed quantities rather than from ESMs. As a result, these observations should include the transient effect of all feedbacks that are currently operating in the real Earth system, including the effect of permafrost carbon loss. Furthermore, the inclusion of the ZEC parameter in our framework additionally allows for the slow-response component of permafrost feedbacks to be captured as part the additional warming or cooling occurs in the transition from increasing to net-zero CO2 emissions.

### Data sources for input parameters and sampling method

Here, we describe the methods and data sources used to derive the distribution of each input parameter used to estimate the TCRE and RCB (for details of each distribution see Supplementary Tables S1 and S2; Supplementary Figs. S1 and S2).

Historical cumulative CO2 emissions (E) and their uncertainty (±1 σ range) for the period from 1870 to 2019 are taken from the 2019 Global Carbon Project estimate23, and include cumulative CO2 emissions from both fossil fuels and land-use change. This estimate is based on energy and industry statistics, as well as land-use book-keeping methods23. Here, we fitted a Gaussian distribution to the mean and 1 standard deviation range of 640 ± 65 GtC (2350 ± 240 GtCO2) reported by ref. 23.

For $$\Delta T_{anth}$$ we use the method of ref. 22 to estimate the anthropogenic contribution to observed warming for the year 2019 (relative to the 1850–1900 base period), by removing natural variability and natural forcing contributions using a statistical model18. We used the mean of temperature observations from three datasets that are interpolated to a high spatial coverage (HadCRUT-CW55, GISTEMP56 and Berkeley Earth57), resulting in a median estimate of $$\Delta T_{anth}$$ of 1.18 °C in 2019. This estimate is 0.06 °C higher than that produced using the four observational datasets used by SR1.5 (GISTEMP, NOAA58, HadCRUT-CW and HadCRUT59) owing to the incomplete spatial coverage in the HadCRUT and NOAA datasets. In addition, the most recent version of HadCRUT-CW includes an upward revision of observed temperature estimates due to improved algorithms to account for biases related to the transition in marine temperature measurement instruments60. Our distribution of $$\Delta T_{anth}$$ accounts for instrumental uncertainty in the HadCRUT dataset (which we assume is representative of instrumental uncertainty in other datasets). We note however that we do not account for other potential sources of uncertainty, such as those related to spatial interpolation methods or systematic instrumentation changes. We also do not explicitly account for uncertainty in dataset choice; rather we have selected observational temperature products that are based on different input measurements of SST and land surface temperature, thus providing three (mostly independent) lines of evidence for the estimates of the observed warming. Furthermore, using these spatially interpolated observational datasets means that our estimate of anthropogenic warming is a global mean surface temperature (GMST) metric that is only 0.03 °C less than a global surface air temperature (GSAT) metric based on only air temperatures rather than blended air and surface ocean temperatures. We therefore argue that our representation of anthropogenic warming is a reasonable compromise between GMST and GSAT, and is also consistent with the temperature goal of the Paris Agreement4,27,28,61.

The current non-CO2 forcing fraction ($$f_{nc}$$) and its uncertainty are taken from an empirical distribution generated using the FaIR climate model emulator18, using updated forcing estimates for aerosols62 and for CO2 and methane63,64. We also used FaIR to estimate the future non-CO2 forcing fraction ($$f_{nc}^ \ast$$), so as to generate an internally consistent set of historical and future forcing fractions. We used 411 net-zero CO2 emission scenarios from the SR1.5 scenario database25,26, and FaIR was run using a 1000-member perturbed parameter ensemble for climate sensitivity, carbon cycle feedbacks and present-day effective radiative forcing. These simulations were constrained to observed temperature changes, so that the scenario ensembles do not contain combinations of climate sensitivity and forcing that are incompatible with observed warming to date. We note that we could equally have used a different emulator (e.g. MAGICC20) for this purpose, though as shown in a recent intercomparison, there is little difference in the forcing distributions generated by different model emulators when driven by consistent forcings65.

A key assumption in our framework is that we can approximate the fraction of historical warming caused by non-CO2 emissions using their fractional effective radiative forcing, which should hold well as long as this forcing fraction does not change rapidly over time50. Given that the non-CO2 forcing fraction has increased over the past decade as a result of decreased aerosol emissions, using recent forcing fractions would overestimate the amount of warming at 2019 caused by non-CO2 emissions, and consequently underestimate the amount of warming caused by CO2 and the resulting TCRE. We therefore used the average non-CO2 forcing fraction for 1990–2019 as a more robust proxy for the 2019 non-CO2 fraction of anthropogenic warming (see Supplementary Fig. S5). The resulting median estimate for $$f_{nc}$$ is 0.14, based on a combined non-CO2 forcing of 0.32 W/m2 and total anthropogenic forcing of 2.30 W/m2 (using the 1990–2019 multi-scenario constrained ensemble average). Of this combined non-CO2 forcing, aerosols contribute −1.03 W/m2 and non-CO2 greenhouse gases (including ozone and stratospheric water vapour from methane oxidation) contribute +1.40 W/m2.

Where $$f_{nc}$$ represents the geophysical uncertainty associated with current estimates of effective radiative forcing, $$f_{nc}^ \ast$$ is meant to reflect uncertainty associated with future emission scenarios. In the SR1.5 scenario database however, future non-CO2 forcing values vary due to both current geophysical uncertainty (affecting the current non-CO2 forcing estimate) as well as future scenario variation in non-CO2 emissions. To separate these two effects, we first defined $$f_{nc}^ \ast$$ as a function of $$f_{nc}$$ based on the linear relationship across all scenarios that reached net-zero CO2 emissions during this century (see Table 1 and thick solid line in Fig. 3a indicating the ordinary least squares fit to all scenarios). This resulted in the equation for $$f_{nc}^ \ast$$ used in the ‘main case’ and shown in Table 1:

$$f_{nc}^ \ast = 0.3081f_{nc} + 0.14 + {\mathrm{offset}}.$$
(13)

This linear approximation allows us to exclude combinations of $$f_{nc}$$ and $$f_{nc}^ \ast$$ that are not covered by the range of SR1.5 scenarios, and to introduce a constant offset value to allow us vary the representation of $$f_{nc}^ \ast$$ to reflect socioeconomic uncertainty independently from the geophysical uncertainty in $$f_{nc}$$. To estimate a representative offset value, we performed a similar fit to the ensemble of simulations for each individual emissions scenario from each model in the SR1.5 scenario database to gauge the portion of the vertical spread of points that can be explained purely by non-CO2 scenario variation. Based on the 5–95% range of the intercepts of across all individual scenario fits, we selected an offset value of ±0.05 to capture the plausible effect of non-CO2 scenario variation on the difference between current and future non-CO2 variation.

To quantify the effect of socioeconomic uncertainty associated with this set of ambitious mitigation pathways, we therefore adjusted the intercept of the main-case regression line by ±0.05 (dashed lines in Fig. 3a) to capture the variation in $$f_{nc}^ \ast$$ across scenarios at a given value of $$f_{nc}$$. This offset therefore reflects human choices leading to socioeconomic uncertainty rather than geophysical uncertainty. Notably, this applies most strongly to emissions of short-lived climate forcers, where decreased emissions of methane and black carbon have the potential to decrease future non-CO2 forcing between now and the time of net-zero CO2 emissions and lead to larger RCBs. Similarly, decreased emissions of reflective aerosols will increase future non-CO2 forcing and lead to smaller RCBs. To reflect this element of human decision-making, we adopted the constant offset approach described above rather than including this socioeconomic uncertainty as part of the formal distribution of the RCB. We do note however that the vertical spread of all points in Fig. 3a is slightly larger than what can be explained by only scenario variation. This suggests that there may be an additional geophysical forcing uncertainty affecting future non-CO2 forcing, that reflects different possible mixes of individual species leading to the same total present-day non-CO2 forcing. We do not currently account for this additional geophysical uncertainty in our analysis, but acknowledge that it may introduce an additional shift of the median carbon budget (and an increase in spread) relative to the results presented in Fig. 2.

Finally, our distribution of $$\Delta T_{ZEC}$$ is based on the coordinated model intercomparison project ZECMIP24,54. In ZECMIP, ESMs and ESMs of Intermediate Complexity (EMICs) estimated the temperature change that occurred after CO2 emissions were set to zero at the point that cumulative emissions reached 750 and 1000 GtC (i.e. 2750 and 3670 GtCO2) in a scenario with prescribed CO2 concentration increases of 1% per year. We use the ZEC50 values from these experiments, which represents the change in temperature 50 years after zero emissions. For the 750 GtC and 1000 GtC experiments (which correspond approximately to the range of TCBs for temperature limits between 1.5 and 2 °C), the mean ZEC50 across models was −0.03 °C (for 750 the GtC experiment) and −0.06 °C (for the 1000 GtC experiment), with a model spread (5–95% range) in both cases of ±0.3 °C (ref. 24). However, since most of the models included in this ensemble did not include the effect of permafrost carbon feedbacks, we shifted the mean ZEC estimate from ZECMIP upward, to be centered on 0 °C, based on the difference in the 50-year ZEC in the UVic ESCM (ref. 66) between model versions with and without permafrost carbon feedbacks. This resulted in a ZEC distribution of 0 ± 0.3 °C as the mean and 5–95% uncertainty range for $$\Delta T_{ZEC}$$ (see Supplementary Fig. S4).

Finally, we derived distributions of the TCRE and RCBs, including their respective medians and uncertainty ranges, by sampling the distributions of input parameters in Eqs. (1) and (3). These inputs are based on either empirical distributions or normal distributions representing the spread of uncertainty in each of the parameters. Here, we assume that the uncertainty ranges in the input distributions are uncorrelated. While these quantities are obviously related in a coupled system, (e.g. warming being a function of $$f_{nc}$$ and E), we are not concerned with how one quantity is correlated with, or changes as a result of changes in other quantities, because our main interest is the uncertainty associated with the value of a particular quantity. As these uncertainty estimates come from different observations or models, and are based on uncertainties around today’s values, they can be assumed to have uncorrelated uncertainty ranges, and thus, a direct sampling method is justified. We do note, however, that the tails of the distributions in figures. 1 and 2 are more sensitive to the assumptions of our framework; we show 5–95% ranges to allow comparison with other studies, though we acknowledge that this range is less robust than the median and 33–67% ranges that we have reported for the RCB distribution.