The sensitivity of the Earth's climate to radiative forcing, and the extent to which its value will affect our estimates of future global warming, remains one of the key uncertainties in long-range climate forecasting. Climate sensitivity is usually characterised as the change in Global Mean Surface Air Temperature (GMSAT) in response to a perturbation in the net radiative flux at the top of the atmosphere, referred to as radiative forcing.

Two metrics commonly used to quantify the climate sensitivity are the Transient Climate Response (TCR) and the Equilibrium Climate Sensitivity (ECS). They are defined as follows: TCR is the rise in GMSAT as CO₂ increases by 1 % annually, calculated at a time when CO₂ concentrations have reached double their initial value. ECS is the long-term increase in GMSAT in response to an instantaneous doubling of atmospheric CO₂, relative to pre-industrial concentrations. TCR values are lower than those of ECS due to the longer timescales required for the deep ocean to reach equilibrium . Both metrics can be computed from Earth System Models (ESMs). Ensembles of ESM simulations are collated within the Coupled Model Inter-comparison Project (i.e CMIP5 , CMIP6 ). This project serves as a foundational component of climate science, providing critical input to both scientific understanding and policy development regarding how climate change will evolve for a range of potential future scenarios of atmospheric greenhouse gas concentrations. In the most recent phase, CMIP6, the spread in the computed values of both TCR and ECS is actually wider than that of its predecessor, CMIP5 . This increase suggests that the newer ESMs have, when considered collectively, increased uncertainty in the projected levels of future global warming.

The Sixth Assessment Report of the Intergovernmental Panel on Climate Change (IPCC AR6, ) provides very likely (at most 90 % confidence) ranges for these quantities of 1.2 to 2.4 K for TCR, and 2.0 to 5.0 K for ECS. Although the majority of CMIP6 models fall within this range, a notable subset do not, with more high-sensitivity models exceeding the upper bound than low-sensitivity models falling below the lower bound (; Table ). This has led to speculation that the true climate sensitivity of Earth could be larger than previously assessed . As expected, models that simulate stronger warming trends in recent decades have higher TCR values, and project larger future temperature increases . ESMs in the CMIP6 ensemble provide projections of future GMSAT corresponding to Shared Socioeconomic Pathway (SSP) scenarios, each of which specifies a distinct potential future trajectory of radiative forcing. Even under the intermediate SSP2-4.5 scenario – which reflects approximately current policies and entails lower radiative forcing than high-emissions scenarios (SSP3-7.0 and SSP5-8.5) – several high-sensitivity CMIP6 models simulate GMSAT increases exceeding 3 °C above pre-industrial levels by mid-century.

There is ongoing debate about the validity of these high-sensitivity models. The emergent constraint technique aims to reduce such uncertainties by searching for an inter-ESM relationship between an aspect of future climate change and a measurable contemporary change in the real climate. Measurements of the current variable along with this regression can constrain the quantity associated with a future climate . The TCR summary metric has proven more amenable to constraint than ECS, owing to its closer alignment with observed climate trends over the historical period, and because ECS by definition corresponds to a climate system in equilibrium, which is not presently the case. Studies that have constrained TCR using emergent constraint methods have consistently shown that high-sensitivity models are less consistent with historical warming . found that the most likely value of TCR is 1.7 ± 0.7 K (90 % confidence), which excludes the higher-sensitivity CMIP6 models. found a very similar estimate for TCR of 1.6 ± 0.4 K (66 % confidence), and went on to show that observationally-constrained warming projections tend to align more closely with the CMIP5 ensemble median than those from CMIP6 models.

In contrast, the true value of equilibrium climate sensitivity (ECS) remains a subject of active scientific debate. An assessment by , which informed the ECS assessment in the IPCC AR6, constrained the range of effective climate sensitivity to between 2.3 and 4.5 K, with a best estimate near 3 K. This study found values below 2 K difficult to reconcile with historical observations, while, critically, values above 4.5 K were also considered unlikely (although not impossible). Paleoclimate studies examining past periods, such as the late Miocene, find evidence that is consistent with ECS values of around 4 K . Constraints on ECS using observational records tend to be lower , with central estimates of ∼ 3 K. Future work that integrates overlapping ranges from multiple lines of evidence on the likely ECS could yield a more tightly constrained estimate of climate sensitivity .

Anthropogenic aerosol forcing continues to be a major source of uncertainty in climate sensitivity estimates. The net radiative effect of aerosols has been difficult to quantify due to their cooling effect and highly complex interactions with clouds. Models with strong aerosol cooling and high sensitivity can produce warming trajectories similar to those of models with weaker cooling and lower sensitivity , making attribution challenging . Historically, global SO₂ aerosol emissions increased rapidly during the mid-20th century, peaking during the 1970s, before subsequently declining from 1980 onwards as a result of improved air quality regulations. This happened to coincide with rapid increases in greenhouse gas emissions, and this combined effect is thought to have contributed to the recent acceleration in observed warming . Present-day estimates of aerosol effective radiative forcing lie between -1.6 and -0.6 W m⁻² , consistent with an inferred ECS of around 2.2 K (90 % range: 1.6–3.0 K), when accounting for pattern effects . However, the influence of aerosols on cloud radiative feedback processes continue to be a major obstacle to constraining climate sensitivity. A recent study by argues that the positive feedback associated with the albedo change suggests ECS values of 4.5 K, a level more consistent with paleoclimate based constraints rather than those constrained by observations over recent decades.

The recent observed warming has further emphasised the need to constrain estimates of climate sensitivity. The years 2023 and 2024 were the warmest on record, with 2024 marking the first time global temperatures exceeded 1.5 K above pre-industrial levels (HadCRUT5, NOAAGlobalTemp, ERA5, Berkeley Earth, ). However, this short-term peak doesn't imply that the running mean of global warming has exceeded 1.5 K, as this surpassing coincided with a strong El Niño event during 2023 and 2024. It is suggested that such peaks due to ENSO may become more pronounced under climate change . Nevertheless, the occurrence of these warm years, in addition to the larger TCR estimates from CMIP6 models in comparison to CMIP5, has raised questions regarding whether the Earth's climate sensitivity could in fact be greater than current estimates. This study uses updated observational data to refine the estimates of climate sensitivity produced by , and to narrow the uncertainty in expected background levels of future warming for time-periods pertaining to the near future (2030), mid-century (2050) and late-century (2090) . We evaluate the extent to which the observed warming from 2020 to 2024 has influenced the likely ranges of TCR and future warming and compare these updated estimates with the likely ranges reported by the IPCC.

To constrain TCR and future warming, we apply the emergent constraint procedure used by and . The emergent constraint technique requires identifying an observable climate variable x that exhibits substantial variability within an ensemble of ESMs (such as CMIP6), and demonstrates a statistically significant relationship, f(x), with a target variable y. Since x is an observable quantity, it can be empirically determined from contemporary measurements. Consequently, f can be used to impose an observationally-informed constraint on y, based on the measurement of x. Such constraints are termed `emergent' because the relationship f arises from the collective behaviour of the model ensemble, and cannot be diagnosed from any individual climate model . In this study, the climate observable x is defined as the change in a statistic of GMSAT relative to a specified baseline year, while the future climate variable y that we wish to constrain is either the TCR or projected future warming at a certain year.

Ideally, the emergent relationship between x and y should be supported by an underlying theoretical framework which characterises their association . In this case, simple one- and two-box climate models provide a theoretical framework predicting a linear relationship between the change in GMSAT and both TCR and future warming . The relationship between TCR and the temperature anomaly ΔT is given by: 1TCR=sΔT+η where s≡Q2×/ΔQ is defined as the ratio of the radiative forcing associated with a doubling of CO₂, Q2×, to the radiative forcing at the time of observation, ΔQ. For the purposes of model fitting, we include an intercept term η of the TCR axis to account for systematic biases, regression dilution and model misspecification.

Models that have demonstrated a stronger warming trend in the past are likely to simulate a greater warming in the future . Radiative forcing over the last 50 years has been dominated by the emission of greenhouse gases, and therefore, a similar linear relationship is also expected for future warming under the SSP scenarios: 2ΔT(Future)=kΔT(Past)+η. Here, k is the proportionality constant that relates past and future warming for each SSP scenario. For a period under which the warming rate is constant, the temperature difference over equal-length intervals should result in k=1. However, since ΔT is time dependent, we normalise ΔT(Past) per decade and use this as the observable. Thus, the ΔT(Future) is directly proportional to the ΔT(Past)/10yr, and the proportionality coefficient k will be scaled by 10yr, which can be interpreted as a decadal warming timescale.

The emergent constraint on the TCR produced in this analysis is largely in agreement with the central TCR estimate of the IPCC AR6 Chapt. 7, but is slightly larger (by ∼ 0.1 K) than that of , and . The likely range in TCR in our analysis was narrower than that of both and , and is very similar to that of as well as the IPCC AR6 Chap. 7. The increase in the central estimate when evaluated using observational data from 1975 to 2019 and 1975 to 2024 is less than the difference between the central estimates in this study to both and . The primary reason for this discrepancy is that the observational temperature anomaly datasets used in this study differ sufficiently to affect the calculation. As noted by , the HadCRUT5 dataset includes upward revisions to temperature anomalies relative to its predecessor HadCRUT4, particularly after the start of the 21st century. These revisions lead to a increased observed warming trend in comparison to and and, consequently, a higher estimated TCR. Since HadCRUT4 has now been superseded by HadCRUT5, it was not used in this study. The differences between the two datasets are likely a result of using updated sea and land temperature measurement techniques, more sophisticated statistical methods to fill observational gaps, and a revised uncertainty evaluation for certain past years. This highlights a degree of sensitivity inherent to the emergent constraints approach, as it's highly dependent on robust observational data.

An additional reason for the discrepancy arises from the difference in the choice of model realisation compared to that of . The approach in this study simply used one ensemble member run per model, the one which had a prescribed set (r1ixpxfx) of initialisation conditions (available in all ESMs with the exception of the EC-Earth3 model). The hierarchical Bayesian approach used in allowed different realisations to be combined into the overall fit. However, the same study showed that it gave nearly identical uncertainty ranges to the method using one ensemble member and OLS. A third source of the discrepancy arises from differences in the time periods used to derive the constraint. This study uses all years from 1975–2024 compared to those used in and , which considered 1975–2019 and 1981–2017, respectively. Evaluating warming over a longer time period allows for a clearer trend to emerge, provided that the warming continues in a manner consistent with those previous years.

An additional source of uncertainty in our emergent constraints is the spatial pattern effects of sea surface temperatures (SST), whereby this effect potentially weakens the relationship between observed warming and climate sensitivity (and hence future warming) . However, studies have shown that the magnitude of emergent bias depends on the choice of SST dataset and the boundary conditions used to drive the forcing . Most analyses of the pattern effect have primarily focused on ECS (rather than TCR), which quantifies climate sensitivity under very long timescales.

As mentioned in Sect. , our comparison of the internal variability between the pre-industrial control runs and the detrended historical runs suggests that our estimate of internal variability in the detrended observational time series could be underestimated, by up to a maximum factor of ∼ 2.7 (Supplement Sect. S3, largest model ratio for CNRM-CM6). Multiplying our observational uncertainty due to internal variability by this factor causes a slight broadening of the likely TCR range. We find that our 90 % confidence range of TCR would change from 1.28–2.33 K to a maximum range of 1.18–2.42 K, for the model with this most extreme ratio of control to historical temperature variability. This small impact to such a large increase in the assumed uncertainty in the observed warming trend is because the uncertainty in the emergent constraint on TCR is predominantly due to the uncertainty in the emergent relationship across the models, rather than the uncertainty in the variability of the observed warming trend.

The emergent constraints on the future temperature anomalies since the pre-industrial era by mid-century and late-century are, unsurprisingly, largely in agreement with those of as well as that of Chap. 4 of the IPCC AR6 . The observable used to produce the emergent constraint in both this study and was the decadal warming rate, whereas that of IPCC AR6 was derived by combining scenario projections, observational constraints, and updated assessments of ECS and TCR. As shown in Fig. , the central estimate of the future warming projection is consistently within the low to mid part of the model range.

It is likely that the 1.5 °C warming level as set out by the 2015 Paris agreement will be exceeded within the next 10 years, given the recent warming trend , and the failure to reduce global CO₂ emissions in line with the lower forcing scenarios SSP1-1.9 and SSP1-2.6 . However, our results show that, for both TCR and future warming, the record global warming of 2023 and 2024 does not justify upward-revision of likely ranges. Therefore, the constraints presented here suggest that avoiding 2 °C of global warming, although challenging, remains possible.