### 2.1 Punctuated Evolution of Anthropogenic Emissions

The chain of events leading to increased *E* was initiated in the 16th century by a transition from wood to coal in the U.K. [*Nef*, 1977]. Such a transition to an abundant energy source fueled the industrial revolution during the 1800. However, anthropogenic emissions began to appreciably impact climate after the second industrial revolution [*Mokyr*, 1998], when new technological developments increased carbon emissions by many energy-intensive sectors (e.g., transportation [*Fuglestvedt et al*., 2008]). Emerging economies (mostly China and India [*Le Quéré et al*., 2013; *Li et al*., 2016]) soon followed a similar path with a delay of ∼60 years. This, together with an increasing exploitation of oil reserves (see Appendix S1, Supporting Information), partly explains the observed punctuated evolution in *c*_{c} (Figure 1b) from 1950 onward.

The description of such punctuated evolution of the human-energy-climate system starts by noting that the diffusion of innovations, that is, the fraction of population using a new technology, can be described by a logistic model [*Swan*, 1973]. In turn, the global average of per capita CO_{2} emissions varies proportionally to the number of individuals with access to carbon-emitting technologies. Hence, a logistic dynamics for *c*_{c} can be expressed as *dc*_{c}(*t*)/*dt* = *r*_{c}*c*_{c}(*t*)[*K*_{c} − *c*_{c}(*t*)], where *r*_{c} is a technology diffusion rate and *K*_{c} is a long-term per capita emission capacity representing the ability of an “average individual” to access energy resources that contribute to *E*. At the global level, adoption of new technologies exhibits a delay *τ*_{c} [*Fanelli and Maddalena*, 2012] associated with the time required for the formulation and implementation of policies, for infrastructure construction, for overcoming the resiliency of competing sectors (e.g., fossil fuel industry against renewable energies [*Showstack*, 2016]), and for bridging technological gaps among countries. The introduction of new energy production technologies thus contributes to continually changing the asymptotic value *K*_{c} leading to a punctuated evolution of *c*_{c}. Such delayed dynamics can be described by a time-delay capacity *K*_{c}(*t* − *τ*_{c}) that links the current *K*_{c} to the history of technological development at *t* − *τ*_{c}. The *K*_{c}(*t* − *τ*_{c}) may be an increasing function, if the spreading of new technologies promotes CO_{2} emissions (as in the case of coal and oil use being progressively adopted globally in the 19th and 20th centuries), or a decreasing one, if the technologies being introduced can be generically labeled as “green.”

### 2.2 Beyond-Peak Anthropogenic Emissions

Beyond “peak” emissions (assumed to be *t* > *t*_{m}), existing scenarios impose declining global CO_{2} emission rates, thereby implicitly assuming efficient spreading of renewable energy sources [*Barthelmie and Pryor*, 2014] and even negative emission strategies [*Smith*, 2015]. However, these technologies are developed in the world's leading economies and, similar to the historically observed spreading of industrialization, it is reasonable to assume that their global diffusion will occur with a delay depending on climate policies and international trade [*Popp*, 2011], as well as intrinsic societal inertia. To investigate the role of delays in *c*_{c} on future global warming scenarios, the coupled dynamics of *c*_{c}(*t*) and *N*(*t*) are used to construct realistic scenarios for *E*(*t*) in which the diffusion of climate-friendly or green technologies offsets global fossil fuel consumption, but with lags. To a leading order, per capita emissions are expected to follow a path in which the timescale of technological diffusion is commensurate to past technological dynamics, with *K*_{c}(*t* − *τ*_{c,m}) a decreasing function and *τ*_{c,m} a new time delay. Therefore, we start our analysis by initially assuming that the timescale of the progressive global diffusion of new technologies is an intrinsic property of the technological transfer process, and is thus valid for both the diffusion of carbon-emitting technologies (as observed in the past) and the spreading of green technologies (to be projected into the future). We then evaluate what changes in the coupled human-energy-climate system dynamics would be required to meet global warming targets by constructing emission scenarios with different delays *τ*_{c,m}. Changes in *E* resulting from the compound dynamics of *N* and *c*_{c} are related to global climate temperature by means of a well-studied zero-dimensional coupled carbon [*Garrett*, 2012] and energy [*Knutti and Hegerl*, 2008] models as described next. By zero-dimensional, we mean a model that does not explicitly resolve spatial variability so that all budget equations are formulated for the entire globe composed of land–ocean–atmosphere reservoirs. On the one hand, a zero-dimensional approach does not describe in detail local/regional patterns of innovation, anthropogenic emissions, and climatic conditions. On the other, delays in globally averaged per capita emissions *c*_{c} arise from spatiotemporal gradients in the development and adoption of technologies. Such local/regional heterogeneities are thus captured by our approach through the global dynamics of technological transfer, which we model explicitly. This motivates our choice of a simple zero-dimensional model.

### 2.3 Human-Energy-Climate System Model

#### 2.3.1 Per Capita Emissions

While *c*_{c} has been nearly steady over the last decade [*Garrett*, 2011, 2012], a plethora of factors influence its dynamics when viewed over a 100 years time frame. Here, a delayed differential equation (DDE) accounting for lags in diffusion of technologies is proposed and is given by:

dcctdt=rccctKctâˆ’Ï„câˆ’cctfortâ‰¤tm,(1) where the delayed per capita emissions capacity *K*_{c} is defined as [*Parolari et al*., 2015]:

Kctâˆ’Ï„c=Ac+âˆ«tâˆ’Ï„ctBcâ€²tâ€²cctâ€²dtâ€²,(2) where *A*_{c} is a background capacity and Bcâ€²t is a time-delayed kernel. The simplest representation of delayed dynamics is obtained by choosing Bcâ€²t=BcÎ´tâˆ’Ï„c with *δ* being a Dirac delta function [*Yukalov et al*., 2009; *Parolari et al*., 2015], leading to:

Kctâˆ’Ï„c=Ac+BcÂ·cctâˆ’Ï„c,(3) where *B*_{c} defines an increase in the ability of each individual to emit CO_{2} relative to the history of the human-energy-climate system at time *t* − *τ*_{c} [*Yukalov et al*., 2009; *Parolari et al*., 2015]. The parameter *B*_{c} represents the impact of past policies and technology adoption at *t* − *τ*_{c} on the per capita emission capacity at time *t*, accounting for the exploitation of energy resources, the development of new technologies, and their delayed diffusion across political boundaries or along socioeconomic gradients. It is to be noted that functions other than Bcâ€²t=BcÎ´tâˆ’Ï„c can also be used by replacing the *δ* with smooth functions weighing differently the entire history from *t*_{c} to *t* (i.e., the function integrates to unity) while still reproducing punctuated growth with some “smearing.” However, such functions require additional parameters that cannot be readily inferred from the limited data here, though the model still allows for their usage. The resulting DDE for *c*_{c}(*t*) is known to exhibit rich dynamics (e.g., exponential and punctuated growth, self-generated oscillations, finite-time singularities, and death [*Yukalov et al*., 2009; *Kaack and Katul*, 2013; *Parolari et al*., 2015]), providing a novel framework for the description of past (*t* ≤ *t*_{m}), as well as projected (i.e., *t* > *t*_{m}), human–energy–climate interactions (Figures 1a and 1b). Equation (1) must be solved numerically using Equation (3) for *t* > *τ*_{c} and *K*_{c} = *A*_{c} + *B*_{c} · *c*_{c}(*t*_{0}) for *t* ≤ *τ*_{c} (where *t*_{0} is the initial time).

#### 2.3.2 Emission Scenarios

Beyond *t*_{m}, adoption of climate-friendly technologies is assumed to out-weight fossil fuel consumption and spread globally but with a lag. To investigate the effects of reduced *E* (through reduced *c*_{c}) on the climate system but maintaining the accepted *N* projections, observed *c*_{c} increases in the past are assumed to follow a punctuated evolution into the future with drops instead of jumps. Hence, future *E* scenarios through *c*_{c} (i.e., for *t* ≤ *t*_{m}) can be described by a transformation of Equation (1). Defining the transformation T:cctâ†’câ€¾ct=cc*âˆ’cct that preserves continuity of *c*_{c} at *t* = *t*_{m}, and changing the sign of the right-hand side of Equation (1) for *t* > *t*_{m} (to indicate the reduction in carbon emissions associated with the adoption of green technologies), leads to:

dcctdt=âˆ’rc,mcc*âˆ’cctKctâˆ’Ï„c,mâˆ’cc*âˆ’cctfort>tm,(4) where cc*=cct0+cctm and Kctâˆ’Ï„c,m=Ac+Bc,mÂ·cc*âˆ’cctâˆ’Ï„c,m for *t* > *t*_{m} (defined by Equation (3) otherwise), being *B*_{c,m} the relative *K*_{c} decrease, and *τ*_{c,m} a new time-delay. The transformation *T* is performed to ensure that, if mitigation is effective in reducing emissions as much as past fossil fuel consumption was effective in emitting carbon (i.e., same set of model parameters *r*_{c,m} = *r*_{c}, *A*_{c,m} = *A*_{c}, *B*_{c,m} = *B*_{c}, *τ*_{c,m} = *τ*_{c}), preindustrial conditions are reached within the same time frame of industrialization (see Appendix S1). The parameter *B*_{c,m} encodes the reduction of per capita emissions due to the introduction of low-carbon emitting technologies, resulting in a punctuated decay of *c*_{c} to finite-time zero-emissions (*B*_{c,m} ≥ 1) or to a stationary level (0 < *B*_{c,m} < 1). However, if green technologies fail to offset fossil fuel consumption due to economic (un)feasibility [*Koningstein and Fork*, 2014] effects, resembling rebound [*Greening et al*., 2000; *Garrett*, 2012] can occur (punctuated oscillations for *B*_{c,m} < 0). Even though the solution of Equation (4) allows negative carbon emissions, negative per capita emissions are unlikely within the simulation time frame (1875–2100) and *c*_{c} is set to be nonnegative but still allowing *dc*_{c}/*dt* < 0.

#### 2.3.3 Population Dynamics

Population *N* is described by the logistic model [*Kaack and Katul*, 2013]:

dNtdt=rNNtKNâˆ’Nt,(5) where *r*_{N} is the growth rate and *K*_{N} is a finite carrying capacity assumed to be constant. The debate remains open on whether finite environmental resources (i.e., a finite carrying capacity) can sustain the observed faster-than-exponential growth [*Johansen and Sornette*, 2001; *Kaack and Katul*, 2013; *Parolari et al*., 2015] Faster than exponential dynamics leads to finite-time singularities, clocking a possible regime shift—or a Doomsday [*Von Foerster et al*., 1960]—in the mid-21st century (see Figure 1). A dynamical carrying capacity can also be used to account for degradation or improved resourcefulness on a densely populated planet together with lags between social and ecological dynamics [*Parolari et al*., 2015]. Following the same approach used for *c*_{c} leads to a DDE for *N*(*t*) with *K*_{N} = *A*_{N} + *B*_{N} · *N*(*t* − *τ*_{N}).

#### 2.3.4 Carbon Balance

The temporal dynamics of global *CO*_{2} emissions from fossil-fuel burning is defined as before with *E*(*t*) = *c*_{c}(*t*) · *N*(*t*). The balance between *E* and global net sinks leads to a global budget for atmospheric *CO*_{2} [*Garrett*, 2012]:

dCO2tdt=Î¼Etâˆ’ÏƒcCO2,TÂ·Î”CO2t,(6) where *μ* is a conversion factor accounting for instantaneous dilution of *CO*_{2} emissions in the total atmospheric mass (i.e., 1 ppmv *CO*_{2} = 2.13 Pg emitted carbon C with 1 unit of C corresponding to 3.667 units of atmospheric *CO*_{2} [*Garrett*, 2012]), *σ*_{c} = *σ*_{c,l} + *σ*_{c,o} is a sink rate accounting for land (*σ*_{c,l}) and ocean (*σ*_{c,o}) carbon uptake, and *ΔCO*_{2}(*t*) = [*CO*_{2}(*t*) − *CO*_{2,0}] is a departure from a preindustrial baseline value *CO*_{2,0} [*Garrett*, 2011, 2012]. Combining data for ocean and land sinks from 1980s and 1990s with a preindustrial equilibrium concentration of 275 ppmv, *Garrett* [2012] provide an estimate for *σ*_{c} of 0.0155 per year. The carbon balance in Equation (6) is a model of maximum simplicity [*Garrett*, 2012] though the sink term *σ*_{c} is regulated by complex carbon-cycle feedbacks occurring over several spatiotemporal scales [*Cox et al*., 2000; *Lenton*, 2000; *Le Quéré et al*., 2013]. The uptake rate of atmospheric *CO*_{2} by land and ocean sinks did decline approximately by one third over the 1959–2012 period [*Raupach et al*., 2014] due to nonlinear responses of the land–ocean system to warming and increasing *CO*_{2}. Many carbon-cycle climate models suggest that the terrestrial carbon sink depends on a trade-off between photosynthesis and respiration: the biosphere can provide a negative feedback between increasing *CO*_{2} and temperature until a threshold temperature is reached when biological respiration (increasing exponentially with *T*) exceeds the *CO*_{2} fertilization effect [*Heimann and Reichstein*, 2008]. The ocean carbon sink depends on the concentration gradient of *CO*_{2} between the atmosphere and the ocean but also on the solubility of *CO*_{2}. The *CO*_{2} partial pressure increases exponentially with sea surface temperature [*Lenton*, 2000], thus reducing the *CO*_{2} gradient driving the carbon sink from the atmosphere to the ocean. Therefore, increasing atmospheric *CO*_{2} will enhance the sink but such a negative feedback is expected to saturate due to warming [*Lenton*, 2000]. These mechanisms are accounted for using a linear dependence between land carbon uptake on temperature and the saturation effect of increasing *CO*_{2} on the ocean sink given by:

Ïƒc,l=Ïƒc,l*1âˆ’Î²lÎ”T,(7) Ïƒc,o=Ïƒc,o*KoKo+Î”CO2,(8) where Ïƒc,l* and Ïƒc,l* are the baseline sink rates, *β*_{l} is a fitting parameter, and *K*_{o} a half-saturation constant.

#### 2.3.5 Surface Temperature

A zero-dimensional global energy-balance can be used to determine changes in the Earth surface temperature *ΔT* [*Schlesinger*, 1986]:

CTdÎ”Ttdt=Î”Qâˆ’G0âˆ’11âˆ’fÎ”Tt,(9) where *C*_{T} is the heat capacity of the upper ocean, *ΔQ* is the radiative forcing, *G*_{0} is the zero-feedback gain of the climate system and *f* is a factor accounting for all the feedback mechanisms occurring within the Earth system (e.g., water vapor increase with warming, changes in lapse rate, albedo, and clouds) [*Schlesinger*, 1986; *Knutti and Hegerl*, 2008]. A typical value of the amplifying feedback *f* is 0.65 with a standard deviation of 0.13 [*Knutti and Hegerl*, 2008]. Accounting for the logarithmic dependence of the radiative forcing on atmospheric *CO*_{2} [*Myhre et al*., 1998], the equilibrium solution of Equation (9) is [*Schlesinger*, 1986; *Jones et al*., 2003; *Knutti and Hegerl*, 2008]:

Î”Teqt=S2Ã—CO2ln2Â·lnCO2tCO2,0,(10) where the climate sensitivity S2Ã—CO2 for a doubling of atmospheric *CO*_{2} is defined as [*Knutti and Hegerl*, 2008]:

S2Ã—CO2=Î”T01âˆ’f,(11) where Î”T0=G0Î”Q2Ã—CO2= 1.2°C the blackbody no-feedback response to the radiative forcing Î”Q2Ã—CO2 resulting from a *CO*_{2} doubling [*Schlesinger*, 1986; *Knutti and Hegerl*, 2008].The actual response of the climate system is given by the transient solution of Equation (9) [*Schlesinger*, 1986]:

Î”Tt=Î”Teqt1âˆ’eâˆ’t/Ï„e,(12) where *τ*_{e} = *C*_{T}*G*_{0}/(1 − *f*) the e-folding time required for the climate system to reach equilibrium. This lag between the equilibrium warming *ΔT*_{eq} and the actual climate response is mainly due to the thermal inertia of the ocean [*Schlesinger*, 1986]. A wide range of *τ*_{e} values varying between 10 and 100 years has been estimated by energy balance, radiative-convective, and general circulation models. For the climate response to an abrupt increase in *CO*_{2} concentration, coupled atmosphere–ocean models suggest that *τ*_{e} ∼50–100 years due to the transport of *CO*_{2}-induced surface heating into the interior of the ocean [*Schlesinger*, 1986]. The evolution of surface temperature is then calculated as *T*(*t*) = *T*_{0} + *ΔT*(*t*), starting from *T*_{0} = *T*(1875) ≈ 286.9°K.

#### 2.3.6 Simulations Setup

The resulting sets of differential equations for the zero-dimensional model are solved numerically with initial conditions set at year 1875, the transition from the first to the second industrial revolution. The human-climate-system model is calibrated using historical CO_{2} concentration and surface temperature data as well as simulation results derived from fully coupled and spatially explicit global carbon-climate models [*Cox et al*., 2000]. Historical data on population growth, anthropogenic emissions, global mean atmospheric CO_{2} concentration, and surface temperature have been assembled from the literature (see Appendix S1 for details [*MacFarling Meure et al*., 2006; *Maddison*, 2010; *Boden et al*., 2011; *United Nations*, *Department of Economic and Social Affairs*, *Population Division*, 2015]). The authors digitized emission scenarios and projected CO_{2} and temperature from the existing studies [*Jones et al*., 2003; *Meinshausen et al*., 2009].