Metastable cosmic strings produce a gravitational wave background that is best modeled with three parameters (string tension Gμ plus independent time scales t_LB and t_NC), yielding a compact analytical spectrum when t_LB greatly exceeds t_NC.
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Balkenhol et al.,Inflation at the End of 2025: Constraints on r and ns Using the Latest CMB and BAO Data,2512.10613
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abstract
Inflation elegantly provides initial conditions for the standard model of cosmology, while solving the horizon, flatness, and magnetic monopole problems. Inflationary models make predictions for the tensor-to-scalar ratio $r$ and the spectral index $n_s$ of initial density fluctuations. In light of relevant data releases this year, we present constraints on these two parameters using the latest cosmic microwave background (CMB) and baryon acoustic oscillation data (BAO) available. Using data from Planck, the South Pole Telescope, Atacama Cosmology Telescope, and BICEP/Keck experiments, we derive $n_s=0.9682\,\pm\,0.0032$ and a 95\% upper limit of $r<0.034$. This upper limit on $r$ is consistent with the official BICEP/Keck result given the numerical precision of the analyses and our choice to impose the self-consistency relation for single field slow-roll inflation on the tensor power spectrum; the $r$ constraint is not impacted by the additional CMB data. While adding DESI BAO data to the CMB data has a negligible impact on $r$, the $n_s$ constraint shifts upward to $0.9728\,\pm\,0.0029$, which favours monomial inflaton potentials with $N_\star \!\sim 50$ over Starobinsky $R^2$ or Higgs inflation with $N_\star = 51$ and $N_\star = 55$, respectively. This shift is caused by marginally significant differences between the CMB and DESI data that remain unexplained in the context of the standard model. We show that a class of polynomial $\alpha$-attractor models can predict the CMB as well as the CMB+DESI $n_s$ results. % with $N_\star =47.1$ and $N_\star=55.1$, respectively. While future data will improve our sensitivity to $r$, robust $n_s$ constraints are just as crucial to differentiate between inflation models. We make the data needed to reproduce the new CMB and BAO results and visualisation tools for $r$-$n_s$ figures available https://github.com/Lbalkenhol/r_ns_2025 .
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representative citing papers
For monomial inflationary potentials with k≥4, the inflaton regenerates from the thermal bath after reheating because its amplitude-dependent mass vanishes asymptotically.
A one-parameter family of α-attractor potentials interpolates between exponential and polynomial attractor predictions, letting n_s scan the range preferred by CMB+DESI.
The paper introduces new exponential and polynomial supergravity ξ-attractor models in the Jordan frame with non-minimal coupling and shows that Palatini gravity with independent affine connection has no supergravity embedding.
Non-minimal coupling of the inflaton to the Holst invariant in metric-affine gravity induces quasi-pole kinetics that generate an exponential plateau potential and Starobinsky-equivalent predictions regardless of the bare potential.
EDE models increase inferred α_s from CMB data, strengthening tension with USR PBH models that predict negative running.
Einstein-Cartan pseudoscalaron inflation coupled to type-I seesaw neutrinos makes nonthermal leptogenesis a necessary mechanism for the baryon asymmetry, yielding ns ~ 0.97, r ~ 0.004 and nB/s ~ 8.7e-11 for gamma ~ -1/100 and lightest Majorana mass ~ 10^13 GeV.
Certain inflation models produce right-handed neutrinos via gravitational effects sufficient for leptogenesis to explain the baryon asymmetry, testable by inflationary gravitational waves.
New ξ-attractors with non-minimal coupling and non-canonical kinetics yield Einstein-frame exponential and polynomial potentials whose ns spans 1-2/N to 1-1/N and r can reach zero as ξ grows, fitting Planck, BICEP/Keck, ACT, SPT, and DESI data, plus a supergravity realization.
Bubble collisions during a first-order phase transition at the end of inflation can generate the observed dark matter abundance in a restricted region of parameter space via direct production and spectator decays.
High-quality axion models with N_DW=1 and dark matter abundance requirement restrict the gauge breaking scale to 1.6e11-1e16 GeV, yielding a band of gravitational wave signals from two-step phase transitions consistent with current observations.
Generalized no-scale models with R=2/(3α) for α≠1 or non-minimal gauge couplings allow unsuppressed inflaton decays, producing calculable reheating temperatures and (n_s,r) predictions.
A supersymmetric U(1)B-L extension with smooth hybrid inflation constrains the scalar spectral index to 0.972-0.974 by requiring non-thermal dark matter relic abundance to match observations.
Constrains inflationary tensor parameters to fit the EPTA DR2 signal under CMB, BBN and LVK bounds, favoring radiation-era horizon re-entry but requiring low reheating temperatures.
Warm inflation with linear dissipation and quartic potential is realized in Weyl geometric gravity, incorporating the Weyl vector to produce a smooth transition to radiation domination with observables consistent with ACT data.
Generic string-loop corrections to the Kähler potential replace near-minimum slow-roll in LVS blow-up inflation with a power-law plateau at larger fields, yielding higher r still consistent with current CMB/BAO data.
Review summarizing inflation theory, its observational successes, and a data-driven tension with Starobinsky inflation addressed by higher-order curvature corrections.
String theory offers classes of inflationary models from type IIB Kaehler moduli and dynamical dark energy models, with a working axion hilltop quintessence construction that stresses initial conditions.
citing papers explorer
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New gravitational-wave templates for metastable cosmic strings: Loop breaking versus network collapse
Metastable cosmic strings produce a gravitational wave background that is best modeled with three parameters (string tension Gμ plus independent time scales t_LB and t_NC), yielding a compact analytical spectrum when t_LB greatly exceeds t_NC.
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Inflaton Regeneration via Scalar Couplings: Generic Models and the Higgs Portal
For monomial inflationary potentials with k≥4, the inflaton regenerates from the thermal bath after reheating because its amplitude-dependent mass vanishes asymptotically.
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Unification of polynomial and exponential cosmological attractors
A one-parameter family of α-attractor potentials interpolates between exponential and polynomial attractor predictions, letting n_s scan the range preferred by CMB+DESI.
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Jordan Frame in Supergravity and Cosmology
The paper introduces new exponential and polynomial supergravity ξ-attractor models in the Jordan frame with non-minimal coupling and shows that Palatini gravity with independent affine connection has no supergravity embedding.
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Quasi-pole inflation in metric-affine gravity
Non-minimal coupling of the inflaton to the Holst invariant in metric-affine gravity induces quasi-pole kinetics that generate an exponential plateau potential and Starobinsky-equivalent predictions regardless of the bare potential.
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Running into tension: primordial black holes from ultra-slow-roll inflation, spectral running, and the Hubble tension
EDE models increase inferred α_s from CMB data, strengthening tension with USR PBH models that predict negative running.
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Einstein-Cartan pseudoscalaron inflation, reheating and nonthermal leptogenesis
Einstein-Cartan pseudoscalaron inflation coupled to type-I seesaw neutrinos makes nonthermal leptogenesis a necessary mechanism for the baryon asymmetry, yielding ns ~ 0.97, r ~ 0.004 and nB/s ~ 8.7e-11 for gamma ~ -1/100 and lightest Majorana mass ~ 10^13 GeV.
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Nonthermal leptogenesis via cosmological gravitational particle production is tested by inflationary gravitational waves
Certain inflation models produce right-handed neutrinos via gravitational effects sufficient for leptogenesis to explain the baryon asymmetry, testable by inflationary gravitational waves.
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New Exponential and Polynomial $\xi$-attractors
New ξ-attractors with non-minimal coupling and non-canonical kinetics yield Einstein-frame exponential and polynomial potentials whose ns spans 1-2/N to 1-1/N and r can reach zero as ξ grows, fitting Planck, BICEP/Keck, ACT, SPT, and DESI data, plus a supergravity realization.
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Dark Matter Production from Bubble Collisions during a First-Order Phase Transition at the End of Inflation
Bubble collisions during a first-order phase transition at the end of inflation can generate the observed dark matter abundance in a restricted region of parameter space via direct production and spectator decays.
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Probing High-Quality Axions with Gravitational Waves
High-quality axion models with N_DW=1 and dark matter abundance requirement restrict the gauge breaking scale to 1.6e11-1e16 GeV, yielding a band of gravitational wave signals from two-step phase transitions consistent with current observations.
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Reheating in No-Scale Models of Inflation
Generalized no-scale models with R=2/(3α) for α≠1 or non-minimal gauge couplings allow unsuppressed inflaton decays, producing calculable reheating temperatures and (n_s,r) predictions.
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$U(1)_{B-L}$ Dark Matter Constrains Smooth (SUSY) Hybrid Inflation
A supersymmetric U(1)B-L extension with smooth hybrid inflation constrains the scalar spectral index to 0.972-0.974 by requiring non-thermal dark matter relic abundance to match observations.
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Inflationary interpretation of the gravitational-wave signal in the European Pulsar Timing Array DR2 with constraints
Constrains inflationary tensor parameters to fit the EPTA DR2 signal under CMB, BBN and LVK bounds, favoring radiation-era horizon re-entry but requiring low reheating temperatures.
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Warm inflation in Weyl geometric gravity
Warm inflation with linear dissipation and quartic potential is realized in Weyl geometric gravity, incorporating the Weyl vector to produce a smooth transition to radiation domination with observables consistent with ACT data.
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Loop Blow-up Inflation: An Overview
Generic string-loop corrections to the Kähler potential replace near-minimum slow-roll in LVS blow-up inflation with a power-law plateau at larger fields, yielding higher r still consistent with current CMB/BAO data.
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Inflation in a nutshell: From basics to latest advances
Review summarizing inflation theory, its observational successes, and a data-driven tension with Starobinsky inflation addressed by higher-order curvature corrections.
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Recent progress on inflation and dark energy from string theory
String theory offers classes of inflationary models from type IIB Kaehler moduli and dynamical dark energy models, with a working axion hilltop quintessence construction that stresses initial conditions.