Recognition: unknown
Constraints on the inflationary vacuum and reheating era from NANOGrav
Pith reviewed 2026-05-08 16:24 UTC · model grok-4.3
The pith
NANOGrav data favor alpha-vacuum initial conditions for inflation and radiation-like reheating while narrowing the allowed alpha range.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Under the assumption that the NANOGrav signal is generated by primordial tensor perturbations, the 15-year dataset prefers nt = 2.20^{+0.36}_{-1.2} and a radiation-like reheating equation of state omega_re = 0.33^{+0.14}_{-0.36}. The same data select non-Bunch-Davies alpha-vacua over the usual Bunch-Davies state and restrict the numerical range of the vacuum parameter alpha. Allowing alpha to become frequency-dependent past a threshold frequency supplies a minimal way to reduce the required blue tilt of the tensor spectrum.
What carries the argument
The alpha-vacuum, a one-parameter deformation of the Bunch-Davies mode functions that alters the normalization and phase of tensor perturbations and thereby reshapes the primordial gravitational-wave energy-density spectrum.
If this is right
- The tensor spectrum must be blue-tilted with nt near 2.2 and the reheating phase must be radiation-like with omega_re close to one-third.
- The vacuum parameter alpha is forced into a narrow numerical interval by the present data.
- Making alpha frequency-dependent above a cutoff frequency removes the need for an extreme blue tilt while still fitting the observed spectrum.
- Future gravitational-wave detectors operating at higher frequencies can directly test whether alpha indeed varies with scale.
Where Pith is reading between the lines
- If alpha-vacua are required by data, the pre-inflationary quantum state of the universe must have differed from the usual assumption, which could leave imprints in other observables such as the CMB power spectrum at the largest scales.
- A frequency-dependent alpha supplies a concrete mechanism that could be realized in models where the initial vacuum state changes across different physical scales.
- The derived bounds on reheating temperature and equation of state can be used to test concrete models of the end of inflation once the vacuum choice is fixed.
Load-bearing premise
The common red noise signal is generated by tensor modes produced during inflation rather than by astrophysical sources or other cosmological processes.
What would settle it
A future pulsar-timing-array measurement that finds a red-tilted tensor spectrum (nt less than zero) or a Hellings-Downs correlation pattern whose amplitude and shape are incompatible with any alpha-vacuum spectrum would rule out the claimed preference.
read the original abstract
NANOGrav and various pulsar timing array experiments recently reported evidence for a common red noise signal across millisecond pulsars. This signal exhibits Hellings-Downs inter-pulsar correlation patterns, providing compelling evidence for a stochastic gravitational wave background (SGWB) signal. In general, such a background can come from several astrophysical and cosmological phenomena. Assuming such SGWB has an inflationary origin, we use latest NANOGrav 15-year dataset to constrain the inflationary parameters e.g., tensor spectral index ($n_t$), tensor-to-scalar ratio ($r$), and explore the implications for the reheating phase through constraints on the reheating equation of state ($\omega_{\text{re}}$) and reheating temperature ($T_{\text{re}})$. We find the preference for an extremely blue-tilted tensor spectrum $n_t=2.20^{+0.36}_{-1.2}$ and the radiation-like reheating scenario $\omega_{\text{re}}=0.33^{+0.14}_{-0.36}$. Despite having no concrete evidence for the nature of the primordial vacua, the computation of gravitational wave (GW) sourced by tensor perturbations assumes the inflationary vacuum to be a Bunch-Davies vacuum. In this work, we examine modifications to the GW spectrum originating from the non-Bunch-Davies primordial vacuum. We find that NANOGrav observations favour a specific type of non-Bunch-Davies vacuum, known as the alpha-vacuum. Furthermore, our analysis demonstrates that the observations strikingly narrow down the range of the parameter $\alpha$ characterizing the vacua. On top of that, we find that a frequency-dependent parametrization of the vacuum parameter $\alpha$ beyond a threshold frequency can yield a minimal solution to alleviate the blue-titled issue. Finally, we highlight the possibility of testing such frequency dependence of $\alpha$ by probing the GW spectrum through future GW experiments.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript assumes the NANOGrav 15-year SGWB signal has an inflationary origin and fits the 15-year dataset for the tensor spectral index nt, tensor-to-scalar ratio r, reheating equation of state ω_re and temperature T_re. It further examines non-Bunch-Davies alpha-vacua, reporting a preference for a specific range of the vacuum parameter α and proposing a frequency-dependent parametrization of α to mitigate the extremely blue-tilted spectrum (nt ≈ 2.2).
Significance. Conditional on the inflationary-origin assumption, the work provides timely constraints on non-standard primordial vacua and reheating using the latest PTA data, with the frequency-dependent α suggestion offering a potential reconciliation mechanism. Explicit credit is due for exploring alpha-vacua beyond the default Bunch-Davies case and for highlighting future GW probes.
major comments (3)
- [Abstract and §1] Abstract and §1: The reported constraints on nt, ω_re, and α (including its frequency-dependent form) are derived exclusively under the explicit assumption of an inflationary SGWB origin. No quantitative model-comparison test (Bayes factor or evidence ratio) against astrophysical alternatives such as SMBH binaries or cosmic strings is provided, rendering the constraints inapplicable if the origin differs.
- [Data-analysis section (likely §3)] Data-analysis section (likely §3): The posterior values nt = 2.20^{+0.36}_{-1.2} and the narrowed α range are obtained by direct fitting, yet the explicit likelihood function, priors on nt/r/α/ω_re, and handling of NANOGrav systematics or alternative data cuts are not specified. This prevents verification of whether the blue-tilt and α preferences survive reasonable variations.
- [Results section on frequency-dependent α (likely §4.3)] Results section on frequency-dependent α (likely §4.3): The claim that a frequency-dependent parametrization of α beyond a threshold frequency yields a minimal solution to the blue-tilt issue requires the explicit functional form, the numerical value of the threshold frequency, and a demonstration that the modified spectrum remains consistent with CMB-scale constraints on nt.
minor comments (2)
- [Theory section] The precise definition of the alpha-vacuum state and the modified tensor power spectrum (how α enters the mode functions or Bogoliubov coefficients) should be stated with an equation in the theory section before the fitting results.
- [Figures] Figures showing posterior contours for nt, α, and ω_re should overlay the standard Bunch-Davies case and indicate the prior ranges used.
Simulated Author's Rebuttal
We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments below and propose revisions where appropriate to improve the clarity and robustness of our analysis.
read point-by-point responses
-
Referee: [Abstract and §1] The reported constraints on nt, ω_re, and α (including its frequency-dependent form) are derived exclusively under the explicit assumption of an inflationary SGWB origin. No quantitative model-comparison test (Bayes factor or evidence ratio) against astrophysical alternatives such as SMBH binaries or cosmic strings is provided, rendering the constraints inapplicable if the origin differs.
Authors: We fully agree that our constraints are derived under the assumption of an inflationary origin for the SGWB, which is clearly stated in the abstract ('Assuming such SGWB has an inflationary origin') and Section 1. The paper does not perform a quantitative model comparison (e.g., Bayes factor) against astrophysical sources because its scope is to explore the implications within the inflationary paradigm, including non-Bunch-Davies vacua. To address this, we will revise the abstract and introduction to more explicitly emphasize the conditional nature of the results and discuss the implications if the origin is different. We will also add a brief note on the possibility of future model comparisons using PTA data. revision: partial
-
Referee: [Data-analysis section (likely §3)] The posterior values nt = 2.20^{+0.36}_{-1.2} and the narrowed α range are obtained by direct fitting, yet the explicit likelihood function, priors on nt/r/α/ω_re, and handling of NANOGrav systematics or alternative data cuts are not specified. This prevents verification of whether the blue-tilt and α preferences survive reasonable variations.
Authors: We appreciate this observation. Upon review, the data analysis details in Section 3 will be supplemented in the revised manuscript with the explicit likelihood function employed for the NANOGrav 15-year dataset, the specific prior distributions chosen for the parameters (nt, r, α, ω_re, T_re), and additional information on the treatment of systematics and data selection cuts. This will enable independent verification and assessment of the robustness of the reported posteriors, including the blue-tilted nt and preferred α range. revision: yes
-
Referee: [Results section on frequency-dependent α (likely §4.3)] The claim that a frequency-dependent parametrization of α beyond a threshold frequency yields a minimal solution to the blue-tilt issue requires the explicit functional form, the numerical value of the threshold frequency, and a demonstration that the modified spectrum remains consistent with CMB-scale constraints on nt.
Authors: We concur that additional specifics are required to substantiate the frequency-dependent α proposal. In the revised version, we will explicitly define the functional form of the frequency-dependent α (e.g., α(f) = α0 for f < f_th and a different form for f > f_th), provide the numerical value of the threshold frequency f_th used in our calculations, and include an analysis or plot demonstrating that the resulting tensor spectrum is consistent with CMB constraints on nt at the relevant scales. This will strengthen the claim that it offers a minimal solution to the blue-tilt tension. revision: yes
Circularity Check
No significant circularity; constraints are standard posterior fits to external data under an explicit assumption
full rationale
The paper explicitly assumes the NANOGrav SGWB has an inflationary origin, adopts a standard tensor power spectrum modified by an alpha-vacuum parametrization, and performs a fit to the 15-year dataset to obtain posteriors on nt, r, omega_re, T_re and alpha (including a frequency-dependent extension). These steps constitute ordinary parameter estimation against independent observational input rather than any self-definitional loop, fitted quantity renamed as prediction, or load-bearing self-citation that reduces the central result to the paper's own inputs by construction. The derivation chain remains self-contained once the stated assumption is granted.
Axiom & Free-Parameter Ledger
free parameters (3)
- tensor spectral index nt =
2.20^{+0.36}_{-1.2}
- reheating equation of state ω_re =
0.33^{+0.14}_{-0.36}
- alpha-vacuum parameter α
axioms (2)
- domain assumption The detected common red noise is a stochastic gravitational wave background of inflationary origin
- standard math Tensor perturbations during inflation source the GW spectrum via standard mode functions modified only by the choice of vacuum
Reference graph
Works this paper leans on
-
[1]
J. Antoniadis et al.,The International Pulsar Timing Array second data release: Search for an isotropic gravitational wave background,Mon. Not. Roy. Astron. Soc.510(2022) 4873 [2201.03980]
-
[2]
A. Zic et al.,The Parkes Pulsar Timing Array third data release,Publ. Astron. Soc. Austral. 40(2023) e049 [2306.16230]
-
[3]
Tarafdar et al.,The Indian Pulsar Timing Array: First data release,Publ
P. Tarafdar et al.,The Indian Pulsar Timing Array: First data release,Publ. Astron. Soc. Austral.39(2022) e053 [2206.09289]
-
[4]
H. Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,Res. Astron. Astrophys.23(2023) 075024 [2306.16216]
work page internal anchor Pith review arXiv 2023
-
[5]
Hellings and G.s
R.w. Hellings and G.s. Downs,UPPER LIMITS ON THE ISOTROPIC GRAVITATIONAL RADIATION BACKGROUND FROM PULSAR TIMING ANALYSIS,Astrophys. J. Lett. 265(1983) L39
1983
-
[6]
S. Vagnozzi,Implications of the NANOGrav results for inflation,Mon. Not. Roy. Astron. Soc. 502(2021) L11 [2009.13432]
-
[7]
Vagnozzi, JHEAp39, 81 (2023), arXiv:2306.16912 [astro-ph.CO]
S. Vagnozzi,Inflationary interpretation of the stochastic gravitational wave background signal detected by pulsar timing array experiments,JHEAp39(2023) 81 [2306.16912]
-
[8]
I. Ben-Dayan, U. Kumar, U. Thattarampilly and A. Verma,Probing the early Universe cosmology with NANOGrav: Possibilities and limitations,Phys. Rev. D108(2023) 103507 [2307.15123]. – 15 –
-
[9]
K. Fujikura, S. Girmohanta, Y. Nakai and M. Suzuki,NANOGrav signal from a dark conformal phase transition,Phys. Lett. B846(2023) 138203 [2306.17086]
-
[10]
A. Addazi, Y.-F. Cai, A. Marciano and L. Visinelli,Have pulsar timing array methods detected a cosmological phase transition?,Phys. Rev. D109(2024) 015028 [2306.17205]
- [11]
- [12]
- [13]
- [14]
- [15]
- [16]
-
[17]
S.-P. Li and K.-P. Xie,Collider test of nano-Hertz gravitational waves from pulsar timing arrays,Phys. Rev. D108(2023) 055018 [2307.01086]
-
[18]
Y. Gouttenoire,First-Order Phase Transition Interpretation of Pulsar Timing Array Signal Is Consistent with Solar-Mass Black Holes,Phys. Rev. Lett.131(2023) 171404 [2307.04239]
-
[19]
M. Ahmadvand, L. Bian and S. Shakeri,Heavy QCD axion model in light of pulsar timing arrays,Phys. Rev. D108(2023) 115020 [2307.12385]
-
[20]
D. Wang,Constraining Cosmological Phase Transitions with Chinese Pulsar Timing Array Data Release 1,2307.15970
- [21]
- [22]
-
[23]
N. Kitajima and K. Nakayama,Nanohertz gravitational waves from cosmic strings and dark photon dark matter,Phys. Lett. B846(2023) 138213 [2306.17390]
- [24]
-
[25]
Superheavy quasistable strings and walls bounded by strings in the light of NANOGrav 15 year data,
G. Lazarides, R. Maji and Q. Shafi,Superheavy quasistable strings and walls bounded by strings in the light of NANOGrav 15 year data,Phys. Rev. D108(2023) 095041 [2306.17788]
-
[26]
A. Eichhorn, R.R. Lino dos Santos and J.L. Miqueleto,From quantum gravity to gravitational waves through cosmic strings,Phys. Rev. D109(2024) 026013 [2306.17718]
-
[27]
N. Kitajima, J. Lee, K. Murai, F. Takahashi and W. Yin,Gravitational waves from domain wall collapse, and application to nanohertz signals with QCD-coupled axions,Phys. Lett. B 851(2024) 138586 [2306.17146]
- [28]
-
[29]
Y. Gouttenoire and E. Vitagliano,Domain wall interpretation of the PTA signal confronting black hole overproduction,Phys. Rev. D110(2024) L061306 [2306.17841]
-
[30]
D. Chowdhury, G. Tasinato and I. Zavala,Dark energy, D-branes and pulsar timing arrays, JCAP11(2023) 090 [2307.01188]
-
[31]
Singling out SO(10) GUT models using recent PTA results,
S. Antusch, K. Hinze, S. Saad and J. Steiner,Singling out SO(10) GUT models using recent PTA results,Phys. Rev. D108(2023) 095053 [2307.04595]
-
[32]
M. Yamada and K. Yonekura,Dark baryon from pure Yang-Mills theory and its GW signature from cosmic strings,JHEP09(2023) 197 [2307.06586]
-
[33]
S. Ge,Stochastic gravitational wave background: birth from string-wall death,JCAP06 (2024) 064 [2307.08185]
-
[34]
S. Basilakos, D.V. Nanopoulos, T. Papanikolaou, E.N. Saridakis and C. Tzerefos, Gravitational wave signatures of no-scale supergravity in NANOGrav and beyond,Phys. Lett. B850(2024) 138507 [2307.08601]
- [35]
-
[36]
Li,Probing the high temperature symmetry breaking with gravitational waves from domain walls,Nucl
X.-F. Li,Probing the high temperature symmetry breaking with gravitational waves from domain walls,Nucl. Phys. B1018(2025) 117036 [2307.03163]
- [37]
-
[38]
E. Babichev, D. Gorbunov, S. Ramazanov, R. Samanta and A. Vikman,NANOGrav spectral indexγ=3 from melting domain walls,Phys. Rev. D108(2023) 123529 [2307.04582]
-
[39]
G.B. Gelmini and J. Hyman,Catastrogenesis with unstable ALPs as the origin of the NANOGrav 15 yr gravitational wave signal,Phys. Lett. B848(2024) 138356 [2307.07665]
-
[40]
Z. Zhang, C. Cai, Y.-H. Su, S. Wang, Z.-H. Yu and H.-H. Zhang,Nano-Hertz gravitational waves from collapsing domain walls associated with freeze-in dark matter in light of pulsar timing array observations,Phys. Rev. D108(2023) 095037 [2307.11495]. [43]NANOGravcollaboration,The NANOGrav 15 yr Data Set: Constraints on Supermassive Black Hole Binaries from t...
-
[41]
J. Ellis, M. Fairbairn, G. H¨ utsi, J. Raidal, J. Urrutia, V. Vaskonen et al.,Gravitational waves from supermassive black hole binaries in light of the NANOGrav 15-year data,Phys. Rev. D 109(2024) L021302 [2306.17021]. [46]NANOGravcollaboration,The NANOGrav 15 yr Data Set: Search for Signals from New Physics,Astrophys. J. Lett.951(2023) L11 [2306.16219]
-
[42]
Bardeen,Gauge-invariant cosmological perturbations,Phys
J.M. Bardeen,Gauge-invariant cosmological perturbations,Phys. Rev. D22(1980) 1882
1980
-
[43]
Kodama and M
H. Kodama and M. Sasaki,Cosmological Perturbation Theory,Prog. Theor. Phys. Suppl.78 (1984) 1
1984
-
[44]
K.A. Malik and D. Wands,Cosmological perturbations,Phys. Rept.475(2009) 1 [0809.4944]
work page Pith review arXiv 2009
-
[45]
Cosmological Backgrounds of Gravitational Waves,
C. Caprini and D.G. Figueroa,Cosmological backgrounds of gravitational waves,Class. Quant. Grav.35(2018) 163001 [1801.04268]
-
[46]
Stochastic Gravitational Wave Backgrounds,
N. Christensen,Stochastic Gravitational Wave Backgrounds,Rept. Prog. Phys.82(2019) 016903 [1811.08797]. – 17 –
-
[47]
The Cold Dark Matter Density Perturbation
A.R. Liddle and D.H. Lyth,The Cold dark matter density perturbation,Phys. Rept.231 (1993) 1 [astro-ph/9303019]. [53]Planckcollaboration,Planck 2018 results. VI. Cosmological parameters,Astron. Astrophys. 641(2020) A6 [1807.06209]
work page Pith review arXiv 1993
- [48]
-
[49]
Inflation Dynamics and Reheating
B.A. Bassett, S. Tsujikawa and D. Wands,Inflation dynamics and reheating,Rev. Mod. Phys. 78(2006) 537 [astro-ph/0507632]
work page Pith review arXiv 2006
- [50]
- [51]
-
[52]
Y. Mambrini and K.A. Olive,Gravitational Production of Dark Matter during Reheating, Phys. Rev. D103(2021) 115009 [2102.06214]
-
[53]
A. Banerjee and D. Chowdhury,Fingerprints of freeze-in dark matter in an early matter-dominated era,SciPost Phys.13(2022) 022 [2204.03670]
-
[54]
N. Bernal and Y. Xu,WIMPs during reheating,JCAP12(2022) 017 [2209.07546]
- [55]
-
[56]
D. Chowdhury and and S. Show,SIMP dark matter during reheating,JCAP01(2025) 101 [2410.02871]
-
[57]
M. Kawasaki, K. Kohri and N. Sugiyama,MeV scale reheating temperature and thermalization of neutrino background,Phys. Rev. D62(2000) 023506 [astro-ph/0002127]
-
[58]
S. Hannestad,What is the lowest possible reheating temperature?,Phys. Rev. D70(2004) 043506 [astro-ph/0403291]
-
[59]
Bounds on very low reheating scenarios after Planck
P.F. de Salas, M. Lattanzi, G. Mangano, G. Miele, S. Pastor and O. Pisanti,Bounds on very low reheating scenarios after Planck,Phys. Rev. D92(2015) 123534 [1511.00672]
work page Pith review arXiv 2015
-
[60]
M. Gerbino, K. Freese, S. Vagnozzi, M. Lattanzi, O. Mena, E. Giusarma et al.,Impact of neutrino properties on the estimation of inflationary parameters from current and future observations,Phys. Rev. D95(2017) 043512 [1610.08830]
-
[61]
Bunch and P.C.W
T.S. Bunch and P.C.W. Davies,Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting,Proc. Roy. Soc. Lond. A360(1978) 117
1978
-
[62]
A. Ashoorioon, K. Dimopoulos, M.M. Sheikh-Jabbari and G. Shiu,Reconciliation of High Energy Scale Models of Inflation with Planck,JCAP02(2014) 025 [1306.4914]
- [63]
-
[64]
S. Maity,ACT-ing on inflation: Implications of non bunch-Davies initial condition and reheating on single-field slow roll models,Phys. Lett. B870(2025) 139913 [2505.10534]
work page internal anchor Pith review Pith/arXiv arXiv 2025
-
[65]
Choice of Quantum Vacuum for Inflation Observables
M. Wood-Saanaoui, R. O. Ramos and A. Berera,Choice of Quantum Vacuum for Inflation Observables,Symmetry18(2026) 399 [2602.22116]
work page internal anchor Pith review arXiv 2026
- [66]
-
[67]
J.-O. Gong and M. Sasaki,Squeezed primordial bispectrum from general vacuum state,Class. Quant. Grav.30(2013) 095005 [1302.1271]. – 18 –
-
[68]
A. Aravind, D. Lorshbough and S. Paban,Non-Gaussianity from Excited Initial Inflationary States,JHEP07(2013) 076 [1303.1440]
-
[69]
S. Bahrami and ´E.´E. Flanagan,Primordial non-Gaussianities in single field inflationary models with non-trivial initial states,JCAP10(2014) 010 [1310.4482]
-
[70]
S. Kundu,Non-Gaussianity Consistency Relations, Initial States and Back-reaction,JCAP 04(2014) 016 [1311.1575]
-
[71]
P.D. Meerburg and M. M¨ unchmeyer,Optimal CMB estimators for bispectra from excited states,Phys. Rev. D92(2015) 063527 [1505.05882]
-
[72]
A. Ashoorioon, R. Casadio and T. Koivisto,Anisotropic non-Gaussianity from Rotational Symmetry Breaking Excited Initial States,JCAP12(2016) 002 [1605.04758]
-
[73]
A. Shukla, S.P. Trivedi and V. Vishal,Symmetry constraints in inflation,α-vacua, and the three point function,JHEP12(2016) 102 [1607.08636]
- [74]
-
[75]
A. Naskar and S. Pal,Generic 3-point statistics with tensor modes in light of Swampland and Trans-Planckian Censorship Conjecture,Eur. Phys. J. C82(2022) 900 [2003.14066]
-
[76]
S. Kanno and M. Sasaki,Graviton non-gaussianity inα-vacuum,JHEP08(2022) 210 [2206.03667]
- [77]
- [78]
- [79]
-
[80]
J. Fumagalli, G.A. Palma, S. Renaux-Petel, S. Sypsas, L.T. Witkowski and C. Zenteno, Primordial gravitational waves from excited states,JHEP03(2022) 196 [2111.14664]
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.