Gravitational caloric theory: From early dark energy to a wide variety of gravitational phenomena

S. X. Tian

REVIEW 3 major objections 4 minor 289 references

Gravitational caloric theory: From early dark energy to a wide variety of gravitational phenomena

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A new gravity theory ties early dark energy to the matter-radiation transition and offers an inflation alternative.

desk verdict A serious vector-tensor theory with a careful dynamical-systems core; the ESHU inflation alternative is not established, but the paper deserves a real referee. read the letter →

arxiv 2608.07318 v1 pith:VS6VRWUV submitted 2026-08-07 gr-qc

S. X. Tian This is my paper · ORCID

classification gr-qc MSC 83D0583F0583C35 PACS 04.50.Kd98.80.-k04.30.-w

keywords gravitationalcalorictheoryearlydarkenergycoincidenceproblemself-tuningmechanismstatichotUniversehorizonwavepolarizationsmodifiedgravity

topics Dark Energy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Gravitational caloric theory (GCT) introduces a vector field sourced by the total fluid equation of state, so that the same physics that marks matter-radiation equality also triggers a burst of early dark energy. The paper argues this removes the EDE coincidence problem because no hand-set energy scale is needed. It also identifies a broader family of energy-cancelling solutions in which the field offsets ordinary energy sources, including a cosmological constant, and proposes an Early Static Hot Universe that can inflate the comoving horizon without inflation. If the theory is right, modified gravity can address several cosmological puzzles at once while still recovering general relativity in the Solar System and predicting luminal gravitational waves.

What carries the argument

The central object is the caloric vector field $S_\mu$, with an effective gravitational tensor $S_{\mu\nu} = \nabla_\mu S_\nu + \nabla_\nu S_\mu + \dots$ whose linear terms make the field act like a source of spacetime curvature. The cosmological trigger is the source term $c_1 (\partial w_f)^2$ in the vector equation, which is nonzero only while the total fluid equation of state changes during the radiation-matter transition. The argument is carried by a two-dimensional dynamical system in the variables $x_1 = S_0/H$ and $x_3 = \varrho_f/H$; stability of the critical point $F_1$ ensures EDE dissipation, a critical point at infinity $I_3$ organizes the ESHU dynamics, and centre manifold theory extracts the asymptotic behaviour of the Λ-cancelling and ESHU attractors.

What would settle it

A Solar System or gravitational-wave test that measures a PPN parameter different from $\gamma=1$ or a gravitational-wave speed different from $c$ on a background where the caloric field cannot be set to zero would falsify the weak-field recovery. Directly, a cosmological perturbation calculation showing that the caloric EDE source excites unstable or observationally excluded matter fluctuations at matter-radiation equality would falsify the EDE mechanism.

Watch

Extended reading notes

Core claim

GCT modifies gravity from the fluid side by coupling a vector field $S_\mu$ to the standard energy-momentum tensor through a source term proportional to $(\partial w_f)^2$, where $w_f$ is the total fluid equation of state. In the radiation- and matter-dominated eras this source vanishes and the field decays to a stable critical point, while during the radiation-matter transition it drives a transient EDE component whose peak occurs near equality. The same field admits Λ-cancelling solutions that asymptote to a linearly expanding universe, and, through a critical point at infinity, an Early Static Hot Universe in which the field offsets hot gas so that a large comoving Hubble radius is generated. A Minkowski-space perturbation analysis singles out $c_2 = 1$ for stability, recovers the Newtonian limit with $\gamma_{\rm PPN}=1$ under asymptotically flat boundary conditions, and yields six gravitational-wave polarizations propagating at the speed of light.

Load-bearing premise

The weak-field recovery of general relativity assumes a Minkowski background with $S_\mu=0$ and standard asymptotically flat boundary conditions, but the paper itself notes that asymptotically flat exact solutions are non-generic, so real cosmological or compact-object backgrounds may not admit these boundary conditions.

Editorial extensions

If this is right

A quantitative cosmological perturbation analysis can test whether the caloric EDE model resolves the Hubble tension without violating CMB constraints.
The ESHU scenario, if viable, would provide a thermal alternative to inflation in which primordial fluctuations could arise from ordinary high-temperature gas rather than vacuum quantum fluctuations.
The self-tuning Λ-cancelling solution offers a concrete dynamical framework for studying the old cosmological constant problem, although the paper notes it does not yet describe a full hot Big Bang history.
For $c_2=1$, GCT predicts all six gravitational-wave polarizations at light speed, giving a clear observational signature for future GW detectors.
The exact spherically symmetric solutions include wormholes and naked singularities but no black holes beyond Schwarzschild, motivating a search for a no-hair theorem or a dynamical collapse obstruction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same $w_f$-trigger mechanism could be applied to the $S_8$ tension by modifying the Poisson equation on cosmological scales, as the paper hints but does not develop.
The Λ-like behaviour of $S_\mu$ seen in both cosmology and local static solutions suggests that the caloric field might be a unified stand-in for dark energy and an effective cosmological constant, a connection the paper leaves speculative.
A testable extension would be to compute the primordial power spectrum generated by thermal fluctuations in ESHU and compare its scalar tilt and non-Gaussianity with CMB observations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this is a new vector-tensor theory, not a rehash. The field S_mu is sourced by the fluid EoS through c1 (dw_f)^2, the standard fluid stays covariantly conserved, and the background cosmology reduces to a 2D dynamical system that they analyze thoroughly. The EDE trigger genuinely fires at the radiation-matter transition, and the 10% amplitude shown in Fig. 5 is derived, not just claimed. That part holds up internally.

The dynamical-systems work is the real strength: Poincare compactification, centre manifold theory for I_3 and I_4, the Abel reduction in the spherically symmetric sector, and the exact solutions (wormhole, mirror Schwarzschild, charged metric) are careful and cross-checked numerically. The weak-field section is standard but done properly: for c2=1 you recover GR in the static limit, get gamma_PPN=1, and six luminal polarizations. The authors also disclose what they haven't done — no black holes beyond Schwarzschild, no Lagrangian, and they flag the boundary-condition caveat themselves.

The soft spots are exactly where you'd find them. First, the EDE mechanism is built into the source term: c1 (dw_f)^2 vanishes in constant-EoS eras and peaks at the transition, so the coincidence problem is solved by construction rather than by independent dynamics. That's a real circularity, though the paper is transparent about it.

Second, the ESHU alternative to inflation is the weak link. The abstract says it 'offers an alternative to inflation,' but the body and Table IV are more honest: n=3 cannot solve the horizon problem (the integral in Eq. 3.34 is finite), and the n=2/n=1 cases need x3_max > e^70, extreme initial conditions, and are 'not robust if dw_f/dt != 0.' Since the real universe has time-varying w_f through the transition, the advertised exponential CHR decrease may not occur. The single-side attractor in Fig. 8b shows the mechanism isn't generic either.

Third, the self-tuning branch is explicitly incomplete — it can't enter the standard hot Big Bang. That's a major limitation, but they say so.

Who is this for? People working on modified gravity, EDE model-building, or vector-tensor theories. It's a construction paper with a lot of mathematics to chew on. It deserves a serious referee — the claims are specific and the limitations are disclosed, but the advertised ESHU claim should be tempered in any revision.

Referee Report

3 major / 4 minor

Summary. The paper introduces gravitational caloric theory (GCT), a vector-tensor extension of general relativity in which a new vector field S_mu is sourced by fluid thermodynamic quantities and coupled to gravity while preserving covariant conservation of the standard fluid energy-momentum tensor. The main cosmological claims are: (i) early dark energy triggered by the radiation-matter transition through the source term c1(∂w_f)^2, with the total fluid equation of state addressing the EDE coincidence problem (Sec. III A); (ii) a Λ-cancelling self-tuning solution (Sec. III B); and (iii) an Early Static Hot Universe (ESHU) that purportedly offers an alternative to inflation by generating a large comoving Hubble radius (Sec. III C). The paper also performs gauge-invariant linear perturbations about Minkowski spacetime, finding c2=1 stability, recovery of the Newtonian limit with γ_PPN=1, and six luminal gravitational-wave polarizations, and analyzes static spherically symmetric solutions, obtaining wormholes, naked singularities, effective Λ behavior, and two new exact solutions. The mathematical derivations are detailed and internally consistent, but the central physical claims are not established at the level asserted in the abstract and conclusions.

Significance. If the EDE trigger were parameter-free and the ESHU mechanism robust, this paper would open a genuinely new direction in modified gravity, and the dynamical-systems toolkit—Poincaré compactification, centre manifold theory, and attractor analysis—is applied carefully and usefully. The exact static solutions and the explicit decoupling of the linearized perturbation equations are also valuable as technical contributions. However, the significance as advertised is currently limited by three load-bearing gaps: the EDE amplitude is set by an order-20 free parameter rather than predicted, the ESHU horizon-solving cases require extreme initial data and are explicitly not robust to time-varying w_f, and the weak-field recovery of general relativity depends on Minkowski background boundary conditions that the paper itself shows are non-generic in the exact spherically symmetric sector. The paper is a serious exploratory study, but the central claims need substantial revision or additional analysis before they can be accepted.

major comments (3)

[III A 3, Eq. (3.15), Fig. 5] The EDE amplitude is not a prediction of the theory. Eq. (3.15) shows Ω_EDE proportional to c1, and the numerical demonstration in Fig. 5 reaches the 10% level by choosing c1=20. This violates the paper's own 'strong rule' in Sec. I A that no additional parameters set the EDE energy scale and that all parameters should be dimensionless and of order unity. Furthermore, the source term c1(∂w_f)^2 is introduced in Sec. II A specifically to vanish in constant-EoS eras and peak at the transition, so the timing of the EDE trigger is encoded in the field equations by construction rather than derived from independent physics. The paper should either demonstrate that a 10% EDE amplitude can be obtained with O(1) parameters and generic initial conditions, or explicitly state that GCT offers a trigger mechanism but not a parameter-free resolution of the EDE coincidence problem.
[III C, Table IV, Sec. VI] The claim that ESHU 'offers an alternative to inflation' is not supported by the paper's own analysis. Table IV and the discussion around Eq. (3.34) show that the n=3 case fails to solve the horizon problem. For the n=2 and n=1 cases, solving the horizon problem requires x3,max > e^70 (Eqs. (3.38) and (3.48)), which entails extreme initial conditions, and Table IV explicitly states 'Not robust if ẇ_f ≠ 0'. Since the real early universe has a time-varying w_f through the radiation-matter transition (Eq. (3.13)), the advertised power-law or exponentially decreasing comoving Hubble radius may not be realized. In addition, Fig. 8 case 1b shows that even at constant w_f a second branch of initial conditions does not follow the I3→F1 attractor, so the mechanism is not generic. The abstract and Sec. VI should be revised to present ESHU as a speculative possibility with known obstructions, not as an established inflationary alternative.
[IV C, V B 3, VI] The weak-field recovery of general relativity, used to claim consistency with Solar System and gravitational-wave constraints, relies on a Minkowski background with S_mu=0 and standard asymptotically flat boundary conditions. The paper itself cautions in Sec. IV and Sec. VI that asymptotically flat exact solutions are non-generic in the explored spherically symmetric sector, and that changing boundary conditions can alter the physical implications. If the true cosmological or compact-object background does not admit those boundary conditions, the Newtonian limit, γ_PPN=1, and the six-polarization result may not apply to the real Universe. Since these constraints are used to validate the theory, the domain of validity of the Minkowski-based analysis must be established, or the claims must be correspondingly qualified, before they can be regarded as tests of GCT.

minor comments (4)

[II A] The text refers to 'Porca theory'; this should be 'Proca theory'.
[I C] There is a typo 'explity i → z_i' in the technical background; it should read 'explicit y_i → z_i projection'.
[I A] The sentence contains a duplicated phrase: 'new independent and precise measurements from the from the TRGB-SBF project'.
[Fig. 5] The horizontal axis label 'N-foldingnumbere' and several other axis labels contain missing spaces or run-together words; they should be cleaned up for readability.

Circularity Check

2 steps flagged · score 6.0 of 10

The EDE coincidence 'resolution' is built into the source term by construction, and the ESHU inflation alternative is an imposed parameter regime rather than an emergent prediction.

self definitional [Sec. II A (source term in Eq. 2.1b) and Sec. III A 3]
"We now turn to the source term of the EDE. This must correctly represent the cosmic radiation-matter transition. Considering the dimensions, a suitable choice is a term like (∂ μwf)2 ... The source term vanishes in both the radiation- and (pressureless) matter-dominated eras due to the constancy of wf, while being nonzero during the transition. ... The source term peaks exactly at matter-radiation equality, whereas the total Ω EDE attains its maximum very near that epoch. ... This result is sufficient to resolve the EDE coincidence problem."

The source term c1(w'_f)^2 is introduced in Sec. II A with the explicit design goal of being nonzero only during the radiation-matter transition, and w_f is parametrized in Eq. (3.13) with N_eq set to matter-radiation equality. The later demonstration that Ω_EDE peaks near equality and the assertion that this resolves the coincidence problem simply read back that input. The timing of EDE is imposed through the definition of the source term, not derived from independent dynamics; the 'resolution' is therefore self-definitional.
other [Sec. III C 3, Eqs. (3.39)-(3.42) and discussion after Eq. (3.47)]
"In summary, the ESHU dynamics with n=1 can be realized by imposing {˜c35 >0, α>0, ˜λ I3,− <0, β c.m. >0} in conjunction with Eqs. (3.39), (3.41) and (3.42). ... To address the horizon problem, r h is required to decrease during the late-ESHU era, which in turn imposes β c.m. >1."

The exponentially decreasing comoving Hubble radius in Eq. (3.47) is not derived from an independent physical mechanism; it is produced by choosing c4, c6, and α through Eqs. (3.39)-(3.42) so that β_c.m. > 1, where β_c.m. > 1 is exactly the condition for the CHR to decrease. The paper then presents this imposed behavior as the basis for claiming that ESHU 'offers an alternative to inflation.' The advertised inflation-like outcome is an input to the parameter construction rather than a prediction from the theory.

full rationale

The central EDE claim is circular in the precise sense that the trigger, c1(∂ w_f)^2, is defined to vanish in constant-EoS eras and peak at the radiation-matter transition; the numerical peak of Ω_EDE near equality and the 'resolution' of the coincidence problem are therefore consequences of the chosen ansatz, not independent outputs. The ESHU n=1 mechanism is similarly an engineered parameter regime: the parameter constraints are imposed to make β_c.m. > 1, and the exponential CHR decrease follows by construction. However, the self-citation to Tian & Zhu is not the main circular step, and much of the paper is self-contained: the Minkowski perturbation analysis (Sec. IV), the stability selection c2 = 1, the Abel-equation reduction and exact spherically symmetric solutions (Sec. V), and the critical-point/centre-manifold mathematics are derived from the stated field equations with stated assumptions and checked numerically/series-wise. The paper also honestly reports that n=3 ESHU fails to solve the horizon problem and that n=2 and n=1 require extreme x3,max and are not robust to time-varying w_f; these are strength-of-claim issues rather than circularity. Overall, because the EDE coincidence resolution and the headline ESHU inflation alternative reduce by construction to inputs of the model, while substantial auxiliary analyses remain independent, the appropriate circularity score is 6.

Assumptions & free parameters 7 free parameters · 6 assumptions · 1 invented entities

The central claim rests on the existence and dynamics of a new vector field S_mu, whose couplings and source terms are chosen by hand to produce EDE, self-tuning, and ESHU. Seven dimensionless parameters and the ESHU initial condition are effectively free inputs. The theory also depends on background assumptions about boundary conditions and the form of the field in symmetric configurations, several of which are flagged by the authors as unproved or potentially non-generic.

free parameters (7)

c1 = 20 in Fig. 5
Controls the EDE amplitude; chosen so Omega_EDE reaches about 10%, not derived from data or first principles.
c2 = varies: 1, 2.5, 0.5, -5
Order-unity dimensionless parameter; Minkowski stability selects c2=1, while ESHU examples use other values to satisfy critical-point conditions.
c3 = varies: -2, 2, -1, -896/73
Together with c5 sets the saddle condition for I3 and the ESHU realizations; values are chosen by hand.
c4 = -186956805/2579236 for n=1 ESHU
Fixed by Eq. (3.41) to enforce the zero-eigenvalue condition needed for n=1; engineered to produce the desired attractor.
c5 = varies: -3, 2.5, -2, -13
Part of the stability constraints for F1 and I3; chosen for each application.
c6 = -76932660/644809 for n=1 ESHU
Fixed by Eq. (3.39) with alpha from Eq. (3.42); rational values are chosen to satisfy the n=1 constraints exactly.
ESHU initial x3,max = 1000 in numerical evolutions; > e^70 required for horizon problem
The numerical examples use x3,max = 1000 to make comparisons feasible, while the analytic n=2 result requires x3,max > e^70, an extreme initial condition.

assumptions (6)

ad hoc to paper The GCT field equations (2.1) are directly constructed and assumed to define a consistent classical field theory without a Lagrangian.
Sec. II B states that no Lagrangian was found; the equations are asserted as the theory.
ad hoc to paper The standard fluid energy-momentum tensor is covariantly conserved, and J_mu includes only fluid components, excluding the cosmological constant.
Sec. II A calls the exclusion of dark energy from J_mu an assumption without physical motivation.
ad hoc to paper The vector-field source term c1(∂w_f)^2 is introduced to encode the radiation-matter transition.
Sec. II A motivates this term because it vanishes in constant-EoS eras and is nonzero during the transition, which is precisely the EDE trigger.
domain assumption The radiation-matter transition is modeled by w_f(N) = (1/3)/(1 + e^{N-N_eq}) with N_eq = -8.13.
Sec. III A 3 uses this standard fitting function for the matter-radiation equality epoch, citing Planck values.
domain assumption Linear perturbation theory around Minkowski assumes S_mu = 0 in the background and standard asymptotically flat boundary conditions.
Sec. IV states these assumptions; Sec. VI notes asymptotically flat solutions are non-generic in GCT, so this may be inappropriate.
ad hoc to paper Static spherically symmetric configurations assume S_mu = [alpha(r), beta(r), 0, 0].
Sec. V A 1 says the full generality of this form for spherical symmetry is not proved.

invented entities (1)

Caloric vector field S_mu
purpose: Mediates the fluid-gravity coupling, sources early dark energy, enables self-tuning of Lambda, and realizes the Early Static Hot Universe.
No direct detection is provided. Falsifiable handles exist in principle, such as six gravitational-wave polarizations and primordial fluctuation signatures, but they are not measured and the theory's parameters are adjusted to produce the desired phenomenology.

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Cite this review

Pith. "Pith review of Gravitational caloric theory: From early dark energy to a wide variety of gravitational phenomena." pith.science (2026). https://pith.science/paper/VS6VRWUV

@misc{pith2026260807318,
  author       = {Pith},
  title        = {Pith review of: Gravitational caloric theory: From early dark energy to a wide variety of gravitational phenomena},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VS6VRWUV}},
  note         = {Machine review of arXiv:2608.07318}
}

abstract

Gravitational caloric theory (GCT) extends general relativity by introducing a vector field $S_\mu$ that is sourced by and non-minimally coupled to the fluid sector, while preserving covariant conservation of the standard fluid energy-momentum tensor. The initial motivation is to trigger early dark energy (EDE) using the total fluid equation of state that encodes the cosmic radiation-matter transition, thereby addressing the associated coincidence problem. The EDE dynamics are analyzed in detail by recasting the background evolution equations as a two-dimensional dynamical system. Poincar\'e compactification probes the global structure of this system and illustrates the role of critical points at infinity in shaping the phase portrait. There are two cosmological applications beyond EDE, both belonging to a broader family of energy-cancelling solutions in which conventional energy components (e.g., the cosmological constant $\Lambda$ or an ideal fluid) preferentially excite $S_\mu$ rather than sourcing spacetime curvature. First, the $\Lambda$-cancelling solution corresponds to the so-called self-tuning mechanism proposed to address the old cosmological constant problem. A major limitation is that our present realization cannot enter the standard hot Big Bang phase and therefore should not be regarded as a complete self-tuning solution. Second, inspired by a nontrivial critical point at infinity in the EDE phase portrait, we propose an \textit{Early Static Hot Universe} in which $S_\mu$ offsets the gravitational effect of ordinary hot gas, yielding quasi-static expansion with a decreasing comoving Hubble radius that can address the horizon problem. This scenario offers an alternative to inflation. This paper opens several avenues for further investigation of GCT, including cosmology, gravitational waves, and compact objects. (Abstract abridged to meet arXiv limits.)

Figures

Figures reproduced from arXiv: 2608.07318 by the authors.

**Figure 2.** FIG. 2. Phase portrait of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. Phase portraits of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: Right shows that, in the far future, the contribution of Λ to the Hubble expansion rate is almost exactly canceled by the energy density of Sµ. We refer to this as the Λ-cancelling solution. This subsubsection is devoted to a detailed analysis of its properties [PITH…

**Figure 6.** Figure 6: FIG. 6. Phase portrait of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: FIG. 7. Saddle region of the critical point [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: FIG. 8. ESHU dynamics obtained by integrating Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]

**Figure 9.** Figure 9: FIG. 9. Radial structure of the caloric de Sitter (red) and anti-de Sitter (blue) spacetimes obtained by numerically integrating [PITH_FULL_IMAGE:figures/full_fig_p034_9.png]

**Figure 10.** Figure 10: FIG. 10. Global classification of the vacuum solutions with [PITH_FULL_IMAGE:figures/full_fig_p037_10.png]

**Figure 11.** Figure 11: FIG. 11. Solutions of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p038_11.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

•Signature: (−,+,+,+)

Riemannian geometry •Metric:g µν. •Signature: (−,+,+,+). •Christoffel symbols: Γλ µν = 1 2 gλα(∂νgµα +∂ µgνα −∂ αgµν). •Riemann curvature tensor: Rρ λµν =∂ µΓρ λν −∂ νΓρ λµ + Γρ αµΓα λν −Γ ρ ανΓα λµ. •Ricci tensor and Ricci scalar: Rµν =R α µαν,andR=g µνRµν. •Einstein tensor: Gµν =R µν − 1 2 gµνR. •Partial derivative:∂ µ. •Covariant derivative:∇ µ and ∇µS...

[2] [2]

11 exhibit mirror symmetry about theη= 0 plane [265]

Mirror of the Schwarzschild metric The red, green, and blue trajectories in Fig. 11 exhibit mirror symmetry about theη= 0 plane [265]. This im- 38 λ µ η singularity at r= 0 or + ∞ singularity at r =rc Limit of blue line: µ→ const.⁄= 0 η→±∞ Limit of red line: η→ const.⁄= 0 µ→−∞ Schwarzschild metric Mirror of Schwarzschild Charged metric (+) Charged metric ...

[3] [3]

Inspired by the series anal- ysis in Sec

An exact charged solution Here aGuess & Checkstrategy is employed to con- struct new exact solutions. Inspired by the series anal- ysis in Sec. V C 1 and the exact solutions in Sec. V C 2, consider the ansatzµ= [2(α−e λ)]−1 with an undeter- mined constantα. Substituting this form into Eq. (5.8a) determinesη(λ), after which Eq. (5.8b) provides a self- cons...

2022

[4] [4]

At a fixed physical spacetime pointpwe write xµ(p)7→˜xµ(p) =x µ(p) +ξ µ(p),(A.1) whereξ µ =O(ϵ) is of the same order as the first-order perturbations

Gauge transformations In linear perturbation theory, a gauge transforma- tion is an infinitesimal coordinate transformation (pas- sive viewpoint). At a fixed physical spacetime pointpwe write xµ(p)7→˜xµ(p) =x µ(p) +ξ µ(p),(A.1) whereξ µ =O(ϵ) is of the same order as the first-order perturbations. We decompose each field into a prescribed background (not a...

[5] [5]

Let (x 1, x2)∗ be a critical point in a finite region, i.e.,P ∗ =Q ∗ = 0

Linear stability theory In the main text we only need planar systems, which we write as x′ 1 =P(x 1, x2), x′ 2 =Q(x 1, x2), (B.1) where ′ ≡d/dN. Let (x 1, x2)∗ be a critical point in a finite region, i.e.,P ∗ =Q ∗ = 0. The Jacobian matrix is J= " ∂P/∂x 1 ∂P/∂x 2 ∂Q/∂x 1 ∂Q/∂x 2 # ,(B.2) and linear stability follows from the eigenvalues ofJ∗. For a 2×2 rea...

[6] [6]

A convenient way to study them is the Poincar´ e compactification [124]

Coordinate transformations for infinity Critical points may also occur at infinity. A convenient way to study them is the Poincar´ e compactification [124]. For definiteness we assume thatPandQin Eq. (B.1) are bivariate polynomials, and denote their highest-degree homogeneous parts byP m(x1, x2) andQ m(x1, x2) (al- lowing one of them to vanish if the degr...

[7] [7]

[115, 122] for rigorous statements)

Centre manifold theory We summarize the center-manifold method for a two- dimensional system with one vanishing eigenvalue (see Refs. [115, 122] for rigorous statements). Throughout this subsection we follow the (z 1, z0) ordering, i.e.,a= [a1, a0]T with indices 1 and 0 denoting the first and sec- ond components, respectively. Letz= [z 1, z0]T and suppose...

[8] [8]

3/(2−4c 2) 0 # ,(B.18) and J∗ = −3 ˜c2

Shifting the critical point to the origin with ˜z=z−z ∗ gives d˜z dσ =J ∗˜z+f( ˜z),(B.13) wheref= [f 1(˜z1,˜z0), f0(˜z1,˜z0)]T satisfiesf i(0,0) = 0 and∂f i/∂˜zj|(0,0) = 0 fori, j∈ {1,0}, i.e.,fstarts at quadratic order. Diagonalize the linear part by intro- ducing eigenvectorsv 1 (forλ) andv 0 (for 0), defining S∗ = [v1,v 0], and transforming tou≡[u 1, u...