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Adiabatic initial conditions perturb every species by the same local time shift, so their relative number-density ratios are initially unperturbed. Photon acoustic modes then begin predominantly as displacement modes with cosine phase.

Equivalently, $\delta {\rho }_i/{\dot{\bar{\rho }}}_i$ is independent of the component $i$. The non-adiabatic pressure perturbation then vanishes, and the comoving curvature perturbation is conserved on superhorizon scales when anisotropic stress and gradient terms can be neglected.

For separately conserved barotropic components, an adiabatic cosmological perturbation has equal ${\delta }_N/(1+w_N)$. Pressureless matter and a $w=-1/3$ string component therefore have ${\delta }_s=2{\delta }_c/3$ on superhorizon scales. In the supplied perfect-fluid synchronous equations, neglecting spatial gradients gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$ for $S={\delta }_s-2{\delta }_c/3$. Both $S$ and its independent decaying-mode amplitude must vanish to select the growing adiabatic solution. At finite wavenumber, pressure gradients generate departures from this leading long-wavelength relation.

In Synchronous gauge in cosmology with coordinates comoving with cold dark matter, the linearized Einstein field equations and matter conservation give the displayed density equation. Primes denote conformal time derivatives. The radiation term contains its pressure contribution and cannot be dropped on superhorizon scales merely because radiation oscillates after cosmological horizon entry.

Neglecting metric perturbations, the rescaling $f=a\delta \varphi$ removes the expansion friction from the scalar perturbation action after a boundary term is discarded. Its canonical momentum is $f^{\prime }$, and each real Fourier oscillator has time-dependent squared frequency $k^2+a^2{V}^{\prime \prime }-a^{\prime \prime }/a$. The oscillator can become inverted on superhorizon scales; the canonical commutation relations still apply.

During a decelerating expansion the comoving Hubble radius grows. A fixed comoving wavenumber then passes from a superhorizon scale to a shorter scale when $k\simeq aH$. In radiation domination, $aH=1/\tau$, so this occurs at $k\tau \simeq 1$. Here horizon entry refers to Hubble radius crossing; a true causal horizon instead depends on a conformal-time integral.

For a light scalar in the Bunch-Davies vacuum with metric perturbations neglected, the mode $e^{-ik\tau }(1-i/(k\tau ))/\sqrt{2k}$ gives the displayed equal-time spectrum. The dimensionless power tends to $H^2/(4{\pi }^2)$ on superhorizon scales. The exact massless infinite-volume correlation has an infrared divergence; a finite-volume or finite-duration infrared regulator, and smearing for coincident products, makes the correlators well defined.

The second term in the equal-time two-point function of a de Sitter scalar is independent of time for constant $H$. It dominates on superhorizon scales and has constant dimensionless power spectrum, giving a scale-invariant inflationary power spectrum. The infrared qualification of a scale-invariant covariance still applies to the formally infinite-volume coincident variance.

For the regular growing mode in a purely matter-dominated flat background, $h=-2{\delta }_c$ and ${\delta }_c=C{\tau }^2$. The synchronous Einstein constraint $\mathcal{H}{h}^{\prime }+k^2{h}^{-}/3=3{\mathcal{H}}^2{\delta }_c$ with $\mathcal{H}=2/\tau$ gives $h^{-}=60C/k^2$. Hence $\zeta =h^{-}/6+{\delta }_c/3=10C/k^2+C{\tau }^2/3$, conserved to leading order on superhorizon scales. Identifying its early constant value gives the displayed density transfer; it includes a $k^2$ factor that is factored out of the conventionally normalized matter transfer function.

The non-adiabatic pressure perturbation isOn superhorizon scales it sources the evolution of the comoving curvature perturbation through a convention-dependent signed relation proportional to $H\delta P_{\mathrm{nad}}/(\bar{\rho }+\bar{P})$.

In deep radiation domination, $a\propto \tau$ and $a^{\prime }/a=1/\tau$. Neglect ${\Omega }_c{\delta }_c$ against the radiation gravity source, so the supplied CDM density equation in a matter-radiation universe becomesOn superhorizon scales omit the $k^2{\delta }_r/3$ pressure-gradient term. Under adiabatic initial conditions, ${\delta }_r=(4/3){\delta }_c$ then givesThe Euler-Cauchy equation characteristic polynomial is $p^2-4$, soThese are the general growing and decaying adiabatic solutions in this approximation. A regular primordial growing mode discards the divergent ${\tau }^{-2}$ solution. Although its conformal-time growth is again quadratic, now ${\delta }_c\propto a^2$, rather than ${\delta }_c\propto a$ in matter domination.

If “general” is taken without imposing adiabaticity, both independent relative-entropy integration constants must also be retained. The gradient-free radiation equation givesSubstitution into the matter equation producesThis is the general solution of the leading supplied coupled system, described by the superhorizon radiation-era entropy integration constants. Setting both $S_0$ and $S_1$ to zero recovers the adiabatic result. Regularity and the full initial Einstein field equations constraints can further restrict which of these leading solutions represents a prescribed physical initial mode.

On scales well inside the horizon, radiation pressure drives oscillations at frequency approximately $k/\sqrt{3}$. Their amplitude is bounded at leading order, and their gravitational forcing of a slowly varying matter contrast can be averaged away. With this approximation, as well as negligible matter self-gravity,hence the logarithmic CDM growth after radiation-era horizon entry isThis is the Mészáros effect. It does not mean that ${\delta }_r$ itself vanishes. For a rapidly oscillating source $R_{\gamma }\cos (\omega \tau +\phi )$, the matter equation has a particular responseIts leading second derivative reproduces $3{\delta }_r/{\tau }^2$; the remaining terms are smaller by powers of $(k\tau {)}^{-1}$. Thus the suppressed cold-matter forcing by rapid radiation oscillations is only order ${\delta }_r/(k\tau {)}^2$ and does not change the secular logarithm. Radiation forcing near horizon entry determines $C_0,C_{\log }$ and cannot be averaged away there. Neither this smoothing approximation nor the neglect of matter gravity remains uniformly valid near equality.

The perturbations are density contrasts, ${\delta }_c=\delta {\rho }_c/{\bar{\rho }}_c$ and ${\delta }_r=\delta {\rho }_r/{\bar{\rho }}_r$, evaluated in the common comoving synchronous gauge. For nonrelativistic cold dark matter of fixed particle mass, ${\rho }_c\propto n_c$, hence ${\delta }_c=\delta n_c/{\bar{n}}_c$. Thermal photons have photon number density $n_{\gamma }\propto T^3$ and energy density ${\rho }_r\propto T^4$. A small local temperature perturbation therefore givesFor adiabatic initial conditions, the ratio $n_{\gamma }/n_c$ is unperturbed, so $\delta n_{\gamma }/{\bar{n}}_{\gamma }=\delta n_c/{\bar{n}}_c$. Combining these equations provesEquivalently ${\delta }_r/(1+w_r)={\delta }_c/(1+w_c)$, with $w_r=1/3$ and $w_c=0$. The relation applies before causal pressure gradients can separate the two species on superhorizon scales; it need not remain true after cosmological horizon crossing.

A superhorizon density contrast is gauge dependent. The stated growth laws describe the growing adiabatic mode in the comoving-gauge density contrast (and the corresponding usual synchronous growing mode), not an arbitrary coordinate density perturbation. Work with a flat, single-fluid background with constant $w=p/\rho$, negligible anisotropic stress, $c=1$, and a comoving Fourier wavenumber $K$.

Here is a derivation from the linear gravitational constraints and evolution equation. For an adiabatic mode, the scalar gravitational potential $\Phi$ obeysPrimes denote conformal time derivatives. The Friedmann equation gives $a\propto {\tau }^{2/(1+3w)}$ and $\mathcal{H}=2/[(1+3w)\tau ]$, so the bracket vanishes. On superhorizon scales, $K\tau \ll 1$, the leading solutions are a constant potential and a decaying solution proportional to ${\tau }^{-(5+3w)/(1+3w)}$. Keeping the growing adiabatic branch therefore makes $\Phi$ time independent to leading order.

Combining the time-time and time-space linear Einstein constraints eliminates the coordinate velocity and gives the comoving density constraintThis is the relativistic comoving constraint, not a subhorizon Newtonian approximation. Since $\rho \propto a^{-3(1+w)}$, it implies $\Delta \propto (a^2{H}^2{)}^{-1}\propto a^{1+3w}$. Also $a\propto t^{2/[3(1+w)]}$ in cosmic time, whenceBoth epochs here mean their respective leading growing solution; matching through matter-radiation equality requires the full two-component evolution. The constant superhorizon curvature or potential is consistent with this growth because the comoving density perturbation carries the additional factor $(K/aH{)}^2$.

In the incoming Bunch-Davies vacuum, only $⟨a_{\mathbf{k}}{a}_{\mathbf{q}}^{†}⟩={\delta }_{\mathbf{kq}}$ contributes to the coincident vacuum expectation value. ThusReplacing the box sum by $\mathcal{V}\int d^{3}k/(2\pi {)}^3$ givesThe subhorizon vacuum contribution is the flat-space zero-point term; at fixed comoving $k$ it redshifts as $a^{-2}$. The other term gives frozen scalar power on superhorizon scales. Specifically, the dimensionless cosmological power spectrum per logarithmic interval isIts late-time value is independent of $k$, producing a scale-invariant inflationary power spectrum. Each fixed Fourier mode freezes separately as $\tau \to 0$.

The coincident integral is formal: the vacuum part diverges at large $k$, and an unlimited scale-invariant part diverges at small $k$. The conclusion concerns the power in each finite logarithmic band. A finite inflationary history supplies a largest generated wavelength, as described by the infrared qualification of a scale-invariant covariance; taking the late-time limit inside an unregulated integral would not be justified.

Use the spatial-potential convention of this paper and assume $\bar{\rho }+\bar{P}\ne 0$. A passive time displacement changes the spatial scalar by $\tilde{\Psi }-\Psi =\bar{N}H{\xi }^0$ and changes the energy density byThe two contributions to the uniform-density curvature perturbation therefore cancel:A spatial scalar displacement changes $E$ but not this combination. Hence $\zeta$ is gauge invariant at linear order. Pure cosmological-constant matter, for which $\bar{\rho }+\bar{P}=0$, does not supply a energy density clock and this formula is not defined for that component alone.

Now take the definite constant-$w$ barotropic equation of state and use physical-time differentiation $\mathcal{D}={\bar{N}}^{-1}{\partial }_t$. Put $A=\bar{\rho }+\bar{P}$. Background stress-energy conservation gives $\mathcal{D}A=-3H(1+w)A$. Differentiating $\zeta =-\Psi +\delta \rho /(3A)$, then substituting the two supplied perturbation equations, givesThus the lapse and expansion perturbations cancel without imposing a gauge. In terms of the non-adiabatic pressure perturbation $\delta P_{\mathrm{nad}}=\delta P-w\delta \rho$,For the stipulated barotropic equation of state, $\delta P=w\delta \rho$, so the first term vanishes. For a Fourier mode, $\Delta =-k^2/a^2$. For regular long-wavelength solutions the remaining gradients are suppressed when $k\ll aH$. Consequently $\dot{\zeta }\simeq 0$ on superhorizon scales, with exact conservation in the leading zero-gradient limit. It is not an exact finite-$k$ identity. Entropy perturbations or a singular gradient expansion can invalidate that conclusion; the supplied adiabatic assumption is what removes the unsuppressed pressure source.

This superhorizon conservation of uniform-density curvature is useful because inflationary fluctuations leave the horizon during cosmic inflation and may re-enter only much later. In an adiabatic single-clock evolution, the conserved uniform-density curvature perturbation transports their initial amplitude through reheating and the later radiation in cosmology and matter eras without solving every short-distance process. It then determines the initial conditions for large-scale structure and Cosmic microwave background anisotropy. Extra fluctuating fields or fluids can introduce isocurvature perturbations and convert them into curvature, so conservation must be checked rather than assumed during such transitions.

Take the regular growing matter mode ${\delta }_c=C{\tau }^2$ and $h=-2C{\tau }^2$. With $\mathcal{H}=2/\tau$, the constraint givesConsequentlyHere ${\zeta }_{\ast }=10C/k^2$ is its early, superhorizon constant value. The fractional time variation is of second gradient order, andThus the primordial curvature is conserved to leading order on superhorizon scales. This is the matter-era density transfer from primordial curvature.

A mode entering after equality evolves in the growing matter solution, givingThis assumes matter-dominated growth to ${\tau }_0$, as in the model. A subsequent dark-energy era would multiply by the appropriate late-time growth suppression. The initial radiation-era adiabatic curvature passes through equality unchanged at leading superhorizon order, so the same ${\zeta }_{\ast }$ is the primordial input.

For $k\gg k_{\mathrm{eq}}$, entry occurs in radiation domination at ${\tau }_h\sim k^{-1}$. The entry contrast is of order ${\zeta }_{\ast }$ because its superhorizon value scales as $k^2{\tau }_h^2{\zeta }_{\ast }$. In the question's stagnation approximation it stays of this order until equality, then grows by $({\tau }_0/{\tau }_{\mathrm{eq}}{)}^2$. Thusindependent of $k$ to that approximation. The precise coefficient needs matching through the equality transition; the horizon convention, including the use of wavelength $2\pi /k$, also changes the matching definition of $k_{\mathrm{eq}}\sim {\tau }_{\mathrm{eq}}^{-1}$ by a fixed factor.

One continuous sharp-equality model, normalizing the plateau by matching the low-$k$ branch at $k_{\mathrm{eq}}$, is the sharp-equality cold-dark-matter transfer approximationThis is a matching approximation, not an exact solution through equality. The function requested here transfers curvature directly to density. It differs from the conventional normalized matter transfer $\mathcal{T}(k)$, where one factors out $k^2{\tau }_0^2/10$: that normalized function tends to one at low $k$ and to $(k_{\mathrm{eq}}/k{)}^2$ in the stagnation model at high $k$.

For the dimensional primordial spectrum $P_{\zeta }=A/k^3$, linear deterministic transfer gives $P_c=|T_{\delta }{|}^2{P}_{\zeta }$. Hence this sharp model yieldsThe spectrum turns over near equality. Retaining the slow radiation-era logarithmic growth refines the high-$k$ transfer to $T_{\delta }\propto ({\tau }_0/{\tau }_{\mathrm{eq}}{)}^2\ln (k/k_{\mathrm{eq}})$, up to constants in the logarithm and matching factors, and therefore refines the small-scale spectrum to $P_c\propto k^{-3}{\ln }^2(k/k_{\mathrm{eq}})$. This is the logarithmic small-scale matter transfer correction to the question's constant-growth idealization. Multiplication by $k^3/(2{\pi }^2)$ would give dimensionless power, whose large-scale slope is $k^4$ rather than the dimensional spectrum's $k$.

For a fixed coordinate cell, the spatial volume element isThus its perturbation relative to the background expansion, or its perturbed comoving volume, is $\Delta V_c/V_c=h/2$. In the cold-matter rest frame each such cell contains a fixed amount of matter. Since density is mass divided by physical volume,Equivalently, stress-energy conservation gives ${\delta }_c^{\prime }=-h^{\prime }/2$; the stated relation fixes the spatially dependent integration constant by the initial mass-coordinate convention. A contraction of comoving volume gives a positive density contrast.

In the matter era $a\propto {\tau }^2$, so $\mathcal{H}=2/\tau$ and the matter density fraction is approximately one. The evolution equation reduces toFor ${\delta }_c\propto {\tau }^p$, the indicial equation is $p(p-1)+2p-6=(p-2)(p+3)=0$. ThereforeThe growing mode is proportional to $a$, and the decaying mode to $a^{-3/2}$. These are the density solutions after the comoving coordinate choice; the residual time mode found in part (i) explains why a similar power in an unfixed density variable cannot be interpreted in isolation.

There is a normalization omission in the printed density parameter if ${\bar{\rho }}_c$ denotes physical density. Its usual definition in conformal time isThe printed expression lacks $a^2$, unless its density is already rescaled by that factor. The matter-era statement ${\Omega }_c\simeq 1$ and the growth equation use the physical density fraction above.

On superhorizon scales during radiation domination, the regular adiabatic cosmological perturbation has approximately constant primordial curvature. The comoving density perturbation is gradient-suppressed, of order $(k/\mathcal{H}{)}^2$ times that curvature. Since $\mathcal{H}\propto {\tau }^{-1}$, it grows as ${\delta }_c\propto k^2{\tau }^2$ for fixed mode amplitude. This is the regular superhorizon solution in the specified slicing, not causal collapse driven by communication across the entire wavelength.

After radiation-era horizon entry, the dominant radiation develops pressure-supported acoustic oscillations. Its potential no longer provides a steady growing gravitational source, while cold matter is too small a fraction of the total density to produce rapid self-gravitating growth. The result is the Mészáros effect: growth is strongly suppressed until equality. The supplied stagnation approximation replaces that slow evolution by a constant. More accurately, ignoring the rapidly oscillating source at leading order gives ${\delta }_c^{\prime \prime }+{\delta }_c^{\prime }/\tau \simeq 0$, with a constant and a logarithmic solution. Thus radiation-era subhorizon growth is at most slow logarithmic growth, rather than the matter-era power law; treating it as constant is the explicit approximation used for the simple transfer estimate below.

For $W=\bar{\rho }+\bar{P}\ne 0$, the conserved uniform-density curvature in the spatial-metric convention $1-2\Phi$ isThe source's plus sign in front of both terms is not gauge invariant and cannot yield the claimed conservation. Choosing the opposite overall sign, $\Phi -\delta \rho /(3W)$, is equally valid, but changing only one term is necessary. A passive time displacement $t\mapsto t+T$ gives $\tilde{\Phi }=\Phi +\bar{N}HT$ and $\widetilde{\delta \rho }=\delta \rho +3\bar{N}HWT$, so their combination above is unchanged. This is the uniform-density curvature sign test.

The lapse perturbation in the perturbed energy-conservation source is $\Psi$, not the spatial-curvature perturbation $\Phi$. Indeed the perturbation of the proper-time density derivative is $\mathcal{D}\delta \rho -\Psi \mathcal{D}\bar{\rho }$, while the normal expansion changes by $-\kappa$. The consistent equation isFor a constant barotropic equation of state, $\delta P=w\delta \rho$ and $\mathcal{D}W=-3H(1+w)W$. Differentiate ${\zeta }_{\mathrm{ud}}$ and cancel the density terms:More generally, $\delta P_{\mathrm{nad}}=\delta P-(\mathcal{D}\bar{P}/\mathcal{D}\bar{\rho })\delta \rho$ givesRegular scalar modes have gradient terms suppressed by $(k/aH{)}^2$. Thus adiabatic uniform-density curvature is conserved to leading order on superhorizon scales, ${\dot{\zeta }}_{\mathrm{ud}}\simeq 0$. This is a long-wavelength limit, not exact finite-wavenumber conservation; it assumes a regular gradient expansion and excludes $W=0$, where this variable is undefined. In contrast the literal printed variable obeys $\mathcal{D}{\zeta }_{\mathrm{printed}}=2\mathcal{D}\Phi +O(k^2)$ even after repairing the energy equation, and generally evolves across changes of the background equation of state.

There are three practical benefits. First, uniform-density curvature is gauge invariant, so matching it does not depend on coordinate slicing. Second, superhorizon conservation of uniform-density curvature transports primordial amplitudes through changes in the expansion history without solving each short-scale dynamical equation. Third, when a mode re-enters the horizon it supplies initial conditions for density growth and Cosmic microwave background anisotropy; non-adiabatic pressure is also an explicit diagnostic of when that simple transport fails.

For the growing adiabatic mode in an epoch of constant $w$, the superhorizon potential is constant. The Friedmann equation gives $4\pi G{a}^2(\bar{\rho }+\bar{P})=\frac{3}{2}(1+w){\mathcal{H}}^2$, so the given relation impliesThus ${\varphi }_{\mathrm{rad}}=-2\mathcal{R}/3$ and ${\varphi }_{\mathrm{mat}}=-3\mathcal{R}/5$. Photon continuity, neglecting its gradient term on superhorizon scales, gives $({\Theta }_0-\varphi {)}^{\prime }=0$. The radiation-era initial condition fixes ${\Theta }_0-\varphi =\mathcal{R}$. Through the matter-radiation transition it follows that ${\Theta }_{0\ast }=\mathcal{R}+{\varphi }_{\mathrm{mat}}=2\mathcal{R}/5$. Since $\psi =\varphi$, the matched large-scale emission perturbation isKeeping the radiation value of ${\Theta }_0$ unchanged across the transition would give the wrong coefficient. This is the Sachs-Wolfe radiation-to-matter matching for the paper's sign convention for comoving curvature perturbation.

Neglect the Integrated Sachs-Wolfe effect, the superhorizon-suppressed Doppler term, observer monopole, and any observer kinematic dipole. Put ${\mathbf{x}}_0=0$ by statistical homogeneity, and defineThe plane-wave expansion on ${\mathbf{x}}_{\ast }=-{\chi }_{\ast }\mathbf{e}$ gives angular multipolesStatistical isotropy and orthonormal spherical harmonics then imply $⟨a_{\ell m}{a}_{{\ell }^{\prime }{m}^{\prime }}^{\ast }⟩={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}{C}_{\ell }$. Radial integration supplies $(4\pi {)}^2(2{\pi }^2)/(2\pi {)}^3=4\pi$, giving the large-angle angular power spectrumThis assumes adiabatic growing modes, matter domination at emission, negligible anisotropic stress, and wavenumbers that are outside the Hubble radius then. It treats recombination as instantaneous, omits late potential evolution and rescattering, and omits lensing at this linear order. The formal primordial contribution is defined for $\ell >0$; the observed dipole has a large additional kinematic contribution, so the clean large-angle primordial comparison is usually $\ell \ge 2$. A scale-invariant spectrum also yields the Sachs-Wolfe plateau $\ell (\ell +1)C_{\ell }=2\pi {\mathcal{P}}_{\mathcal{R}}/25$ for $\ell \ge 1$.

Because density is a scalar field, $\tilde{\rho }(\tilde{x})=\rho (x)$. Write both sides as background plus perturbation and expand the background time shift:The coordinate shift acting on an already first-order perturbation contributes only at second order. Thus the density perturbation gauge transformation isThe uniform-density curvature perturbation in the paper's sign convention isThe derivative on ${\bar{\rho }}^{\prime }$ is essential and is present in the PDF; the TeX transcription drops it. The scalar-curvature combination transforms aswhile the density term changes by $-\mathcal{H}T$. Hence the two time-slicing changes cancel:On a uniform-density slice, $\delta \rho =0$ and this variable is the signed spatial-curvature perturbation. The construction assumes ${\bar{\rho }}^{\prime }\ne 0$; a pure constant-density cosmological constant does not define such a time slicing.

In Newtonian gauge in cosmology, $E=B=0$ and $C=-\Phi$. The background cosmological perfect-fluid continuity equation gives ${\bar{\rho }}^{\prime }=-3\mathcal{H}(\bar{\rho }+\bar{P})$, so, writing $D=\bar{\rho }+\bar{P}$,Set the cosmological adiabatic sound speed $c_a^2={\bar{P}}^{\prime }/{\bar{\rho }}^{\prime }$. Then $D^{\prime }=-3\mathcal{H}D(1+c_a^2)$. Differentiate the previous expression and insert the perturbed energy-conservation equation:Consequently the curvature evolution equation in this sign convention isThe last equality defines the total energy-frame velocity by $\mathbf{q}=D\mathbf{v}$. For adiabatic cosmological perturbations, the non-adiabatic pressure perturbation vanishes. On superhorizon scales, the stated suppression of the velocity divergence makes the remaining gradient term negligible. Thus $\zeta$ is conserved to leading order in $k/\mathcal{H}$. The positive sign of the pressure source follows from the paper's definition of $\zeta$ and must not be replaced by a formula using the opposite curvature convention.

Superhorizon conservation of uniform-density curvature lets one carry a primordial perturbation from its inflationary generation through otherwise complicated eras and predict the initial conditions for later Cosmic microwave background anisotropy and structure growth. It relies on adiabaticity: an isocurvature perturbation can source curvature evolution, so the same conclusion is not automatic in an arbitrary multifield model.

In the constant-$H$ de Sitter approximation, $a=-1/(H\tau )$ with $\tau <0$, so $a^{\prime \prime }/a=2/{\tau }^2$. The massive rescaled mode obeysFor a fixed mass in the specified open interval, at sufficiently late superhorizon scales with $|k\tau |\ll 1$, the $k^2$ term is negligible compared with the inverse-square term in the limiting equation. Try $v\propto (-\tau {)}^p$. The indicial equation is $p(p-1)+m_{\chi }^2/H^2-2=0$, givingThus both independent branches of the massive de Sitter superhorizon scalar mode are present inThe coefficients are constant in time and fixed by initial conditions. Writing $-k\tau >0$ avoids ambiguous fractional powers of the negative number $k\tau$; a chosen complex phase in the printed notation can be absorbed into the coefficients.

For the specified mass range, $1/2<\nu <\sqrt{5}/2<3/2$. The physical fluctuation is $\delta {\chi }_{\mathbf{k}}=v_{\mathbf{k}}/a=-H\tau v_{\mathbf{k}}$, so its two branches behave as $(-\tau {)}^{3/2-\nu }$ and $(-\tau {)}^{3/2+\nu }$. Both exponents are strictly positive. Therefore both physical-field branches decay as $\tau \to 0^{-}$:The slower branch of $v$ itself grows because $1/2-\nu <0$, but the expanding scale factor more than cancels that growth. A spatially homogeneous background field of this mass satisfies the corresponding $k=0$ equation and also has these decaying powers.