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A coasting fluid has pressure $p=-\rho /3$. The cosmological perfect-fluid continuity equation makes its density scale as $a^{-2}$. If it alone fills a flat expanding universe, the scale factor grows linearly in physical time. Its background density has the same scale-factor dependence as a spatial-curvature term, although the fluid does not itself create spatial curvature.

In a spatially flat universe containing separately conserved pressureless matter and a coasting fluid with $w=-1/3$, the ratio $\eta ={\rho }_s/{\rho }_m$ is proportional to the scale factor. If the coasting component is modeled by the barotropic fluid perturbation equations with no anisotropic stress, its contribution to the trace gravitational source vanishes because $1+3w=0$. The matter growing mode isA decaying solution is $D_{-}=\sqrt{1+\eta }/{\eta }^{3/2}$. The reduction of order formula with integrating factor ${\eta }^{3/2}\sqrt{1+\eta }$ produces the integral solution. It has $D=\eta [1-4\eta /7+O({\eta }^2)]$ during matter domination and tends to $5/2$ during coasting-fluid domination. Thus growth freezes rather than continuing in proportion to the scale factor. The background geometry is flat despite the same background expansion law as an open dust universe.

A flat universe with separately conserved pressureless matter and a coasting fluid satisfies $a^{\prime 2}=H_0^2[{\Omega }_{m}a+(1-{\Omega }_m)a^2]$. For both densities positive, choosing the Big Bang at zero conformal time gives $a={\Omega }_m[\cosh (\alpha \tau )-1]/[2(1-{\Omega }_m)]$, with $\alpha =H_0\sqrt{1-{\Omega }_m}$. Physical time follows by integrating $dt=ad\tau$. The hyperbolic parametrization is formally like the open dust solution, but the spatial geometry here is flat.

There are two normalization problems in the printed energy identity. In conformal time, the physical density of a homogeneous scalar is $T_{00}/a^2$, and its kinetic contribution contains ${\varphi }^{\prime 2}$, not one power of ${\varphi }^{\prime }$. The conformal density of a canonical scalar field follows directly from the stress-energy tensor:Writing $K={\varphi }^{\prime 2}/(2a^2)$, the condition $K-V=-(K+V)/3$ gives $V=2K$, henceThe reconstruction is consistent with this corrected identity; the unsquared printed identity cannot hold as an energy relation.

The constant-equation-of-state density scaling for this coasting fluid gives ${\rho }_{\varphi }={\rho }_{X0}{a}^{-2}$, where ${\rho }_{X0}=3M^2{H}_0^2{\Omega }_X$. Choose the branch on which $\varphi$ increases. ThenKeeping all three density parameters explicitly, the Friedmann equations in conformal time giveFor ${\Omega }_X>0$, defineThe square root is $\sqrt{(a+b{)}^2-A^2}$. Integrating on the expanding branch giveswhere $a({\varphi }_0)=1$. Thus the scalar-field reconstruction of a coasting component isSubstitution into the potential gives the requested present-field parametrization:The domain is the branch where $a>0$ and the hyperbolic argument is positive; reversing the direction of the field changes the sign of $B$. The actual discriminant condition is ${\Omega }_m^2/4>{\Omega }_r{\Omega }_X$. The supplied stronger bound suffices for physical density fractions $0<{\Omega }_X\le 1$. No density parameter has been eliminated using the flatness sum.

As a check, ${\varphi }^{\prime }$ is constant and $V_{,\varphi }=-2V(a^{\prime }/a)/{\varphi }^{\prime }$. Since $a^{2}V={\varphi }^{\prime 2}$ along this solution, ${\varphi }^{\prime \prime }+2(a^{\prime }/a){\varphi }^{\prime }+a^2{V}_{,\varphi }=0$ identically. The reconstructed field therefore satisfies its equation of motion as well as the Friedmann equations and required equation of state.

Separate conservation of a fluid with fixed $w=P/\rho$ gives $\dot{\rho }+3H(1+w)\rho =0$, hence $\rho \propto a^{-3(1+w)}$. The $w=-1/3$ component therefore has ${\rho }_Q\propto a^{-2}$. Flatness today fixes its normalization to ${\rho }_{Q0}=3H_0^2(1-{\Omega }_{M0})/(8\pi G)$, soThis is a coasting fluid: its contribution to $\rho +3P$ is zero, so it does not itself drive accelerated expansion. Its background Friedmann contribution resembles curvature, although the actual spatial geometry remains flat.

Choose the Big Bang at $\tau =t=0$. For $0<\Omega <1$, the expanding singular solution of the Riccati equation is $\mathcal{H}=\beta \coth (\beta \tau /2)$. Integrating $a^{\prime }/a=\mathcal{H}$ gives $a=C{\sinh }^2(\beta \tau /2)$. The Friedmann constraint fixes $C=\Omega /(1-\Omega )$, rather than leaving an independent amplitude. ThusNow $dt=ad\tau$. Integrating from the singular origin givesThis derives the flat matter-coasting-fluid Friedmann solution. For small $\tau$, $a\propto {\tau }^2$ and $t\propto {\tau }^3$, hence $a\propto t^{2/3}$. At late time $a\propto t$, as expected when the coasting fluid dominates.

Set $a=1$ in the parametric solution. With $b=1-\Omega$,ConsequentlyThe endpoint values are obtained by limits. A particularly direct proof of the age bound avoids subtracting nearly equal terms: from the Friedmann equation,For nonnegative component densities, $0\le \Omega \le 1$ and $a\le \Omega +(1-\Omega )a\le 1$. The integrand lies between $\sqrt{a}$ and $1$. Integrating therefore givesEquality holds for pure matter and pure coasting fluid, respectively. These are the age of a flat matter-coasting-fluid universe and the coasting-fluid age bound; the positivity range is necessary for the stated universal inequality.

Use natural units and conformal time $dt=ad\tau$. The conformal Hubble parameter is $\mathcal{H}=a^{\prime }/a=aH$. For the barotropic equation of state $P=(\gamma -1)\rho$, the Friedmann acceleration equation givesThe Friedmann equation gives $8\pi G{a}^2\rho /3={\mathcal{H}}^2+k$. Hence the constant-equation-of-state conformal Riccati equation isFor $\alpha \ne 0$, separation and integration of this Riccati equation yield, after choosing the origin of conformal time,The open positive-energy density branch uses the hyperbolic cotangent, rather than the cotangent. Integrating $a^{\prime }/a=\mathcal{H}$ giveswhere the positive constants are fixed by the energy density normalization. These formulas apply on branches where the bases are positive. For $\alpha >0$, take the Big Bang at $\tau =0$. The cosmic time is thenThe integrals also give the general answer after selecting a suitable reference time when a finite Big Bang endpoint does not exist. The exceptional coasting fluid $\gamma =2/3$ has $\alpha =0$, so $\mathcal{H}=h$ is constant: $a=A{e}^{h\tau }$ and $t=(A/h)e^{h\tau }+t_{\ast }$. If $h=0$, instead $a=A$ and $t=A\tau +t_{\ast }$. A flat fluid with $\gamma =0$ is the constant-density de Sitter spacetime case, for which the flat time integral is logarithmic.

During matter domination, $\gamma =1$ and $\alpha =1/2$. Absorbing numerical factors into separate positive constants gives the closed matter-dominated Friedmann solution, the flat Einstein-de Sitter universe and the open matter-dominated Friedmann solution:To compare the age of a dust universe without a cosmological constant, specify the normalization. At fixed present Hubble parameter $H_0$, with $x=a/a_0$ and present matter cosmological density parameter ${\Omega }_m>0$, the Friedmann equation givesThe integrand decreases strictly with ${\Omega }_m$ for $0<x<1$. Therefore age ordering of dust universes at fixed Hubble parameter gives, on the expanding branches,The final bound is approached by the empty Milne universe. This compares models with the same $H_0$; fixing instead the present scale factor and energy density gives a different comparison.

Write $\mathcal{H}=a^{\prime }/a$ for the conformal Hubble parameter and ${\theta }_N=i\mathbf{k}\cdot {\mathbf{v}}_N$ for the velocity divergence of a Fourier mode. Work in the synchronous gauge comoving with the cold dark matter, so ${\mathbf{v}}_C=0$. This remains consistent with its pressureless momentum equation ${\mathbf{v}}_C^{\prime }+\mathcal{H}{\mathbf{v}}_C=0$. Its density equation then gives ${\delta }_C^{\prime }=-h^{\prime }/2$.

For the string component, $1+3w_S=0$. Thus it makes no contribution to the trace gravitational source in the supplied perfect-fluid model. Substitute $h^{\prime }=-2{\delta }_C^{\prime }$ and $h^{\prime \prime }=-2{\delta }_C^{\prime \prime }$ in that trace equation to obtainThe absence of a pressure-gradient term reflects the zero cosmological sound speed of the cold matter.

For the $w_S=-1/3$ coasting fluid, the continuity and momentum equations becomeThe second equation follows by taking $i\mathbf{k}\cdot$ the vector equation; in particular, $i\mathbf{k}\cdot i\mathbf{k}=-k^2$, which fixes the pressure sign. Differentiate the first equation and use ${\theta }_S={\delta }_C^{\prime }-3{\delta }_S^{\prime }/2$:Eliminating ${\delta }_C^{\prime \prime }$ with the cold-matter equation givesThese are equations for the specified idealized barotropic string fluid. A cosmic string network with additional anisotropic stress need not obey the same pressure response.

The Jeans length is the physical wavelength at which the restoring pressure term and self-gravity balance. In the elementary static-fluid approximation,Wavelengths longer than ${\lambda }_J$ are unstable to self-gravitating collapse, whereas shorter ones have restoring sound oscillations. Expansion and relativistic effects modify this elementary criterion. Ideal cold dark matter has $c_s=0$, hence no positive pressure-supported Jeans length in these equations. For the specified negative-pressure barotropic string fluid, $c_s^2=-1/3$; there is no real restoring-pressure Jeans threshold.

For $\eta \gg 1$, the matter growth equation has leading termsThe gravitational term is one inverse power smaller than the derivative coefficients. Dropping it at leading order gives $({\eta }^2{D}_{\eta }{)}_{\eta }=0$, hence a constant mode and a decaying ${\eta }^{-1}$ mode. A more controlled check is possible: direct substitution gives $D_{-}=\sqrt{1+\eta }/{\eta }^{3/2}$ as an exact solution. Reduction of order then gives the growing mode, normalized by $D_{+}/\eta \to 1$ at early times,The integral grows as $\eta +O(\log \eta )$, while the prefactor is ${\eta }^{-1}[1+O({\eta }^{-1})]$. Thuswith corrections of order $\log \eta /\eta$. The perturbation freezes because the matter fraction tends to zero while the coasting fluid controls the expansion.

To describe the requested cold-dark-matter transfer function, its normalization must be specified. The early matter-era Hamiltonian constraint, in the negative-expansion convention for $K$, gives $-4k^2\Psi /a^2-4H\kappa =16\pi G\delta \rho$. For the growing comoving dust mode, the continuity equation gives $\kappa ={\dot{\delta }}_C=H{\delta }_C$ in proper time. Since $16\pi G\bar{\rho }=6H^2$, this yields ${\delta }_C=-2k^2\Psi /[5(aH{)}^2]$. On early superhorizon scales $\Psi$ is the conserved curvature amplitude, up to the chosen overall sign, and $(aH{)}^2\propto a^{-1}$. This establishes the $k^{2}a$ initial growing behavior rather than assuming identical horizon-entry amplitudes at all epochs. For regular adiabatic modes outside the horizon during early matter domination, $A_k=\mathcal{A}{k}^2{\mathcal{R}}_{\ast }(k)$, with a $k$-independent constant $\mathcal{A}$ and primordial curvature amplitude ${\mathcal{R}}_{\ast }$. The matter equation above has no $k$, so at fixed late timeConsequently the density-per-curvature transfer is proportional to $k^2{D}_{+}$ on scales entering after equality. The conventional shape transfer, which factors out $k^2$ and the common growth factor, is a plateau, $T(k)\simeq 1$ on these scales. Comparing with a matter-only universe that continues growing as $\eta$ instead gives a common amplitude suppression $D_{+}/\eta \simeq 5/(2\eta )$, still independent of $k$. A decline caused solely by later horizon entry would contradict the supplied scale-independent matter equation: the regular synchronous growing solution already evolves outside the Hubble radius. This is the scale-independent matter suppression by a coasting fluid.

There is also a limiting entry scale. Since $a^2{\rho }_S$ is constant, define $k_{\ast }^2=(8\pi G/3)a^2{\rho }_S$. ThenA mode enters the Hubble radius only if $k>k_{\ast }$; $k\gg k_{\ast }$ enters during matter domination, whereas $k\downarrow k_{\ast }$ enters arbitrarily late. Modes $k\le k_{\ast }$ never enter in this ideal coasting future. The requested post-equality entrants occupy $k_{\ast }<k<k_{\mathrm{eq}}$. The following original diagram distinguishes the two transfer conventions and marks this restricted interval; it does not attempt to model modes entering during radiation domination.

The adiabatic initial conditions for separately conserved components have equal ${\delta }_N/(1+w_N)$. Therefore the radiation relation is replaced here byTo check dynamically, put $S={\delta }_S-2{\delta }_C/3$. At $k=0$, subtracting two-thirds of the matter equation from the string equation gives $S^{\prime \prime }+2\mathcal{H}{S}^{\prime }=0$. Selecting $S=0$ and no independent relative-velocity mode preserves the relation. Thus the growing adiabatic string density contrast follows the matter mode and is also approximately frozen at late times to leading order in gradients. This is the adiabatic matter-string density relation.

At late times $\mathcal{H}\to k_{\ast }$, while ${\Omega }_C$ and ${\delta }_C^{\prime }$ vanish. The homogeneous string equation becomesFor $k\gg k_{\ast }$, one solution grows rapidly with exponent approximately $k/\sqrt{3}-k_{\ast }$, instead of undergoing radiation-like acoustic oscillations. This is a gradient instability of a negative-pressure perfect fluid. For $k\ll k_{\ast }$, the growing exponent is only $k^2/(6k_{\ast })$, while the other exponent is approximately $-2k_{\ast }$. The leading zero-gradient approximation is frozen, but at fixed nonzero superhorizon $k$ a slow instability can accumulate over sufficiently long conformal time. The strict gradient limit and infinite-future limit therefore need not commute.

These are the apparent behaviors of the equations given. They do not establish rapid collapse of a realistic cosmic string network: tension, anisotropic stress and the constitutive sound response were excluded in replacing the network by this barotropic perfect fluid. Once the modeled string perturbation becomes nonlinear, the linear calculation itself ceases to apply.

Normalize the present scale factor to one. For $0<\Omega <1$, the parametric solution givesThus the age of a flat matter-coasting-fluid universe isA direct age integral proves the bound more transparently than manipulating inverse hyperbolic functions. Since $\dot{a}=H_0\sqrt{\Omega /a+b}$,For $0\le a\le 1$ and $0\le \Omega \le 1$, one has $a\le \Omega +ba\le 1$, henceIntegrating givesEquality on the left is the pure pressureless matter universe, $\Omega =1$; equality on the right is the pure coasting fluid, $\Omega =0$. These are all physically allowed density fractions for the nonnegative two-fluid model. The integrand decreases with $\Omega$, so the coasting-fluid age bound is approached continuously between those endpoints. In the closed expression the apparent $b=0$ singularities cancel using $\mathrm{arsinh}x=x-x^3/6+\cdots$; as $\Omega \to 0$, the term $\Omega \log (1/\Omega )$ vanishes. The limits are respectively $H_0{t}_0=2/3$ and $1$.