Modified Friedmann Dynamics

Cosmic expansion does not by itself identify the gravitational field equations. A modified theory can reproduce a prescribed scale factor while assigning the same history to new fields, curvature operators, or a different branch of a constraint. This page derives a concrete example—metric $f(R)$ gravity—to show where modified Friedmann equations come from, which data they require, and how their general-relativistic limit is checked.

Helpful background. Einstein Equations supplies the FLRW tensor reduction and conservation identity used below; Gravitational Actions and Boundary Terms supplies the metric-variation method; Formulations of General Relativity supplies the initial-value distinction between evolution and constraint equations; Gravity Beyond General Relativity supplies the theory-space and effective-field-theory scope; Causality and Global Structure supplies the distinction between a regular local branch and a globally admissible spacetime; and Gravitational Radiation and Waveforms supplies the tensor-sector baseline needed when the same cosmology is later tested with waves.

Curvature as a dynamical variable

Consider the Jordan-frame action

$$S=\frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,f(R)+S_{\mathrm m}[g,\psi], \qquad F(R)\equiv \frac{df}{dR}.$$

Matter is minimally coupled to $g_{\mu\nu}$, so $\nabla_\mu T^{\mu\nu}=0$. Varying the metric gives

$$F R_{\mu\nu}-\frac12 f g_{\mu\nu} -\nabla_\mu\nabla_\nu F+g_{\mu\nu}\Box F =8\pi G T_{\mu\nu}.$$

The derivatives of $F$ are the important novelty. Because $F$ depends on $R$, the metric equation is fourth order unless the extra scalar degree of freedom is displayed explicitly. Taking the trace exposes that scalar:

$$3\Box F+FR-2f=8\pi G T.$$

Thus $R$ is no longer fixed algebraically by the matter trace. It propagates, subject to a branch choice when the map $F(R)$ is not globally invertible. The metric field and trace equations appear in Sotiriou and Faraoni 2010, § II.A, Eqs. (6) and (8), and their scalar–tensor reformulation in Sotiriou and Faraoni 2010, § III.A, Eqs. (50)–(63).

The two Friedmann equations

For a spatially flat FLRW metric, $ds^2=-dt^2+a^2(t)d\boldsymbol{x}^2$, define $H=\dot a/a$. The curvature scalar is

$$R=6\bigl(\dot H+2H^2\bigr).$$

For a perfect fluid, the $00$ and spatial-trace components of the metric equation become

$$3F H^2 =8\pi G\rho+\frac12(FR-f)-3H\dot F,$$ $$-2F\dot H =8\pi G(\rho+p)+\ddot F-H\dot F.$$

Matter conservation remains

$$\dot\rho+3H(\rho+p)=0.$$

The trace is a linear combination of the two metric equations, rather than a third independent metric equation. There is also a differential consistency relation. Put every term of the first Friedmann equation on the left and call the result $C$. Using matter conservation, $F=f_R$, and $R=6(\dot H+2H^2)$ gives

$$\dot C=3H\left[2F\dot H+8\pi G(\rho+p)+\ddot F-H\dot F\right].$$

Thus a preserved constraint implies the second Friedmann equation wherever $H\ne0$. At a bounce or turnaround, $H=0$, this division is unavailable: retain the evolution equation itself. This identity is a useful sign check and explains why independently integrating every displayed equation would overdetermine the system.

It is tempting to move every non-Einstein term to the right-hand side and call it an effective dark-energy density. One possible definition is

$$8\pi G\rho_{\mathrm{eff}} =\frac12(FR-f)-3H\dot F+3(1-F)H^2.$$

Then $3H^2=8\pi G(\rho+\rho_{\mathrm{eff}})$. This is bookkeeping rather than a new material fluid: a different division by $F$, or a conformal change of variables, reallocates terms. The invariant content is the original field equation together with its matter-coupling prescription.

Initial data, branches, and fixed points

The first Friedmann equation constrains the initial values of $a$, $H$, $F$, $\dot F$, and $\rho$. The trace equation evolves $F$; the relation $F=f_R(R)$ then selects $R$ only on a chosen invertible branch. At points where $f_{RR}=0$, the scalar description changes character and one must return to the original equations rather than divide by $f_{RR}$.

A vacuum de Sitter solution has constant $H$, hence constant $R=12H^2$. Its algebraic condition is

$$F(R_{\mathrm{dS}})R_{\mathrm{dS}}-2f(R_{\mathrm{dS}})=0.$$

Linearizing the trace equation about that solution gives the scalaron mass

$$m_F^2=\frac{F-Rf_{RR}}{3f_{RR}} \bigg|_{R=R_{\mathrm{dS}}}.$$

On a branch intended to connect continuously to attractive general relativity, $F\gt0$ avoids reversing the effective gravitational coupling, while $m_F^2\gt0$ prevents a long-wavelength tachyonic instability of the constant-curvature point. These conditions are local; neither proves that the full cosmological trajectory is stable. The de Sitter stability conditions and scalaron mass are reviewed in De Felice and Tsujikawa 2010, §§ 4.1 and 5.1, Eqs. (4.79)–(4.80) and (5.2), PDF.

General-relativistic and dimensional checks

Set $f(R)=R-2\Lambda$. Then $F=1$ and $\dot F=0$, so the equations reduce to

$$3H^2=8\pi G\rho+\Lambda, \qquad -2\dot H=8\pi G(\rho+p).$$

This is the strongest elementary sign check: the term $(FR-f)/2$ must become $+\Lambda$. Dimensions provide another. In units with $c=1$, $R$, $H^2$, and $8\pi G\rho$ all have dimension $L^{-2}$, while $F$ is dimensionless when $f$ has the dimension of $R$. Finally, inserting the two modified Friedmann equations into the differentiated constraint verifies the contracted Bianchi identity and catches an incorrect sign in the $H\dot F$ term.

The example also shows a general lesson. A background equation is not a complete cosmological theory. One must still count perturbative degrees of freedom, test their kinetic and gradient terms, specify the valid cutoff, and state which solution branch is followed.

What an expansion history can and cannot establish

Suppose an observer supplies a smooth function $H(a)$. The first modified Friedmann equation can then be read as a differential reconstruction equation for $f(R)$ along that particular history, provided $R(a)$ is monotonic on the interval of interest. That reconstruction does not select a unique fundamental theory. Different functions can agree over the sampled curvature range, and an added homogeneous solution can change the off-trajectory behavior. The same $H(a)$ may also be produced by a scalar–tensor model or by general relativity with a suitable material stress tensor.

The background nevertheless performs essential consistency work. It fixes the coefficients that enter the quadratic perturbation action, locates zeros of $F$, and identifies epochs at which a chosen branch ceases to be invertible. A model that crosses $F=0$ cannot be rescued by assigning the divergent terms to an “effective fluid”; the original gravitational kinetic coefficient has degenerated. Conversely, a smooth background with $F\gt0$ is only a necessary condition. A negative scalar gradient coefficient, a tensor instability, or a cutoff below the scales being evolved can still invalidate it.

For that reason, background calculations should be reported with the defining action, matter coupling, branch, initial constraint, and general-relativistic limit—not just with an effective equation-of-state curve. Those items are sufficient for another reader to reproduce the reduction and to carry its result into the perturbation analysis without guessing which theory was meant.

Exercises

For $f(R)=R+\alpha R^2$ in vacuum, find all constant-curvature solutions and evaluate the scalaron mass about the Minkowski branch.

Solution

Here $F=1+2\alpha R$. The fixed-point condition is

$$(1+2\alpha R)R-2(R+\alpha R^2)=-R=0,$$

so the only finite constant-curvature vacuum is $R=0$. For $\alpha\ne0$, $f_{RR}=2\alpha$ and the mass formula gives

$$m_F^2=\frac{1}{6\alpha}.$$

Thus the Minkowski scalar mode is non-tachyonic for $\alpha\gt0$. At $\alpha=0$ the theory is GR and the propagating scalar is absent; one must not divide by $f_{RR}=0$ and interpret an infinite mass as an extra physical mode. This conclusion is local and does not by itself establish global stability or the validity of the quadratic truncation at arbitrarily high curvature.

References

• De Felice, Antonio, and Shinji Tsujikawa. “$f(R)$ Theories.” Living Reviews in Relativity 13, 3 (2010). DOI. Open PDF
• Sotiriou, Thomas P., and Valerio Faraoni. “$f(R)$ Theories of Gravity.” Reviews of Modern Physics 82, 451–497 (2010). DOI. Open PDF