In string theory low-energy $n$-dimensional gravity is described by an action of the following form: $$S^{(n)}=\frac{1}{2\kappa^{(n)}}\int d^nx\sqrt{-G}e^{-2\Phi}\Big(\mathcal{R}+4\partial_\mu\Phi\partial^\mu\Phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\Big)\tag{1}$$ where $H_{\mu\nu\rho}=\partial_\mu B_{\nu\rho}+\partial_\nu B_{\rho\mu}+\partial_\rho B_{\mu\nu}$ (see David Tong's Lectures on String Theory ch.$7$ p.$168$ eq.$7.16$, using natural units $\hbar=c=1$ and metric signature $-$,$+$,$+$,$+$).

There are three massless bosonic fields: the space-time metric $G_{\mu\nu}$, the anti-symmetric tensor field $B_{\mu\nu}$ and the scalar dilaton field $\Phi$.

The string coupling parameter is given dynamically by the dilaton field: $$g_s=e^\Phi.\tag{2}$$ To a first approximation at tree level (see David Tong's Lectures on String Theory ch.6 p.128) the strength of the gravitational coupling between strings is given by $$\phi=\frac{1}{g_s}.\tag{3}$$ Substituting Eqns.$(2,3)$ into Eqn.$(1)$, performing a dimensional reduction to $4$-d, ignoring the $B_{\mu\nu}$ field and adding a cosmological constant $\Lambda$, we obtain the following $4$-d action $$S^{(4)}=M^2_{Pl}\int d^4x\sqrt{-g}\Big(\frac{\phi^2\mathcal{R}}{2}+2\partial_\mu\phi\partial^\mu\phi-\Lambda\Big),\tag{4}$$ where the reduced Planck mass $M_{Pl}=1/\sqrt{8\pi G^{(4)}}$. By setting $\phi=1$ one recovers the Einstein-Hilbert action with a cosmological constant $\Lambda$. It is interesting to note that the $\Lambda$ term breaks the scale-invariance symmetry of the action in Eqn.$(4)$ allowing for time evolution of the universe.

In order to find a cosmological solution we use the FLRW metric in a generalized form whose line element is given by $$ds^2=-dt^2+a^2(t)\gamma_{ij}(u)du^i du^j\tag{4.5}$$ where $t$ is the timelike coordinate, $a(t)$ is the scale factor and $\gamma_{ij}$ is the metric on a maximally symmetric $3$-space with coordinates $u^1,u^2,u^3$.

We wish to find the cosmological equations of motion of the action in Eqn.$(4)$. In order to do this we first transform to a time coordinate $\eta$ with an arbitrary lapse function using $dt=N(\eta)d\eta$. The equations of motion can be derived from generalized Euler-Lagrange equations, using time $\eta$, for the dilaton field $\phi,\phi'$ $(5)$, lapse function $N,N'$ $(6)$ and scale factor $a,a',a''$ $(7)$. After transforming back to time $t$ the equations of motion are: \begin{eqnarray} \ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}+\frac{3}{2}\phi\Big(\frac{\ddot{a}}{a}+\frac{\dot{a}^2}{a^2}+\frac{k}{a^2}\Big) &=& 0,\tag{5}\\ \phi^2\Big(\frac{\dot{a}^2}{a^2}+\frac{k}{a^2}\Big)+\frac{2}{3}\dot{\phi}^2+2\frac{\dot{a}}{a}\dot{\phi}\phi+\frac{\Lambda}{3} &=& 0,\tag{6}\\ \phi^2\Big(2\frac{\ddot{a}}{a}+\frac{\dot{a}^2}{a^2}+\frac{k}{a^2}\Big)+4\frac{\dot{a}}{a}\dot{\phi}\phi+2\ddot{\phi}\phi+\Lambda &=& 0.\tag{7} \end{eqnarray}

Scale change leads to effective universe expansion

Now I make the assumption that the expansion of the universe is entirely due to the increase in the dilaton field $\phi(t)$ so that we have $$\phi(t)=a(t).\tag{8}$$ As $\phi(t)$ increases the masses of all particles increase and therefore their sizes decrease according to the uncertainty principle of quantum mechanics. This decrease of the length of rulers leads to an effective increase in the scale factor $a(t)$.

By substituting Eqn.$(8)$ into Eqns.$(5,6,7)$ we find a unique vacuum solution given by: \begin{eqnarray} a(t) &=& H_0\ t,\tag{9}\\ k &=& -3H_0^2,\tag{10}\\ \Lambda &=& -2H_0^2\tag{11}\\ \end{eqnarray} where $H_0$ is the Hubble constant.

The negative $k$ and $\Lambda$ terms could be hard to interpret. However I assume that what is important is the "kinetic" energy density given by the term $M_{Pl}^2\ \dot{a}^2/a^2$ so that: $$\rho_{k,\Lambda}(t)=\frac{M_{Pl}^2}{t^2}.\tag{12}$$

Thus we obtain a spatially open universe that only seems to be expanding because particle sizes are decreasing. The cosmological redshift is then explained by the increase of energies of present-day absorbing atoms relative to past emitting atoms.

In fact the effective linear expansion of space given by $a(t)=H_0\ t$ for an empty universe with a cosmological constant $\Lambda$ is a very good fit to the cosmological expansion data for the actual universe according to researchers like Fulvio Melia (see observational support for his $R_h=ct$ model). It is certainly true that the linearly expanding vacuum solution of the present model is much better "behaved" than the exponentially expanding vacuum solution of standard general relativity.

The energy scale $E_{k,\Lambda}$ for the kinetic energy density $\rho_{k,\Lambda}$, given in Eqn.$(12)$, at the present time $t_0=1/H_0$ is approximately

$$E_{k,\Lambda}(t_0) = \rho^{1/4}_{k,\Lambda}(t_0) = H^{1/2}_0 M^{1/2}_{Pl},\tag{12.1}$$

$$\log{E_{k,\Lambda}(t_0)} = \frac{1}{2}(\log{H_0}+\log{M_{Pl}}).\tag{12.2}$$

Therefore the kinetic energy $E_{k,\Lambda}(t_0)$ of a model universe with a cosmological constant is halfway between the Hubble constant and the Planck mass on a logarithmic scale in accordance with the following diagram by David Tong in his Quantum Field Theory Lectures, ch.$0$, p.$5$.

energy scales

Possible experimental test

Consider two charged massive spheres both with mass $m$ and charge $q$ such that their gravitational attraction is balanced by their electrostatic repulsion. diag

Thus, using $8\pi G=M^{-2}_{Pl}$, we have $$\frac{q^2}{4\pi\epsilon_0 d^2}=\frac{G m^2}{d^2}=\frac{m^2/M^2_{Pl}}{8\pi d^2},\tag{12}$$ $$\frac{2q^2}{\epsilon_0}=\frac{m^2}{M^2_{Pl}}.\tag{13}$$ The balance of forces only depends on a ratio of masses which does not change as the dilaton field $\phi(t)$ increases. Therefore the separation $d$ between the centers of the spheres does not change. But as $\phi(t)$ increases the size of the atoms in the spheres decrease so that the radii $r$ decrease. Let us assume that we measure the distance $x=d-2r$ between the spheres by comparing it with $r$ so that we have $$\frac{x}{r}=\frac{d}{r}-2.\tag{14}$$ If the increasing dilaton field $\phi(t)$ causes masses to increase and length scales to decrease then our measurement of the separation distance ratio $x/r$ should increase. This effect should occur for a table-top experiment with a fixed large mass and a small mass on a frictionless track. The fractional change in the effective separation distance would be roughly $10^{-10}$ per year so that it would have to be measured using laser interferometry.

  • $\begingroup$ You might be interested in a similar proposal physics.stackexchange.com/questions/620794/… it predicts that the observed matter density would be between 0.25 and 0.333 as observed, also physics.stackexchange.com/questions/634875/… $\endgroup$ Oct 30, 2021 at 10:19
  • $\begingroup$ @JohnHunter In considering changes of scale I think one should use natural units such that $c$ and $\hbar$ are set to dimensionless unity. By doing this you ensure that your theory automatically obeys the fundamental principles of special relativity and quantum mechanics. $\endgroup$ Oct 30, 2021 at 22:08
  • $\begingroup$ @ John Eastmond , thankyou, it's late here now, but I'll have a think about your suggestion. $\endgroup$ Oct 31, 2021 at 0:28
  • $\begingroup$ @JohnHunter For instance you want time/space, energy/time frequency and momentum/spatial frequency to be equivalent. Setting $\hbar=c=1$ ensures that automatically. $\endgroup$ Oct 31, 2021 at 11:14
  • 1
    $\begingroup$ I don't see a question here, just a presentation of a (possibly non-mainstream) theory. What exactly do you want to know here? $\endgroup$
    – ACuriousMind
    Nov 13, 2021 at 12:34

1 Answer 1


There's nothing to explain. The time evolution of $a(t)$ isn't a mystery; it's predicted by general relativity if you assume that the nongravitational content of the universe is a certain mixture of matter and radiation and a cosmological constant.

The spacetime geometries describable with FLRW coordinates are just the geometries that satisfy the cosmological principle (homogeneity and isotropy). The shape is related to the matter content by the GR field equation, as it would be in any other spacetime. $a(t)$ is just a way of describing the shape, and shouldn't be taken as evidence that there's anything else going on. On the contrary, if you introduced another field to explain it, you would have to explain why the matter that we know to be present doesn't have the effect that GR says it should.

You might be able to get rid of the cosmological constant, and maybe some of the dark matter, if you were successful, but fitting all the existing experimental data is a very high bar.

  • $\begingroup$ It's true my model is just a vacuum solution. I'm arguing that it is quite realistic as the effect of the vacuum swamps the influence of matter and radiation. $\endgroup$ Oct 30, 2021 at 22:25
  • $\begingroup$ The effective linear expansion of space given by $a(t)=H_0\ t$ is actually a very good fit to the cosmological expansion data according to researchers like Fulvio Melia (see support for his $R_h=ct$ model in arxiv.org/abs/1608.02070). $\endgroup$ Oct 30, 2021 at 22:30

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