The age of the universe with vacuum energy

The key equation from Lecture 18 is the second Friedmann equation, in which pressure appears alongside mass density:

Guth stresses a point that is easy to misread. The pressure here has no mechanical effect at all. It is the same everywhere, and a pressure only pushes things when it is larger on one side than the other. Its effect is purely gravitational: in Einstein's equations, pressure is itself a source of gravity. Positive pressure attracts; negative pressure repels.

The vacuum's energy density cannot change with time — the vacuum is the lowest energy density the laws of physics allow, and as far as we know those laws do not change. That alone fixes its pressure:

Splitting density and pressure into a normal part and a vacuum part, $\rho = \rho_n + \rho_\text{vac}$ and $p = p_n + p_\text{vac}$, the vacuum pressure disappears from the equations because it is known in terms of $\rho_\text{vac}$:

In the first equation a negative times a negative gives a positive acceleration.

Every term on the right of the first-order equation falls as $a$ grows — except $\rho_\text{vac}$, which is constant. The only way a non-zero vacuum energy can fail to take over is if a closed universe recollapses first. Otherwise, eventually $H^2$ is a constant:

Guth says our universe is approaching such an exponentially expanding phase now; we are not there yet.

A student asks whether this means the universe simply expands forever into nothingness. Yes: ordinary matter thins out to nothing while the vacuum energy stays. There is one twist. Our vacuum might be a false vacuum — one that behaves like a vacuum for a long time but can decay. Then most of future spacetime would still expand exponentially, but decays would occur here and there, punching growing spherical holes into the background: a "Swiss cheese" universe. Guth promises to return to this later in the course.

The first calculation of the day: the age of the universe in terms of quantities astronomers can measure, now that vacuum energy is one of the ingredients alongside non-relativistic matter and radiation. The strategy is simple. We know how each term in the Friedmann equation depends on time, and we can measure each term today; so we integrate backwards until the scale factor reaches zero — the Big Bang.

Start from

with the known time dependence of each density (a subscript 0 always means "today"):

The constants are arranged so that each density equals its present value at $t = t_0$.

Switching to Ω's

Cosmologists prefer to quote densities as fractions of the critical density — the total density that makes $k = 0$ and space flat. For any ingredient $x$ (matter, radiation or vacuum):

so that, for example, $\rho_m(t) = \frac{3H_0^2}{8\pi G}\left[\frac{a(t_0)}{a(t)}\right]^3\Omega_{m,0}$, with the fourth power for radiation and no power for the vacuum. To make the curvature term look like the others, define

and, because the ratio $a(t_0)/a(t)$ keeps coming back, name its inverse:

Guth notes that $x$ is exactly what Barbara Ryden's textbook calls the scale factor, since she normalises $a$ to 1 today.

The Friedmann equation in one line

With an abbreviation $F(x)$ that Guth uses only for this lecture, the Friedmann equation becomes

[On the board Guth first wrote $x^2$ in the denominator and corrected it to $x^4$ on p. 8.] The powers of $x$ in $F$ look unfamiliar because $x^4$ has been factored out, but their differences are right: matter falls one power of $x$ more slowly than radiation, and so on.

Now put $t = t_0$, i.e. $x = 1$. The left side is $H_0^2$, which cancels the $H_0^2$ on the right, and every power of $x$ becomes 1:

This is the simplest way to think of $\Omega_{k,0}$: it is $1$ minus all the real ingredients. In this language the total $\Omega$ is always 1, and whatever is not in matter, radiation or vacuum energy is assigned to curvature.

Take the square root of the Friedmann equation. Since $a(t_0)$ is a constant, $\dot a/a = \dot x/x$:

Adding up all the time intervals from the Big Bang ($a = 0$, so $x = 0$) to today ($x = 1$) gives the age:

Integrating only up to some $x < 1$ gives the age of the universe when the scale factor had that value.

Guth calls this the state-of-the-art formula: its only ingredients are $H_0$ and the matter, radiation and vacuum Ω's (curvature follows from the sum rule), and it is exactly what the Planck team used to announce the age of the universe. [He quotes their result as 13.9 billion years here; the 2013 Planck value was 13.8 billion years, close to the "13.7, 13.8" he gives a few minutes later.]

In general the integral has to be done numerically, and that is how it is usually done. Two special cases can be done by hand. One is matter plus curvature with no vacuum energy — the closed and open universes of Lectures 9 and 10. The other is a flat universe with only matter and vacuum energy: $\Omega_{\text{rad},0} = \Omega_{k,0} = 0$, so $\Omega_{m,0} + \Omega_{\text{vac},0} = 1$. Guth gives the answer without deriving it; it splits into three cases:

The first case, with negative vacuum energy, is not our universe, but the calculation can be done. The middle case is the familiar flat matter-dominated universe. The last case is, to a good approximation, our universe: curvature is unmeasurably small, and radiation matters only near the radiation-dominated era. Although the answer has to be written in three pieces, it is one smooth analytic function across all of them, just like the age of a matter-only universe.

This last form is not in the lecture: it is the same integral taken only up to $x$, which Guth points out can be done; at $x = 1$ it reduces to the $\tanh^{-1}$ form above.

Guth recalls the matter-only result (slide 7, the age of open, closed and flat universes against $H_0$). For a Hubble constant of 67–70 km/s/Mpc and any $\Omega$ between about 0.1 and 1, the ages come out of order 10 billion years — not old enough for the oldest stars — and only about 9 billion years if $\Omega = 1$.

Adding vacuum energy changes everything. Slide 8 shows the age against $H_0$ for flat universes with matter and vacuum energy, for several values of $\Omega_m$. Ryden's benchmark model and the favoured point from the WMAP seven-year data lie almost on top of each other, and Guth adds that the Planck point (not yet on the posted slide) (just under 70 km/s/Mpc, about 0.7 vacuum energy and 0.3 matter) lies there too. The age is 13.7–13.8 billion years, perfectly consistent with the oldest stars. An age problem that troubled cosmology for many years vanishes once dark energy is included.

A student notices that Guth said "dark energy". The terms are not quite synonyms:

• Vacuum energy has a precise meaning: the energy of the vacuum, which by definition never changes.
• Dark energy is the more ambiguous name for whatever is driving the acceleration, since we don't know for sure what it is. The other conceivable candidate is a very slowly evolving scalar field, of the same type as the field that drives inflation, but at a much lower energy and changing much more slowly. Observers are working hard to tell the two apart.

As of 2013, no time variation had been found: everything known was consistent with pure vacuum energy.

Other questions: Is the amount of dark energy related to the amount of dark matter? Not as far as anyone knows — the two differ by a factor of roughly two and a half, with no known connection. Do we know what dark energy is? Probably, Guth says: there is a good chance it is vacuum energy, and its phenomenology is understood completely. But what gives the vacuum its energy, and especially how large that energy should be, is another matter: there, he says, we are totally clueless — a subject he leaves for the end of the lecture or the next one.

• Pressure has no mechanical effect when it is uniform, but it is a source of gravity; the vacuum's negative pressure gives a repulsion, so $\ddot a$ gains $+\frac{8\pi}{3}G\rho_\text{vac}a$, and vacuum energy eventually dominates, with $a \propto e^{H_\text{vac}t}$.
• With $\Omega_x = \rho_x/\rho_c$, $\Omega_{k,0} = -kc^2/(a^2(t_0)H_0^2)$ and $x = a/a(t_0)$, the Friedmann equation reads $H^2 = H_0^2F(x)/x^4$, and the Ω's add up to 1.
• The age is $t_0 = \frac{1}{H_0}\int_0^1 x\,dx/\sqrt{F(x)}$ — the formula behind the Planck age. It is usually integrated numerically.
• For a flat universe with matter and vacuum energy, $t_0 = \frac{2}{3H_0}\tanh^{-1}\!\sqrt{1-\Omega_{m,0}}\,/\sqrt{1-\Omega_{m,0}}$; with $\Omega_m \approx 0.3$ this gives about 13.7–13.8 billion years, and the age problem disappears.
• "Dark energy" is whatever drives the acceleration; "vacuum energy" is one specific, constant candidate, consistent with all data so far.