9 Inflation: Implementation with a Scalar Field

Recall that we’re looking for an equation of state such that , and that in Section 6 we showed that this is possible using a scalar field where the potential $V(\phi )$ dominates over the (unfortunately-named) “kinetic” term ${\dot{\phi }}^2/2$.

Is it possible to sustain a scalar field in such a state? To start answering that question, we need the equation of motion for the field itself. By applying the Euler-Lagrange equations (232) to the action (229), we found Eq. (234), which can be rewritten as:

$$g^{\mu \nu }{\nabla }_{\mu }({\partial }_{\nu }\phi )+\frac{\mathrm{d}V}{\mathrm{d}\phi }=0.$$

(249)

Using the Christoffel symbols (5) for the flat FRW cosmology, the equation of motion for the homogeneous case where ${\partial }_i\phi =0$ can be expanded as:

$$\boxed{\ddot{\phi }+3H\dot{\phi }+V^{\prime }=0},$$

(250)

where we define $\dot{\phi }\equiv \mathrm{d}\phi /\mathrm{d}t$ and $V^{\prime }\equiv \mathrm{d}V/\mathrm{d}\phi$ (note in particular that the $V$ derivative is with respect to the field value, not with respect to time).

☞ Exercise 9A Check that you can derive Eq. (250) starting from the general equation of motion (234). You will need to use the Christoffel symbols (5) for the flat FRW metric, and the covariant derivative for covectors is given by (71). By changing the independent variable to the conformal time $\eta$, show that the equation of motion can be written in the following useful form: $$\frac{{\mathrm{d}}^2\phi }{\mathrm{d}{\eta }^2}+2aH\frac{\mathrm{d}\phi }{\mathrm{d}\eta }+a^2{V}^{\prime }=0.$$ (251)

Returning to equation (250); in a non-expanding spacetime, $H=0$, we correctly recover (226) for the homogeneous case when ${\nabla }^2=0$. Once expansion is included, the additional term looks just like a friction term if we temporarily imagine $\phi$ to represent the coordinates of a ball rolling down a hill defined by height $V(\phi )$. As mentioned in Section 2, this is just an analogy, and is only valid in the homogeneous limit. Nonetheless thinking of the field value as the physical position of a ball and $V(\phi )$ as its height will later help provide intuition for the behaviour of the solution.

Finally, we also have the Einstein field equations (from the variation with respect to the metric) which gives us the Friedmann equation:

$$\boxed{H^2=\frac{8\pi G}{3}\left[\frac{1}{2}{\dot{\phi }}^2+V(\phi )\right]}.$$

(252)

One could also derive the acceleration equation from the space-space part of the Einstein equation, but remember this would turn out to be redundant because the equations of motion, conservation of energy-momentum and the Einstein equations are all interlinked. (We discussed this before in the context of the behaviour of perfect fluid universes.)

One way to get inflation is to trap a field in a false vacuum (left panel of Figure 11), i.e. $V(\phi )\ne 0$ but $\dot{\phi }=0$. The energy density is constant since $\phi$ is a constant. Consequently,

$$\frac{\dot{a}}{a}=\sqrt{\frac{8\pi G\rho }{3}}=\mathrm{const}.$$

(253)

We obtain exponential expansion with $H\propto {\rho }^{1/2}=\mathrm{const}$. If the field is trapped for $60$ $e$-foldings, $H(t_e-t_b)>60$, the horizon problem is solved (see Section 2). Eventually the field quantum tunnels out of the false vacuum and inflation ends with a period of “reheating” where the residual energy in the field is converted into particles through couplings to the fields responsible for dark and standard-model matter. (The terminology is a bit unhelpful but refers to the idea that the energy is being turned into a thermal bath of particles – so “heating” – and the universe may have contained such a thermal bath before inflation started too – so “reheating”).

This overall scenario is known as “old inflation”, and is the original way that Alan Guth formulated the idea in a seminal 1981 paper. Sadly it was later realised that his idea doesn’t quite work because localized regions tunnelling into the true vacuum get overwhelmed by the bulk of the spacetime which continues inflating. Even if small bubbles of true vacuum form, they can never coalesce in order for the universe as a whole to move to the true vacuum.

In “New Inflation”, Andrei Linde, Paul Steinhardt and Andreas Albrecht postulated that the scalar field instead rolls slowly toward the true vacuum in a nearly flat potential (right panel of Figure 11). If the potential is flat enough, the friction term $3H\dot{\phi }$ in equation (250) can keep $\dot{\phi }$ – and hence the kinetic energy – very small indeed. This means that the potential energy continues to dominate and inflation goes ahead. Eventually the field reaches the bottom of the hill and reheating occurs in the same way as in old inflation.

There is a closely related version of inflation called “chaotic inflation”. This is motivated by asking why the field would have started near $\phi =0$ in the new inflation scenario. More plausibly, $\phi$ should start from random values of order the Planck scale (giving the term “chaotic”). The resulting differences for our purposes are very minor, however; in any patch of the universe, one ends up with a slow-roll solution just like in the new inflation scenario (although perhaps approaching the minimum of $V(\phi )$ from the right-hand-side instead of the left-hand-side as depicted in Figure 11). We won’t need to worry about this distinction in the present course.

In the following discussion, for compatibility with the professional inflation literature we will set $8\pi G\equiv M_{\mathrm{Pl}}^{-2}\equiv 1$.

Equation (248) shows that $w=-1$ inflation occurs if the field is evolving slowly enough that the potential energy dominates over the kinetic energy (this is known as the “slow-roll” limit):

$${\dot{\phi }}^2\ll V(\phi )$$

(254)

However, that’s not quite enough on its own. You also want the second time derivative of $\phi$ to be small enough to allow this slow-roll condition to be maintained for a sufficient time period. Thus, inflation also requires

$$|\ddot{\phi }|\ll |3H\dot{\phi }|\text{ and }|\ddot{\phi }|\ll |V^{\prime }|.$$

(255)

Satisfying these conditions requires the smallness of two dimensionless quantities known as slow-roll parameters. We start by defining these as follows:

	${ϵ}_V(\phi )$	$\equiv$	${\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{V^{\prime }}{V}}\right)}^2$		(256)
	${\eta }_V(\phi )$	$\equiv$	${\displaystyle \frac{V^{\prime \prime }}{V}},$		(257)

where $V^{\prime \prime }\equiv {\mathrm{d}}^{2}V/\mathrm{d}{\phi }^2$. (Just to be clear: ${\eta }_V$ is not at all related to the conformal time $\eta$. Sometimes you will see slow-roll parameters written as just $ϵ$ and $\eta$; in this course I’ll try hard always to append the subscript $V$’s to prevent any ambiguity.) Typically, the slow-roll regime is stated as being the condition that

$$\boxed{{ϵ}_V,|{\eta }_V|\ll 1}\text{.}$$

(258)

Unfortunately this packs away the original motivation, equations (254) and (255), behind the scenes. The condition we’ve written down is about the potential $V(\phi )$ only – not directly about the dynamical solution $\phi (t)$. For that reason conditions (258) are sometimes known as the flatness conditions (referring to the shape of the potential, not the shape of the universe) and ${ϵ}_V$ and ${\eta }_V$ as flatness parameters. Now we need to see what the relevance of ${ϵ}_V$ and ${\eta }_V$ are to a solution for $\phi (t)$.

Here’s the plan: we’ll show that if a slow-roll solution exists in a given potential, then the slow-roll parameters in the relevant part of that potential must indeed be small. From equation (255) and (250) we have

$$3H\dot{\phi }=-V^{\prime }\text{.}$$

(259)

We can also take the Friedmann equation, (252), in tandem with the condition (254), to derive the slow-roll relation

$$3H^2=V(\phi )\text{.}$$

(260)

Rearranging the equation (259), we find that

$${\dot{\phi }}^2={\left(\frac{V^{\prime }}{3H}\right)}^2=\frac{V^{\prime 2}}{3V}\text{,}$$

(261)

where for the second equality I’ve used equation (260). Using (254) one last time tells us that

$$\frac{{\dot{\phi }}^2}{V(\phi )}\ll 1\Rightarrow \frac{V^{\prime 2}}{3V^2}=\frac{2{ϵ}_V}{3}\ll 1$$

(262)

and we have shown that ${ϵ}_V\ll 1$ must hold, just as we wanted to. Next, let’s get an equivalent result for ${\eta }_V$.

📝 Exercise 9B Show that the existence of a slow-roll solution implies that $|{\eta }_V|\ll 1$. Hint: Start by showing that $\dot{H}=V^{\prime }\dot{\phi }/6H$. Then expand $\ddot{\phi }/V^{\prime }$ in terms of the potential alone by repeatedly making use of equations (259) and (260) and using your expression for $\dot{H}$ when needed. Finally note that $\left|\ddot{\phi }/V^{\prime }\right|\ll 1$ is the slow-roll condition (255), and so prove the required result.

If the “flatness conditions” (${ϵ}_V\ll 1$, ${\eta }_V\ll 1$) are valid, then slow-roll inflation is possible. However, we have not shown the opposite – i.e. we have not shown that slow-roll inflation necessarily occurs when a potential obeys the flatness conditions.

It does turn out that in such a potential, slow-roll is an attractor solution: you can find solutions that at a given instant of time do not obey slow-roll, but they quickly revert back to the slow-roll solution. It’s easy to understand why from our “ball on the hill” analogy; slow-roll corresponds to the friction-limited regime where the ball is rolling at its terminal velocity. Start it faster, and friction will slow it down. Start it slower, and gravity will speed it up.

To show more formally that slow-roll is an attractor is a bit fiddly and is usually demonstrated via the Hamilton-Jacobi formalism. We’ll not need it in this course – what we really care about is what we did show formally: to reiterate, we can’t have slow-roll inflation if we don’t have flatness in the potential. Because of this link, the flatness conditions are sometimes also called the slow-roll conditions.

Slow-roll inflation must end when the flatness conditions (258) are violated:

$${ϵ}_V({\phi }_{\mathrm{end}})\approx 1.$$

(263)

The number of $e$-folds before inflation ends is

	$N(\phi )$	$\equiv$	$\ln {\displaystyle \frac{a_{\mathrm{end}}}{a_{\mathrm{start}}}}$		(264)
		$=$	${\displaystyle {\int }_{a_{\mathrm{start}}}^{a_{\mathrm{end}}}}{\displaystyle \frac{1}{a}}da={\displaystyle {\int }_{t_{\mathrm{start}}}^{t_{\mathrm{end}}}}Hdt$
		$\approx$	${\displaystyle {\int }_{{\phi }_{\mathrm{end}}}^{{\phi }_{\mathrm{start}}}}{\displaystyle \frac{V}{V^{\prime }}}d\phi .$

This can be evaluated for any given potential, giving us a handle on the number of $e$-folds supported. Combined with the minimal required $N$ that you derived in Exercise 2, we have derived another constraint on the admissable types of potential $V(\phi )$.

☞ Exercise 9C Consider inflation in a quadratic potential $V(\phi )=m^2{\phi }^2/2$. If ${\phi }_{\mathrm{end}}=0$, what value must ${\phi }_{\mathrm{start}}$ take to achieve the minimal $N=60$ $e$-folds? What about if one sets ${\phi }_{\mathrm{end}}$ using the condition (263)?

• •

  Inflation with $P=-\rho$ (i.e. $w=-1$) can be achieved using a scalar field $\phi$ where the potential energy $V(\phi )$ dominates over the kinetic energy ${\dot{\phi }}^2/2$ (see also Section 8);

• •

  But one must somehow keep the potential dominant for a sufficiently long period;

• •

  “Old” inflation achieved this by trapping the scalar field in a false vacuum state, so that inflation would only end when quantum tunnelling finds the true vacuum;

• •

  However, this picture is problematic because the random tunnelling process only creates tiny bubbles of non-inflating spacetime, whereas today’s whole observable Universe must have stopped inflating roughly simultaneously;

• •

  “New” and “chaotic” inflation use instead a classical slow-roll trajectory where the inflaton field gradually moves down towards the true vacuum (in such a way that ${\dot{\phi }}^2$ remains small);

• •

  A necessary requirement on the potential $V(\phi )$ to permit such a regime is that the two flatness parameters are small, ${ϵ}_V\ll 1$ and $|{\eta }_V|\ll 1$ – these relate to the first and second derivatives of the potential and are defined by equation (256) and (257);

• •

  Slow-roll inflation must break down somewhat before inflation ends, because $V^{\prime }$ will become small so ${ϵ}_V$ will become large – this is the start of the ‘reheating’ epoch where the energy in the inflaton field is transferred into the standard model (and dark matter) sector;

• •

  This break-down point is defined by ${ϵ}_V({\phi }_{\mathrm{end}})\simeq 1$; if we also somehow know a starting field value ${\phi }_{\mathrm{start}}$, the number of $e$-folds of slow-roll inflation can be calculated directly from the shape of the potential.