Classical universes of the no-boundary quantum state

We analyze the origin of the quasiclassical realm from the no-boundary proposal for the Universe’s quantum state in a class of minisuperspace models. The models assume homogeneous, isotropic, closed spacetime geometries, a single scalar field moving in a quadratic potential, and a fundamental cosmological constant. The allowed classical histories and their probabilities are calculated to leading semiclassical order. For the most realistic range of parameters analyzed, we find that a minimum amount of scalar field is required, if there is any at all, in order for the Universe to behave classically at late times. If the classical late time histories are extended back, they may be singular or bounce at a finite radius. The ensemble of classical histories is time symmetric although individual histories are generally not. The no-boundary proposal selects inflationary histories, but the measure on the classical solutions it provides is heavily biased towards small amounts of inflation. However, the probability for a large number of e-foldings is enhanced by the volume factor needed to obtain the probability for what we observe in our past light cone, given our present age. Our results emphasize that it is the quantum state of the Universe that determines whether or not it exhibits a quasiclassical realm and what histories are possible or probable within that realm.

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Figure 1

The complex solutions that provide the steepest descents approximation to the NBWF are found by integrating the field equations along a broken contour $C_B(X)$ in the complex $\tau$ plane. In order for the solutions to behave classically at late times, one ought to tune the tangent $\gamma$ of the phase of $\varphi$ at the South Pole and the turning point $X$ of the contour. We show the tangent $\gamma$ (left) and the turning point $X$ (right) here as a function of the absolute value ${\varphi }_0$ of $\varphi$ at the South Pole. There is a qualitative difference between $\mu <3/2$ models on the one hand where $\gamma$ remains finite for all ${\varphi }_0$ (dotted curve), and $\mu >3/2$ models on the other hand where $\gamma$ diverges at a critical value ${\varphi }_0^c$ (remaining curves). In the latter case there is no combination $(X,\gamma )$ for which the classicality conditions at a large scale factor hold when ${\varphi }_0<{\varphi }_0^c$: the ensemble of possible classical histories is restricted to a bounded surface in phase space. The right panel shows the critical value ${\varphi }_0^c$ increases slightly with $\mu$, for fixed $m$, and tends to 1.27 as $\Lambda \to 0$, independently of the value of $m^2$. From top to bottom, the different curves show $\gamma$ and $X$ for $\mu =3/4$, $33/20$, $9/4$, 3, and $\gamma$ for $m^2=0.05$, $\Lambda =0$ in the solid curve in the left panel.

Figure 2

The real and imaginary part of the scalar field $\varphi$ for a typical complex solution that provides the semiclassical approximation to the NBWF. This solution has $\mu =3/4$ and ${\varphi }_0=2$. It is shown along a broken contour $C_B(X)$ in the complex $\tau =(x,y)$ plane that runs first along the $x$ axis from the South Pole at $x=0$ to a value $x=X$, and then vertically in the $y$ direction. The turning point $X$ is the largest value of $x$ plotted in the left-hand two figures. It and the imaginary part of $\varphi$ at the SP are determined by the requirement that the imaginary part of the action becomes constant with increasing $y$ (cf. Fig. 1). This is necessary for classicality at late times and it implies that the imaginary part of $\varphi$ decays rapidly to zero with increasing $y$.

Figure 3

The real and imaginary part of the scale factor for the same complex solution as in Fig. 2, along the same broken contour $C_B(X)$ in the complex $\tau$ plane. The imaginary part of $a(\tau )$ rapidly decays to zero along the $y$ axis as a consequence of the classicality conditions, whereas the real part grows exponentially for some time.

Figure 4

Left panel—The ratio of the gradient squared of the real to the imaginary part of the action plotted along the $y$ axis, for the complex solution that behaves classically at large scale factor, with $\mu =3/4$ and ${\varphi }_0=2$. For $y<2$ the ratio still significantly deviates from zero, not because $I_R$ varies in the $y$ direction (as can be seen in the right panel) but because the gradient of $I_R$ in the $X$ direction is not small. Right panel— The real part of the action of this complex solution rapidly stabilizes along the $y$ axis.

Figure 5

The asymptotic value of the real part of the action of the complex solutions that behave classically at large scale factor plotted as a function of ${\varphi }_0$ and for $\mu =3/4$. This determines the relative probabilities predicted by the NBWF for the corresponding classical Lorentzian histories. The upper curve shows the prediction of the perturbation theory for small $\varphi$ around the empty de Sitter space with cosmological constant $\Lambda$.

Figure 6

The asymptotic value of the real part of the action of the complex solutions that behave classically at late times plotted as a function of ${\varphi }_0$, in three $\mu >3/2$ models (left) and in a scalar field model with quadratic potential and $\Lambda =0$ (right). The values of $\mu$ and $m^2$ are as in Fig. 1. The action tends to a finite value at the lower bound ${\varphi }_0^c$ that arises from the classicality conditions, and it goes to zero as $\sim 1/{\varphi }_R^2(0)$ at large ${\varphi }_0$.

Figure 7

The no-boundary wave function predicts that all histories that behave classically at late times undergo a period of inflation at early times as shown here by the linear growth of the instantaneous Hubble constant $\widehat{h}={\widehat{a}}_{,t}/\widehat{a}$ in five representative classical histories for $\mu =3$ and for ${\varphi }_0$ between 1.3 and 4.

Figure 8

Left panel—The number of e-foldings $N$ of inflation driven by the scalar field (as opposed to the cosmological constant of the background) in the classical histories predicted by the NBWF, for five different models. From top to bottom, the different curves correspond to $\mu =3/4$, 3, $9/4$, $33/20$ and $\Lambda =0$, $m^2=0.05$. Right panel—Detail of the left panel showing the regime around the critical value ${\varphi }_0^c$ in the $\mu >3/2$ and pure scalar field models. The lower bound on ${\varphi }_0$ that arises from classicality implies a lower bound on the number of e-foldings.

Figure 9

The scale factor ${\widehat{a}}_b$ (left) and the scalar field ${\widehat{\varphi }}_b$ (right) at the bounce of the classical histories predicted by the NBWF. The values of $\mu$ and $\Lambda$ are as in Fig. 8. When $\mu <3/2$ the histories always bounce at a minimum radius in the past. By contrast for $\mu >3/2$ there is a transition from bouncing to initially singular at a critical value ${\varphi }_0^s\approx 1.5$. Above that ${\widehat{\varphi }}_b\approx {\varphi }_0$.

Figure 10

A “phase diagram” that summarizes some of the early universe properties of the Lorentzian histories predicted by the NBWF. For $\mu <3/2$ there is one classical history associated with each value of ${\varphi }_0$ and the universe bounces in the past for all ranges of ${\varphi }_0$. For $\mu >3/2$, however, there are no classical histories for ${\varphi }_0<{\varphi }_0^c$ (bottom curve). Between ${\varphi }_0^c$ and ${\varphi }_0^s$ (top curve) the Lorentzian histories have an initial singularity. Finally, for ${\varphi }_0>{\varphi }_0^s$ the Lorentzian solutions bounce at a nonzero minimum radius in the past. The limiting values of ${\varphi }_0^c$ and ${\varphi }_0^s$ when $\Lambda \to 0$, for fixed $m$, are 1.27 and 1.54, respectively.

Figure 11

The ratio of the energy density ${\rho }_b$ in the scalar field at the bounce over the vacuum energy density ${\rho }_V$, for the allowed Lorentzian histories for $\mu =3/4$ (left) and for (right, from top to bottom) $\mu =3$, $9/4$, and $33/20$. When $\mu >3/2$ there is a minimum matter density needed at early times for the history to exhibit classical behavior at late times.

Figure 12

A “phase diagram” that combines some of the early and late time properties of the Lorentzian histories predicted by the NBWF. Below the solid curve the homogeneous isotropic classical universes predicted by the NBWF recollapse to a big crunch, whereas above the curve the universes continue to expand forever.

Figure 13

The reality of the NBWF implies the ensemble of allowed classical histories is time symmetric. The individual classical bouncing histories, however, are generally not time symmetric about the bounce. A natural measure of the amount of time asymmetry of the individual histories is provided by the quantity $\eta =({\widehat{\varphi }}_{,t}{)}_b/{\widehat{\varphi }}_b$. The left panel shows $\eta$, as a function of ${\varphi }_0$, for $\mu =3/4$. In the right panel we plot $\eta$ for (from top to bottom) $\mu =33/20$, $9/4$, and 3. One sees that for $\mu >3/2$, $\eta$ diverges when ${\varphi }_0\to {\varphi }_0^s$ where ${\varphi }_0^s$ separates singular from bouncing solutions.

Figure 14

A gallery of classical Lorentzian histories. The scale factor $\widehat{a}(t)$ (left) and the scalar field $\widehat{\varphi }(t)$ (right) in the classical histories labeled by ${\varphi }_0=1.32$, for four different values of $\mu$. The value of $\mu$ decreases from top to bottom taking the values $\mu =100$, 96, $9/4$, $33/20$. If $m^2$ is fixed these correspond to increasing $\Lambda$. These are four qualitatively different cosmologies, ranging from initially singular histories that recollapse again (top) to eternally expanding universes that bounce in the past (bottom). At intermediate values of $\mu$ (2nd row) the NBWF is consistent with the standard picture of inflationary cosmology, consisting of a short period of inflation as the scalar field rolls down from high up the potential followed by an era of oscillation representing particle creation and ensuing matter domination. Eventually the cosmological constant takes over to drive a second (future-eternal) phase of exponential expansion.

Figure 15

To account for the different possible locations in the universe of the Hubble volume that contains our data, one ought to multiply the relative probabilities for classical histories coming from the NBWF (left) by a volume factor, to obtain the probability (right) for what we observe in our past light cone. The resulting volume-weighted probability distribution favors a large number of e-foldings when the ensemble is restricted to universes that last sufficiently long.

Figure 16

The complex extremizing solution $G$. $G(\tau ,\overline{\mu })$ is the complex solution of the linearized equation for the scalar field which is regular at the origin and equal to 1 there. All other field extrema are multiples of $G$. Three plots are shown for $\overline{\mu }$ equalling 0.25, 0.50, and 0.75. Each uses the NBI contour in the $\tau$ plane that extends horizontally from the origin along the real axis to $x=\pi /2$ and then vertically in the imaginary ($y$) direction. Along this particular contour the unperturbed metric makes a smooth transition between a Euclidean instanton and a Lorentzian de Sitter metric. $G$ is real along the real part of the contour but complex along the imaginary component. The real parts of $G$ are indicated by solid lines, the imaginary parts by dashed lines. The curves are in fact continuous along the contour with appropriate matching conditions for the derivatives reflecting the change in direction of the contour at $x=\pi /2$, $y=0$. (cf. Fig. 2.).

Figure 17

The complex extremizing solution $G(\tau ,\overline{\mu })$ for $\overline{\mu }$ equaling 1.25, 1.50, and 1.75. The figure is otherwise the same as Fig. 16.

Figure 18

The parameter $\gamma \equiv \mathrm{Im}({\varphi }_0)/\mathrm{Re}({\varphi }_0)$ plotted along the vertical contour $x=\pi /2$. This is the ratio required to have $\varphi$ real along this contour. On the left the curves reading from bottom to top for $\overline{\mu }$ equal to 0.2, 0.4, 0.6, and 0.8. The ratio becomes constant at large $y$. On the right the single value $\overline{\mu }=1.5$ is plotted showing the generic lack of stabilization at large $y$. (cf. Fig. 1.).

Figure 19

The first figure shows Lorentzian histories of the scalar field for various values of $\overline{\mu }$. The value of $\widehat{\chi }/{\varphi }_0$ is plotted vertically. Starting from the top on the left and moving downward the values of $\overline{\mu }$ are 0.2, 0.4, 0.6, and 0.8. The solutions are not generally time symmetric although the ensemble of Lorentzian solutions is time symmetric. The second figure shows the time-asymmetry parameter $\eta$ (6.2) as a function of $\overline{\mu }$. (cf. Fig. 13.)

Figure 20

Curves of constant real part of the action in minisuperspace. When the conditions for classicality are satisfied these will be classical Lorentzian solutions for large $b$. That will be the case for the curves on the left with $\overline{\mu }<1$, but not those on the right for which $\overline{\mu }>1$. Since the real part of the zeroth order action contributes an overall additive constant $-\pi /(2H^2)$ it is convenient to label the constant action curves by the value of $(2H^2/3\pi )I^{(2)}$, [cf. (a17)]. Reading from left to right the values of $(2H^2/3\pi )I^{(2)}$ shown are 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 for the $\overline{\mu }=0.5$ on the left. For $\overline{\mu }=1.5$ on the right the values are 0.01, 0.1, 1, 10, 100. Note that the $b$ axis starts from 1 which is the value of $b$ at the bounce.

Figure 21

Curves of the constant real part of the action in the complex $\tau$ plane. The end point values $(X,Y)$ provide a set of coordinates for minisuperspace that are alternatives to $(b,\chi )$. The unperturbed curve where $b$ is real is the vertical curve at $X=\pi /2$ along which $2H^2{I}_R/3\pi =-1/3$. Perturbations in that curve along which the perturbing matter action is constant are shown here for several values of $\overline{\mu }$. On the left are curves where $\overline{\mu }$ has the values 0.25, 0.50, and 0.75 reading up from the lowest to highest. On the right curves corresponding to 1.0 to 1.4 in steps of 0.1 reading left to right. The perturbative calculation of these curves is only valid when they remain close to the vertical line at $X=\pi /2$. For that reason no scale is indicated for the $X$ axis. For $\overline{\mu }<1$ the curves of constant $I_R$ approach classical solutions. For $\overline{\mu }>1$ the oscillation at large $y$ is an indication that they do not approach classical solutions.

Figure 22

The classicality ratios $C{l}_b$ and $C{l}_{\chi }$ plotted for $\mu =0.75$ (top pair) and $\mu =3$ (bottom pair) in minisuperspace $(b,\chi )$ where $Y\equiv {cosh}^{-1}(b)$. In lowest order perturbation theory $C{l}_b$ is proportional to ${\chi }^2$, and the value $\chi =0.1$ was used for these illustrations. $C{l}_{\chi }$ is independent of $\chi$ in leading order. The classicality conditions are well satisfied for $Y\gtrsim 2$ when $\mu =0.75$. For $\mu =3$ it is not satisfied at all because $C{l}_{\chi }$ never drops much below unity. There are no classical histories for $\mu >3/2$.

Figure 23

The real part of the action along the classical Lorentzian solution with ${\varphi }_0=1$ for several values of $\overline{\mu }$. The curves on the left reading bottom to top range from $\overline{\mu }=.3$ to $\overline{\mu }=0.9$ in steps of 0.1. On the right the range from bottom to top is $\overline{\mu }=1.1$ to $\overline{\mu }=1.7$ in steps of 0.1. The marked qualitative difference between the $\overline{\mu }<1$ and $\overline{\mu }>1$ is important for classical predictions. On the left the real part approaches a constant at large $t$. Its gradient in this direction is small compared to the gradient of $S$. The curves approach the integral curves for classical, Lorentzian solutions with probabilities proportional to $\exp (-2I_R)$. For $\overline{\mu }>1$ the real part of the action does not approach a constant, its gradient remains comparable to the gradient of $S$, and, as a consequence, classical behavior is not predicted for the scalar field.

Figure 24

The real part of the perturbative Euclidean action $I_R$ (a14) is plotted for three values of $\overline{\mu }<1$ where classical behavior is predicted. The relative probabilities for classical Lorentzian histories labeled by different values of ${\varphi }_0$ are $\exp (-2I_R)$. Reading from left to right the three values of $\overline{\mu }$ are 0.75, 0.50, and 0.25.