Gravitons and Temperature Fluctuation Correlations from Inflation

Here we consider the spatially flat Friedman-Robertson-Walker spacetime with metric

where $t$ and $\eta$ are the comoving and conformal time coordinates, respectively. We are interested in treating gravitons propagating on this background spacetime. It was shown by LifshitzLifshitz that in the transverse, tracefree gauge, which eliminates all gauge freedom, each of the two independent polarization states is equivalent to a massless, minimally coupled scalar field which satisfies

The positive norm plane wave solutions of this equation may be taken to be

where $\ell_{P}$ is the Planck length and we use the normalization for graviton modes, which is a factor of $\sqrt{16\,\pi}\,\ell_{P}$ times that for the scalar field, Here $\chi_{k}(\eta)$ satisfies

where $R(\eta)$ is the scalar curvature.

First consider the spacetime of the inflationary model, deSitter space, where we may take

where $H$ is the Hubble parameter and $\eta_{0}<\eta<\eta_{R}<0$. Here $\eta_{0}$ denotes the time when inflation began and $\eta_{R}$ that when it ends in reheating. Here the scalar curvature is $R=12\,H^{2}$. The solutions of Eq. (4) may be expressed in terms of Hankel functions as

where $|c_{2}(k)|^{2}-|c_{1}(k)|^{2}=1.$

If we set $c_{1}=0$ and $c_{2}=1$, the resulting mode function

defines the Bunch-Davies vacuum state of deSitter space BD78 . In the case of a massive scalar field, it is the unique deSitter invariant state, and is analogous to the Minkowski vacuum, the unique Lorentz invariant state in flat spacetime. However, the massless scalar field has a infrared divergent two-point function, $\langle\varphi(x)\,\varphi(x^{\prime})$, so the Bunch-Davies state is not well defined for either the massless scalar or the graviton fields. There is a class of states which are free of infrared divergences FordParker:1977b ; FV86 . If we require that $c_{1}(k)+c_{2}(k)\rightarrow 0$ as $k\rightarrow 0$, then the two-point function is infrared finite. This class may be said to be Bunch-Davies-like if we also have $c_{1}(k)\approx 0$ and $|c_{2}(k)|\approx 1$ for $k\gtrsim k_{c}$ for some small cutoff scale $k_{c}$. We can view this scale as set by some initial condition before the onset of inflation. If $k_{c}$ is sufficiently small that the associated modes have not entered the horizon since the end of inflation, then they seem to play no role in present cosmological observations. In this case, we may take Eq. (7) as the initial graviton mode before reheating.

The creation of gravitons at the end of inflation may be viewed as gravitational particle creation described by a Bogolubov transformation. Here we take the in-modes to be the Bunch-Davies-like modes given by Eqs. (3) and (7). In the subsequent radiation dominated phase, the scalar curvature vanishes, $R=0$, so the positive unit norm mode functions become

We are interested in very long wavelength gravitons modes which are far outside the horizon at the end of inflation, so $k\ll H$. These modes are not sensitive to the finite duration of reheating, so we assume a sharp transition in which the scale factor is given by Eq. (5) for $\eta<-H^{-1}$ and by

for $\eta>-H^{-1}$. Note that we take the reheating time to be $\eta_{R}=-H^{-1}$ and have set $a(\eta_{R})=1$.

The mode functions in the in and out regions are related by a Bogolubov transformation of the form

where

The Bogolubov coefficients may be found by using Eq. (10) and its first time derivative at $\eta=\eta_{R}$, using the fact that both $a(\eta)$ and $a^{\prime}(\eta)$ are continuous at this point. The results AH86 ; A88 are

and

The mean number of gravitons created in a given mode is $\langle N_{k}\rangle=|\beta_{k}|^{2}$. Here that number is very large. The energy spectrum of created gravitons is scale invariant in the sense that the graviton energy density in wave number interval $d^{3}k$ is proportional to $k\,|\beta_{k}|^{2}\,d^{3}k\propto dk/k$, which independent of the choice of scale for $k$. Note that in a spatially flat expanding universe, linear three-momentum is conserved, so the gravitons are created in pairs with equal and opposite wave vectors, ${\bf k}$ and ${-\bf k}$

The quantum particle creation process is being described in the Heisenberg picture, so the quantum state of the system, $|\psi_{0}\rangle$, does not change in time. However, its description is very different in the deSitter space in-region from that in the radiation dominated out-region. In the in-region, it is one of several infrared finite choices of vacuum state. In the out-region, it is a highly excited state with very large numbers of particles in modes for which $k\ll H$.

The quantum state of the created particles in the out-Fock space may be shown to be

where $\gamma_{k}=\beta_{k}/\alpha_{k}$. Here $\{n_{\bf k}\}$ denote a set of occupation numbers; for each mode ${\bf k}$ there is non-negative integer $n_{\bf k}$ which gives the number of particles in mode ${\bf k}$ and the number of particles in mode ${-\bf k}$. In our enumeration, we may require $k_{z}\geq 0$ to avoid over counting. The overall constant $c_{0}$ satisfies

where the latter form follows from Eq. (11). This quantum state is a multi-mode squeezed vacuum state, containing all possible numbers of pairs of correlated particles. Strictly, Eq. (14) applies to the massless scalar field. For gravitons, we need to include the polarization among the mode labels. This will be explicitly addressed in Sect IV.1, where a sum over polarizations will be performed.

In the single mode case, only one type of pair is present. Here we have

where $|n,n\rangle$ denotes a state containing $n$ pairs, with one member of the pair in mode ${\bf k}$, and the other in mode ${-\bf k}$.

The expectation value of the field operator vanishes in this state,

due to the fact that $\langle m,m|a_{\bf k}|n,n\rangle\propto\langle m,m|n-1,n\rangle=0$ for all integers $m$ and $n$. This holds for both the one-mode case of Eq. (16), and the multi-mode case of Eq. (14). As a result, the state $|\psi_{0}\rangle_{1}$ does not describe a classical wave.

It is well known in quantum theory that outcomes of observations can depend crucially upon the details of the question which is addressed, or the specific measurement which is undertaken. At least in the cosmological case, we can only observe one member of a pair of correlated gravitons. This can be seen if we adopt a set of wave packet mode functions, rather than plane waves with a precisely defined wave vector. Consider a set of wave packets with a finite but small bandwidth, $\Delta k\ll k=|{\bf k}|$, where ${\bf k}$ is the peak wave vector of a packet. These packets will have a spatial extent of order $\Delta x\approx 1/\Delta k$.

In the wave packet basis, the correlated pairs of gravitons are created into a pair of packets peaked about values of ${\bf k}$ with equal magnitude, but opposite sign. This means that the two packets leave the region of their creation in opposite directions. A given observer will be able to observe at most one of these gravitons. This is illustrated in Fig, 1 for the case of gravitons which influence the redshifts of CMB photons.

Suppose we are making measurements of local field operators in a spacetime region ${\cal R}$, and using a complete set of wave packets with mode labels $\{j\}$. It is convenient to divide the set $\{j\}$ into two subsets, $\{j_{1}\}$ and $\{j_{2}\}$, where the $j_{1}$ modes are nonzero in ${\cal R}$ , and the $j_{2}$ modes vanish there and do not contribute to local expectation values in ${\cal R}$. Now we can rewrite Eq. (14) as

Equation (16) is the special case where there is one $j_{1}$ mode and one $j_{2}$ mode.

If $i$ and $\ell$ are two modes in set $j_{1}$, then the expectation values in state $|\psi_{0}\rangle$ of the associated creation and annihilation operators are

but

This implies that expectation values of linear field operators vanish in this state, but those of quadratic operators can be nonzero. Furthermore, the effects of the state $|\psi_{0}\rangle$ in ${\cal R}$ are equivalent to a density matrix.

Let $a$ and $a^{\dagger}$ be associated with any given mode in set $j_{1}$, and define a coordinate by

and a conjugate momentum by

Then $\langle x\rangle=\langle p\rangle=0$  , but the ubcertainties are nonzero: $\Delta x=\sqrt{x^{2}}=\Delta p$. Furthermore the uncertainty relation becomes

In general, $|\psi_{0}\rangle$ is not a state of minimum uncertainty, unlike the Stoller squeezed states, and will have very large uncertainty when $|\beta_{j}|\gg 1$. This implies large fluctuations, as evidenced the large variance of linear field operators.

There is a simple analog involving a squeezed state of photons which is illustrated in Fig. 2. A strong classical electromagnetic wave in a nonlinear material with a nonzero second or third order susceptibility creates a time dependent effective index of refraction for vacuum electromagnetic modes. This leads to mixing of positive and negative frequencies and quantum creation of photons, an effect analogous to the quantum creation of gravitons in inflation discussed in Sect. II. The photons are emitted in correlated pairs in opposite directions from the material in a quantum state of the form of $|\psi_{0}\rangle$ given in Eq. (18).

In this subsection, we focus on the case of a single massless, minimally coupled scalar field $\varphi(x)$, which is equivalent to one polarization state of the graviton field in the transverse tracefree gauge. Assume that we are in the radiation dominated era, when the mode functions may be given by Eq. (8) and the field operator may be expressed as

where the $a_{\bf k}$ satisfy the relations in Eqs. (19) and (20) in a state of the form of $|\psi_{0}\rangle$. Next define a correlation operator by

and a correlation function by

Recall that $\langle\varphi(x)\rangle=0$, so $C(x,x)$ is the variance of the fluctuations of $\varphi(x)$. If $|\beta_{j}|^{2}\gg 1$, then

where we ignore a term of order $|\beta_{j}|^{0}$.

Consider the case where a single mode $j$ gives the dominant contribution to $C(x,x^{\prime})$, so that $C(x,x^{\prime})\propto F_{j}(x)$, and the correlation function has the form of a classical wave as a function of $x$ for fixed $x^{\prime}$. This is similar to the case of a coherent state. However, unlike a highly excited coherent state, here the fluctuations of the correlation operator are very large:

so the fractional variance of $K$ is of order one

The fluctuations are illustrated in Fig. 3. Because of the large fractional fluctuations, the correlation effects can only be observed in an ensemble average.

The graviton field operator may be expanded in terms of mode functions as

where ${\rm e}^{\mu\nu}({\bf k},\lambda)$ is the polarization tensor for mode ${\bf k},\lambda$. The expectation value of $h^{\mu\nu}$ in a coherent state for mode ${{\bf k},\lambda}$ is a classical gravitational wave. Here we use a linear polarization basis, so $\lambda=+,\times$, and the polarization tensors are implicitly defined by Eqs. (30) and (38).

The graviton correlation function may be expressed as

Here we have used Eqs. (8) and (20), as well as the convention that $\sum_{\bf k}\rightarrow\int d^{3}k$ in the continuum limit.

If we use Eq. (13), with $|\beta_{k}|^{2}\propto k^{-4}$, then the integral in Eq. (44) will diverge at the lower limit. This issue is most likely distinct from the infrared divergence discussed in Sect. II.1 which required a cutoff at $k=k_{e}$. The latter was required because gravitons and massless minimally coupled scalar fields necessarily break deSitter symmetry. The cutoff at $k=k_{e}$ arises from a memory of pre-inflationary conditions which is was not erased by a finite amount of inflation. The infrared divergence carried by $|\beta_{k}|^{2}$ is most likely due to a breakdown of the particle interpretation of the out-Fock space. Recall that $|\beta_{k}|^{2}$ is interpreted as the mean number of gravitons in mode $k$ in the out-region. However, it is not clear that graviton number is meaningful for modes whose wavelength is far greater than the horizon size. We propose to modify Eq. (44) by restricting the integration to modes with $k>k_{0}$, where $k_{0}$ is the wave number associated with the horizon size at the time when the gravitons interact with the CMB photons. Thus, Eq. (44) is replaced by

The angles involved in the $d\Omega_{k}$ integration are illustrated in Fig. 4. First note that $\cos\chi=\hat{\bf k}\cdot\hat{\bf q}=\cos\theta\,\cos\phi$. Thus $\sin^{2}\chi=1-(1-c^{2})\,\cos^{2}\phi$, with $c=\cos\theta$ and we have

Now we can write

where we define a reduced correlation function by

which is normalized by $K(0,0)=1$. The reduced correlation function may be expressed in terms of the functions $C_{n}$ and $S_{n}$ are defined by

and

These functions may be expressed in terms of trigonometric and cosine-integral functions and are given in the Appendix.

In this subsection, we deal with the simpler case where ${\bf r}=0$, but $\Delta\eta=\eta-\eta^{\prime}$ is nonzero. This is the case where the CMB photons arrive from the same direction, but were emitted at different times and hence different redshifts. In this case there is no angle dependence inside the cosine function in Eq. (48), and we find

which is plotted in Fig. 5. This function describes the temperature fluctuations correlations for photons arriving from the same direction, but different redshifts, with the pre-factor of $1/[a(\eta)\,a(\eta^{\prime})]$ in Eq. (45) removed.

Note that the region in which the inflationary gravitons interact with the CMB photons is actually a last scattering shell of finite thickness. This thickness is determined by the time required for the electrons and protons in the primordial plasma to combine into neutral hydrogen atoms. It has been estimated by Hadzhiyska and Spergel HS19 to be of order $\Delta\eta\approx 19\,{\rm Mpc}$, which is equivalent to $\Delta z\approx 90$ in redshift.

When ${\bf r}\not=0$, the argument of the cosine function in Eq. (45) depends upon $\theta$. In this case, we have

We may now write Eq. (48) as

with $\tau=k_{0}\,\Delta\eta$ and $\rho=k_{0}\,r$.

The case of spatial separation alone, $K(0,r)$ is plotted in Fig. 6, and the general case is illustrated in Fig. 7.

We have seen in the previous subsection that the temperature fluctuations produced by a graviton bath display distinctive correlations and anti-correlations, both in space and in time. The anti-correlations mean that a region whose temperature is above average, is more likely to be near a region which is cooler than average, and vice versa. The separation between the hotter and the cooler region can be either in angle or in redshift. The magnitude of the expected angular or redshift depends upon the unknown parameter $k_{0}$, and is given by $k_{0}\,\Delta r\approx 3$ for spatial or angular separations and by $k_{0}\,\Delta\eta\approx 2$ for temporal or redshift separations. Note that the ratio of these two expected separations, $\Delta r/\Delta\eta\approx 1.5$, is independent of $k_{0}$. As was emphasized in Sec. III.4 and illustrated in Fig. 3, correlation functions are averages of highly fluctuating quantities. This means that the temperature anti-correlations can only be observed by averaging over several pairs of regions.