A Spatial gauge freedom

On Separate Universes

Abstract

The separate universe conjecture states that in General Relativity a density perturbation behaves locally (i.e. on scales much smaller than the wavelength of the mode) as a separate universe with different background density and curvature. We prove this conjecture for a spherical compensated tophat density perturbation of arbitrary amplitude and radius in CDM. We then use Conformal Fermi Coordinates to generalize this result to scalar perturbations of arbitrary configuration and scale in a general cosmology with a mixture of fluids, but to linear order in perturbations. In this case, the separate universe conjecture holds for the isotropic part of the perturbations. The anisotropic part on the other hand is exactly captured by a tidal field in the Newtonian form. We show that the separate universe picture is restricted to scales larger than the sound horizons of all fluid components. We then derive an expression for the locally measured matter bispectrum induced by a long-wavelength mode of arbitrary wavelength, a new result which in standard perturbation theory is equivalent to a relativistic second-order calculation. We show that nonlinear gravitational dynamics does not generate observable contributions that scale like local-type non-Gaussianity , and hence does not contribute to a scale-dependent galaxy bias on large scales; rather, the locally measurable long-short mode coupling assumes a form essentially identical to subhorizon perturbation theory results, once the long-mode density perturbation is replaced by the synchronous-comoving gauge density perturbation. Apparent -type contributions arise through projection effects on photon propagation, which depend on the specific large-scale structure tracer and observable considered, and are in principle distinguishable from the local mode coupling induced by gravity. We conclude that any observation of beyond these projection effects signals a departure from standard single-clock inflation.

a]Liang Dai, b]Enrico Pajer, c]Fabian Schmidt \affiliation[a]Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA \affiliation[b]Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands \affiliation[c]Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany

1 Introduction and summary of results

An accurate theoretical description of the nonlinear large-scale structure is one of the major goals of theoretical cosmology. Although it is by nature a difficult problem, nonlinear structure formation is a key ingredient toward decoding the initial cosmological perturbations in the primordial Universe and testing the large-scale dynamics of gravitation. This is especially important given the plethora of current or incoming large-scale structure surveys aiming at measurements of unprecedented accuracy. A general relativistic treatment is essential if perturbations on scales close or comparable to the horizon are being considered. Since the initial conditions of structure (e.g., due to inflation) are generally specified when they are outside the horizon, this issue necessarily arises when attempting to connect structure in the observable universe to the initial conditions.

One of the primary reasons of why we would like to connect late-time observables of large-scale structure to the initial conditions at nonlinear order is the study of interactions during inflation, in particular through primordial non-Gaussianity. Specifically, the interaction between long-wavelength and short-wavelength perturbations is an interesting diagnostic of inflationary physics, as it can distinguish between one or more light degrees of freedom during inflation. This can be most directly measured from the squeezed configuration of the three-point (bispectrum) or higher-point functions, where “squeezed” means that one Fourier mode has a wavelength much longer than all the other modes. For single-clock inflation, model-independent squeezed-limit consistency relations are known to hold (1); (2); (3); (4); (5); (6); (7); (8); (9), the simplest representative of which reads , where is the amplitude of local-type non-Gaussianity in the primordial (Newtonian gauge) potential. Physically, they encode the existence of a single preferred clock. A detection of local-type primordial non-Gaussianity that departs from the unique single-field prediction would prove the existence of more than one light field during inflation.

In terms of the matter three-point function, local-type non-Gaussianity schematically leads to a squeezed limit of 1

(1)

where corrections are suppressed by and we have assumed so that the transfer function is unity. While consistency relations are a statement about transformation properties under diffeomorphisms, in a local observer’s frame the diffeomorphism freedom is completely fixed by the observer’s proper time and distance standards. As shown in (10) (see also (11); (12); (13); (14)), this implies that squeezed-limit consistency relations drop out of locally measurable quantities. Non-single-field models which violate the consistency relation will on the other hand contribute to physical observables and can be measured in LSS surveys through the bispectrum or scale-dependent bias  (15).

Thus, an important question to ask is whether nonlinear gravitational evolution of the matter density field, neglected in Eq. (1), will lead to contributions of -type to matter statistics. Investigating this requires a relativistic treatment of perturbations at second order since both the long- and short-wavelength modes have to be followed through horizon crossing. This has been done for example in (16); (17); (18). Unfortunately, standard higher-order perturbative approaches for nonlinear mode-coupling are often plagued by unphysical gauge artifacts due to the freedom to parameterize perturbations in different coordinate systems (e.g., (19)).

An alternative approach to this problem is based on the intuition that a local observer who has access only to short comoving distances , which are much smaller than the scale of variation of the long-wavelength modes, would interpret local small-scale physics within a Friedmann-Lemaître-Robertson-Walker (FLRW) background, i.e. a “separate universe” (20). To formulate this intuition rigorously, one starts with a coarse-grained universe in which small-scale inhomogeneities with greater than a fixed comoving scale are first smoothed out. One then studies how small-scale structure with evolves in this “local background” spacetime, instead of the “global” background universe. To solve small-scale clustering in this modified background, one can take a Lagrangian perspective — if non-gravitational forces are negligible at large distances, one can follow inertial observers in the coarse-grained universe and their vicinity. As initially proposed in (10) and rigorously proven in (21), one can locally construct a frame, Conformal Fermi Coordinates (CFC), valid across a comoving distance of order at all times, which realizes this physical picture. In CFC, the observer interprets the small-scale structure around her as evolving in an FLRW universe that is modified due to long-wavelength modes with comoving wavenumber , with corrections quadratic in the spatial distance to the observer

(2)

where is the local scale factor and is the local spatial curvature. The fact that the metric is valid over a fixed comoving scale is very important. This means that we can apply the CFC frame even when the small-scale modes are outside the horizon. This is crucial in order to connect to the initial conditions given by inflation. Therefore, within CFC one can keep track of any general relativistic effect that could arise as either long- or short-wavelength perturbations cross the horizon.

This “separate universe” approach enjoys at least two advantages. On the one hand, gauge freedom is eliminated because the calculation directly yields what the CFC observer physically measures. On the other hand, it easily relates to small-scale structure formation in homogeneous FLRW backgrounds, which we know very well how to handle. An important requirement for this is that indeed corresponds to a locally observable scale factor. This can be achieved easily by using the local velocity divergence as described in (21) and Sec. 3 below.

The separate universe picture strictly holds if the local spacetime is indistinguishable from an unperturbed FLRW universe within a small region of space but for an extended (or even infinite) duration of time. We find that this is in general only true if the observer and all components of matter with equation of state are comoving. This implies that the separate universe picture is restricted to scales larger than the sound horizon of all relevant fluids.

One of the main results of this work is the precise form of the CFC metric Eq. (2) and the relation of the local scale factor and curvature to the long-wavelength perturbations, which we assume to be linear. The latter relation becomes simplest in synchronous-comoving (sc) gauge, where we obtain (for a flat global cosmology) the surprisingly concise relations

(3)

where is the long-wavelength density perturbation, is the scale factor of the background universe, and is the linear growth factor normalized to during matter domination. Furthermore, the tidal corrections in Eq. (2) can be written as

(4)

where are simply related to the scalar potentials in the conformal Newtonian gauge through

(5)

and an analogous relation holds for in terms of . Note that involve the trace-free part of the second spatial derivatives of and , i.e. the Newtonian tidal tensor, and thus vanish for an isotropic perturbation.

These results can be summarized in words as:

On scales larger than the sound horizon of the fluid, the effect of a long-wavelength mode as measured locally is completely captured by a modified local scale factor and spatial curvature, Eq. (3), and a pure tidal field, Eq. (5).

Ref. (10) showed that the consistency relation for single-clock inflation, transformed into CFC, states that there is no primordial correlation between long- and short-wavelength potential perturbations. This provides the trivial initial conditions for the small-scale fluctuations in CFC. Integrating the growth of small-scale density perturbations in the presence of a long-wavelength mode , we then obtain the leading contribution from nonlinear gravitational evolution, valid for arbitrarily small ,

(6)

Here, is the linear small-scale density field (i.e. in the absence of long-wavelength modes), while is the long-wavelength density perturbation (in synchronous gauge), and is proportional to the tidal tensor. Eq. (6) shows that the locally observable small-scale density in single-clock inflation has no contributions that scale as local-type non-Gaussianity (this holds not just for quadratic but equally for higher order terms ). This implies that there is no scale-dependent bias of large-scale structure tracers even when taking into account nonlinear gravitational evolution fully relativistically. We further derive the leading squeezed-limit matter bispectrum from Eq. (6) [Eq. (112)] which proves this point. Specifically, this is the bispectrum which would be seen by a “central observer” if synchronous observers distributed on the past lightcone of the central observer communicated their local densities and power spectra of small-scale fluctuations. This bispectrum is suppressed over local-type non-Gaussianity by a factor of .

In order to provide predictions for correlation functions measured from Earth, we need to include photon propagation (“projection”) effects. These correspond to mapping the locally measurable observables in physical space to the observer’s coordinates of measured photon redshift and arrival direction. These effects do contain contributions that correspond to order unity . However, we stress that projection effects are purely kinematical, not dynamical, and depend on the details of the tracer considered. Thus, they are in principle distinguishable from the locally measurable local-type non-Gaussianity, which, as we have shown, is not generated by gravitational evolution and must come from the initial conditions. A detailed treatment of projection effects is beyond the scope of this paper. For recent literature on this subject, see (10); (22); (23); (24).

The remainder of this paper is organized as follows. In Sec. 2, we demonstrate the concept of “separate universe” by presenting a full general relativistic proof that a compensated spherical top-hat region of overdensity embedded in a FLRW universe behaves exactly as a separate FLRW universe (with different spatial curvature). In Sec. 3 we specialize to the CFC for a free-falling observer in the presence of a long-wavelength scalar metric perturbation at linear order. We discuss in Sec. 4 the sufficient and necessary conditions for which local FLRW expansion is exact, i.e. for which the spacetime is truly locally indistinguishable from an FLRW universe. The explicit calculation of the locally measurable squeezed matter bispectrum for an Einstein-de Sitter (EdS) universe () with single-clock initial conditions is presented in Sec. 5. Concluding remarks are given in Sec. 6.

Regarding our notation, up to including Sec. 3 we distinguish between the global comoving position and the CFC comoving position , so as to highlight the distinction between the global coordinate system and the CFC. After that we will exclusively use spatial CFC coordinates and denote them simply by . On the other hand, we always explicitly distinguish between the CFC times and the global times .

2 Proof of the separate universe conjecture for a compensated tophat

The “separate universe conjecture” states that a spherically symmetric perturbation in an FLRW background (taken to be CDM in this section) behaves like a separate FLRW universe with different matter density and curvature. This holds up to higher spatial derivative corrections, but at all times. We now provide a proof of this statement for a spherical compensated tophat, or just tophat for short (see figure 1), without making any assumptions about the amplitude or wavelength of the perturbation (i.e. we do not assume that it is subhorizon). Higher derivative corrections are avoided by assuming a homogeneous density perturbation.

Consider a “background” FLRW universe described in comoving spherical coordinates by

(7)

where we have allowed for spatial curvature . The label “” stands for “outside”, as it will become clear shortly. Note that is the area radius, so that the proper surface area of a 2-sphere of radius is .2 This will become very useful in the following. We assume that the universe is filled with dust of uniform density and a cosmological constant. Now consider cutting out a sphere of comoving radius from the FLRW background (where “” stands for “comoving”), and collapsing the matter in the sphere into a point mass . If we choose to be constant in time, then is conserved.3 The authors of (25) first showed for a matter-dominated universe that there is a unique solution consisting of a Schwarzschild metric interior to smoothly matched to the FLRW background Eq. (7) at for all times (more precisely, the metric is continuously differentiable at the boundary). This result has since been generalized to include a cosmological constant (27); (26), in which case the point mass metric becomes the Schwarzschild-de Sitter solution:

(8)

where the label “” stands for “Schwarzschild”. Hence, the matter surrounding the shell is oblivious to the fact that the interior has been collapsed to a black hole, while a test particle anywhere inside is oblivious to the surrounding homogeneous matter. Note that by matching the angular part of the metric, we can immediately identify . Moreover, is the area radius for . In the interior coordinates, the boundary is thus set by . Using this relation, it is then easy to show that at the boundary the geodesic equation derived from the metric Eq. (8) exactly matches that given by the FLRW background Eq. (7), which is simply  const, once the Friedmann equation for is inserted. This is of course a consequence of the metric being continuously differentiable at the boundary.

Consider now the case where we do not collapse the matter inside into a black hole, but compress it to a finite radius (where “” stands for “inside”), maintaining a homogeneous density (see Fig. 1). Birkhoff’s theorem (28), generalized to include a cosmological constant, states that the unique spherically symmetric vacuum solution to Einstein’s equation is Schwarzschild-de Sitter. Thus, Eq. (8) still describes the spacetime outside of the mass, . We now perform the exact same matching as derived by (25); (27); (26), but inverting outside and inside. In fact, nothing in the matching is particular to the case of vacuum inside a homogeneous matter distribution; it equally applies to a homogeneous density distribution inside vacuum. Thus, there exists a unique FLRW solution of the form Eq. (7) with density that smoothly matches to the Schwarzschild-de Sitter metric at , where is determined by mass conservation.

Figure 1: Illustration of the setup used to prove the separate universe approach.

Moreover, we can use the geodesic motion that follows from the metric Eq. (8) to derive the scale factor in the interior FLRW solution. The geodesic equation for Eq. (8) and purely radial motion, where is the proper time, becomes

(9)

which can be integrated to give

(10)

where we used . Inserting Eq. (8), we obtain

(11)

This can be turned into a Friedmann equation by introducing a scale factor different from the “background” . The normalization of the scale factor is arbitrary, and we can choose to normalize to where is some reference time and . Dividing Eq. (11) by , we obtain

(12)

We have relabeled the integration constant suggestively as . This is indeed the Friedmann equation for a CDM universe with background matter density and curvature . Assuming that the perturbation was initialized with a very small amplitude at an early time when the effect of is negligible, one can then subtract the background FLRW equation following from Eq. (7), and linearize in at . As shown in Sec. 4 below (see also (29)), this yields a relation between the initial (linear) density perturbation and curvature

(13)

where for simplicity we have assumed that the background FLRW universe is flat (although this is not a necessary assumption).

Thus, we have proved that, at least for the case of a compensated tophat density profile, a spherically symmetric perturbation in a CDM background evolves exactly as a separate curved CDM universe. We have not assumed that the perturbation is small or that the scales are subhorizon. Indeed, neglecting for simplicity, the Schwarzschild-de Sitter metric in the vacuum surrounding the perturbation cannot be perturbatively approximated as Minkowski if

(14)

since . Thus, for a horizon-scale perturbation, the vacuum exterior is far from Minkowski. Nevertheless, the separate universe description is exact. In fact, an observer in the vacuum region surrounding the overdensity would see this separate universe as a black hole.

One might wonder how this can be applied to an underdensity (void) rather than an overdensity. In this case, the inner FLRW solution which we have called now becomes the background , while the outer, less dense FLRW solution becomes the “perturbation” . From the perspective of an observer inside either FLRW solution, the situation is completely symmetric, as they cannot tell by any local measurement whether they are embedded in a larger “background” universe.

In Sec. 4, we will return to the separate universe picture in the context of general scalar perturbations in a cosmology with multiple fluids.

3 Scalar perturbations in the CFC frame

In this section, we describe how long-wavelength adiabatic scalar perturbations are treated in the CFC frame. This provides the basis for reconsidering the separate universe ansatz in the following sections.

3.1 Review of Conformal Fermi Coordinates (CFC)

We first briefly review the basic concept of Conformal Fermi Coordinates (CFC), which was first introduced in (10) and defined rigorously in (21). Consider an observer free-falling in some spacetime. We are mostly considering applications in the cosmological context, in which this spacetime is approximately FLRW, but this does not have to be the case. Her trajectory is a timelike geodesic , whose tangent vector we will denote as . One can construct a frame where the spatial origin is always located on this geodesic, and in which the metric takes the form4

(15)

Thus, in these coordinates the metric looks like an unperturbed FLRW metric in the vicinity of the observer’s trajectory at all times, up to corrections that go as the spatial distance from the origin squared. Note that a metric of the form Eq. (15) means that an observer at the origin is free-falling, with proper time given by

(16)

Briefly, this frame can be constructed as follows. For now, let be a positive scalar field in the neighborhood of G. We introduce the conformal metric

(17)

We then define a “conformal proper time” along through

(18)

where denotes the point on the central geodesic which has proper time . This defines our time coordinate . At some point along we can construct an orthonormal tetrad , where is again the tangent vector to , with . The tetrad is defined at all other points on by parallel transport, so that this condition is preserved. A point corresponding to CFC coordinates is then located as follows. First, we move to the point on specified by . We then construct a spatial geodesic of the conformal metric which satisfies , and whose tangent vector at is given by

(19)

The point is then found by following this geodesic for a proper distance . As we show in (21), this yields a metric in the form Eq. (15). Moreover, the leading corrections are given by

(20)

where is the Riemann tensor of the conformal metric, evaluated in the CFC frame at on the central geodesic.

Note that if we now were to choose , the CFC would reduce exactly to the ordinary Fermi Normal Coordinates (30). Then, the metric perturbations would contain the leading corrections due to Hubble flow , restricting the validity of the coordinates to subhorizon scales. Instead, we will choose so that it captures the locally observable expansion of the Universe, which automatically extends the validity of the coordinates to the spatial scale of perturbations which can be superhorizon. The well-defined prescription that fixes up to a multiplicative constant will be given in Sec. 3.2.1.

3.2 Scalar perturbations around an FLRW universe

We now turn to an FLRW universe with scalar perturbations in the conformal-Newtonian (cN) gauge,

(21)

In the absence of anisotropic stress we have ; however, we will not make that assumption in this section. Before we apply the CFC construction, we note that we are implicitly performing a coarse-graining of the metric Eq. (21) on some scale . This is because in the actual universe, scalar perturbations exist on all scales, so that without any coarse-graining the metric Eq. (15), which assumes that the corrections are small, is only valid on infinitesimally small scales. In the following, and will thus denote coarse-grained metric perturbations.

Further, we will work to linear order in and . The perturbative expansion in should not be confused with the power expansion in . The former expansion is valid as long as , and is chosen here for simplicity; the CFC construction also works for spacetimes that differ strongly from FLRW. The latter expansion on the other hand is good if is smaller than the typical variation scale for , which is the fundamental expansion parameter in CFC. Following standard convention, spacetime indices are assumed to be raised and lowered with the metric , while latin indices are raised and lowered with . Correspondingly, denotes the flat-space Laplacian

(22)

CFC construction

We begin with deriving the tetrad. Consider a free-falling observer traveling along the time-like central geodesic . His 4-velocity is given by

(23)

where the 3-velocity is considered as first-order perturbation.5 obeys the geodesic equation, given by

(24)

where a prime denotes derivative with respect to conformal time . Our choice of the spatial components of the orthonormal tetrad is

(25)

which we have aligned with the global coordinate axes for simplicity and without loss of generality.

We now want to determine by defining the locally observable expansion rate of the Universe to match the CFC Hubble rate . Following (21), consider the convergence of the geodesic congruence along the central geodesic . This is a locally measurable quantity, which intuitively corresponds to the change in time of the proper volume of a bundle of geodesic trajectories in the neighborhood of the central geodesic . In an unperturbed FLRW universe, we have

(26)

Thus, we define at all points along the geodesic

(27)

In terms of cN gauge quantities the convergence of a geodesic congruence is given by

(28)

The local Hubble parameter is then given by

(29)

Note that Ref. (35) in their construction using the standard Fermi Normal Coordinates (FNC) only include the last, velocity-divergence term, which dominates when the long-wavelength mode is deep inside the horizon. However, the second term here is a necessary correction on (super-)horizon scales, since on those scales does not correspond to physical peculiar velocities.

Consider, for example, the motion of two nearby fluid elements in the presence of a superhorizon long-wavelength mode. For adiabatic initial conditions, the two fluid elements will then have vanishing relative peculiar velocity, which scales as , i.e. the fluid elements have a constant, infinitesimal coordinate separation . The physical separation is . The proper time interval is . The physical relative velocity , i.e. the rate of change of the proper distance between them, then reads

(30)

where the last equality holds since we neglect terms of order . Therefore, the relative velocity and satisfy exactly the local version of Hubble’s law, showing that Eq. (29) is indeed the proper expression for the local Hubble rate.

Since so far we have only fixed , the local scale factor is defined up to a multiplicative constant. In other words, we have a residual freedom to rescale , which keeps the proper time unchanged. Of course, this constant is arbitrary and should cancel out of any proper observable. We will here fix the constant in order to make our results of the next sections more transparent. Specifically, we demand that

(31)

which means that at early times, the local scale factor-proper time relation is the same as that in the unperturbed background cosmology. The ratio of scale factors at a fixed spacetime point is then at early times given by

(32)

where is the asymptote as of the potential along the geodesic. A direct integration gives

(33)

In order to obtain the leading corrections to the CFC metric (either via Eq. (20) or an explicit coordinate transformation of the metric), we also need the second derivative . This can be obtained easily from Eq. (27),

(34)

where again all quantities are evaluated on .

CFC metric

We can now derive the remaining corrections to the CFC metric, making use of the geodesic equation for . This yields in terms of the conformal Newtonian gauge perturbations

(35)
(36)
(37)

Let us consider a spherically symmetric configuration about the CFC observer. It amounts to setting , and to zero, and replacing with and the same for . We then find that and vanish, while reduces to

(38)

where we have suggestively introduced

(39)

The tensorial structure distorts the proper distance only along directions perpendicular to the radial direction, as guaranteed by our construction of CFC (recall that is defined as the geodesic distance in the conformal metric). We are free to choose a different radial coordinate (see App. A for a discussion of the residual gauge freedom in the spatial component of the CFC metric)

(40)

In terms of this new radial coordinate, the area radius of the spacetime (see Sec. 2) is given by . Under this rescaling, the CFC metric simply becomes (valid to )

(41)

The spatial part of the metric is the familiar stereographic parameterization of a curved, homogeneous space.

We now go back to general anisotropic configurations of . In the next section, we will see that, in the context of scalar perturbations considered here, it is necessary that all cosmic fluids are comoving with velocity for the separate universe picture to hold. The comoving condition implies that in fact vanishes, by way of the time-space Einstein equation. Therefore, we will set to zero in the remainder of the paper. We can again use the residual gauge freedom described in App. A.2 to bring into a more familiar form. After some algebra, this yields

(42)

We have thus put the CFC metric into the conformal Newtonian form. The metric perturbations enter in two distinct ways. First, the isotropic part of the perturbation (proportional to and themselves as well as their time derivatives) modifies the background and leads to spatial curvature. The anisotropic part, which is completely determined by

(43)

and the analogous , enters as a tidal field. Note that since the CFC frame was constructed based on local observables, no gauge modes yield any observable imprint in the CFC metric. Further, it is important to emphasize that no subhorizon assumption has been made about either or the CFC metric itself; Eq. (42) is valid on arbitrarily large scales as long as the corrections are small. We will turn to the interpretation of the CFC metric in the next section.

4 Separate universe revisited

In the previous section we derived the CFC frame metric for scalar perturbations, which appears to be of the FLRW form with tidal corrections. However, in order to test whether an observer along the central geodesic would indeed interpret local measurements as an FLRW universe in the absence of tidal corrections, we need to derive the equation for and compare it to the Friedmann equations. Further, we need to verify that the “curvature” is indeed constant. Throughout we will set the tidal corrections in Eq. (42) to zero in keeping with our linear order treatment.

Let us consider an observer traveling along the central geodesic who performs local measurements. Suppose further this observer knows about General Relativity. She will compare the local Hubble rate with the Friedmann equations, the first of which reads

(44)

where is the local rest-frame matter density, coarse-grained over the region considered as CFC patch. Using Eq. (29), is given in terms of global quantities by

(45)

where is the density perturbation in cN gauge. On the other hand, the spatial curvature which can be inferred from, e.g. shining light rays and measuring angles between them, is given in terms of global quantities by Eq. (39). We thus need to verify whether  const, and under which conditions. Only in that case would the observer actually interpret the results of local measurements as an FLRW universe.

Let us assume that the Universe is filled with multiple uncoupled fluids labeled by . Each fluid has a homogeneous equation of state , with possibly time-dependent . The homogenous part of each fluid evolves as , which implies

(46)

where and is the total (homogeneous) energy density. The acceleration is given by

(47)

In the global cN-gauge coordinates, we let each component have a fractional density perturbation and peculiar velocity , where is the velocity potential. Note that denotes the velocity of the central geodesic throughout. We now write down the time-time and time-space Einstein equations, which have the sum of all fluids as sources,

(48)
(49)

Using that

(50)

Eqs. (48)–(49) can be inserted into the first two terms of Eq. (45) to yield

(51)

This is Eq. (39), showing that the quantity appearing in the local Friedmann equation is indeed the spatial curvature. Thus, Eq. (45) provides an equivalent relation between the long-wavelength adiabatic perturbation in conformal-Newtonian gauge and the local curvature.

4.1 Conservation of curvature

In the FLRW solution, the spatial curvature is constant. This is thus a necessary condition that needs to be satisfied for the separate universe picture to hold. To study this, we allow each fluid to have a pressure perturbation in addition to the density perturbation . In the rest frame of the fluid, we have

(52)

where the rest-frame speed of sound does not necessarily equal the adiabatic sound speed

(53)

because the pressure perturbation can be non-adiabatic. Each fluid then evolves according to (31)

(54)
(55)

Using Eqs. (46)–(47), Eqs. (48)–(49), and Eqs. (54)–(55) (keeping general), we can directly compute the time derivative of Eq. (45).6 We also assume that follows the peculiar motion of the non-relativistic matter and therefore satisfies the geodesic equation (the version of Eq. (55)). We obtain

(56)

The final result is obtained after lengthy but straightforward algebra. The second term vanishes if long-wavelength perturbations do not source anisotropic stress .

If the Universe is only composed of non-relativistic matter () plus a possible cosmological constant (), and the CFC observer co-moves with the matter fluid along the central geodesic, is conserved and the observer cannot distinguish the spacetime from a curved CDM universe using any local measurements (that is, measurements over a scale much smaller than the wavelength of the perturbations ). This applies to any isotropic configuration around the geodesic, and is not restricted to the tophat considered in Sec. 2.

If there are extra fluids with other values of in the Universe, as postulated in many of the alternative cosmologies, in general does not evaluate to zero. If all fluids co-move with the CFC observer , however, does remain a constant. Since this situation requires all fluids to follow geodesic motion, non-gravitational forces have to be negligible. This will be true when the sound horizon is much smaller than the wavelength of interest, i.e.  (32). This further generalizes our considerations of Sec. 2.

Assuming that the CFC observer co-moves with the average cosmic fluid such that , we can re-write Eq. (45) using Eqs. (48)–(49) to find

(57)

where is the gauge-invariant curvature perturbation on comoving slices. Notice that we have made no assumption about the decaying adiabatic mode (the second adiabatic solution, always present in the small limit (33)). Its contribution cancels out of this equation. The conservation of is hence directly related to the conservation of and it is valid also in the presence of a decaying adiabatic mode. Note that the conditions for  const are different than for  const, where is the curvature perturbation on uniform density slices. The latter is conserved if pressure is only a function of energy density (34), while the conservation of does not exclude non-adiabatic pressure. Thus, , not , is the locally observable curvature, and the separate universe picture relies on conservation of the former, not the latter. Note that in (35); (29), the comoving curvature perturbation was denoted with , specifically . Thus, our result Eq. (57) agrees with those references. Eq. (57), together with the discussion after Eq. (29) above, thus confirms that superhorizon perturbations cannot affect the cosmology as observationally determined by measurements within the horizon (36).

Finally, one can also verify that the second Friedmann equation is satisfied by , which is shown in App. C. Unlike the first Friedmann equation, this does not impose any conditions on the fluid components. Indeed, it is a simple consequence of the Raychaudhuri equation applied to the CFC metric Eq. (42).

4.2 Relation to synchronous-comoving gauge

It is illuminating to connect the curvature , Eq. (57) to the synchronous-comoving (sc) gauge. Following (38), we write the metric in global coordinates as

(58)

where . Here and in the following, we drop the subscript on the spatial CFC coordinates, since the global coordinates will not appear explicitly anymore and so there is no ambiguity. The time-time component of the Einstein equation is

(59)

where is the density perturbation in synchronous-comoving gauge. Further, the comoving condition implies that . Note that the quantity on the l.h.s. of Eq. (59) is , i.e. we have

(60)

That is, the spatial metric potential in cN gauge is related to the density perturbation in sc gauge by the Newtonian Possion equation. This relation holds on all scales, a fact which is being used in N-body simulations (37).

One can easily verify that the local Hubble rate for the metric Eq. (58) is given by

(61)

Let us now consider a spherically symmetric perturbation in sc gauge. Straightforward evaluation of the spatial part of the conformal Riemann tensor, together with then yields

(62)

and thus, matching to the spatial curvature (Eq. (38)),

(63)

Let us verify again the local Friedmann equation,

(64)

Multiplying the Friedmann equation by , using Eq. (61), and subtracting the background contribution, we obtain

(65)

which is exactly the time-time component of the Einstein equation Eq. (59) (multiplied by ).

Let us now restrict to an EdS universe. The continuity equation for pressureless matter in sc gauge reads (38)

(66)

We thus have