

A consequence of non-Gaussian perturbations on the Sachs-Wolfe effect is studied. For a particular power spectrum, predicted Sachs-Wolfe effects are calculated for two cases: Gaussian (random phase) configuration, and a specific kind of non-Gaussian configuration. We obtain a result that the Sachs-Wolfe effect for the latter case is smaller when each temperature fluctuation is properly normalized with respect to the corresponding mass fluctuation ${\delta M\over M}(R)$. The physical explanation and the generality of the result are discussed.

<span>**I. INTRODUCTION**</span>

It is a challenging problem for cosmologists to explain the large scale structures. It is generally believed that some microphysical processes produced small amplitude perturbations in the early universe, and that the present large scale structures grew by gravitational instability of these perturbations. Perturbations are called Gaussian (G) perturbations if the phases of each fourier mode are random, and they are called non-Gaussian (NG) perturbations if otherwise.

G initial fluctuations are well motivated in the context of inflationary cosmology. During the inflationary phase, quantum fluctuations of a scalar field, which drives the inflation, produces G fluctuations \[1-4\]. There have been many, both analytical and numerical, studies of evolution of G fluctuations \[5-7\]. An evolutionary scenario requires further assumptions about the type of matter in the universe, (e.g., Cold Dard Matter (CDM) or Hot Dark Matter (HDM)), and the initial power spectrum of the fluctuations $p(k) \equiv
{\vert \delta_k \vert}^2$. In some cases even $\Omega \equiv \rho /\rho_{critical}$ and $\bf \wedge$, the cosmological constant, are left as free parameters.

Statistical methods are typically used to make comparisons between predictions of a model and the observations. Here a further assumption about the relationship between the distribution of matter and the distribution of observed galaxies needs to be made \[8-10\], i.e., biased galaxy formation.

For a model to be branded a success it must fulfill at least two requirements. First, it must be capable of reproducing the observed large scale structures. Second, its predicted Cosmic Microwave Background Radiation (CMBR) must agree with the observations \[11-14\]. There are several mechanisms contributing to the CMBR temperature fluctuations, but for fluctuations on large angular scales ($\theta \gg 1^o$) only two are important. The first is the doppler effect coming from our pecular velocity with respect to the CMBR rest frame. The second is the Sachs-Wolfe effect, which is caused by fluctuations in gravitational potentials on the last scattering surface \[15\]. With the recent results from the Cosmic Background Explorer (COBE) \[14\], some G models are on the process of being ruled out. For example, HDM, $\Omega = 1$, with the Harrison-Zel'dovich power spectrum predicts too large ${\Delta T\over T}$ in direct conflict with COBE \[16,17\]. An alternative to Gaussian models are whole class of non-Gaussian (NG) models. Examples of NG models are Cosmic Strings, Monopoles, and Textures, which are the remnants of phase transitions at early universe \[18-23\]. Production of some NG fluctuations can even be accomodated within the context of inflationary scenario \[24-26\]. There are also mechanisms for generating NG perturbations without resorting to 'exotic' physics of the early universe, for example the explosion scenario of Ostriker and Cowie \[27\], and Ikeuchi \[28\].

Detailed N body studies for evolution of NG perturbations have been done \[29-33\], and their corresponding Sachs-Wolfe effects were calculated \[34,35\]. In this paper, we would like to do a comparative study of Gaussian vs. non-Gaussian fluctuations in the context of Sachs-Wolfe effect. The question we would like to address is whether the predicted Sachs-Wolfe effect can be reduced by describing the primordial density perturbations by a particular NG field configuration instead of a G field configuration. That is, for a fixed power spectrum $p(k) = {\vert \delta_k\vert}^2$, we would like to consider the consequences of two types of field configurations, ${\delta \rho\over \rho}(\vec x)
= \sum \vert \delta_k\vert e^{i\phi (k)}e^{ik\cdot x}$. The first is the G case where all the phases $e^{i\phi(k)}$ are random. The second is a NG case where the phases are correlated in some way. For each case we will calculate the predicted ${\Delta T\over T}$ due to the Sachs-Wolfe effect and compare the magnitudes after normalizing each ${\Delta T\over T}$ by the corresponding mass fluctuation ${\delta M\over M}(R)$ at some scale $R$.

Even for a fixed power spectrum it is obvious that the set of all possible NG field configurations is an infinite dimensional parameter space. There have been some analytical work on the parametrization of NG perturbations ( Moscardini et al. \[36\], Weinberg et al. \[37\]). Because their parametrization is in the 1 point probability function space and not in fourier or configuration space their method is not directly applicable here. The reason is that every possible field configuration described by a different set of phases $e^{i\phi (k)}$ gives a probability function $P[\delta ]$, but each probability function does not give a unique field configuration. Therefore, in this paper, we will first study the consequences of a very special NG field configuration, i.e., $e^{i\phi (k)}$ are all +1 (or all -1). Clearly the resulting configuration ${\delta \rho\over \rho}(\vec x)$ is a spherical overdensity (spherical void). The difference between this NG and the G configuration is most accute at the center of this object, where all the amplitudes of different wavelength fluctuations add linearly in the NG case.

In this paper, for this NG field configuration, we have done a model calculation for the power spectrum $p(k) \sim \delta (k - k_o)$. As will be shown in detail, we arrive at the result that the predicted Sachs-Wolfe effect for this NG case, when properly normalized by mass fluctuation, is smaller than the corresponding prediction for the G case.

The layout of the paper is as follows. In section 2, some notations and basic formuli are quoted from literature. In section 3, ${\Delta T\over T}$ and ${\delta M\over M}(R)$ are calculated for the power spectrum $p(k) \sim \delta (k - k_o)$ as G fluctuations. In sections 4 and 5, for the same power spectrum, we calculate ${\Delta T\over T}$ and ${\delta M\over M}(R)$ again, respectively, but now as NG fluctuations. In section 6, the calculated temperature fluctuations for the two cases are normalized with respect to the corresponding mass fluctuation, and they are compared. In section 7, physical explanation of the result is given and some comments and generalization are given.

In the model calculations the scale factor of the universe and the density perturbations were assumed to evolve as in flat CDM universe.

<span>**II. PRELIMINARIES**</span>

By definition, a spatial perturbation is just a fourier sum of fluctuations on all wavelengths. $${\delta \rho\over \rho}(\vec x) = \sum_{\vec k}\vert \delta_{\vec k}\vert
e^{i\phi (\vec k)}e^{i{\vec k}\cdot {\vec x}}.$$

A power spectrum is defined as $$p(\vec k) \equiv {\vert \delta_{\vec k}\vert }^2
\longrightarrow
 {\left( {L\over 2\pi}\right) }^3{\vert \delta_{\vec k}\vert }^2d^3k
{\rm \ continuum \ limit}.$$

$L$ is an arbitrary length which is large compare to any length scale in the problem. If the fluctuations were produced by some microphysical processes in the early universe, then, a priori, there should be no preferred direction. Therefore, rotational invariance dictates that $p(\vec k) = p(\vert k\vert)$.

A model power spectrum we will use is

$$p(k) = \left\{ \begin{array}{ll}
\delta_o^2  &\vert k_o - {\Delta k\over 2}\vert \leq k
\leq \vert k_o + {\Delta k\over 2}\vert \\
 0  &{\rm otherwise.}
\end{array}
\right.$$

We will not use the Dirac delta function, i.e., $p(k) \sim \delta (k - k_o)$, because for a NG case we will need to evaluate integrals of the form $\int \vert \delta_k \vert d^3k\{ ... \}$, and square root of a delta function is tricky to handle. We will assume that $\Delta k$ is thin enough so that $\int f(k)dk \approx f(k_o)\Delta k$ for any reasonable function $f(k)$.

It turns out that this is the simplest power spectrum with which we can demonstrate the essential differences between the G and a particular NG configuration, which we will next discuss.

With this power spectrum, we will consider two different field configurations, i.e., choice of phases.

$$e^{i\phi (k)} = \left\{ \begin{array}{ll}
&{\rm  random, Gaussian \ case \ (G)} \\
&{\rm all +1 \ (or \  all -1), \ a \ non-Gaussian \ case \ (NG)}
\end{array} \right.$$

The physics behind generating a CMBR fluctuation from a density fluctuation is easy to understand. Simply stated, a matter perturbation distorts space, and therefore it 'distorts' geodesics of photons which were last aaron99@example.net. Because a Newtonian potential is time independent in a linear regime only the perturbation in the Newtonian potential on the last scattering surface is important in the linear Sachs-Wolfe calculation. Contributions from any intermediate distortion of a geodesic is unimportant by the blue and redshift cancellation effect. If a structure becomes nonlinear, then the gravitational potential is no longer constant, and the cancellation is not complete. This higher order effect has been studied by Rees and Sciama \[38\].

The relationship between a matter perturbation and ${\Delta T\over T}$ can be described most succintly by the expression \[15, 39\]

$${\Delta T\over T}(\hat q, \vec x) \equiv
{ T(\hat q, \vec x) - T_o\over T_o} =
{1\over 3}
\left[ \delta\phi(\vec x + \tau_o\hat q) -
 \delta\phi(\vec x) \right],$$

where $ \delta\phi(\vec x)$ is the Newtonian potential at point $\vec x$, $\tau_o = 2H_o^{-1}$, $\hat q$ is a direction in sky, and $T_o = 2.7 K$ is the mean CMBR temperature. From $\nabla^2\delta\phi = 4\pi G\delta \rho$, we get

$$\delta\phi(\vec x)= -{3\over 2}H_o^2\sum_k\delta_k(t_o)k^{-2}
e^{i\vec k\cdot \vec x}.$$

Rigorously, these expressions are valid in time-orthogonal coordinate and for a flat universe. The generalization to an open or closed universe is discussed by Anila and Motta \[40\].

<span>**III. GAUSSIAN FLUCTUATIONS**</span>

For G fluctuations, because of the random phase of each $\delta_k$ (2.1), only statistical information can be studied, e.g., the angular correlation function of ${\Delta T\over T}$. Using (2.5) and (2.6) we get

$$\begin{aligned}
C(\hat q_1 \cdot \hat q_2)
&\equiv
 &\langle {\Delta T\over T}(\hat q_1,\vec x)
 {\Delta T\over T}(\hat q_2,\vec x)\rangle_{space} \nonumber \\
&=
&\pi H_{o}^{4}{\left( {L\over 2\pi }\right)^3}
 \int {dk\over k^2}p(k) \left( j_o(k\tau_o \vert \hat q_1 - \hat q_2 \vert )
 - {\rm monopole - dipole} \right).\end{aligned}$$

The monopole (i.e., spherically symmetric) term is subtracted out because it is unobservable for a fixed observer. The dipole term is also subtracted out because in principal it is impossible to distinguish an intrinsic dipole anisotropy from the observer's pecular velocity. The monopole and the dipole terms can be identified by using the following identity, which is proven in the appendix.

$$j_o(\vert A\hat A - B\hat B\vert ) = \sum_l(2l+1)j_l(A)
j_l(B)p_l(\hat A\cdot \hat B)$$

$j_o$ and $p_l$ are Spherical Bessel and Legendre functions, respectively. Setting $A=k\tau_o$, $\hat A=\hat q_1$, $B=k\tau_o$, and $\hat B=\hat q_2$ we identify the monopole and the dipole terms to be $ j_o^2(k\tau_o)$ and $3j_1(k\tau_o)^2p_1(\hat q_1\cdot
\hat q_2)$, respectively.

Using a thin $\Delta k$ approximation to evaluate the integral for the power spectrum (2.3), we finally get \[15,41\]

$$C(\hat q_1 \cdot \hat q_2)
= \pi H_o^4{\left( {L\over 2\pi }\right)}^3
  {\Delta k\over k_o^2}\delta_o^2 \left( {\sin {\varsigma k_o}\over
\varsigma k_o} - j_o^2(k_o\tau_o) - 3j_1(k_o\tau_o)^2p_1(\hat q_1 \cdot
\hat q_2)\right),$$

with $\varsigma \equiv \tau_o\vert \hat q_1 - \hat q_2 \vert$. The overall normalization $\delta_o$ must be fixed. This is usually done by relating it to either ${\delta M\over M}(R)$ or mass autocorrelation function $\xi (\vec r) \equiv \langle {\delta \rho\over \rho}(\vec x + \vec r)
 {\delta \rho \over \rho}(\vec x)\rangle_{space}.$ Normalizing $\delta_o$ by $\xi (\vec r)$ would be subjecting the model to more stringent requirements, which would be appropriate for a more realistic power spectrum like $p(k) \sim k^n $. Therefore, for our simplistic models, we choose to normalize it by ${\delta M\over M}(R)$.

Again for the G case, we will just quote the well known result relating the power spectrum and ${\delta M\over M}(R)$ \[39\].

$$\langle \vert {\delta M\over M}(R)\vert^2\rangle_{space} =
{\left( {L\over 2\pi }\right)}^3
 \int d^3k p(k) W(kR)$$

where $W(kR)$ is a window function. We will use the top down window function

$$W(y) = 9y^{-6}\left( \sin y - y\cos y\right)^2.$$

Evaluating the expression for the power spectrum (2.3) we get

$$\langle \vert {\delta M\over M}(R)\vert^2 \rangle_{space} =
4\pi {\left( {L\over 2\pi }\right)}^3
 k_o^2\Delta k \delta_o^2 W(k_oR).$$

<span>**IV. ${\Delta T\over T} $ FOR NON-GAUSSIAN FLUCTUATIONS**</span>

In this section, we would like to recalculate the temperature fluctuation for the same power spectrum (2.3) but now for the NG configuration (2.4). Recall that in the G case the best one can do was to evaluate <span>*rms*</span> of various quanties of interest. For a NG case this is no longer necessary. For NG perturbations it is possible to get an explicit expression for a temperature fluctuation as a function of position $\vec x$ in a direction $\hat q$. Using the fact that $e^{i\phi (k)} = 1$, it is trivial to do the angular sum $d\Omega_k$ in Eq. (2.6).

$${\Delta T\over T}(\hat q, \vec x,\tau_o)
= -2\pi H_{o}^{2}{\left( {L\over 2\pi }\right)}^3
 \int dk p(k)^{1/2}
\left( j_o(k\vert \vec x + \tau_o\hat q\vert ) - {\rm monopole - dipole}
 \right).$$

Using a thin $\Delta k$ approximation and Eq. (3.2) with $A = k\tau_o$, $\hat A = \hat q$, $B = kr$, and $\hat B = \hat x$, to identify the monopole and the dipole terms, we get

$${\Delta T\over T}(\hat q, \vec x,\tau_o)
= -2\pi H_{o}^{2}{\left( {L\over 2\pi }\right)}^3
\Delta k_o \delta_o
\left( \begin{array}{l}
{\sin {\varsigma k_o}\over \varsigma k_o}  - j_o(k_o\tau_o)j_o(k_or) \\
 + 3j_1(k_o\tau_o)j_1(k_or)p_1(\hat q\cdot \hat x)
\end{array} \right),$$

where now $\varsigma \equiv \vert \tau_o \hat q + r\hat x \vert $. It should be noted that this is the result for the specific NG configuration (2.4), and other NG configurations will give different results.

<span>**V. ${\delta M\over M}(R)$ FOR NON-GAUSSIAN FLUCTUATIONS**</span>

We would now like to derive a simple expression for a mass fluctuation at some scale R that is appropriate for NG fluctuations. Again, for a NG perturbation it is no longer necessary to evaluate <span>*rms* </span>of quantities of interest. By definition (2.1), for (2.4) $${\delta \rho\over \rho }(\vec x) =  {\left( {L\over 2\pi }\right)}^3
 \int k^2dk d\Omega_k \vert \delta_k\vert
\left[ {e^{i\phi (k)}\over 1}\right]e^{i\vec k\cdot \vec x}.$$

The angular integral $d\Omega_k$ is simple to evaluate.

$${\delta \rho\over \rho }(r) = 4\pi {\left( {L\over 2\pi }\right)}^3
 \int k^2dk  \vert \delta_k\vert
{\sin (kr)\over kr}.$$

Defining ${\delta M\over M}(R)$ as an excess mass within a sphere of radius $R$ centered on a point where the density contrast is maximum, we get

$${\delta M\over M}(R) =
4\pi {\left( {L\over 2\pi }\right)}^3
 \int k^2dk \vert \delta_k \vert \sqrt{ W(kR)},$$

where $W(kR)$ is the familiar top down window function (3.5). Evaluating this for power spectrum (2.3) we get

$${\delta M\over M}(R) =
4\pi {\left( {L\over 2\pi }\right)}^3
  k_o^2\Delta k \delta_o \sqrt{ W(k_oR)}.$$

<span>**VI. COMPARISON OF TWO TYPES OF PERTURBATIONS**</span>

We will now use the simplest method to compare the magnitudes of two ${\Delta T\over T}$ as predicted by each model. We will first write each ${\Delta T\over T}^{\rm NG(G)}$ in terms of the corresponding ${\delta M\over M}(R)^{\rm NG(G)}$. Then we will compare the magnitudes of ${\Delta T\over T}^{\rm NG}$ and ${\Delta T\over T}^{\rm G}$ by setting ${\delta M\over M}(R)^{\rm NG} = {\delta M\over M}(R)^{\rm G}$. Using (3.3) with (3.6), we get for the G case

$$\left ( {\Delta T\over T}\right)_{rms}^G(\hat q_1, \hat q_2)
= {1\over 2}W(k_oR)^{-1/2}H_o^2k_o^{-2}
\left( {\delta M\over M}(R)\right)_{rms}^G
 \left\vert {\sin {\varsigma k_o}\over
\varsigma k_o} - j_o^2(k_o\tau_o) - 3j_1(k_o\tau_o)^2p_1(\hat q_1 \cdot
\hat q_2)\right\vert^{1/2}$$

with $\varsigma = \tau_o\vert \hat q_1 -  \hat q_2\vert$. For the NG case using (4.2) with (5.4), we get

$${\Delta T\over T}^{NG}(\hat q, \vec x)
= - {1\over 2}W(k_oR)^{-1/2}H_o^2k_o^{-2}{\delta M\over M}^{NG}(R)
\left( \begin{array}{l}
{\sin {\varsigma k_o}\over \varsigma k_o}  - j_o(k_o\tau_o)j_o(k_or) \\
 + 3j_1(k_o\tau_o)j_1(k_or)p_1(\hat q\cdot \hat x)
\end{array} \right)$$

with $\varsigma = \vert \tau_o\hat q + r\hat x\vert$. The graph of ${\Delta T\over T}$ for G case is shown in Figure 1. The graphs of ${\Delta T\over T}$ for the NG case are shown in Figures 2(a)-2(f) for various values of $k_or$. It is interesting to note that for the NG case the temperature anisotropy depends on position in space. This is because the information about the positions of mass perturbations is not lost in the ${\Delta T\over T}$ calculation for a NG case, whereas in the corresponding calculation for the G case they are lost by incorporating the <span>*rms*</span> methods.

It should be noted that we are comparing the magnitudes of <span>*rms*</span> of autocorrelation function for ${\Delta T\over T}$ with the local temperature fluctuation; therefore, the angular parameter $\theta$ that appears in the figures for both G and NG have different meanings.

In graphing ${\Delta T\over T}'s$, a common normalization was used for both G and NG cases. From the figures we can conclude that, for the models described by (2.3) and (2.4), the predicted Sachs-Wolfe effect for the NG case is about $ 3\cdot 10^{-3}$ smaller than for the G case. Actually this conclusion is true as long as $\vert \tau_o\hat q - r\hat x \vert
\gg k_o^{-1}$. In other words, the position of spherical overdensity (void) does not intersect the locus of last scattering surface in relation to the observer at $\vec x$.

<span>**VII. DISCUSSION**</span>

Before getting into the discussion of the generality of the result, some physical explanations are in order. First, the difference between G and the NG configuration (2.4) on ${\delta \rho\over \rho}(\vec x)$ is obvious. For the NG case, there are regions in space where the amplitudes of different wavelength fluctuations are adding coherently. In these regions the magnitudes of ${\delta \rho\over \rho}(\vec x)$ are larger than the <span>*rms*</span> of ${\delta \rho\over \rho}(\vec x)$, which is a relevant quantity for the G case. With this understanding, the conclusion that the Sachs-Wolfe effect for the NG configuration (2.4) is smaller than the Sachs-Wolfe effect for the G configuration may seems contrary to Eq. (2.5). After all, if matter is more 'clumpy' in the NG configuration (2.4), than the maximum difference in the Newtonian potential should be larger for the NG case. And therefore, ${\Delta T\over T}$ for the NG configuration should also be larger. The loophole in this argument lies in two facts. First, (2.5) needs to be normalized because of the quantity $\delta_o$ (see (2.3), (3.3), (4.2)). Second, in calculating the Sachs-Wolfe effect $l = 0$ (monopole) term of ${\delta T\over T}$ is subtracted out because it is unobservable. For the NG configuration (2.4) under consideration, it is easy to see that regions in space where ${\Delta T\over T}(\hat q, \vec x)$ is much larger than the <span>*rms*</span> of ${\Delta T\over T}(\hat q, \vec x)$ are spherically symmetric. Therefore, when the $l=0$ term is subtracted out the enhancement of going from a linear sum to the 'random walk' sum largely disappears.

We would now like to give a proof that the NG configuration (2.4) among all possible NG configurations gives the smallest Sachs-Wolfe effect for the power spectrum (2.3). As stated in the introduction, even for a fixed power spectrum, the set of all NG field configurations is an infinite parameter space. For the argument we choose to parametrize a NG configuration in a following way. A general phase of a fluctuation $\delta_k$ can be expanded in terms of spherical harmonics

$$e^{i\phi (\vec k)} = \sum_{l,m}Y_{l,m}(\hat k)G_{l,m}(\vert k\vert ).$$

For the power spectrum (2.3) we are considering, $\vert k\vert$ is constant, so $G_{l,m}(k)$ are just constants. Since $e^{i\phi }$ is a unit norm, it implies that ${\sum \vert G_{l,m}\vert }^2 = 4\pi$ or $ \vert G_{l,m}\vert \leq
\sqrt{4\pi}$. In this general NG field configuration, it is easy to show that for fixed power spectrum the new mass fluctuation is (Eqs. (5.1)-(5.4))

$$\left. {\delta M\over M}(R)\right\vert_{NG}^{new} =
{G_{o,o}\over \sqrt{4\pi }}
\left.{\delta M\over M}(R)\right\vert_{NG}^{old}.$$

By old we mean the previously discussed case $e^{i\phi} = 1$. We are being little cavalier here. Even though the resulting density configuration ${\delta \rho\over \rho }(\vec x)$ may not be spherically symmetric, we are defining $ {\delta M\over M}(R)$ as an excess mass within a sphere of radius $R$ centered on a point where the density contrast is maximum.

The change in the Newtonian potential is more complicated. Inserting (7.1) into (2.6) and after $d\Omega_k$ we get

$$\delta\phi(\vec x + \tau_o \hat q)
= -6\pi H_{o}^{2}{\left( {L\over 2\pi }\right)}^3
\Delta k \delta_o
\sum_{l,m}i^lY_{l,m}(\hat \varsigma)j_l(k_o\varsigma)G_{l,m}$$

with $\vec \varsigma \equiv \vec x + \tau_o\hat q$, and ${\Delta T\over T}(\hat q, \vec x) = {1\over 3}[\delta \phi(\vec \varsigma ) -
{\rm monopole}]$ with (see (2.5))

$${\rm monopole \ term } =
{\int \delta\phi(\vec \varsigma)d\Omega_{\hat q}\over
\int d\Omega_{\hat q}} = \delta \phi(\vec x)j_o(k_o r).$$

Denoting quantities with the $\delta_o$ and $G_{l,m}$ dependence extracted by a hat (e.g., $\hat A$) we get from (7.3), (7.4), and (2.5)

$${\Delta T\over T}(\hat q, \vec x) = {\delta_o\over 3}\sum_{l,m} G_{l,m}\left[
 \widehat{\delta\phi}_{l,m}(\vec \varsigma) -
 \widehat{\delta\phi}_{l,m}(\vec x)j_o(k_or)
 \right].$$

And from (5.4) and (7.2) we get ${\delta M\over M} = \delta_o G_{oo}{\widehat{\delta M}\over M}$. Eliminating $\delta_o$, we need to find the extremum of

$${\Delta T\over T}(\hat q, \vec x) = {G_{oo}^{-1}\over 3}
{\delta M\over M}({\widehat{\delta M}\over M})^{-1}
\sum_{l,m} G_{l,m}\left[
 \widehat{\delta\phi}_{l,m}(\vec \varsigma) -
 \widehat{\delta\phi}_{l,m}(\vec x)j_o(k_or)
\right]$$

subject to the constraint ${\sum \vert G_{l,m}\vert }^2 = 4\pi$. With the method of lagrange multiplier, we get ${\sum \vert G_{l,m}\vert }^2 = G_{oo}^2$, or $e^{i\phi} = G_{oo}Y_{oo} = \pm 1$. It is easy to check that the extremum is a minimum because another configuration, e.g., $G_{oo} \approx 0$, gives a larger value. It should be noted that the variational method alone does not tell us that the field configuration $e^{i\phi} = \pm 1$ gives a Sachs-Wolfe effect that is smaller than for the G configuration. A remark on the measure of generality of the result is possible. ${\Delta T\over T }$ given by Eq. (7.6) is a smooth linear function of $G_{l,m}$ except for $G_{oo}$. And therefore, combined with the restriction of ${\sum \vert G_{l,m}\vert }^2 = 4\pi$, the conclusion will be unchanged as long as $G_{oo} \approx \pm \sqrt{4\pi}$.

The result for a more realistic power spectrum $p(k) \sim k^n$ will be discussed elsewhere \[42\]. The decrease in the Sachs-Wolfe effect persists but to a different degree.

In light of the result that there are NG field configurations that can reduce the Sachs-Wolfe effect, we could turn the analysis on its head. By properly incorporting void evolution, we could have addressed the question of the largest possible void compatible with COBE given a NG density fluctuations (here $e^{i\phi} \approx -1$). This would be the NG counterpart of the work by Blumenthal et al. \[43\].

And finally, even though in our analysis we have assumed that perturbations and the scale factor of the universe evolved as that of flat FRW universe, the decrease in the Sachs-Wolfe effect will obviously persist in other cosmologies but to different degrees.

<span>**IIX. CONCLUSION**</span>

In this paper, we have taken a power spectrum, which was essentially a delta function, and have shown that there is a set of non-Gaussian field configurations for which the Sachs-Wolfe effect is smaller than the corresponding effect for the Gaussian configuration case. This set of non-Gaussian perturbations is characterized by having a large monopole term , $G_{oo} \approx \pm \sqrt{4\pi}$, in the $Y_{l,m}$ decomposition of the phases, i.e., $e^{i\phi (\vec k)} =
\sum_{l,m}Y_{l,m}(\hat k)G_{l,m}(\vert k\vert ) \approx \pm 1 .$

The conflict between the recent COBE measurements of CMBR and some HDM Gaussian models can potentially be salvaged by the combination of a non-Gaussian fluctuation model and HDM.

<span>**APPENDIX**</span>

We would like to prove an identity.

$$j_o(\vert A\hat A - B\hat B\vert ) = \sum_l(2l+1)j_l(A)
j_l(B)p_l(\hat A\cdot \hat B)$$

consider

$$\begin{aligned}
\int e^{i(A\hat A - B\hat B)\cdot \hat C}d\Omega_c
&=& \int\sum_l i^l(2l+1)j_l(\vert A\hat A - B\hat B\vert)
 p_l((A\hat A - B\hat B)\cdot \hat C)d\Omega_c \\
&=& 4\pi j_o(\vert A\hat A - B\hat B\vert ) \\
&=& \int\sum_{l,l'}\left[ i^l(2l+1)j_l(A)
 p_l(\hat A\cdot \hat C)\right]
\left[ (-i)^{l'}(2l'+1)j_{l'}(B)
 p_{l'}(\hat B\cdot \hat C)\right] d\Omega_c    \\
& &  {\rm using} \int p_l(\hat A\cdot \hat C)
p_{l'}(\hat B\cdot \hat C)d\Omega_c = {4\pi \over 2l+1}
p_l(\hat A\cdot \hat B)\delta_{l l'} \\
&=& 4\pi \sum_l(2l+1)j_l(A)j_l(B)p_l(\hat A\cdot \hat B).
\ \ Q.E.D \end{aligned}$$

<span>**ACKNOWLEDGEMENTS**</span>

I would like to thank George Field, T. Piran, G. Efstathiou, Hardy Hodges, Helmut Zaglauer, for discussions and valuable comments, Dalia Goldwirth for bring to my attention a pedagogical review of Sachs-Wolfe effect by G. Efstathious, and R. Brandenberger for carefully reading the original manuscript and for his many constructive criticisms.

<span>**FIGURE CAPTIONS**</span>

1.  Graph of the square root of the temperature autocorrelation function $\vert \left< {\Delta T(\theta_o)\over T}
     {\Delta T(\theta_o +\theta)\over T}\right>\vert^{1/2}$, Eq. (6.1), as function of $\theta$. $H^{-1} = 3000MPC$ and $2\pi k_o^{-1} = 40MPC$ was used to imitate the peak in the power spectrum for a HDM. The normalization is arbitrary.

2.  Graph of the temperature fluctuation ${\Delta T\over T}(\hat q, \vec x)$, Eq. (6.2), as function of $\theta=cos^{-1}(\hat q\cdot ~\hat x)$ for various values of $r = \vert \vec x \vert$. Figures 2(a)-2(f) correspond to $k_or = [0, 2.0, 4.0, 6.0, 8.0, 10.0]$, respectively. The values of $H_o$, $k_o$, and the overall normalization are the same as those of Fig. 1.

<span>**REFERENCES**</span>

1.  A. H. Guth, and S. Y. Pi, Phys. Rev. Lett. <span>**49**</span>, 1110 (1982).

2.  S. W. Hawking, Phys. Lett. <span>**B 115**</span>, 295 (1982).

3.  A. A. Starobinsky, Phys. Lett. <span>**B 117**</span>, 175 (1982).

4.  J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. <span>**D 28**</span>, 679 (1983).

5.  G. Efstathiou, C. S. Frenk, S. D. M. White, and M. Davis, Mon. Not. R. Astron. Soc. <span>**235**</span>, 715 (1988).

6.  T. Suginohara, and Y. Suto, Astrophys. J. <span>**371**</span>, 470 (1990).

7.  J. F. Beacom, et al., Astrophys. J. <span>**372**</span>, 351 (1991).

8.  M. Davis, G. Efsthathious, C. S. Frenk, and S. D. M. White, Astrophys. J. <span>**292**</span>, 371 (1985).

9.  C. Park, Mon. Not. R. Astron. Soc. <span>**251**</span>, 167 (1991).

10. N. Kaiser, Astrophys. J. <span>**284**</span>, L9 (1985).

11. P. Meinhold, and P. Lubin, Astrophys. J. <span>**370**</span>, L11 (1991).

12. S. S. Meyer, E. S. Cheng, and L. A. Page, Astrophys. J. <span>**371**</span>, L7 (1991).

13. G. F. Smoot, et al., Astrophys. J. <span>**371**</span>, L1, (1991).

14. G. F. Smoot, et al., Astrophys. J. Lett. (submitted).

15. R. K. Sachs, and A. M. Wolfe, Astrophys. J. <span>**147**</span>, 73 (1967).

16. E. W. Kolb, and M.S. Turner, <span>*The Early Universe*</span> (Addison-Wesley Publishing Company, 1990).

17. E. L. Wright, et al., Astrophys. J. Lett. (submitted).

18. T. W. B. Kibble, J. Phys. <span>**A9**</span>, 1387 (1976).

19. Ya. B. Zel'dovich, Mon. Not. R. Astro. Soc. <span>**192**</span>, 663 (1980).

20. A. Vilenkin, Phys. Rep. <span>**121**</span>, 263 (1985).

21. N. Turok, Phys. Rev. Lett. <span>**63**</span>, 262 (1989).

22. M. Barriola, and A. Vilenkin, Phys. Rev. Lett. <span>**63**</span>, 341 (1989).

23. D. P. Bennett, and S. H. Rhie, Phys. Rev. Lett. <span>**65**</span>, 1709 (1990).

24. T. J. Allen, B. Grinstein, and M. B. Wise, Phys. Lett. <span>**B 197**</span>, 66 (1987).

25. D. S. Salopek, J. R. Bond, and J. M. Bardeen, Phys. Rev. <span>**D 40**</span>, 6 (1989).

26. L. A. Kofman, and A. D. Linde, Nucl. Phys. <span>**B 282**</span>, 555 (1987).

27. J. P. Ostriker, and L. L. Cowie, Astrophys. J. <span>**243**</span>, L127 (1981).

28. S. Ikeuchi, Publ. Astron. Soc. Japan <span>**33**</span>, 211 (1981).

29. R. J. Scherrer, A. L. Melott, and E. Bertschinger, Phys. Rev. Lett. <span>**62**</span>, 379 (1989).

30. C. Park, D. N. Spergel, and N. Turok, Astrophys. J. <span>**372**</span>, L53 (1991).

31. A. K. Gooding, C. Park, D. N. Spergel, N. Turok, J. R. Gott, Astrophys. J. (in press).

32. R. Cen, J. P. Ostriker, D. N. Spergel, N. Turok, Astrophys. J. (in press).

33. M. J. West, D. H. Weinberg, A. Dekel, Astrophys. J. <span>**353**</span>, 329 (1990).

34. J. Traschen, N. Turok, and R. Brandenberger, Phys. Rev. <span>**D 34**</span>, 919 (1986).

35. N. Turok, and D. N. Spergel, Phys. Rev. Lett. <span>**64**</span>, 2736 (1990).

36. L. Moscardini, S. Matarrese, F. Lucchin, and A. Messina, Mon. Not. R. Astro. Soc. <span>**248**</span>, 424 (1991).

37. D. H. Weinberg, and S. Cole, Mon. Not. R. Astro. Soc. (submitted).

38. M. J. Rees, and D. W. Sciama, Nature, <span>**217**</span>, 511 (1968).

39. P. J. E. Peebles, <span>*The Large Scale Structure of the Universe*</span> (Princeton University Press, Princeton, 1980).

40. A. M. Anile, and S. Motta, Astrophys. J. <span>**207**</span>, 685 (1976).

41. E. Martinez-Gonzales, and J. Sanz, Astrophys. J. <span>**347**</span>, 11 (1989).

42. J. H. Kung, (in preparation).

43. G. R. Blumenthal, et al., Astrophys. J. <span>**338**</span>, 225 (1992).

{'source': 'arXiv_out/gr-qc9208011'}